Robust Nearly-Efficient Estimation of Large Panels with Factor Structures

TL;DR

This paper introduces a robust, nearly-efficient GLS-based estimator for large panel data models with latent factor structures, achieving bias correction without requiring factor estimation.

Contribution

It develops a bias-adjusted GLS estimator that is robust to unknown factor structures and preserves efficiency in large panels, extending existing methods.

Findings

Estimator is nearly efficient and bias-adjusted.

Method performs well in Monte Carlo simulations.

Empirical application demonstrates practical usefulness.

Abstract

This paper studies estimation of linear panel regression models with heterogeneous coefficients, when both the regressors and the residual contain a possibly common, latent, factor structure. Our theory is (nearly) efficient, because based on the GLS principle, and also robust to the specification of such factor structure because it does not require any information on the number of factors nor estimation of the factor structure itself. We first show how the unfeasible GLS estimator not only affords an efficiency improvement but, more importantly, provides a bias-adjusted estimator with the conventional limiting distribution, for situations where the OLS is affected by a first-order bias. The technical challenge resolved in the paper is to show how these properties are preserved for a class of feasible GLS estimators in a double-asymptotics setting. Our theory is illustrated by means of Monte Carlo exercises and, then, with an empirical application using individual asset returns and firms' characteristics data.

Tables

Panel A
	t-ratios			GLS estimates
parameter:	10-th quantile	average	90-th quantile	10-th quantile	average	90-th quantile
intercept	-182.9	-32.6	125.2	-0.123	-0.024	0.063
Mkt	1507.2	2488.4	3568.2	0.061	0.010	0.014
SIZE	-239.5	-83.6	77.1	-0.217	-0.069	0.073
BM	-118.4	14.7	153.5	-0.063	0.006	0.076
DVOL	-12.1	118.9	272.4	-0.01	0.038	0.085
PRICE	-266.2	-126.7	4.6	-0.088	-0.044	0.001
YLD	-301.9	-103.6	59.9	-0.150	-0.051	0.010
RET23	-289.1	-143.4	-26.4	-0.021	-0.012	-0.002
RET46	-264.9	-139.9	-14.2	-0.022	-0.012	-0.001
RET712	-234.4	-121.4	-18.7	-0.019	-0.011	-0.001
Panel B
test statistic:	10-th quantile		average		90-th quantile
$F_{\gamma }$	1444899.20		5135622.34		9905946.98
$F_{\beta }$	82958.69		296084.32		612749.73
$F_{\beta ,\gamma }$	7690568.38		20110261.61		37936804.13

Panel A
	t-ratios			GLS estimates
parameter:	10-th quantile	average	90-th quantile	10-th quantile	average	90-th quantile
intercept	-210.6	-36.6	133.5	-0.116	-0.021	0.069
Mkt	1473.1	2325	3349.8	0.006	0.009	0.0128
SMB	-541.1	256.6	1072.5	-0.003	0.003	0.010
HML	-340.2	348.3	1068.7	-0.002	0.002	0.006
SIZE	-252.3	-91.7	81.6	-0.205	-0.068	0.065
BM	-138.3	14.6	169.3	-0.06	0.004	0.076
DVOL	-12.8	125.1	278.6	-0.003	0.036	0.081
PRICE	-303.3	-140.4	-2.1	-0.089	-0.042	-0.001
YLD	-327.1	-113.7	72.7	-0.148	-0.050	0.009
RET23	-305.8	-155.2	-30.8	-0.021	-0.011	-0.002
RET46	-316.6	-160.3	-25.8	-0.021	-0.011	-0.002
RET712	-272.5	-141.3	-28.5	-0.018	-0.010	-0.002
Panel B
test statistic:	10-th quantile		average		90-th quantile
$F_{\gamma }$	3148154.05		8766878.36		16270455.79
$F_{\beta }$	87651.83		315163.52		634901.58
$F_{\beta ,\gamma }$	12894492.39		29647189.57		55497464.04

Panel A
	t-ratios			GLS estimates
parameter:	10-th quantile	average	90-th quantile	10-th quantile	average	90-th quantile
intercept	-208.7	-36.6	129.1	-0.120	-0.021	0.075
Mkt	1303	2172.2	3207.8	0.006	0.009	0.012
SMB	-605.5	259.9	1143.2	-0.003	0.002	0.010
HML	-326.7	202.5	863.8	-0.003	0.001	0.007
RMW	-440.7	-57.1	344.4	-0.006	-0.001	0.006
CMA	-488.1	-0.661	484.4	-0.007	0.0001	0.007
SIZE	-255.8	-90.3	78.5	-0.206	-0.067	0.062
BM	-141.7	16.4	163.8	-0.063	0.005	0.075
DVOL	-11.1	124.7	266.5	-0.002	0.036	0.081
PRICE	-300.1	-138.9	0.813	-0.087	-0.042	-0.001
YLD	-319.1	-113.1	71.9	-0.151	-0.050	0.009
RET23	-304.6	-154.5	-26.7	-0.021	-0.011	-0.002
RET46	-318.8	-158.8	-21.7	-0.021	-0.011	-0.001
RET712	-265.6	-141.1	-31	-0.018	-0.010	-0.002
Panel B
test statistic:	10-th quantile		average		90-th quantile
$F_{\gamma }$	4930084.6		11721601.5		20418779.1
$F_{\beta }$	83573.5		303480.1		569989.4
$F_{\beta ,\gamma }$	14961844.2		34707178.6		60856307.5

Equations

$${\mathrm{y}}_i=\mathrm{D}{\mathbf{\alpha }}_i+{\mathrm{X}}_{\mathbf{i}}{\mathbf{\beta }}_{\mathbf{i}}+{\mathbf{u}}_{\mathbf{i}},$$

$${\mathrm{u}}_{\mathbf{i}}=\mathbf{F}{\mathbf{b}}_i+{\mathbf{\epsilon }}_i,\text{with}{\mathbf{\Xi }}_{\mathbf{i}}:=\mathbb{E}{\mathbf{\epsilon }}_{\mathbf{i}}{\mathbf{\epsilon }}_{\mathbf{i}}^{\prime },$$

$${\mathbf{X}}_i=\mathbf{D}{\mathbf{\Delta }}_i+\mathbf{F}{\mathbf{\Gamma }}_i+{\mathbf{V}}_i,$$

$${\mathbf{\epsilon }}_t=\mathbf{R}{\mathbf{a}}_t,\text{for}t=1,\dots T,$$

$$a_{it}=\sum _{s=0}^{\infty }{\varphi }_{is}{\eta }_{i,t-s},\underset{i}{\sup }\sum _{s=0}^{\infty }{s}^2|{\varphi }_{is}|<\infty ,\text{with}{\varphi }_{i0}=1,$$

$$\underset{i}{\inf }|{\varphi }_i(z)|>\kappa ,|z|\le 1,\text{where}{\varphi }_i(z)=\sum _{s=0}^{\infty }{\varphi }_{is}{z}^s.$$

$$\underset{i_1}{\sup }\underset{t_1}{\sup }\sum _{i_2\cdots i_{\ell }=1}^N\sum _{t_2\cdots t_h=1}^T|{\mathrm{cum}}_h({\epsilon }_{i_1{t}_1},{\epsilon }_{i_2{t}_2}\cdots ,{\epsilon }_{i_{\ell }{t}_h})|<\infty ,$$

$$\underset{k_1\cdots k_s}{\sup }\underset{i_1}{\sup }\underset{t_1}{\sup }\sum _{i_2\cdots i_{\ell }=1}^N\sum _{t_2\cdots t_h=1}^T(1+t_j^2)|{\mathrm{cum}}_h(v_{i_1{t}_1{k}_1},\cdots ,v_{i_{\ell }{t}_h,k_s})|\le \infty .$$

$$\frac{\bm{\mathrm{Z}}^{\prime}\bm{\mathrm{Z}}}{T}\xrightarrow{p}\bm{\Sigma_{\bm{\mathrm{Z}}^{\prime}\bm{\mathrm{Z}}}},\quad\textrm{with}\quad\bm{\Sigma_{\bm{\mathrm{Z}}^{\prime}\bm{\mathrm{Z}}}}:=\left[\begin{array}[]{cc}\bm{\Sigma}_{\bm{\mathrm{D}}^{\prime}\bm{\mathrm{D}}}&\bm{\Sigma}_{\bm{\mathrm{D}}^{\prime}\bm{\mathrm{D}}}\\ \bm{\Sigma}_{\bm{\mathrm{F}}^{\prime}\bm{\mathrm{D}}}&\bm{\Sigma}_{\bm{\mathrm{F}}^{\prime}\bm{\mathrm{F}}}\end{array}\right]>0,$$

$${\mathrm{D}}_{\perp }^{\prime }\left(\frac{1}{N}\sum _{i=1}^N{\mathcal{M}}_{{\mathbf{Z}}_{\mathbf{i}}}{\mathrm{u}}_{\mathbf{i}}{{\mathrm{u}}_{\mathbf{i}}}^{\prime }{\mathcal{M}}_{{\mathbf{Z}}_{\mathbf{i}}}\right){\mathrm{D}}_{\perp }=\frac{1}{N}\sum _{i=1}^N{\mathbf{M}}_{({\mathrm{D}}_{\perp }^{\prime }{\mathrm{X}}_{\mathbf{i}})}{\mathrm{D}}_{\perp }^{\prime }{\mathrm{u}}_{\mathbf{i}}{{\mathrm{u}}_{\mathbf{i}}}^{\prime }{\mathrm{D}}_{\perp }{\mathbf{M}}_{({\mathrm{D}}_{\perp }^{\prime }{\mathrm{X}}_{\mathbf{i}})}>0,$$

$${\mathrm{B}}_{\mathbf{N}}:=\frac{1}{N}\sum _{i=1}^N{\mathrm{b}}_{\mathbf{i}}{{\mathrm{b}}_{\mathbf{i}}}^{\prime }>0.$$

$${\mathbf{A}}_{\mathbf{N}}$$

$${\mathbf{\Psi }}_i:={{\mathbf{\Gamma }}_{\mathbf{i}}}^{\prime }\frac{{\mathrm{F}}^{\prime }{\mathcal{M}}_{\mathrm{D}}\mathrm{F}}{T}{\mathbf{\Gamma }}_{\mathbf{i}}+{\mathbf{\Sigma }}_{{\mathbf{V}}_{\mathbf{i}}^{\prime }{\mathbf{V}}_{\mathbf{i}}}.$$

$${\mathcal{M}}_{\mathrm{D}}{\mathrm{y}}_i={\mathcal{M}}_{\mathrm{D}}{\mathrm{X}}_{\mathbf{i}}{\mathbf{\beta }}_{\mathbf{i}}+{\mathcal{M}}_{\mathrm{D}}{\mathrm{u}}_{\mathbf{i}}.$$

$${\hat{\mathbf{\beta }}}_{\mathbf{i}}^{\mathbf{OLS}}:=({{\mathrm{X}}_{\mathbf{i}}}^{\prime }{\mathcal{M}}_{\mathrm{D}}{\mathrm{X}}_{\mathbf{i}}{)}^{-1}{\mathrm{X}}_{\mathbf{i}}{\mathcal{M}}_{\mathrm{D}}{\mathrm{y}}_i.$$

$${\mathcal{M}}_{\mathrm{D}}{\mathrm{S}}_N{\mathcal{M}}_{\mathrm{D}},\text{mbox}setting{\mathrm{S}}_N:=\mathrm{F}{\mathrm{B}}_{\mathbf{N}}{\mathrm{F}}^{\prime }+{\mathbf{\Xi }}_{\mathbf{N}},\text{mbox}with{\mathbf{\Xi }}_{\mathbf{N}}:=\frac{1}{N}\sum _{i=1}^N{\mathbf{\Xi }}_{\mathbf{i}}.$$

$${\hat{\mathbf{\beta }}}_i^{UGLS}:={\left({{\mathrm{X}}_{\mathbf{i}}}^{\prime }{\mathcal{M}}_{\mathrm{D}}{\left({\mathcal{M}}_{\mathrm{D}}{\mathrm{S}}_N{\mathcal{M}}_{\mathrm{D}}\right)}^{+}{\mathcal{M}}_{\mathrm{D}}{\mathrm{X}}_{\mathbf{i}}\right)}^{-1}{{\mathrm{X}}_{\mathbf{i}}}^{\prime }{\mathcal{M}}_{\mathrm{D}}({\mathcal{M}}_{\mathrm{D}}{\mathrm{S}}_N{\mathcal{M}}_{\mathrm{D}}{)}^{+}{\mathcal{M}}_{\mathrm{D}}{\mathrm{y}}_i.$$

$$({\mathcal{M}}_{\mathrm{D}}{\mathrm{S}}_N^{-1}{\mathcal{M}}_{\mathrm{D}}{)}^{+}={\mathrm{D}}_{\perp }{\left({\mathrm{D}}_{\perp }^{\prime }{\mathrm{S}}_N{\mathrm{D}}_{\perp }\right)}^{-1}{\mathrm{D}}_{\perp }^{\prime },$$

$$y_{\mathbf{i}}={\mathrm{D}}_{\perp }^{\prime }{\mathrm{y}}_i,{\mathcal{X}}_{\mathbf{i}}={\mathrm{D}}_{\perp }^{\prime }{\mathrm{X}}_{\mathbf{i}},{\mathbf{ϵ}}_{\mathbf{i}}={\mathrm{D}}_{\perp }^{\prime }{\mathbf{\epsilon }}_{\mathbf{i}},\mathcal{F}={\mathrm{D}}_{\perp }^{\prime }\mathrm{F},u_{\mathbf{i}}={\mathrm{D}}_{\perp }^{\prime }{\mathrm{u}}_{\mathbf{i}},$$

$${\hat{\mathbf{\beta }}}_i^{UGLS}$$

$$y_{\mathbf{i}}={\mathcal{X}}_{\mathbf{i}}{\mathbf{\beta }}_i+u_{\mathbf{i}},\text{with}{u}_{\mathbf{i}}=\mathcal{F}{\mathrm{b}}_{\mathbf{i}}+u_{\mathbf{i}},$$

$${\hat{\mathbf{\beta }}}_{\mathbf{i}}^{\mathbf{GLS}}:={\left({{\mathcal{X}}_{\mathbf{i}}}^{\prime }{{\hat{\mathcal{S}}}_{\mathbf{N}}}^{-1}{\mathcal{X}}_{\mathbf{i}}\right)}^{-1}{{\mathcal{X}}_{\mathbf{i}}}^{\prime }{{\hat{\mathcal{S}}}_{\mathbf{N}}}^{-1}{y}_{\mathbf{i}},$$

$${\hat{\mathcal{S}}}_{\mathbf{N}}:=N^{-1}\sum _{i=1}^N{\hat{u}}_{\mathbf{i}}{{\hat{u}}_{\mathbf{i}}}^{\prime },\text{mbox}with{\hat{u}}_{\mathbf{i}}:=y_{\mathbf{i}}-{\mathcal{X}}_{\mathbf{i}}{\hat{\mathbf{\beta }}}_i^{OLS}={\mathcal{M}}_{{\mathcal{X}}_{\mathbf{i}}}{u}_{\mathbf{i}},$$

$$T^{\frac{1}{2}}({\hat{\mathbf{\beta }}}_i^{OLS}-{\mathbf{\beta }}_i-{\tau }_i^{OLS})\stackrel{d}{\to }\mathcal{N}\left(0,{\mathbf{\Sigma }}_{\mathbf{i}}\right),$$

$${\tau }_i^{OLS}:={{\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }{\mathcal{X}}_{\mathbf{i}}}}^{-1}{\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }\mathcal{F}}{\mathrm{b}}_{\mathbf{i}},$$

$${\mathbf{\Sigma }}_{\mathbf{i}}:={{\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }{\mathcal{X}}_{\mathbf{i}}}}^{-1}{\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }{\mathrm{D}}_{\perp }^{\prime }{\mathbf{\Xi }}_{\mathbf{i}}{\mathrm{D}}_{\perp }{\mathcal{X}}_{\mathbf{i}}}{{\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }{\mathcal{X}}_{\mathbf{i}}}}^{-1},$$

$${\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }\mathcal{F}}$$

$${\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }{\mathcal{X}}_{\mathbf{i}}}$$

$${\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }{\mathrm{D}}_{\perp }^{\prime }{\mathbf{\Xi }}_{\mathbf{i}}{\mathrm{D}}_{\perp }{\mathcal{X}}_{\mathbf{i}}}$$

$$T^{\frac{1}{2}}\left({\hat{\mathbf{\beta }}}_{\mathbf{i}}^{\mathbf{UGLS}}-{\mathbf{\beta }}_i\right)\stackrel{d}{\to }\mathcal{N}\left(0,{\mathbf{\Sigma }}_N^{\star }\right),$$

$${\tau }_i^{OLS}={{\mathbf{\Sigma }}_{{{\mathcal{X}}_{\mathbf{i}}}^{\prime }{\mathcal{X}}_{\mathbf{i}}}}^{-1}{\mathbf{\Sigma }}_{{\mathrm{F}}^{\prime }{\mathcal{P}}_F\mathrm{G}}{\mathrm{b}}_{\mathbf{i}},$$

Full text

Keywords: GLS estimation; panel; factor structure; robustness; bias-correction.

1 Introduction

This paper considers (nearly) efficient estimation of linear panel regression models with heterogeneous coefficients, when both the regressors and the residual contain a common, latent, factor structure. At the same time, our estimation procedure does not require any knowledge of such latent factor structure, not even the maximum possible number of latent factors, let alone the latent factors themselves. This qualifies our procedure as robust. Factor models represent one of the most popular and successful way to capture cross-sectional and temporal dependence, especially when facing a large number of units ($N$) and time periods ($T$), although in our context factors and their loadings represent nuisance parameters.

However, the possibility of a common factor structure in both regressors and residuals, which would typically arise when omitting relevant regressors, leads to endogeneity, making estimation by ordinary least squares (OLS) invalid such that all its asymptotic properties are not holding any longer.

We first consider an unfeasible generalized least squares (UGLS) estimator for the regression coefficients, based on the presumption that the covariance matrix of the residuals, evaluated conditional on the latent factors, is known. It turns out that, regardless of the possibility of endogeneity (that is when regressors and residuals are correlated), the UGLS is $\sqrt{T}$-consistent and asymptotically normal distributed *without * requiring any information on the factor structure, such as the number of factors or the factors themselves and their loadings. This contrasts with the asymptotic bias plaguing the OLS estimator, under the same circumstances. In other words, the UGLS does not only represent a more efficient estimator but provides an *automatic * biased-adjusted estimator with desirable asymptotic properties. This result is due to an important insight, namely the existence of a form of *asymptotic orthogonality * between the common factors, that affect the residuals, and the *inverse * of the residuals’ covariance matrix. Most importantly, such asymptotic orthogonality is manifested at a very fast rate, namely the squared norm of the product between the covariance matrix and the factors is $O(T^{-1})$.

The challenge arises when considering a feasible version of the UGLS estimator. A natural approach, here followed, is to make use of the panel dimension, considering the sample (across $N$) covariance matrix of the OLS residuals, which in turn have been obtained by a (time-series) regression with $T$ observations. Unlike the OLS and UGLS cases, the asymptotic theory for the GLS requires both $N$ and $T$ to diverge. Lack of consistency of the OLS estimator for the regression coefficients unavoidably implies that such $T\times T$ sample covariance matrix is not consistent, element by element, for the true residuals’ covariance matrix but will converge (element by element) instead to a pseudo-true covariance matrix. The surprising, crucial, result here established is that such pseudo-true covariance matrix is also asymptotically orthogonal to the latent factors, and at the same rate of convergence $O(T^{-1})$. Indeed, there is an entire class of matrices, rather than a unique matrix, that is asymptotically orthogonal to the factors. This is the most intriguing aspect of our theory. The technical achievement of this paper is to show that the *feasible * GLS (henceforth GLS) estimator for the regression coefficients, is $\sqrt{T}$-consistent and asymptotically normal, as *both * $N,T$ diverge to infinity. Again, this holds even when the OLS remains an invalid estimator. At the same time, since the pseudo-true value differs in general from the true covariance matrix, the GLS might not be as efficient as the UGLS. However, evaluation of the GLS iteratively, as explained below, permits to make it close to the UGLS estimator.

In summary, the GLS estimator exhibits four main desirable, compelling, properties. First, it permits to carry out inference on the regression coefficients based on conventional asymptotic distributions. In particular, the GLS estimator of the regression coefficients has a mixed-normal asymptotic distribution, implying the possibility of inference by means of chi-squared criteria. Second, as in classical estimation theory, it delivers (nearly) efficient estimation. Third, the GLS estimator does not require any knowledge of the exact number of latent factors, or even an upper bound of such number. In particular, the number of factors can be either smaller, equal or larger than the number of regressors. Fourth, the GLS is computationally easy to handle since it simply requires to perform ($N$) linear regressions, without invoking any nonlinear numerical optimizations. Our approach can be also applied to the dual case of cross-sectional regressions with time-varying coefficients.

This paper belongs to, and extends, two different strands of literature.

First, it has been demonstrated, in various contexts, that efficient estimation techniques not only lead to an improvement of precision but, most importantly, resurrect the required asymptotic properties, in terms of bias, rate of convergence and distribution, in situations where these are not warranted by non-efficient approaches.

In the context of cointegrated systems, Phillips (1991a) and Phillips (1991b) show that use of the efficient, full system, maximum likelihood (ML) goes beyond an efficiency improvement: it solves the well-known issues of specification and inference in cointegrated systems, that plagues unrestricted VAR estimation such as the presence of asymptotic biases and non-standard asymptotic distributions (i.e. the Dickey-Fuller distribution).111Indeed, especially Phillips (1991a) provides a detailed explanation of these properties, namely removing second order bias, dealing with endogeneity, absence of nuisance parameters and, obviously, achievement of full efficiency. Note that ML is asymptotically equivalent to GLS in that set-up. Although our theoretical framework is not one of cointegrated systems, strong analogies emerge with Phillips (1991a) and Phillips (1991b): in both cases, a (local) mixed-normal distribution arises and efficient estimation mitigates the lack of strong exogeneity. Moreover, such deficiency (i.e. lack of exogeneity) is manifested through the form of the residuals’ covariance matrix: non-block diagonality for (triangular) cointegrated systems of Phillips (1991a), Phillips (1991b) and a factor structure such as ours, which also rules out block-diagonality, for our framework. Second, Phillips (1991a) demonstrates how these remarkable properties of efficient estimation are warranted by full system regressions but not by single-equation regressions. Likewise, our method requires the full information arising from the panel, namely one needs both $T$ and $N$ to diverge.

Robinson & Hidalgo (1997) study estimation of time series regression models, when both the regressors and the residual exhibit long-memory, and in fact spectral singularities can arise at any frequency. Under these circumstances, in particular when the spectral singularities of the regressors and residuals arise at the same frequency with sufficient intensity, the OLS estimator is no longer $\sqrt{T}$-consistent and asymptotically normal. However, under the same circumstances, Robinson & Hidalgo (1997) show that a class of weighted least squares estimates, which includes GLS as a special case, has standard asymptotic properties.

An important difference between our approach and Phillips (1991a), Phillips (1991b) and Robinson & Hidalgo (1997) is that their estimation procedure is affected by a second-order bias, that is their estimators are consistent (although with non-standard rate of convergence and asymptotic distribution), whereas in our context a first-order bias arises, leading for instance to inconsistency of the OLS estimator. Therefore, our GLS adjustment appears compelling in our framework.

Second, inference of panel data model with a latent factor structure in the residuals and heteroreneous regression coefficients has been studied, initially, from a purely econometric perspective and, more recently, from an empirical finance angle.

In a linear cross-sectional regression Andrews (2005) shows that, when residuals and regressors share a factor structure, $\sqrt{N}$-consistency of the OLS estimator is preserved only with uncorrelated factor loadings.222Although not spelled out, Andrews (2005) can be readily applied to time-varying coefficients. Within a linear time regression, Pesaran (2006) shows that heterogeneous regression coefficients can be $\sqrt{T}$-consistently estimated OLS by augmenting the regressors with cross-sectional averages of the dependent variable and individual-specific regressors. Ando & Bai (2015) consider a panel model with (sparse) heterogenous coefficients and establish the asymptotics of a penalized OLS estimator. Maintaining the assumption of a common latent factor structure in the residuals of a panel data model with heterogeneous coefficients, Ergemen & Velasco (2017) allow for the possibility that the idiosyncratic innovation is non-stationary, in particular exhibiting long memory.

Motivated by empirical asset pricing, new methods to conduct robust inference on panel data models with a latent factor structure have been recently developed. Giglio & Xiu (2018) derive the asymptotics for a procedure to estimate the risk-premium of an observed factor, robust to the omission of the set of relevant (i.e. priced) factors. Like us, they adopt a double-asymptotics approach. However, Giglio & Xiu (2018) differ from us because they focus on estimation of the parameters of the second-pass regression, that is when the asset-pricing restriction is imposed, whereas we ignore any asset-pricing content (i.e., from the point of view of the two-pass methodology, we focus on the parameters of the first-pass regression). Moreover, their procedure relies on estimating the complete space spanned by the latent factors driving the model whereas our method can avoid this aspect altogether. Gagliardini, Ossola and Scaillet (2018) study the properties of a diagnostic criterion to detect an approximate factor structure in the residuals of large, unbalanced, panel data models. Like us, they consider a double-asymptotic setting and ignore any asset-pricing restrictions on the parameters of the panel data model. Moreover, Gagliardini et al. (2018) method is robust, in the sense that it does not need to explicitly estimate the latent factor structure embedded in the residuals, just like us. However their focus is specifically to check whether the unobserved residuals have a factor structure whereas our method focuses on estimation of the regression coefficients to the observed, possibly heterogenous, regressors.

Unlike the previous papers, the large majority of contributions to this literature focused on the case of constant regression coefficients. Pesaran (2006) shows that a faster rate of convergence is achieved with constant regression coefficients. Bai (2009) considers joint estimation of the constant regression coefficients and of the residuals’ factor structure components through an iterative OLS procedure. The same estimator has been studied by Moon & Weidner (2017) under weaker conditions on the observed regressors. Moon & Weidner (2015) show that Bai (2009) and Moon & Weidner (2017) results hold, with no loss of efficiency, when the exact number of latent factors $M$ is unknown and only an upper bound is specified. Bai & Liao (2017) show that GLS estimation leads to an efficiency improvement over the Bai (2009) and Moon & Weidner (2017) OLS-type estimator. Greenaway-McGrevy et al. (2012) establish $\sqrt{NT}$-asymptotics for the OLS estimator by augmenting the regressors with the principal component estimator of the common factors extracted from the observable data.333Several generalizations of the aforementioned results have been considered. Pesaran & Tosetti (2011) and Chudik & Pesaran (2015) confirm the same asymptotic results of Pesaran (2006) when spatial-dependence in the idiosyncratic component of the innovation’s factor structure as well as dynamic panel, respectively, are allowed for. Karabiyik et al. (2015) show that Pesaran’s estimator retains its asymptotic properties under weaker conditions, allowing for either correlated loadings or for the number of latent factor $m$ to be larger than the number of observables, whereas Westerlund & Urbain (2015) discuss some limitations. Song (2013) extends Bai (2009) to the case of non-constant regression coefficients establishing $\sqrt{T}$-asymptotics when $T/N^{2}\rightarrow 0$. Dynamic panel are permitted. Other contributions to this literature include Holtz-Eakin, Newey & Rosen (1988), Ahn, Hoon Lee & Schmidt (2001), Bai & Ng (2004), Phillips & Sul (2003), Moon & Perron (2003) and Phillips & Sul (2007).

None of these papers address the issue of efficient estimation, except for Bai & Liao (2017)444Bai & Liao (2017) focus on the homogeneous parameter case, unlike us, and considers joint estimation of the latent factors and parameters, generalizing Bai (2009) and Moon & Weidner (2017). More importantly, the motivation of Bai & Liao (2017) differs drastically from ours because they focus on the GLS approach for an efficiency improvement of an estimator that already exhibits the conventional asymptotic properties under Bai & Liao (2017) assumptions, in particular iid-ness across time. In our case, our GLS approach mitigates the first-order bias affecting the OLS estimator, where we allow for both serial and cross-sectional correlation as well as heteroskedasticity of the residuals. , but rather focus on various, ingenious, ways to mitigate the bias induced by the correlation between regressors and innovations. In contrast, our GLS approach allows to tackle both issues, at the same time, without requiring *any * knowledge of the factor structures affecting the regressors and innovations.555In particular, given that we can afford to be completely agnostic about the need to conduct inference on the latent factor structure affecting the model, our work differ, both in terms of focus and in terms of the techniques developed, from the multitude of papers developing inference methods on latent factor structures (on estimating the number of latent factors see Bai & Ng (2002), Hallin & Liska (2007), Amengual & Watson (2007), Onatski (2009), Onatski (2010), Ahn & Horenstein (2013) and on estimating latent factor structures see Forni et al. (2000), Stock & Watson (2002), Bai & Ng (2002), Bai (2009) among others. Our asymptotic distribution theory requires $T^{3}/N^{2}\rightarrow 0$ whereas the milder $T/N\rightarrow 0$ ensures consistency. This relative speed spells out a neat dichotomy in terms of the role of $T$ and $N$: the faster rate of divergence for $N$ is asked for to estimate accurately the (inverse of the) sample-covariance matrix required by the GLS formula, which in turn mitigates the asymptotic bias. Instead, the slower divergence of $T$ controls the asymptotic variance of the GLS estimator, dictating ultimately the estimator’s rate of convergence. Noticeably, the relative speed requested by our estimator differs from the relative speed requested by the alternative procedures described above, suggesting that our result can also be viewed as complementary to the others, for example more suitable to short panels where $N$ is much larger than $T$.666For example, Pesaran (2006) requires $T/N^{2}\rightarrow 0$ for asymptotic normality but the weaker condition $T/N\rightarrow 0$ is required for homogeneous regression coefficients, where the faster $\sqrt{NT}$ -rate of convergence is achieved. Moreover, one needs the number of heterogeneous regressors to be greater than number of latent factors. Bai (2009) shows that the regression coefficients’ estimator is also $\sqrt{NT}$-consistent, when $T/N\rightarrow\kappa>0$ for some constant $\kappa$. Bai (2009) estimator is asymptotically biased, in general, but an asymptotically valid bias-correction is established under slightly stronger conditions. Moon & Weidner (2017) establish $\sqrt{NT}$-asymptotics, again when $N$ and $T$ diverge at the same speed (i.e. $N/T\rightarrow\kappa>0$).

This paper proceeds as follows. Section 2 illustrates the general model and the assumptions required for estimation of regressions with unit-specific parameters. The asymptotic results for the OLS, UGLS and GLS estimators are presented in Section 3. Section 4 describes estimation and inference of the coefficients to common regressors. The technical contributions of the paper are discussed and highlighted in Section 5 Section 6 discusses various issues related to the GLS estimator. In particular, we first explore the case when the regressors and the residuals do not depend on the same set of factors. Second, we discuss the conditions under which the feasible GLS will still work in the context of dynamic panels. Third, despite the inefficiency of the feasible GLS, we explain how substantial efficiency gains can be achieved by a multi-step version of the GLS estimator. Fourth, we describe how consistent estimation of the asymptotic covariance matrix can be obtained. Fifth, we describe how to implement our estimator to cross-sectional regressions with time-varying coefficients. Our theoretical results are corroborated by a set of Monte Carlo experiments described in Section 7. An empirical application, which investigates whether firms’ characteristics are relevant to individual stock returns, is presented in Section 8. Section 9 concludes. The proofs of our theorems are reported in Appendix B, relying on three technical results, enunciated in Appendix A. Appendix C defines some quantities of interest for the construction of the GLS estimator, in particular regarding the (inverse of the) covariance matrix of the residuals. The Supplement contains appendices D-J with the proofs of additional material, that serve out main results.

Hereafter we use the following notation. Let $\bm{A}(R\times C)$ denotes a generic real $R\times C$ matrix with entries $a_{nm}$; in short $\bm{A}=[a_{nm}]_{n,m=1}^{R,C}$, or simply $\bm{A}=[a_{nm}]$ when the matrix’s dimension is clear. Similarly, $\bm{a}$ denotes a generic column vectors of length $R$ with element $a_{n}$; in short $\bm{a}=[a_{n}]_{n=1}^{R}$. The transpose of a $\bm{A}$ is denoted by $\bm{A}^{\prime}$. If $R=C$, $\lambda_{1}(\bm{A})$ and $\lambda_{N}(\bm{A})$ denote the minimum and the maximum eigenvalue of $\bf{A}$, respectively. With $\bm{A}>(\geq)0$ we mean that $\bm{A}$ is positive definite (positive semi positive). Let $\left\|\bm{A}\right\|_{sp}=\sqrt{\lambda_{N}\left(\bm{A}^{\prime}\bm{A}\right)}$ denotes the spectral norm of $\bm{A}$, and $\|\bm{A}\|=\sqrt{{\rm tr}\left(\bm{A}^{\prime}\bm{A}\right)}$, where ${\rm tr}(\cdot)$ denotes the trace, is the Frobenius norm. When $R=C$ we define the column and row norm of $\bm{A}$ as $\|\bm{A}\|_{col}=\max_{1\leq n\leq C}\sum_{n^{\prime}=1}^{C}|a_{nn^{\prime}}|$ and $\|\bm{A}\|_{row}=\max_{1\leq n^{\prime}\leq C}\sum_{n=1}^{C}|a_{nn^{\prime}}|$, respectively. Furthermore, for $C>R$, we use $\mathscr{P}_{\bm{A}}=\bm{A(A^{\prime}A)^{+}A^{\prime}}$, where $\bm{A}^{+}$ denotes the Moore-Penrose generalized inverse of $\bm{A}$ and $\mathscr{M}_{A}=\bm{I}_{C}-\mathscr{P}_{A}$, where $\bm{I}_{C}$ is the identity matrix of dimension $C\times C$. If $\bm{A}$ has full column rank, $\bm{A}_{\bot}$ denotes the $C\times(C-R)$ matrix satisfying $\bm{A}_{\bot}^{\prime}\bm{A}=\bm{0}$, where $\bm{0}$ is a matrix of zeros, and $\bm{A}^{\prime}_{\bot}\bm{A}_{\bot}=\bm{I}_{C-R}$. We use $``{\xrightarrow{p}}"$, $``{\xrightarrow{d}}"$ to denote convergence in probability and convergence in distribution, respectively, and $\mathcal{N}\left(\bm{a},\bm{B}\right)$ denote a random vector normally distributed with mean and covariance matrix equal to $\bm{a},\bm{B}$, respectively. For $\bm{A}(C\times R),\bm{B}(C\times C)$ and $\bm{C}(C\times P)$ being three random matrices, then $\bm{\Sigma}_{A^{\prime}BC}$ denotes the probability limit (when finite) of $C^{-1}\bm{A^{\prime}BC}$ as $C\rightarrow\infty$. For the $C\times R$ random matrices $\bm{A}_{N},\bm{B}_{N}$ that are functions of $N$ we write $\bm{A}_{N}\approx\bm{B}_{N}$ if $\left\|\bm{A}_{N}-\bm{B}_{N}\right\|\xrightarrow{p}0$ when $N\to\infty$. $\operatorname{\mathscr{F}}(\bm{A})$ denotes the sigma-algebra generated by the random matrix $\bm{A}$, $\mathbb{P}(\cdot)$ and $\mathbb{E}(\cdot)$ indicate the probability of an event and the expectation of a random variable, respectively. In the sequel, $\kappa$ denotes a generic, positive constant, which need not to be the same every time we use it.

2 Model: definitions and assumptions

Assume that the observed variables obey a linear regression model with $S$ common observed regressors ${\bf d}_{t}=(d_{t1},\dots,d_{tS})^{\prime}$ and $K$ heterogeneous regressors ${\bf x}_{it}=(\mathrm{x}_{it1}\ldots\mathrm{x}_{itK})^{\prime}$. Following the convenient specification put forward by Pesaran (2006), the model for the $i$th unit can be expressed, in matrix form, as

[TABLE]

for an observed $T\times 1$ vector ${\bf y}_{i}=(\mathrm{y}_{i1},\dots,\mathrm{y}_{iT})^{\prime}$, an observed $T\times S$ matrix ${\bf D}=({\bf d}_{1}\ldots{\bf d}_{T})^{\prime}$ of common regressors, an observed $T\times K$ matrix ${\bf X}_{i}=({\bf x}_{i1}\ldots{\bf x}_{iT})^{\prime}$ of unit-specific regressors, and an unobserved $T\times 1$ vector ${\bf u_{i}}=(\mathrm{u}_{i1}\ldots\mathrm{u}_{iT})^{\prime}$. In turn, the innovation vector satisfies the factor structure:

[TABLE]

for an unobserved $M\times 1$ vector of factor loadings ${\bf b}_{i}$, an unobserved $T\times M$ matrix of common factors ${\bf F}=({\bf f}_{1},...,{\bf f}_{T})^{\prime}$ and an unobserved $T\times 1$ vector of idiosyncratic innovations ${\bm{\varepsilon}}_{i}=(\varepsilon_{i1}\ldots\varepsilon_{iT})^{\prime}$. The unit specific regressors satisfy:

[TABLE]

for an unobserved $S\times K$ matrix of factor loadings ${\bf\Delta}_{i}=(\bm{\delta}_{i1}\ldots\bm{\delta}_{iS})^{\prime}$ with $\bm{\delta}_{il}=(\delta_{il1}\ldots\delta_{ilK})^{\prime}$, an unobserved $M\times K$ matrix of factor loadings ${\bf\Gamma}_{i}=(\bm{\gamma}_{i1}\ldots\bm{\gamma}_{iM})^{\prime}$ with $\bm{\gamma}_{il}=(\gamma_{il1}\ldots\gamma_{ilK})^{\prime}$, and an unobserved $T\times K$ matrix of idiosyncratic innovations ${\bf V}_{i}=(\bm{\mathrm{v}}_{i1}\ldots\bm{\mathrm{v}}_{iT})^{\prime}$ with $\bm{\mathrm{v}}_{it}=(v_{it1},\dots v_{itK})^{\prime}$. The maintained assumption here is that $K$, $S$ and $M$ *do not vary * with $T$ and $N$. Moreover, we do not need to impose *any * relationship between them so that, in particular, $M$ can be either smaller, equal or bigger than $K$. Although model (1) is written as a single regression across time for a given $i$, we assume that in fact a panel of observations $\{{\bf y}_{1}\ldots{\bf y}_{N},{\bf X}_{1}\ldots{\bf X}_{N}\}$ is available and fully used within our methodology.

As explained below, throughout our analysis we always *de-mean * the data by $\mathscr{M}_{\bm{\mathrm{D}}}$. This allows to avoid making any assumptions on $\bm{\Delta_{i}}$. We now present our assumptions which, thank to the detailed specification of model (1)-(3), appear relatively primitive.

Assumption 2.1** (idiosyncratic innovation $\varepsilon_{it}$)**

The $N\times 1$ vector $\bm{\varepsilon}_{t}=({\varepsilon}_{1t}...{\varepsilon}_{Nt})^{\prime}$ satisfies the following equation

[TABLE]

where the $N\times N$ matrix of constants $\bm{R}=[r_{ij}]$ satisfies $\parallel\!\!\bm{R}\!\!\parallel_{row}+\parallel\!\!\bm{R}\!\!\parallel_{col}\,<\infty$, $\inf_{i}\sum_{j=1}^{N}|r_{ij}|>\kappa$ for some $\kappa>0$, and the elements of the $N\times 1$ vector $\bm{a_{t}}=\left(a_{1t},a_{2t},\dots,a_{Nt}\right)^{\prime}$ follow a linear process:

[TABLE]

where the sequence $\{\eta_{it}\}$ is independent and identically distributed across $i$ and $t$ with $\mathbb{E}\eta_{it}=0$ and $\mathbb{E}|\eta_{it}|^{6}<\infty$. Moreover, for every complex number $z\in\mathbb{C}$,

[TABLE]

Remark 2.1

Assumption 2.1 is similar to Assumptions 1 and 2 in Pesaran & Tosetti (2011) and, with same variations, this form of cross-sectional and time dependence has been adopted also by Moon & Weidner (2017), Moon & Weidner (2015) and Onatski (2015). The above assumption turns out to be extremely convenient for establishing the asymptotic distribution of the feasible and unfeasible GLS estimators along the lines of Theorem 1 in Robinson & Hidalgo (1997).

Remark 2.2

Assumption 2.1 implies that, for every $2\leq h,\ell\leq 6$:

[TABLE]

where the summands are the cumulants of order $h$ of $\varepsilon_{i_{1}t_{1}},\varepsilon_{i_{2}t_{2}}\cdots,\varepsilon_{i_{\ell}t_{h}}$.

Remark 2.3

By Brockwell & Davis (1991), Proposition 4.5.3, (6) implies that the eigenvalues of the covariance matrices of $\bm{a_{j}}=(a_{j1},\dots a_{jT})^{\prime}$ are bounded, and greater than $\kappa$ for every $i$. Easy calculations give $\bm{\Xi_{i}}=\sum_{j=1}^{N}r_{ij}^{2}\mathbb{E}\bm{a}_{j}\bm{a}_{j}^{\prime}$, implying that $\inf_{i}\lambda_{1}(\bm{\Xi_{i}})>\kappa$ and $\sup_{i}\lambda_{T}(\bm{\Xi_{i}})<\infty$.

Assumption 2.2** (regressor innovation ${\bf{V}}_{\bm{i}}$)**

The sequence $\{{v}_{itk}\}$ have zero mean, $\sup_{i}\sup_{k}\sup_{t}\mathbb{E}|\!v_{itk}\!|^{12}<\infty$ and they satisfy, for every $2\leq h,\ell,s\leq 14$ and $2\leq j\leq h$:

[TABLE]

Moreover, $\inf_{i}\lambda_{1}\left(\mathbb{E}\bm{\mathrm{v}}_{it}\bm{\mathrm{v}}^{\prime}_{it}\right)>\kappa$, where $\bm{\mathrm{v}^{\prime}_{it}}=(v_{1it},\dots,v_{Kit})$.

Remark 2.4

Assumption 2.2 implies that $T^{-1}{\bf{V}}_{\bm{i}}^{\prime}{\bf{V}}_{\bm{i}}\xrightarrow{p}T^{-1}\sum_{t=1}^{T}\mathbb{E}\bm{\mathrm{v}}_{it}\bm{\mathrm{v}}^{\prime}_{it}=:\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}{\bf{V}}_{\bm{i}}}$, with $\inf_{i}\lambda_{1}\left(\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}{\bf{V}}_{\bm{i}}}\right)>\kappa$. It follows that $\sup_{i}\left\|\left({\bf{V}}_{\bm{i}}^{\prime}{\bf{V}}_{\bm{i}}/T\right)^{-1}\right\|=O_{p}(1)$.

Remark 2.5

The $v_{itk}$ can be interpreted as the high-rank components of the regressors $\mathrm{x}_{itk}$, adopting Moon & Weidner (2015) terminology, as opposed to the $\bm{\mathrm{D}}$ which represent the low-rank components. For instance, if for each $k$ the $v_{itk}$ are generated as $\varepsilon_{it}$ in Assumption 2.1, one obtains $\bm{V_{k}}=\left[v_{kti}\right]_{t,i=1}^{T,N}=O_{p}(\sqrt{\max(N,T)})$ for every $k$ (see the discussion in Moon & Weidner (2015), Appendix 1 and Onatski (2015)). In contrast, $(\sum_{i=1}^{N}\parallel\!\!\bm{\mathrm{D}}\!\!\parallel^{2})^{1\over 2}=(N\parallel\!\!\bm{\mathrm{D}}\!\!\parallel^{2})^{1\over 2}=O_{p}(\sqrt{(NT})$.

Assumption 2.3** (latent and observed factors)**

Set $\bm{\mathrm{Z}}=(\bm{\mathrm{D}},\bm{\mathrm{F}})=[\mathrm{z}_{ti}]$ for $1\leq t\leq T$ and $1\leq j\leq M+S<\infty$. Then,

[TABLE]

and $\bm{\Sigma}_{\bm{\mathrm{D}}^{\prime}\bm{\mathrm{D}}}>0$, $\bm{\Sigma}_{\bm{\mathrm{F}}^{\prime}\bm{\mathrm{F}}}>0$. Moreover, we assume $\mathbb{E}\left\|\bm{\mathrm{z}_{t}}\right\|^{4}<\infty.$ where $\bm{\mathrm{z}_{t}}=(\mathrm{z}_{t,1},\dots,\mathrm{z}_{t,M+S})^{\prime}$

Remark 2.6

Equation (7) implies that $\bm{\mathrm{F}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}>0$ (see Lütkepohl (1996), Result (4), Section 9.11.2).

Remark 2.7

Although not strictly necessary, we are ruling out trending behaviours in $\bm{\mathrm{D}}$ and $\bm{\mathrm{F}}$. However, $\bm{\mathrm{D}}$ and $\bm{\mathrm{F}}$ are allowed to be cross-correlated as well as serially correlated. although not perfectly collinear. For instance, the joint dynamics of $\bm{\mathrm{Z}}$ could be described by a multivariate stationary ARMA.

Assumption 2.4** (regressors)**

For every $i$, the matrix of unit specific regressors $\bm{\mathrm{X}_{i}}$ and the matrix of common regressors $\bm{\mathrm{D}}$ have full row rank. Moreover, setting $\bf{Z}_{i}:=[\bm{\mathrm{D}},\bm{\mathrm{X}_{i}}]$, $N^{-1}\sum_{i=1}^{N}\mathscr{M}_{\bf{Z}_{i}}\bm{\mathrm{u}_{i}}\bm{\mathrm{u}_{i}}^{\prime}\mathscr{M}_{\bf{Z}_{i}}$ has always rank $T-S$ for sufficiently large $N$ and $T$.

Remark 2.8

Assumption 2.4 requires enough cross section heterogeneity of the $\bm{\mathrm{X}_{i}}$’s across individuals. Simple manipulations show that

[TABLE]

implying that the empirical covariance matrix $\bm{\hat{\mathcal{S}}_{N}}$ defined in (19) is invertible.

Assumption 2.5** (loadings $\bm{\mathrm{b}_{i}}$ and ${\bf\Gamma}_{i}$)**

$\bm{\Gamma_{i}}$* and $\bm{\mathrm{b}_{i}}$ are non-random such that $\left\|\bm{\Gamma_{i}}\right\|<\infty$ and $\left\|\bm{\mathrm{b}_{i}}\right\|<\infty$ and, for $N>M$,*

[TABLE]

and

[TABLE]

is positive definite with

[TABLE]

Remark 2.9

Condition (8) implies that the factor structure (2) is strong, as defined in Pesaran & Tosetti (2011). This is commonly assumed in the literature. The technical condition (9) is used in the proof of Theorem 3.2. As shown in Section H.1 in the Supp. Material, Lemma H.1(iv), the matrices in brackets are of full rank. Hence, (9) will be satisfied when there is enough cross-sectional heterogeneity in the sample. Finally, our results will not change if random loadings are assumed (and cross-sectionally independent from other parameters).

Assumption 2.6** (independence)**

The $f_{mt},v_{ksi},\varepsilon_{uj}$ are mutually independent for every $i,j$ and $t,s,u$ and $m,k$.

Remark 2.10

We are not allowing for any correlation between any entries of $\bm{\varepsilon_{j}}$ and $\bm{\mathrm{X}_{i}}$. This rules out the possibility that $\bm{\mathrm{X}_{i}}$ contains a weakly exogenous component, and in this respect we are similar to Pesaran (2006) and Bai (2009). The implications from generalizing this assumption, in particular when considering dynamic panels where one element of $\bm{\mathrm{X}_{i}}$ represents the lagged dependent variable, are discussed in Section 6.2.

Remark 2.11

Assumptions 2.2, 2.3 and 2.6 and Remark 2.6 imply that $T^{-1}\bm{\mathrm{X}_{i}}^{\prime}\bm{\mathrm{X}_{i}}\xrightarrow{p}\bm{\Sigma}_{\bm{\mathrm{X}_{i}}^{\prime}\bm{\mathrm{X}_{i}}}>0$ and $T^{-1}\bm{\mathrm{X}_{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{X}_{i}}\xrightarrow{p}\bm{\Sigma}_{\bm{\mathrm{X}_{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{X}_{i}}}>0$, for every $i$. Hence $\left\|(T^{-1}\bm{\mathrm{X}_{i}}^{\prime}\bm{\mathrm{X}_{i}})^{-1}\right\|=O_{p}(1)$ and $\left\|(T^{-1}\bm{\mathrm{X}_{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{X}_{i}})^{-1}\right\|=O_{p}(1)$ for $T$ large enough.

3 Estimators: definitions and asymptotics

Our main objective is to estimate the heterogeneous slope coefficients $\bm{\beta}_{i}$ of (1). However, estimation of the coefficients ${\bm{\alpha}}_{i}$ of the common regressors is also discussed in Section 4. Hence, without loss of generality, we premultiply both sides of (1) by the projection matrix $\mathscr{M}_{\bm{\mathrm{D}}}$, obtaining

[TABLE]

We consider three different estimators for the parameters ${\bm{\beta}}_{i}$, namely the OLS, the unfeasible and feasible GLS estimators. Regarding the OLS estimator for $\bm{\beta_{i}}$:

[TABLE]

We now consider GLS estimation. Define the cross-sectional averages of the individual covariance matrices of the $\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{u}_{i}}$, conditional on sigma algebra generated by $\bm{Z}$, defined in Assumption 2.3:

[TABLE]

We assume without loss of generality that $\bf{d_{t}}$ includes an element equal to one, i.e. we allow for an intercept term, leading to $\mathbb{E}\bm{\mathrm{u}_{i}}=0$. The presence of $\mathscr{M}_{\bm{\mathrm{D}}}$ could cause some complications in the definition of the GLS estimator since the $\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{u}_{i}}$ have a singular covariance matrix. We show how to solve this issue and obtain a model with a non-singular residual covariance matrix that can be used to construct the GLS estimator.

Proceeding along the lines of Magnus & Neudecker (1988), Section 11 in Chapter 13, one gets the UGLS estimator when the residual covariance matrix to model (11) is singular:

[TABLE]

By Lemma D.2 in the Supp. Material

[TABLE]

where $\bm{\mathrm{D}}_{\bot}$ is the $T\times(T\!-\!S)$ full rank matrix such that $\mathscr{M}_{\bm{\mathrm{D}}}=\bm{\mathrm{D}}_{\bot}\bm{\mathrm{D}}_{\bot}^{\prime}$ where $\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\mathrm{D}}_{\bot}=\bm{I}_{T-S}$. Assumption 2.4 and display (13) imply that the inverse in (14) is well defined for any $T$. By substitution, setting for simplicity

[TABLE]

one obtains

[TABLE]

where we set $\bm{\mathcal{S}}_{N}=\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\mathrm{S}}_{N}\bm{\mathrm{D}}_{\bot}.$ This means that the UGLS has now the more conventional expression of the generalized least squares for the model

[TABLE]

without involving Moore-Penrose matrices. Pre-multiplying the data by $\bm{\mathrm{D}}_{\bot}^{\prime}$ reduces the sample size by $S$ units since now the $\bm{\mathcal{y}_{i}}$ and the $\bm{\mathcal{X}_{i}}$ have $T-S$ rows. Likewise, considering again model (17), an equivalent representation of (12) is $\bm{\hat{\beta}_{i}^{OLS}}=(\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{X}_{i}})^{-1}\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{y}_{i}}$.

Along the same lines, our proposed *feasible * GLS estimator is given by

[TABLE]

where

[TABLE]

for $N$ and $T$ large enough, by Assumption 2.4 and Remark 2.8, $\bm{\hat{\mathcal{S}}_{N}}$ has full rank. The following two theorems enunciates the asymptotic distribution of the OLS, UGLS and GLS estimators, respectively. The proofs are given in Appendixes B.1 and B.2, respectively. Further details are provided in the Supp. Material.

Theorem 3.1

When Assumptions 2.1, 2.2, 2.3, 2.4, 2.5 and 2.6 hold, for any $N$ and as $T\to\infty$

(i) (OLS estimator)

[TABLE]

where

[TABLE]

is the bias term, and the asymptotic covariance matrix equals

[TABLE]

setting

[TABLE]

(ii) (UGLS estimator)

[TABLE]

with $\bm{\Sigma}^{\star}_{N}:=\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}{\bf{V}}_{\bm{i}}}^{-1}\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}\bm{\Xi_{i}}\bm{\Xi^{-1}_{N}}{\bf{V}}_{\bm{i}}}\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}{\bf{V}}_{\bm{i}}}^{-1}$.

Remark 3.1

The OLS estimator is affected by a first-order bias. It will be asymptotically unbiased if either $\bm{\mathrm{b}_{i}}=\bm{0}$ or $\bm{\Gamma_{i}}=\bm{0}$ or, alternatively, for diagonal $\bm{\Sigma}_{\bm{\mathcal{F}}^{\prime}\bm{\mathcal{F}}}$ as well as with $\bm{\Gamma_{i}}$ and $\bm{\mathrm{b}_{i}}$ satisfying $\gamma_{il}b_{il}=0$ for every $l$ and $i$. Essentially, this means that the entries of $\bm{\Gamma_{i}}$ are non zero whenever the corresponding entries of $\bm{\mathrm{b}_{i}}$ are zero, for the same row $l$, and viceversa. More in general, no bias arises if ${\bf b}_{i}$ belongs to the null space of $\bm{\Sigma}_{\bm{\mathcal{F}}^{\prime}\bm{\mathcal{F}}}\bm{\Gamma_{i}}$, assuming $M>K$.

Remark 3.2

One can assume without loss of generality that the same latent factors $\bm{\mathrm{F}}$ enter into $\bm{X}_{i}$ and $\bm{\mathrm{u}_{i}}$. In fact, assume $\bm{\mathrm{u}_{i}}=\bm{\mathrm{G}}\bm{b}_{i}+\bm{\varepsilon}_{i}$ with the rows of $\bm{\mathrm{G}}$ correlated, but not identical to the rows of $\bm{\mathrm{F}}$. Then the bias takes the form

[TABLE]

exploiting the decomposition $\bm{\mathrm{G}}=\mathscr{P}_{F}\bm{\mathrm{G}}+\mathscr{M}_{F}\bm{\mathrm{G}}$. Hence the bias will only be non-zero due to the portion of $\bm{\mathrm{G}}$ correlated with $\bm{\mathrm{F}}$. The same consideration applies to the GLS estimator. In Section 6.1 we explore more in details the implications of having different, yet correlated, factor structures for regressors and innovations.

Remark 3.3

The UGLS estimator is asymptotically unbiased, consistent and asymptotically normal as $T\rightarrow\infty$. Moreover, the UGLS estimator can be efficient in the GLS sense. In particular, when the $\bm{\varepsilon}_{i}$ are not (unconditionally) heteroskedastic, namely $\bm{\Xi_{i}}={\bm{\Xi}}$, then the UGLS asymptotic covariance matrix does not have the sandwich form, unlike for the OLS estimator. One can define the UGLS differently, for instance replacing $\bm{\mathrm{S}}_{N}$ with $\bm{\mathrm{F}}\bm{\mathrm{B}_{N}}\bm{\mathrm{F}}^{\prime}+\bm{\Xi_{i}}$ in (14). However, our definition of the UGLS estimator makes it closer to the population counterpart to the class of feasible GLS estimators here studied.

We now present the main result of the paper.

Theorem 3.2

When Assumptions 2.1, 2.2, 2.3, 2.4, 2.5 and 2.6 hold, as $1/T+T/N\rightarrow 0$,

[TABLE]

and, as $(1/T)+(T^{3}/N^{2})\to 0$, then

[TABLE]

*where *

[TABLE]

with $\bm{\Sigma_{\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{X}_{i}}}}$ defined in (23).

Remark 3.4

The GLS estimator is asymptotically unbiased, consistent and asymptotically normal as both $N,T\rightarrow\infty$ such that $T^{3}/N^{2}\rightarrow 0$. The feasible GLS estimator is not efficient in general. A multi-step generalization achieves substantial efficiency gains, see Section 6.

4 Common regressors

We now consider estimation of the coefficients $\bm{\alpha}_{i}$ to the common regressors $\bm{\mathrm{D}}$ in model (1). A natural generalization of the GLS estimator would be

[TABLE]

where $\bf{Z}_{i}$ has been defined in Assumption 2.4 and

[TABLE]

that is $\bm{\hat{\mathrm{u}}_{i}}$ are the OLS residuals, for $(\hat{\bm{\alpha}}_{i}^{OLS^{\prime}},\hat{\bm{\beta}}_{i}^{OLS^{\prime}})^{\prime}:=(\bf{Z}_{i}^{\prime}\bf{Z}_{i})^{-1}\bf{Z}_{i}^{\prime}\bm{\mathrm{y}}_{i}$. However, we show in Theorem I.1 (Supp. Material, Appendix I) that $\tilde{\bm{\alpha}}_{i}^{GLS}={\bm{0}}_{M}$ and $\tilde{\bm{\beta}}_{i}^{GLS}=\hat{\bm{\beta}}_{i}^{GLS}$ due to a cancellation that occurs as a consequence of $\bm{\mathrm{D}}$ being common across units. If the joint distribution for the estimators of $({\bm{\alpha}}_{i}^{\prime},{\bm{\beta}}_{i}^{\prime})^{\prime}$ is not required, one can estimate the $\bm{\alpha}_{i}$ as the projection of $\bm{\mathrm{y}}_{i}-\bm{\mathrm{X}_{i}}\hat{\bm{\beta}}_{i}^{GLS}$ on $\bm{\mathrm{D}}$ yielding

[TABLE]

Using our theory, its asymptotic distribution follows (see Theorem I.2 ins the Spp. Material for further details). Note that the additional assumption $\bm{\mathrm{F}}^{\prime}\bm{\mathrm{D}}={\bm{0}}$ is required. For example, if we are interested in a model with an intercept term, heterogenous across units, such as $\bm{\mathrm{D}}{\bm{\alpha}}_{i}=\bm{\iota}_{T}\alpha_{i1}+\bm{\mathrm{D}}_{2}\bm{\alpha}_{i2}$, with $\bm{\alpha}_{i}=(\alpha_{i1},\bm{\alpha}_{i2}^{\prime})^{\prime},\bm{\mathrm{D}}=(\bm{\iota}_{T},\bm{\mathrm{D}}_{2})$, then one of the restrictions $\bm{\mathrm{F}}^{\prime}\bm{\mathrm{D}}={\bm{0}}$ is simply $\sum_{t=1}^{T}\bm{f}_{t}=\bm{0}$. If, moreover, a grand-mean is also allowed for, such as $\bm{\mathrm{D}}\bm{\alpha}_{i}=\bm{\iota}_{T}\alpha_{3}+\bm{\iota}_{T}\alpha_{i1}+\bm{\mathrm{D}}_{2}\bm{\alpha}_{i2}$, then the additional restriction $\sum_{i=1}^{N}\alpha_{i1}=0$ is needed. Similar identification conditions are discussed in Bai (2009) and Moon & Weidner (2017).777Most of the papers on estimation of panel regressions with so-called interactive fixed effects, such as ours, focus exclusively on the coefficients to the heterogeneous time-varying regressors. Among the few exceptions, is Bai (2009) who shows that, without further identification assumption, estimation of the coefficient to common regressors is possible only for constant parameters. In contrast, for the case of non-constant coefficients further identification assumptions similar to ours are needed. Moon & Weidner (2017) study the same estimator of Bai (2009) under weaker conditions on the regressors, allowing for instance for pre-determinatedness. Our identification condition for the coefficients to common regressors implies their weaker corresponding assumption. They focus exclusively on the case of constant regression coefficients.

If instead the joint distribution for estimators of $\bm{\alpha}_{i}$ and $\bm{\beta}_{i}$ is required, this can be achieved by a slight modification of our GLS estimator, namely

[TABLE]

for the non-singular matrix

[TABLE]

Non-singularity of $\breve{\bm{S}}_{N}$ follows by augmenting the matrix $\breve{\bm{S}}_{N}$, of rank $T-S$, with the projection matrix $\mathscr{P}_{\bm{\mathrm{D}}}$ of rank $S$. Scaling by $N^{-1}{\rm tr}\left(\tilde{\bm{S}}_{N}\right)$ in not required by the asymptotic theory but could be relevant in finite-samples to ensure the same order of magnitude of the two terms in $\breve{\bm{S}}_{N}$. It turns out that the same identification condition $\bm{\mathrm{F}}^{\prime}\bm{\mathrm{D}}={\bm{0}}$, discussed above, is required. Monte Carlo experiments are reported in Section 7 to assess the small-sample properties of these estimators.

5 Technical contributions

The asymptotics for the GLS estimator requires four key auxiliary results, enunciated in Appendix A, which could be useful in a broader set of statistical problems. The main reason for this complexity is that, unlike most of the existing theoretical results on GLS estimation, we are not restricting the number of free elements of the weighting matrix to be finite. Indeed, in our case the number of free elements of the weighting matrix is $O(T^{2})$ and hence rapidly increasing with $T$. To tackle the curse-of-dimensionality issue, we exploit the approximate factor structure of the weighted matrix, that we write as $\bm{E=FAF^{\prime}+C}$, for (possibly random) $M_{1}\times M_{1}$ matrix $\bm{A}>0$ and $T\times T$ matrix $\bm{C}>0$ for every finite $T$. The inverse has a convenient form thanks to the Sherman-Morrison formula (see Appendix D.1).

Lemma A.1 establishes the asymptotic orthogonality between the inverse of the matrix $\bm{E}$, and the factor $\bm{\mathrm{F}}$. More precisely, when $\bm{A}$ and $\bm{C}$ satisfy a set of mild regularity conditions

[TABLE]

This is a remarkably fast rate given that $\bm{E}^{-1}\bm{\mathrm{F}}$ is $T\times M_{1}$ dimensional, with $M_{1}$ fixed, hence with its number of rows increasing with $T$. It implies that for a large class of $T\times M_{2}$ matrices $\bm{P}$ (satisfying the mild regularity conditions of Lemma A.1) possibly unrelated to both $\bm{E}$ and $\bm{F}$, then $\bm{P}^{\prime}\bm{E}^{-1}\bm{F}=O_{p}(1)$ and, when the entries of $\bm{P}$ have zero mean and are stochastically independent of $\bm{E}$ and $\bm{F}$, then $\bm{P}^{\prime}\bm{E}^{-1}\bm{F}=O_{p}(T^{-\nicefrac{{1}}{{2}}})$. These rates are very different from the usual case, arising when $\bm{E}$ and $\bm{F}$ are unrelated. For example, when $\bm{A}=\bm{0}$, under the same assumptions on $\bm{P}$, one gets that $\bm{P}^{\prime}\bm{E}^{-1}\bm{F}$ is of order $O_{p}(T^{\nicefrac{{1}}{{2}}})$ or $O_{p}(T)$, depending on whether $\bm{P}$ has zero or non-zero mean, respectively, assuming that $\bm{F}$, $\bm{A}$ and $\bm{P}$ are mutually independent.

The asymptotic orthogonality (33) plays a crucial role in establishing the asymptotics for the GLS (and UGLS) estimator. To better understand this, consider the following decomposition of the GLS estimator:

[TABLE]

Since $\bm{\hat{\mathcal{S}}_{N}}=\bm{\mathcal{F}}\bm{\hat{A}_{N}}\bm{\mathcal{F}}^{\prime}+\bm{\hat{\bm{\mathcal{C}}}_{N}}$, for some random matrices $\bm{\hat{A}_{N}},\bm{\hat{\bm{\mathcal{C}}}_{N}}$ (specified in Appendix C) function of both $T$ and $N$, we show that Lemma A.1 applies to the bias term, namely the first term in (34). In particular, by (33) term $\bm{\mathcal{X}_{i}}^{\prime}\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\mathcal{F}}$ is of a smaller order of magnitude (and vanishes asymptotically) than $\bm{\mathcal{X}_{i}}^{\prime}\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\epsilon_{i}}$ as long as $N$ is diverging faster than $T$. Note that both the dimension of $\bm{\hat{\bm{\mathcal{C}}}}_{N}$, as well as its elements, are changing with $T$. The faster rate for $N$ is demanded for by the need to have $\bm{\hat{A}_{N}}$ and $\bm{\hat{\bm{\mathcal{C}}}}_{N}$ with the desired limiting properties.

To save notation we rename $\bm{\hat{\beta}_{i}^{GLS}}$ in (18) as $\bm{\hat{\beta}_{i}(\bm{\hat{\mathcal{S}}_{N}})}$ and define $\bm{\hat{\beta}_{i}(\bm{\mathcal{H}_{N}})}$ the estimator obtained replacing the weighting matrix $\bm{\hat{\mathcal{S}}_{N}}$ with $\bm{\mathcal{H}_{N}}$ in (18). The matrix $\bm{\mathcal{H}_{N}}=\bm{\mathcal{F}}\bm{A_{N}}\bm{\mathcal{F}}^{\prime}+\bm{\mathcal{C}_{N}}$, defined in Appendix B.2 (display (B.22)), is non-stochastic, if $\bm{\mathrm{Z}}$ is fixed, and, most importantly, satisfies the assumption of Lemma A.1. Existing conditions for the asymptotic equivalence of $\bm{\hat{\beta}_{i}(\bm{\hat{\mathcal{S}}_{N}})}$ and $\bm{\hat{\beta}_{i}(\bm{\mathcal{H}_{N}})}$ cannot be applied here. Although the inverse operator of a matrix is an analytic function, one cannot rely on the delta method due to the curse of dimensionality, namely the fact that the elements as well as the size of $\bm{\hat{\mathcal{S}}_{N}}$ are varying with $T$ and $N$. For similar reasons, element-wise convergence of $\bm{\hat{\mathcal{S}}_{N}}$ cannot be combined with the Slutsky’s Theorem, as discussed in Mandy & Martins-Filho (1994). Considering the absolute convergence $\left\|\bm{\hat{\mathcal{S}}_{N}}^{-1}-\bm{\mathcal{H}^{-1}_{N}}\right\|_{sp}$ and use random matrix theory is not a viable option. Even if we were able to obtain the optimal convergence rate $\left\|\bm{\hat{\mathcal{S}}_{N}}-\bm{\mathcal{H}_{N}}\right\|_{sp}=O_{p}(\sqrt{T/N})$ established by for i.i.d. data, it would be not sufficient to obtain $\sqrt{T}\left(\bm{\hat{\beta}_{i}(\bm{\hat{\mathcal{S}}_{N}})}-\bm{\hat{\beta}_{i}(\bm{\mathcal{H}_{N}})}\right)=o_{p}(1)$ without strengthening the Assumptions of Theorem 3.2. The latter convergence requires indeed proving that $T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\left(\bm{\hat{\mathcal{S}}_{N}}^{-1}-\bm{\mathcal{H}^{-1}_{N}}\right)\bm{\mathcal{u}_{i}}=o_{p}(1)$. We would have

[TABLE]

To prove our results we found convenient proceeding in two steps: firstly we prove that $T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\mathcal{u}_{i}}$ is asymptotically equivalent $T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\mathcal{u}_{i}}$, with $\SS$ as defined in Appendix C, equation (C.1) ,and secondly that the latter term is in turn asymptotically equivalent to $T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\mathcal{u}_{i}}$. Lemma A.2, a simple extension of a well known result in matrix algebra, entails that

[TABLE]

implying that $\bm{\mathcal{X}_{i}}^{\prime}\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\mathcal{u}_{i}}$ can be re-written as the (algebric) sum of $\bm{\mathcal{X}_{i}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\mathcal{u}_{i}}$, $\bm{\mathcal{X}_{i}}^{\prime}\bm{\Omega_{N}}^{-1}(\bm{\hat{\mathcal{S}}_{N}}-\SS)\bm{\Omega_{N}}^{-1}\bm{\mathcal{u}_{i}}$ and $\bm{\mathcal{X}_{i}}^{\prime}\bm{\Omega_{N}}^{-1}(\bm{\hat{\mathcal{S}}_{N}}-\SS)\bm{\Omega_{N}}^{-1}(\bm{\hat{\mathcal{S}}_{N}}-\SS)\bm{\Omega_{N}}^{-1}\bm{\mathcal{u}_{i}}$. In the proof we show that the second last term is of order $O_{p}(T/N^{\nicefrac{{1}}{{2}}})$ and that the last term is of order $O_{p}(T^{2}/N)$, respectively. Instead, the first leading term will exhibit the usual $O_{p}(T^{\nicefrac{{1}}{{2}}})$ rate of convergence. For the second and third terms to be asymptotically negligible, in terms of asymptotic distribution, one requires that $T^{3\over 2}/N$ goes to zero as $T$ increases. Unfortunately our approach requires lengthy calculations involving high order cumulants that are bounded using the diagram formula (see Appendix H in the Supp. Material).

A further step necessary to derive the convergence in distribution of the estimator is the derivation of the asymptotic distribution of $T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\mathcal{u}_{i}}$. We first show that the latter term is equivalent to $T^{-\nicefrac{{1}}{{2}}}{\bf{V}}_{\bm{i}}^{\prime}\bm{\mathrm{C}_{N}}^{-1}\bm{\varepsilon_{i}}$. In order to exploit the results in Robinson & Hidalgo (1997), the absolute row/column summability of $\bm{\mathrm{C}_{N}}^{-1}$ needs to be shown. To accomplish this task we first approximate $\bm{\mathrm{C}_{N}}$ with a circulant symmetric matrix, as further discussed in the Supp. Material, Appendix E. By Lemma A.3, that extends a result in Zygmund (1959), we prove that the inverse of the latter matrix has indeed bounded row norm.

6 Discussion and generalizations

In this section we describe various generalizations to our framework. In particular, we explain the consequences of allowing for different, yet related, factor structures in the regressors and residuals, respectively. Then, we show to derive a consistent estimator for the asymptotic covariance matrix of the GLS estimator. We also discuss how to achieve efficiency improvements of the GLS by an iterative procedure. Finally, we explain how our results apply to cross-sectional regressions with time-varying coefficients.

6.1 Different factor structures

So far we have assumed that the unit-specific regressors $\bm{\mathrm{X}_{i}}$ and the true residuals $\bm{\mathrm{u}_{i}}$ of (1) share the same common, latent, factors. We now explore the implications of allowing that possibly different, yet correlated, set of factors affect the regressors and the residuals, respectively. Let us here illustrate the UGLS case and then provide more details in the Supp. Material (Appendix J) for the (feasible) GLS. To simplify the exposition we assume that $\bm{\mathrm{D}}=\bm{0}$, and

[TABLE]

where $\bm{\mathrm{F}}_{1}(T\times M_{1})$ and $\bm{\mathrm{F}}_{2}(T\times M_{2})$ satisfy Assumption 2.3. By Remark A.1,

[TABLE]

Let $\bm{W}:=\bm{\Xi^{-1}_{N}}\bm{\mathrm{F}}$, then

[TABLE]

where $\bm{\mathrm{F}}^{\prime}_{2\bot}\bm{\mathrm{F}}_{2}=\bm{0}$.

It follows that even if $\bm{\mathrm{F}}_{1}\notin\textrm{sp}(\bm{\mathrm{F}}_{2})$, where $\textrm{sp}(\bm{\mathrm{F}}_{2})$ denote the space spanned by $\bm{\mathrm{F}}_{2}$, the UGLS estimator is still consistent but the term $\breve{\bm{\mathrm{F}}}_{1}^{\prime}\bm{\Xi_{i}}^{-1}\bm{\varepsilon_{i}}$, with $\breve{\bm{\mathrm{F}}}_{1}:=\bm{W}_{\bot}\left(\bm{\mathrm{F}}_{2\bot}^{\prime}\bm{W}_{\bot}\right)^{-1}\bm{\mathrm{F}}_{2}^{\prime}\bm{\mathrm{F}}_{1}\in\textrm{sp}(\bm{W}_{\bot})$, will contribute to the asymptotic distribution of the estimator.888We conjecture that one needs to assume on the $\breve{\mathrm{g}}_{t}$ the same conditions assumed by Robinson and Hidalgo (1997) about their $x_{t}$ (see their Condition 7) . In Appendix J (Supp. Material) we show, heuristically, that the FGLS estimator of (37) enjoys the same asymptotic properties stated in Theorem 3.2.

6.2 Dynamic models

Although our set-up allows for dynamics, through the dynamic autocorrelation of either the factors and the idiosyncratic error, our results extend to the case of dynamic panel with factor structure such as

[TABLE]

where $\bm{\mathrm{X}_{i}}$ and $\bm{\mathrm{u}_{i}}$ satisfy (3) and (2), respectively, and we set $\bm{y}_{-1,i}=(\mathrm{y}_{i0},\dots,\mathrm{y}_{iT-1})^{\prime}$, with first-order autoregressive coefficients satisfying $-1<\rho_{i}<1$ for every $i$. We set $\bm{D}=\bm{0}$ to simplify the exposition. Obviously one can re-write (39) as

[TABLE]

It turns out that applying our GLS estimator (18) to specification (40) will still work when further conditions are assumed on the idiosyncratic part of the residuals $\bm{\mathrm{u}_{i}}={\bf F}{\bf b}_{i}+\bm{\varepsilon}_{i}$, namely that the $\varepsilon_{it}$ are i.i.d. across time but have some degree of cross-correlation across $i$. A similar assumption is made by Chudik & Pesaran (2015), Assumption 1, also in the context of dynamic panel data models. Notice that our result is rather strong because we are *not * ruling out that the model residuals $u_{it}$ are dependent across time (and across $i$), through the factors ${\bf f}_{t}$. Moreover, again thanks to the factors ${\bf f}_{t}$, regressors and residuals are correlated and thus we are violating the classical strong-exogeneity assumption typically advocated in a GLS framework.

A technical proof goes beyond the scope, and the page limit, of the present paper but details, corroborated by Monte Carlo simulations, are available upon request.

6.3 Efficiency improvements

The form of the asymptotic covariance matrix of the GLS, indicated in Theorem 3.2, denotes lack of efficiency, unlike for the UGLS estimator case (in the special sense discussed). This arises because although $\bm{\hat{\mathcal{S}}_{N}}$ is *approximated * by the matrix $\bm{\mathcal{H}_{N}}$ defined in Appendix B.2, (in the sense that $\bm{\mathcal{X}_{i}}^{\prime}\bm{\hat{\mathcal{S}}_{N}}\bm{\mathcal{u}_{i}}=\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\mathcal{u}_{i}}+o_{p}(\sqrt{T})$), the latter does not coincide with the true covariance matrix $\bm{\mathcal{S}}_{N}$. This is to be expected since $\bm{\mathcal{H}_{N}}$ is constructed based on the OLS residuals $\bm{\hat{\mathcal{u}}_{i}}=\bm{\mathcal{y}_{i}}-\bm{\mathcal{X}_{i}}\bm{\hat{\beta}}_{i}^{OLS}$ where $\bm{\hat{\beta}}_{i}^{OLS}$ is non consistent for $\bm{\beta}_{i}$. However, a multi-step procedure can be envisaged that could achieve (near) asymptotic efficiency, or more precisely an estimator with an asymptotic distribution arbitrarily close to the UGLS estimator. We shall call the outcome of this procedure the iterated-GLS estimator. The first step would be to construct the GLS estimator as explained in the previous sections, which we now denominate as $\hat{\bm{\beta}}_{i}^{(1)}$. We then construct the associated residuals $\bm{\hat{\mathcal{u}}_{i}}^{(1)}=\bm{\mathcal{y}_{i}}-\bm{\mathcal{X}_{i}}\bm{\hat{\beta}}_{i}^{(1)}$. Notice that now $\bm{\hat{\beta}}_{i}^{(1)}$ is a consistent estimator for $\bm{\beta}_{i}$. The second step entails constructing $\bm{\hat{\mathcal{S}}_{N}}^{(1)}=N^{-1}\sum_{i=1}^{N}\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\hat{\mathcal{u}}_{i}}^{(1)}\bm{\hat{\mathcal{u}}_{i}}^{(1)^{\prime}}\bm{\mathrm{D}}_{\bot}$ and using it to obtain $\bm{\hat{\beta}}_{i}^{(2)}=\left(\bm{\mathcal{X}_{i}}^{\prime}\left(\bm{\hat{\mathcal{S}}_{N}}^{(1)}\right)^{-1}\bm{\mathcal{X}_{i}}\right)^{-1}\bm{\mathcal{X}_{i}}^{\prime}\left(\bm{\hat{\mathcal{S}}_{N}}^{(1)}\right)^{-1}\bm{\mathcal{y}_{i}}.$ In general the $h$th step entails constructing $\bm{\hat{\beta}}_{i}^{(h)}=\left(\bm{\mathcal{X}_{i}}^{\prime}\left(\bm{\hat{\mathcal{S}}_{N}}^{(h-1)}\right)^{-1}\bm{\mathcal{X}_{i}}\right)^{-1}\bm{\mathcal{X}_{i}}^{\prime}\left(\bm{\hat{\mathcal{S}}_{N}}^{(h-1)}\right)^{-1}\bm{\mathcal{y}_{i}}$, where $\bm{\hat{\mathcal{S}}_{N}}^{(h-1)}$ is obtained based on $\bm{\hat{\beta}}_{i}^{(h-1)}$. We conjecture that as $h$ increases, the asymptotic distribution of $\bm{\hat{\beta}}_{i}^{(h)}$ is getting arbitrarily close to the one of the UGLS. This is confirmed by the Monte Carlo experiments presented in Section 7. Although the theoretical analysis of this iterated-GLS is not developed here, techniques along the lines of the ones developed in the current paper would allow to establish the asymptotics. Indeed, since $\bm{\hat{\beta}}_{i}^{(h)}$ is consistent for $\bm{\beta}_{i}$ for any $h\geq 2$, the asymptotics should follow more easily than for the GLS estimator.

6.4 Estimation of asymptotic covariance matrix

Consistent estimation of the GLS asymptotic covariance matrix can be obtained in different ways, depending on the type of heteroskedasticity and correlation assumed for the $\bm{\varepsilon_{i}}$. For instance, using the results of Newey & West (1987), Theorem 2, one obtains the covariance matrix estimator for $\hat{\bm{\beta}}_{i}^{GLS}$

[TABLE]

setting $\hat{\bm{A}}_{ih}:=T^{-1}\sum_{t=h+1}^{T}\hat{u}_{it}^{GLS}\hat{u}_{it-h}^{GLS}\hat{\mathcal{x}}_{it}\hat{\bf\mathcal{x}}_{it-h}^{\prime},\,\,\,h=0,1,...T-1$, where $\hat{\bm{\mathcal{X}_{i}}}:=(\hat{\mathcal{x}}_{i1}...\hat{\mathcal{x}}_{iT})^{\prime}=\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\mathcal{X}_{i}}$ and $\hat{\bm{\mathcal{u}_{i}}}^{GLS}:=(\hat{u}_{1i}^{GLS}...\hat{u}_{iT}^{GLS})^{\prime}=\bm{\mathcal{y}_{i}}-\bm{\mathcal{X}_{i}}\hat{\bm{\beta}}_{i}^{GLS}$ and the bandwidth $n=n(T,N)$ grows slowly with $N$ and $T$. The same approach has been used in Pesaran (2006), eq. (51) and (52). Bai (2009), Section 7, provides estimators of the asymptotic covariance matrix when correlation and heteroskedasticity of either series or cross-section form is allowed for, using Newey & West (1987) and a partial-samplig approach, respectively. Note that these approaches cannot be applied to our case since require constant regression coefficients, involving averaging across both $N$ and $T$. Similar approaches have been used by Moon & Weidner (2017) and Moon & Weidner (2015) under more restrictive dependence assumptions.

Notice that, although $\hat{\bm{\mathcal{u}_{i}}}^{GLS}$ contains a factor structure, as it is evident by from its population counterpart $\bm{\mathcal{u}_{i}}=\bm{\mathcal{F}}\bm{\mathrm{b}_{i}}+\bm{\epsilon_{i}}$, the contribution of $\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\mathcal{F}}\bm{\mathrm{b}_{i}}$ to $\bm{\hat{\mathcal{S}}_{N}}^{-1}\hat{\bm{\mathcal{u}_{i}}}^{GLS}$ is (asymptotically) negligible with respect to the contribution of $\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\epsilon_{i}}$, whose asymptotic variance is require for consistent estimation of the asymptotic covariance matrix of $\bm{\hat{\beta}}_{i}^{GLS}$. An alternative approach consists of estimating the idiosyncratic component of $\hat{\bm{\mathcal{u}_{i}}}^{GLS}$ directly, for example by principal components, yielding $\hat{\bm{\epsilon_{i}}}^{GLS}:=(\hat{\epsilon}_{i1}^{GLS}\cdots\hat{\epsilon}_{iT}^{GLS})^{\prime}$ and then replacing $\hat{\bm{A}}_{ih}$ by $T^{-1}\sum_{t=h+1}^{T}\hat{\epsilon}_{it}^{GLS}\hat{\epsilon}_{it-h}^{GLS}\hat{\bf x}_{it}\hat{\bf x}_{it-h}^{\prime}$ into (41). Preliminary testing for the number of factors $M$ is required in this case, making it less appealing.

Consistent estimation of the asymptotic covariance matrix for $(\breve{\bm{\alpha}}_{i}^{GLS\prime},\breve{\bm{\beta}}_{i}^{GLS\prime})^{\prime}$ of (31) follows along the same lines, leading to:

[TABLE]

where $\breve{\bm{A}}_{ih}={1\over T}\sum_{t=h+1}^{T}\breve{u}_{it}^{GLS}\breve{u}_{it-h}^{GLS}\breve{\bf z}_{it}\breve{\bf z}_{it-h}^{\prime},\,\,\,h=0,1,...T-1$, where $\breve{\bm{\mathrm{Z}}}_{i}=(\breve{\bf z}_{i1}...\breve{\bf z}_{iT})^{\prime}=\breve{\bm{S}}_{N}^{-1}\bm{\mathrm{Z}}_{i}$ and $\breve{\bm{\mathcal{u}_{i}}}^{GLS}=(\breve{u}_{1i}^{GLS}...\breve{u}_{iT}^{GLS})^{\prime}=\bm{\mathrm{y}}_{i}-\bm{\mathrm{D}}\breve{\bm{\alpha}}_{i}^{GLS}-\bm{\mathrm{X}_{i}}\breve{\bm{\beta}}_{i}^{GLS}$.

6.5 Cross-sectional regressions

As an example of a cross-sectional regression with factor structure, consider Andrews (2005) model:

[TABLE]

where $(y_{it},{\bf x}_{it})$ are assumed $i.i.d.$ across units conditional on ${\bf c}_{1t},\,{\bf C}_{2t}$ by Andrews (2005), Assumption 1, with

[TABLE]

with ${\bf c}_{1t},\,{\bf u}_{i}^{*}$ are $d_{1}\times 1$ random vectors and ${\bf C}_{2t},\,{\bf x}_{i}^{*}$ respectively a random matrix of dimension $k\times d_{2}$, with $d_{2}\geq k$, and a random vector of dimension $d_{2}\times 1$ and $\varepsilon_{it}$ and ${\bf v}_{it}$ are $i.i.d.$ innovations across $i$ and $t$, respectively scalar and $k\times 1$, with zero mean and variances $\xi_{i,t}$ and ${\bf\Sigma}_{V_{t}^{\prime}V_{t}}$, respectively. We focus here on Andrews (2005)’s standard factor structure, spelled out in his Assumption SF1, here slightly extended to allow for an idiosyncratic component in both the regression error $u_{it}$ and the regressors ${\bf x}_{it}$ as well as time-variation in parameters, common factors and covariance matrices. The first extension is unavoidable for us since when $\varepsilon_{it}=0\,\,\,a.s.$ our theory does not apply. Model (42)-(43)-(44) can be rewritten as

[TABLE]

where we set ${\bf y}_{t}=(y_{1t}\ldots y_{Nt})^{\prime}$ and ${\bf D}={\bm{\iota}}_{N},{\bf X}_{t}={\bf X}^{*}{\bf C}_{2t}^{\prime}$, with ${\bf X}^{*}=({\bf x}_{1}^{*}\ldots{\bf x}_{N}^{*})^{\prime}$ and parameters ${\bm{\vartheta}}_{t}=({\bm{\alpha}}_{t}^{\prime},{\bm{\beta}}_{t}^{\prime})^{\prime}$, and the ${\bm{u}}_{t}=(u_{1t}\ldots u_{Nt})^{\prime}$ satisfy the factor structure (2)

[TABLE]

with ${\bf f}_{t}={\bf c}_{1t}$, ${\bf B}=({\bf u}_{1}^{*}\ldots{\bf u}_{N}^{*})^{\prime}$ and $\bm{\varepsilon}_{t}=(\varepsilon_{1t}\ldots\varepsilon_{Nt})^{\prime}$.

In analogy with Section 2, our proposed *feasible * GLS estimator is

[TABLE]

setting $\bm{\mathcal{y}_{t}}=\bm{D}_{\bot}^{\prime}\bm{Y}_{t},\bm{\mathcal{X}_{t}}=\bm{D}_{\bot}^{\prime}\bm{X}_{t}$, assuming large enough $N$ and $T$ to ensure invertibility of $\hat{\bm{\mathcal{S}_{T}}}$, given by

[TABLE]

where $\bm{\hat{\beta}}_{t}^{OLS}=\left(\bm{\mathcal{X}_{t}}^{\prime}\bm{\mathcal{X}_{t}}\right)^{-1}\bm{\mathcal{X}_{t}}^{\prime}\bm{\mathcal{y}_{t}}$ and ${\bm{\hat{\mathrm{u}}_{t}}}$ are the OLS estimator and the OLS regression residuals, respectively, of regression (45). Notice that now $\mathscr{M}_{\bm{\mathrm{D}}}=\bm{I}_{N}-\bm{\mathrm{D}}(\bm{\mathrm{D}}^{\prime}\bm{\mathrm{D}})^{-1}\bm{\mathrm{D}}^{\prime}=\bm{D}_{\bot}\bm{D}_{\bot}^{\prime}$ is a $N\times N$ matrix and $\bm{\mathrm{D}}_{\bot}$ is a $N-1\times N$ matrix. Given the duality between $\hat{\bm{\beta}}_{t}^{GLS}$ and $\hat{\bm{\beta}}_{i}^{GLS}$, we conjecture that under a set of regularity conditions analogous to Assumptions 2.1-2.6 one obtains consistency of $\hat{\bm{\beta}}_{t}^{GLS}$ for $1/N+N/T\rightarrow 0$ and asymptotic normality of $\sqrt{N}(\hat{\bm{\beta}}_{t}^{GLS}-{\bm{\beta}}_{t})$ for $1/N+N^{3}/T^{2}\rightarrow 0$. Extension to a more general form of common observed regressors, other than ${\bf D}={\bm{\iota}}_{N}$, can be obtained along the lines of Section 4.

7 Monte Carlo analysis

We conduct a set of Monte Carlo experiments to appreciate the relevance of our asymptotic results for the GLS estimator in finite samples.

7.1 Design

The data generating process is

[TABLE]

where the single regressor satisfies

[TABLE]

Note that the model implies an observed common factor equal to $1$ for all observations. The single regressor is allowed to be contemporaneously correlated with the innovation through one of the latent common factors (whenever $b_{i10}\delta_{i10}\neq 0$). The factor loadings are normally distributed random variables, $i.i.d.$ across unit:

[TABLE]

and the latent common factors and the idiosyncratic components are stationary stochastic processes, mutually independent to each other, satisfying

[TABLE]

where each ${\eta}_{jf,t}\sim NID(0,1)$, mutually independent for $j=1,2,3$, and

[TABLE]

with $\rho_{i\varepsilon}\sim UID(0.05,0.95),\,\rho_{iv}\sim UID(0.05,0.95),\sigma_{i\varepsilon}^{2}\sim UID(0.5,1.5)$ where $NID,UID$ means $iid$ normally and uniformly distributed respectively. Finally, the parameters of interest are constant across replications and equal to $\alpha_{i0}=1,\gamma_{i0}=0.5$ and, assuming $N$ even,

[TABLE]

This Monte Carlo design is a simplified version of Pesaran (2006), designed in such a way that (through (62)) the rank condition in Pesaran (2006), eq. (21), is not satisfied. Pesaran (2006) shows that under this circumstance his individual specific estimator for $\beta_{i0}$ is invalid whereas his pooled estimators for $\beta_{0}=\mathbb{E}\beta_{i0}$ remains consistent.

We consider $2000$ Monte Carlo replications with sample sizes $(N,T)\in\{60,200,600\}\times\{30,100,300\}$, where $N>T$.

The results are summarized in Tables 4,5, and 6, where we report the sample mean and the root mean square error for the estimates of the parameter ${\bf\alpha}_{i0},{\bf\beta}_{i0}$, averaged across the Monte Carlo iterations. We consider four estimators which corresponds to four panels of each table: the GLS, the multi-step GLS (described in Section 6) where the iteration is carried out $J=4$ times, the OLS and the UGLS estimators. In particular, for each of these four estimators, we report the average across all $N$ units of the sample mean (denoted by mean) $MM^{-1}\sum_{m=1}^{MM}\hat{\alpha}_{i}^{m}$ and of the root mean square error (denoted by (denoted by rmse) $\left(MM^{-1}\sum_{m=1}^{MM}(\hat{\alpha}_{i}^{m}-1)^{2}\right)^{1\over 2}$ and the average across the units $i=N/2+1,...,N$ of $MM^{-1}\sum_{m=1}^{MM}\hat{\beta}_{i}^{m}$ and $\left(MM^{-1}\sum_{m=1}^{MM}(\hat{\beta}_{i}^{m}-3)^{2}\right)^{1\over 2}$ with $MM=2,000$. Recall that we assumed that the true intercept coefficients are constant across units whereas the regression coefficients take two different values for the first half and second half of the $N$ units. Here $\hat{\alpha}_{i}^{m}$ and $\hat{\beta}_{i}^{m}$ denote, respectively, the estimates of the intercept and regression coefficients corresponding to the $m$th Monte Carlo iteration for a generic estimator.

7.2 Results

Our comments below apply to each table, with minor differences. Since the GLS and multi-step GLS estimators requires $N\geq T$, each panel is made by a lower triangular matrix. Obviously, the OLS and the UGLS estimator do not require this constraint since they can be also evaluated when $N<T$ but we did not report the results for this case. The upper left panel describes the GLS results. One can see how the bias diminishes as both $N,T$ grow or when $N$ increases for a given $T$. This is because the inverse of the pseudo-covariance matrix is better estimated in these circumstances. In contrast, although still negligible in absolute terms, the bias, if any, tends to increase when $T$ grows for a given $N$. Instead, as expected, the rmse always diminishes when $T$ increases for a given $N$ or when they both increase. In general these results suggest that the bias of the estimates varies mainly with $N$ and their variance varies with $T$. The same pattern is observed with respect to the multi-step GLS results, reported in the upper right panel. The only difference is that now the bias and the rmse are always much smaller than the GLS case. The lower right panel reports the results for the UGLS which is unfeasible in practice since it involves the true covariance matrix ${\bf S}_{N}$. As a consequence, the results do not depend on $N$ but only on $T$. The bias is negligible even for small samples and, for larger sample sizes, it is remarkably comparable to the iterated GLS although the latter exhibit a slightly larger rmse. Finally, the lower left panel reports the OLS results which also do not depend on $N$, as expected. Under our design, the OLS estimator is non-consistent obtaining a bias which is much larger than for any other estimators and, more importantly, only marginally varying as $N$ or $T$ increases. The rmse diminishes suggesting that the variance of the OLS estimator is converging to zero with the squared bias converging to the squared of $\tau_{i}^{OLS}$.

8 Empirical Application: Firms’ Characteristics and Expected Returns

We present an empirical application of our methodology, inspired by asset-pricing theory. According to so-called beta-pricing models, asset returns follow a factor model:

[TABLE]

where $R_{i,t}$ defines the rate of return for asset $i$, in excess of the risk-free rate, and $\bm{\mathrm{D}}_{t}$ is a vector of observed factors, with coefficients $\tilde{\bm{\gamma}}_{i}$. Important, special, cases of model (63) are the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965), when $\bm{\mathrm{D}}_{t}$ is the (scalar) excess market return with $\tilde{\alpha}_{i}=0$ for every $i$, and the Arbitrage Pricing Theory (APT) of Ross (1976), when $\bm{\mathrm{D}}_{t}$ is a vector of possibly non-traded factors.999Focusing on the special case when $\bm{\mathrm{D}}_{t}$ are the excess returns of traded assets, the APT holds when the $\tilde{\alpha}_{i}$, although not zero, satisfy the condition $\tilde{\bm{\alpha}}^{\prime}(var(\bm{\mathrm{u}_{t}}))^{-1}\tilde{\bm{\alpha}}<\infty$, setting $\tilde{\bm{\alpha}}=(\tilde{\alpha}_{1},\cdots,\tilde{\alpha}_{N})^{\prime}$.

Model (63), together with some form of no-arbitrage and some constraints of the covariance matrix of the $u_{i,t}$, implies that expected excess returns $\mathbb{E}(R_{it})$ are linear in the coefficients $\bm{\alpha}_{i}$ only, namely that the $\bm{\mathrm{D}}_{t}$ are the only source of risk (see Corollary 1, Chamberlain (1983)). However, this fundamental paradigm has been challenged empirically. For instance, Daniel & Titman (1997) and Davis et al. (2000) provide strong evidence according to which stocks characteristics, such as market capitalization (size), valuation (book-to-market) and other characteristics do influence expected returns well beyond the betas. One can extend model (63) to allow for characteristics by specifying:

[TABLE]

where now $X_{i,t}$ defines a vector of characteristics associated with the $i$th stock, setting $\bm{\alpha}_{i}=(\tilde{{\alpha}}_{i},\tilde{\bm{\gamma}}_{i}^{\prime})^{\prime}$. Model (64) can be interpreted as, and in fact is equivalent to, our basic model (1). Moreover, it is conceivable that the error term has a factor structure, such as (2), possibly correlated with both the $\bm{\mathrm{D}}_{t}$ and the ${\bm{X}}_{t}$. For instance, this is arises whenever one suspects the possibility of missing, pervasive, factors. Our asymptotic distribution theory can be used to assess whether the $\bm{\alpha}_{i}$ or the $\bm{\beta}_{i}$ or both are significant or not.

We use a data set of monthly observations, from January 1966 to December 1994, of individual asset returns extracted from CRSP and of firms’ characteristics extracted from COMPUSTAT.101010See Brennan et al. (1998) for details. In particular, the eight characteristics that we consider are SIZE (the natural logarithm of the market value of the equity of the firm as of the end of the second to last month), BM (the natural logarithm of the ratio of the book value of equity plus deferred taxes to the market value of equity, using the end of the previous year market and book values)111111As in Fama and French (1993), the value of BM for July of year t to June of year t+1 was computed using accounting data at the end of year t-1., DVOL (the natural logarithm of the dollar volume of trading in the security in the second to last month), PRICE (the natural logarithm of the reciprocal of the share price as reported at the end of the second to last month), YLD (the dividend yield as measured by the sum of all dividends paid over the previous 12 months, divided by the share price at the end of the second to last month), RET2-3 (the natural logarithm of the cumulative return over the two months ending at the beginning of the previous month), RET4-6 (the natural logarithm of the cumulative return over the three months ending three months previously), RET7-12 (the natural logarithm of the cumulative return over the 6 months ending 6 months previously).121212Lagged return variables were constructed to exclude the return during the immediate prior month in order to avoid any spurious association between the prior month return and the current month return caused by thin trading or bid-ask spread effects.

We report the results in Table 1,2 and 3. In particular, we consider three different factor models, depending on the set of common factors. Table 1 refers to the CAPM model augmented with the eight characteristics. We report the average, across the $N=356$ assets, of the GLS estimates $(\breve{\bm{\alpha}}_{i}^{GLS\prime},\breve{\bm{\beta}}_{i}^{GLS\prime})^{\prime}$ in (31) for each regression parameter, together with their $10$th and $90$th percentiles, out of the $N$ assets. Similarly, we report the average, across the $N=356$ assets, of the t-$ratio$s for each regression parameter, together with their $10$th and $90$th percentiles, out of the $N$ assets. Finally, we report the F test statistics corresponding to three different joint hypotheses, namely for all $\bm{\alpha}_{i}=0$, or all $\bm{\beta}_{i}=0$ or both. Again, we report the average across the $N$ assets of the F test statistics, and their $10$th and $90$th percentile. Table 2 refers to the three-factor model of Fama & French (1993), augmented with the eight characteristics, whereby the elements of $\bm{\mathrm{D}}_{t}$ are the market, the small-minus-large (SML) and the high-minus-low (HML) portfolio returns, respectively. Finally, Table 3 refers to the five-factor model of Fama & French (2015), augmented with the eight characteristics, whereby the elements of $\bm{\mathrm{D}}_{t}$, with respect to the three-factor model, are augmented by the profitability (RMW) and investment (CMA) portfolio returns.

Across all the three asset-pricing models, the results strongly indicate that characteristics influences excess returns, and highly significantly so. This emerges both by considering individual t-$ratios$ as well as the F test for the joint hypothesis that the coefficients to the characteristics (i.e. the $\bm{\beta}_{i}$) are all zero. Noticeably, the effects of the common factors, for example the market return for the CAPM, are also strongly significant, across the three asset-pricing models. Indeed, their effects appear unambiguously stronger than for the characteristics, although they are both highly significant.

9 Concluding remarks

This paper proposes a feasible GLS estimator for linear panel with common factor structure in both the regressors and the innovation. We establish our results for time regressions with unit-specific coefficients, and present several generalisations such as dynamic panels, cross-section regressions with time varying coefficients and different factor structures for regressors and residuals. The GLS estimator is consistent and asymptotically normal, when both the cross-section $N$ and time series $T$ dimensions diverge to infinity where, under the same circumstances, the OLS is first-order biased. In summary, the GLS estimator exhibits four main properties: first, it permits to carry out inference on the regression coefficients based on conventional distributions; second, as in classical estimation theory, it delivers (almost) efficient estimation; third, it does not require any knowledge of the exact number of latent factors, or even an upper bound of such number; and fourth, the GLS is computationally easy to handle without invoking any nonlinear numerical optimizations. Our results are corroborated by a set of Monte Carlo experiments and illustrated by an asset-pricing empirical application.

Note to Table 1: Panel A reports t-$ratios$ and parameter estimates corresponding to the CAPM model, augmented with characteristics SIZE, BM, DVOL, PRICE, YLD, RET23, RET46 and RET712:

[TABLE]

where $R_{it}$ defines the excess return on asset $i$, $R_{Mkt,t}$ is the $S\&P500$ excess return and $X_{it}$ the $8\times 1$ vector of characteristics. Panel B reports the $F$ test statistics corresponding to the null hypotheses $H_{0}:\tilde{\gamma}_{i}=0$, $H_{0}:\bm{\beta}_{i}=0$ and $H_{0}:\bm{\beta}_{i}=0,\tilde{\gamma}=0$, given by $F_{\gamma},F_{\beta}$ and $F_{\beta,\gamma}$ respectively.

The data are monthly and makes a panel of monthly observations with $T=348,N=356$. The characteristics have been cross-sectionally standartized. Column 2 to 4 of Panel A report the $10$-th decile, the average and the $90$th decile of the t-$ratios$ across the $N$ assets. Columns 5 to 7 of Panel A report the same quantities with respect to the parameter estimates, using the GLS estimator $(\breve{\bm{\alpha}}_{i}^{GLS\prime},\breve{\bm{\beta}}_{i}^{GLS\prime})^{\prime}$ in (31). Their covariance matrix is estimated using the approach described in Section 6.4. Column 2 to 4 of Panel B report the $10$-th decile, the average and the $90$th decile of the three F test statistics across the $N$ assets.

Note to Table 2: Panel A reports t-$ratios$ and parameter estimates corresponding to the Fama & French (1993) 3-factor model, augmented with characteristics SIZE, BM, DVOL, PRICE, YLD, RET23, RET46 and RET712:

[TABLE]

where $R_{it}$ defines the excess return on asset $i$, $R_{Mkt,t}$ is the $S\&P500$ excess return, $R_{SMB,t}$ is the size factor, $R_{HML,t}$ is the value factor and ${\bf X}_{it}$ the the $8\times 1$ vector of characteristics. Panel B reports the $F$ test statistics corresponding to the null hypotheses $H_{0}:\tilde{\bm{\gamma}}_{i}=0$, $H_{0}:\bm{\beta}_{i}=0$ and $H_{0}:\bm{\beta}_{i}=0,\tilde{\bm{\gamma}}_{i}=0$, given by $F_{\gamma},F_{\beta}$ and $F_{\beta,\gamma}$ respectively, setting $\tilde{\bm{\gamma}}_{i}=(\tilde{\gamma}_{i1},\tilde{\gamma}_{i2},\tilde{\gamma}_{i3})^{\prime}$. For details refer to the notes to Table 1.

Note to Table 3: Panel A reports t-$ratios$ and parameter estimates corresponding to the Fama and French Fama & French (2015) 5-factor model, augmented with characteristics SIZE, BM, DVOL, PRICE, YLD, RET23, RET46 and RET712:

[TABLE]

where $R_{it}$ defines the excess return on asset $i$, $R_{Mkt,t}$ is the $S\&P500$ excess return, $R_{SMB,t}$ is the size factor, $R_{HML,t}$ is the value factor, $R_{RMW,t}$ is the size factor, $R_{CMA,t}$ is the value factor and ${\bf X}_{it}$ the the $8\times 1$ vector of characteristics. Panel B reports the $F$ test statistics corresponding to the null hypotheses $H_{0}:\tilde{\bm{\gamma}}_{i}=0$, $H_{0}:\bm{\beta}_{i}=0$ and $H_{0}:\bm{\beta}_{i}=0,\tilde{\bm{\gamma}}_{i}=0$, given by $F_{\gamma},F_{\beta}$ and $F_{\beta,\gamma}$ respectively, setting $\tilde{\bm{\gamma}}_{i}=(\tilde{\gamma}_{i1},\tilde{\gamma}_{i2},\tilde{\gamma}_{i3},\tilde{\gamma}_{i4},\tilde{\gamma}_{i5})^{\prime}$. For details refer to the notes to Table 1.

Note to Table 4: data are generated according to model

[TABLE]

with regressor $x_{it}=\gamma_{i0}+\delta_{i10}f_{1t}+\delta_{i30}f_{3t}+v_{it}$. Factor loadings are normally distributed random variables, $iid$ across units and mutually independent, satisfying $b_{i10}\sim NID(1,0.2),$     $b_{i20}\sim NID(0,0.2),$ $\delta_{i10}\sim NID(0.5,0.5),$ $\delta_{i30}\sim NID(0,0.5).$ Latent common factors are ${f}_{j,t}=$ $0.5{f}_{j,t-1}$ $+\sqrt{0.5}{\eta}_{jf,t},$ with ${\eta}_{jf,t}$ $\sim NID(0,1)$, mutually independent for $j=1,2,3$, and idiosyncratic innovation are $\varepsilon_{it}=$ $\rho_{i\varepsilon}{\varepsilon}_{it-1}$ $+{\eta}_{i\varepsilon,t}$ with ${\eta}_{i\varepsilon,t}$ $\sim NID(0,\sigma_{i}^{2}(1-\rho_{i\varepsilon}^{2})),$ $v_{it}=$ $\rho_{iv}{v}_{it-1}+{\eta}_{iv,t},$ with ${\eta}_{iv,t}$ $\sim NID(0,(1-\rho_{iv}^{2})),$ with $\rho_{i\varepsilon}$ $\sim UID(0.05,0.95),$ $\rho_{iv}$ $\sim UID(0.05,0.95),$ $\sigma_{i\varepsilon}^{2}$ $\sim UID(0.5,1.5)$, $iid$ across $i=1,...,N$ and mutually independent.

Parameters of interest are constant across replications and equal to $\alpha_{i0}=1,\gamma_{i0}=0.5$ and, assuming $N$ even, $\beta_{i0}=1$ for $i=1,...,N/2$ and $\beta_{i0}=3$ for $i=N/2+1,...,N.$

Panels headed by mean and rmse report, respectively, $N^{-1}\sum_{i=1}^{N}\left(MM^{-1}\sum_{m=1}^{MM}\hat{\alpha}_{i}^{m}\right)$ and

$N^{-1}\sum_{i=1}^{N}\left(MM^{-1}\sum_{m=1}^{MM}(\hat{\alpha}_{i}^{m}-1)^{2}\right)^{1\over 2}$ with $MM=2,000$. Here $\hat{\alpha}_{i}^{m}$ denotes the estimate, based on either the GLS $\tilde{\bm{\alpha}}_{i}^{GLS}$ of Section (4) (top left panel), multi-step GLS with $J=4$ steps (top right panel), OLS $\bm{\hat{\alpha}_{i}^{OLS}}$ (bottom left panel) and UGLS $\bm{\hat{\alpha}_{i}^{UGLS}}$ (bottom right panel) of $\alpha_{i0}$ for the $m$ Monte Carlo iteration.

**Appendices

**

All the proofs and technical details are reported in the three appendixes (Appendixes A,B and C) of the manuscripts and in the seven appendixes of the Supplement (Appendixes D,E,F,G,H,I,J).

In particular, Appendix A states (without proof; see Appendix F) Lemmas A.1 to A.3 and Appendix B contains the proofs to Theorem 3.1 and 3.2 and states (without proofs; see Appendix G) Propositions B.1 to B.17. Finally, Appendix C defines the $\SS$ matrix and associated quantities that characterize the asymptotic distribution of the GLS estimator.

Regarding the Supplement, Appendix D contains needed results of linear matrix algebra (Lemmas D.1 to D.6), Appendix E contains some results used to construct bounds on the inverse of various covariance matrixes (Lemmas E.1 to E.3 and Corollary E.3), Appendix F contains the proofs to Lemmas A.1 to A.3, Appendix G contains the proofs to Propositions B.1 to B.17, Appendix H contains important auxiliary results for the proof of Theorem 3.2 (Lemmas H.1 to H.19), Appendix I formalizes the asymptotic properties of the estimators for the common observed regressors’ coefficient (Theorems I.1 to I.3) and, finally, Appendix J provides some technical details for the case when the regressors and the residuals have different, yet correlated, factor structures.

A Central Lemmas

In the following $m,m_{1},m_{3},m_{3}$ denote positive constants. The proofs of the lemmas stated in this section are provided in Appendix F.

Lemma A.1

Let ${\bf A}(m_{1}\times m_{1})$, ${\bf C}(m_{2}\times m_{2})$, and ${\bf B}(m_{1}\times m_{2})$, $m_{1}>m_{2}$ be random matrices. Set ${\bf E:=BCB^{\prime}+A}$ and assume that $\lambda_{1}\left(\bf{A}\right)=O_{p}(1)$. Assume further that $m_{2}<\infty$, and

1. (a)

$\left\|\frac{\bm{B^{\prime}A^{-1}B}}{m_{1}}\right\|=O_{p}(1)\;$* and $\quad\left\|\left(\frac{\bm{B^{\prime}A^{-1}B}}{m_{1}}\right)^{-1}\right\|=O_{p}(1)$.* 2. (b)

$\left\|\left(\frac{\bm{C^{-1}+B^{\prime}A^{-1}B}}{m_{1}}\right)^{-1}\right\|=O_{p}(1)$. 3. (c)

$\left\|\frac{\bm{B^{\prime}(A^{-1})^{\prime}A^{-1}B}}{m_{1}}\right\|=O_{p}(1)$. 4. (d)

$\left\|\bm{C}\right\|=O_{p}(1)$* and $\left\|\bm{C}^{-1}\right\|=O_{p}(1)$.*

Then,

1. (i)

$\parallel\bm{E}^{-1}\bm{B}\parallel_{2}=O_{p}\left(m_{1}^{-\nicefrac{{1}}{{2}}}\right).$**

Let $\bm{D}$ be a $m_{1}\times m_{3}$ matrix, with $m_{3}<\infty$.

1. (ii)

If $\left\|m_{1}^{-1}\bm{D}^{\prime}\bm{A}^{-1}\bm{B}\right\|_{2}=O_{p}(1)$, then $\left\|\bm{D}^{\prime}\bm{E}^{-1}\bm{B}\right\|=O_{p}\left(1\right).$ 2. (iii)

If $\left\|m_{1}^{-\nicefrac{{1}}{{2}}}\bm{D}^{\prime}\bm{A}^{-1}\bm{B}\right\|_{2}=O_{p}(1)$, then $\left\|\bm{D}^{\prime}\bm{E}^{-1}\bm{B}\right\|=O_{p}\left(m_{1}^{-\nicefrac{{1}}{{2}}}\right)$.

Remark A.1

Let $\bm{\bar{E}}:=\bm{A}^{-1}-\bm{A}^{-1}\bm{B}\left(\bm{B^{\prime}A^{-1}B}\right)^{-1}\bm{B}^{\prime}\bm{A}^{-1}$. Write $\bm{H}:=\bm{A^{-1}B}$; following (Johansen 1996, p.40) we find

[TABLE]

implying that $\bm{\bar{E}}\bm{B}=\bm{0}$. Because

[TABLE]

it follows that the inverse of the matrix $\bm{E}$ approximates a matrix orthogonal to $\bm{B}$.

Lemma A.2

Let $\bm{A},\bm{B}$ nonsingular matrices. Then

[TABLE]

for $J=1,2,\dots.$

Lemma A.3

Let $g(\omega)$ be a periodic (mod $2\pi$) symmetric function defined over $-\pi\leq\omega\leq\pi$ with bounded $r$-th order derivative $g^{(r)}(\cdot)$, some $r\geq 1$. Then the Fourier coefficients $\varsigma_{h}=\int_{-\pi}^{\pi}g(\omega)\cos(h\omega)d\omega$ satisfy:

[TABLE]

B Proof of the main theorems

The proofs of the theorems rely on several propositions, the proofs of which are relegated in Appendix G

B.1 Proof of Theorem 3.1

Proof of part (i)

Rewrite equation (12) as

[TABLE]

Proposition B.1 implies that

[TABLE]

and

[TABLE]

Hence $(\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{X}_{i}})^{-1}\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{F}}\bm{\mathrm{b}_{i}}\bm{\mathrm{b}_{i}}\xrightarrow{p}\bm{\tau_{i}^{OLS}}$, proving $(\ref{eq:bias17})$.

Next, we derive the asymptotic distribution of

[TABLE]

We first show that the first term in (B.4) satisfies

[TABLE]

Note that $\bm{\Gamma_{i}}^{\prime}\bm{\mathrm{F}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\varepsilon_{i}}=\bm{\Gamma_{i}}^{\prime}(-\bm{\mathrm{F}}^{\prime}\bm{\mathrm{D}}(\bm{\mathrm{D}}^{\prime}\bm{\mathrm{D}})^{-1},\bm{I}_{M}){\bm{\mathrm{Z}}^{\prime}\bm{\varepsilon_{i}}},$ with $\bm{\mathrm{Z}}$ defined in Assumption 2.3. Because of (7), to prove (B.5) it suffices to prove that

[TABLE]

Adapting (Robinson & Hidalgo 1997, Proof of Theorem 1), and using equations (4) and (5) we have

[TABLE]

where $\bm{w_{j}}:=T^{-\nicefrac{{1}}{{2}}}\sum_{t=1}^{T}\bm{\mathrm{z}}_{t}a_{jt}$, and $\bm{\mathrm{z}}^{\prime}_{t}=(z_{t1},\dots,z_{t(M+S)})^{\prime}$ is the $t-$th row of $\bm{\mathrm{Z}}$. For $\tau_{0}=\tau_{0}(T)$ yet to be chosen, define

[TABLE]

and let $\bm{w}_{j1}:=\bm{w}_{j}-\bm{w}_{j0}$, $\bm{W}_{j1}:=\bm{W}_{j}-\bm{W}_{j0}$, where

[TABLE]

Write

[TABLE]

Noting that $\mathbb{E}\left\|\bm{W_{j1}}\right\|\leq\mathbb{E}\left\|\bm{w_{j1}}\right\|^{2}$, Propositions B.2 and B.4 implies that, for $\tau_{0}$ increasing suitably with $T$, $\left\|\bm{W_{j}}^{-1/2}\bm{w_{j1}}\right\|=o_{p}(1)$ and $\left\|\bm{W_{j1}}\bm{W_{j0}}^{-1}\right\|=o_{p}(1)$. Hence, $\bm{W_{j}}^{-1/2}\bm{w_{j}}\approx\bm{W_{j0}}^{-1/2}\bm{w_{j0}}$ as $T\to\infty$. Therefore, Bernstein’s Lemma (see Hannan (1970), p. 242), Propositions B.3 and B.4 imply that, for every given $j$, $\bm{w_{j}}\xrightarrow{d}\mathcal{N}(\bm{0},\bm{\mathcal{W}_{j}})$ with $\bm{\mathcal{W}_{j}}$ defined in Equation (B.12). However, the terms $\bm{w_{j}}$ (and their limits) are uncorrelated across $j$’s implying that $T^{-{1\over 2}}\bm{\mathrm{Z}}^{\prime}\bm{\varepsilon_{i}}=\sum_{j=1}^{N}r_{ij}\bm{w_{j}}\xrightarrow{d}\mathcal{N}(\bm{0},\sum_{j=1}^{N}r_{ij}^{2}\bm{\mathcal{W}_{j}}),$ where notice that by easy calculations $\sum_{j=1}^{N}r_{ij}^{2}{\bm{\mathcal{W}}}_{j}=\bm{\Sigma}_{\bm{\mathrm{Z}}^{\prime}\bm{\Xi_{i}}\bm{\mathrm{Z}}}$, proving (B.6). It is worth noting that as $N$ increases, the latter distribution can be made arbitrarily close to $\mathcal{N}\left(\bm{0},\sum_{j=1}^{\infty}r_{ij}^{2}\bm{\mathcal{W}_{j}}\right)$ because of the absolute summability of the $r_{ij}$.

The second term in (B.4) satisfies $T^{-\nicefrac{{1}}{{2}}}{\bf{V}}_{\bm{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\varepsilon_{i}}\approx T^{-\nicefrac{{1}}{{2}}}{\bf{V}}_{\bm{i}}^{\prime}\bm{\varepsilon_{i}}$ (see Proposition B.1(iv)). The proof of the weak convergence,

[TABLE]

is very similar to the proof of (B.6), and hence omitted. By assumption (2.6), the two terms on the RHS of (B.4) are uncorrelated, so by (B.5) and (B.10) its LHS converges weakly to a random variable with mean zero, and variance given in Equation (3.1).

Proposition B.1

2. (i)

$\frac{\bm{\mathcal{F}}^{\prime}\bm{\mathcal{F}}}{T}={\bm{\mathrm{F}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}\over T}\xrightarrow{p}\bm{\Sigma}_{\bm{\mathrm{F}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}}=\bm{\Sigma}_{\bm{\mathrm{F}}^{\prime}\bm{\mathrm{F}}}-\bm{\Sigma}_{\bm{\mathrm{F}}^{\prime}\bm{\mathrm{D}}}\bm{\Sigma}_{\bm{\mathrm{D}}^{\prime}\bm{\mathrm{D}}}^{-1}\bm{\Sigma}_{\bm{\mathrm{D}}^{\prime}\bm{\mathrm{F}}}$* .* 3. (ii)

$\frac{\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{V}}_{\bm{i}}}{T}={{\bf{V}}_{\bm{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}{\bf{V}}_{\bm{i}}\over T}\xrightarrow{p}\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}{\bf{V}}_{\bm{i}}}.$** 4. (iii)

$\left\|\frac{\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{F}}}{T}\right\|=\left\|{{{\bf{V}}_{\bm{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}}\over T}\right\|=O_{p}(T^{-\nicefrac{{1}}{{2}}})$. 5. (iv)

$\left\|{{{\bf{V}}_{\bm{i}}^{\prime}\bm{\varepsilon_{i}}}\over\sqrt{T}}-{{{\bf{V}}_{\bm{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\varepsilon_{i}}}\over\sqrt{T}}\right\|=O_{p}(T^{-\nicefrac{{1}}{{2}}})$. 6. (v)

$\bm{\Sigma}_{\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{X}_{i}}}>0.$**

Proposition B.2

For $\tau_{0}$ increasing suitably with $T$

[TABLE]

Proposition B.3

As $T\to\infty$

[TABLE]

Proposition B.4

As $T\to\infty$

[TABLE]

and $\bm{\Phi}_{j}$ is the $T\times T$ matrix with $(t,s)-$th element equal to $\sum_{u=-\infty}^{\min(t,s)}\phi_{j,t-u}\phi_{j,s-u}=\sum_{v=0}^{\infty}\phi_{j,v}\phi_{j,v+|t-s|}$.

Proof of part (ii)

All the limits below hold as $T\rightarrow\infty$. By simple manipulation of equation (16)

[TABLE]

We first show that estimator is asymptotically unbiased. Proposition B.5 implies

[TABLE]

and

[TABLE]

implying that the the bias term is $O_{p}\left(T^{-\nicefrac{{1}}{{2}}}\right)$.

To complete the proof we need to derive the limiting distribution of the latter term in (B.13). By Proposition B.6, $T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{S}}^{-1}_{N}\bm{\epsilon_{i}}\approx T^{-\nicefrac{{1}}{{2}}}{\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}\bm{\varepsilon_{i}}$. Proposition B.7$(i)$ shows that, $T^{-\nicefrac{{1}}{{2}}}{\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}\bm{\varepsilon_{i}}\approx$ $T^{-\nicefrac{{1}}{{2}}}$ ${\bf{V}}_{\bm{i}}^{\prime}\bm{\bar{\Xi}_{N}}\bm{\varepsilon_{i}}$, where $\bm{\bar{\Xi}_{N}}$, defined in the same proposition, is circulant and symmetric. Hence, as stated by the second part of Proposition B.7, $\|\bm{\bar{\Xi}_{N}}\|_{row}<\infty$, allowing us to exploit again (Robinson & Hidalgo 1997, Propositons 1 and 2).

Similarly to (B.7)-(B.9), we define

[TABLE]

where $\bar{\xi}_{N}(t-s)$, denoting the $(t,s)-$element of $\bm{\bar{\Xi}_{N}}$, satisfies $\bar{\xi}_{N}(h)=\bar{\xi}_{N}(T-h)$, $h=0,1,\dots,T-1$. Write

[TABLE]

and define $\bm{w}^{(\bar{\xi})}_{ij1}:=\bm{w}^{(\bar{\xi})}_{ij}-\bm{w}^{(\bar{\xi})}_{ij0}$, and $\bm{W}^{(\bar{\xi})}_{ij1}:=\bm{W}^{(\bar{\xi})}_{ij}-\bm{W}^{(\bar{\xi})}_{ij0}$, where

[TABLE]

Proceeding as in the proof of the first part of the theorem, Propositions B.8-B.10 allow us to establish that, for any $i,j$ $\bm{w^{(\bar{\xi})}_{ij}}\xrightarrow{d}\mathcal{N}(\bm{0},\bm{\mathcal{W}^{(\bar{\xi})}_{ij}})$, where $\bm{\mathcal{W}^{(\bar{\xi})}_{ij}}$ defined in (B.20) below. It follows that, by (B.15)

[TABLE]

and $\sum_{j=1}^{N}r_{ij}^{2}\bm{\mathcal{W}^{(\bar{\xi})}_{ij}}=\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}\bm{\Xi_{i}}\bm{\Xi^{-1}_{N}}{\bf{V}}_{\bm{i}}}$, as required. The result in (25) is proved by (B.14) and (B.18).

Proposition B.5

2. (i)

$\left\|\bm{\mathcal{F}}^{\prime}\bm{\mathcal{S}}^{-1}_{N}\bm{\mathcal{F}}\right\|=O_{p}(1).$** 3. (ii)

$\left\|\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{S}}^{-1}_{N}\bm{\mathcal{F}}\right\|=O_{p}\left(T^{-\nicefrac{{1}}{{2}}}\right).$** 4. (iii)

$T^{-1}\left(\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{S}}^{-1}_{N}\bm{\mathcal{V}}_{\bm{i}}\right)\xrightarrow{d}\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}{\bf{V}}_{\bm{i}}}>0.$**

Proposition B.6

2. (i)

$\left\|\bm{\mathcal{F}}^{\prime}\bm{\mathcal{S}}^{-1}_{N}\bm{\epsilon_{i}}\right\|=O_{p}\left(T^{-\nicefrac{{1}}{{2}}}\right).$** 3. (ii)

$\left\|{\bm{\mathcal{V}}_{\bm{i}}^{\prime}}{\bm{\mathcal{S}}^{-1}_{N}}{\bm{\epsilon_{i}}}-{\bf{V}}_{\bm{i}}^{\prime}{\bm{\Xi^{-1}_{N}}}{\bm{\varepsilon_{i}}}\right\|=O_{p}(1).$**

Proposition B.7

Let $\bm{\bar{\Xi}_{N}}:=2\pi\bm{P}\bm{G}_{\xi_{N}}^{-1}\bm{P}^{\prime}$, with $\bm{P}$ defined in (E.3) and $\bm{G}_{\xi_{N}}={\rm diag}({\bm{g}}(\xi_{N},\omega))$ is defined in (E.2) with $g_{\xi_{N}}(\omega)=\sum_{h=-\infty}^{\infty}\xi_{N}(h)\cos(h\omega)$, where $\xi_{N}(h)=\xi_{N,ts}$ is the $(t,s)-$entry of the matrix $\bm{\Xi_{N}}$ defined in (13). Then,

1. (i)

$\left\|{\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}\bm{\varepsilon_{i}}-{\bf{V}}_{\bm{i}}^{\prime}\bm{\bar{\Xi}_{N}}\bm{\varepsilon_{i}}\right\|=O_{p}(1)$. 2. (ii)

$\left\|\bm{\bar{\Xi}_{N}}\right\|_{row}<\infty$.

Proposition B.8

For $\tau_{0}$ increasing suitably with $T$

[TABLE]

Proposition B.9

As $T\to\infty$

[TABLE]

Proposition B.10

Let $\bm{\mathcal{W}^{(\bar{\xi})}_{ij}}$ as defined in (B.20) below. Then,

[TABLE]

where the matrix $\bm{\Phi_{j}}$ has been defined in (B.12)

B.2 Proof Theorem 3.2

By Proposition B.11 below

[TABLE]

where

[TABLE]

and $\bm{A_{N}}$ and $\bm{\mathrm{C}_{N}}$ are defined in equations (9) and (28), respectively. The matrix $\bm{\mathcal{H}_{N}}$ can be seen as the FGLS counterpart of $\bm{\mathcal{S}}^{-1}_{N}$. The proof follows closely that of Theorem 3.1 part (ii). However, here we consider the joint asymptotics, for (N,T) diverging simultaneously, as can be appreciated from the inspection of Proposition B.11. For the first term on the LHS of (B.21), proposition B.12 implies that

[TABLE]

The matrix $\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}\bm{\mathrm{C}_{N}}^{-1}{\bf{V}}_{\bm{i}}}=\mathbb{E}\left({\bf{V}}_{\bm{i}}^{\prime}\bm{\mathrm{C}_{N}}^{-1}{\bf{V}}_{\bm{i}}\right)$ is non-stochastic, but does depend on $N$, in general. To complete the proof we need to derive the limiting distribution of the latter term in (B.21). By and Proposition B.13,

[TABLE]

that is, $T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\epsilon_{i}}\approx T^{-\nicefrac{{1}}{{2}}}{\bf{V}}_{\bm{i}}^{\prime}\bm{C^{-1}_{N}}\bm{\varepsilon_{i}}$. Proposition B.14.(i) show that, in turn, $T^{-1/2}{\bf{V}}_{\bm{i}}^{\prime}\bm{C^{-1}_{N}}\bm{\varepsilon_{i}}\approx T^{-1/2}{\bf{V}}_{\bm{i}}^{\prime}\bm{\bar{C}_{N}}\bm{\varepsilon_{i}}$, where $\bm{\bar{C}_{N}}$ is a circulant matrix defined in the same proposition. Hence, the LHS of (B.21) can be further approximated as

[TABLE]

The second part of Proposition B.14 states that the rows of the matrix $\bm{\bar{C}_{N}}$ are absolutely summable, allowing us to exploit again (Robinson & Hidalgo 1997, Propositons 1 and 2).

Similarly to (B.15)-(B.17), we write

[TABLE]

where $\bar{c}_{N}(t-s)=\bar{c}_{N,ts}$ is the $(t,s)$-entry of the $(T\times T)$ matrix $\bm{\bar{C}_{N}}$ and

[TABLE]

with $\phi_{jh}=0$ for $h<0$. By Assumption 2.1 it also follows that, for any $N$

[TABLE]

with $\bm{W}^{(\bar{c})}_{ij}:=\mathbb{E}\left(\bm{w}^{(\bar{c})}_{ij}\bm{w}^{(\bar{c})^{\prime}}_{ij}|\{\operatorname{\mathscr{F}}({\bf{V}}_{\bm{i}})\}\right).$ Proceeding along the lines of Theorem 1, Part (ii) using Propositions B.15-B.16 we establish that,

[TABLE]

The result in display (27) follows from display (B.24) and Proposition B.17.

Proposition B.11

For $1/T+T^{3}/N^{2}\to 0$:

1. (i)

$\left\|T^{-1}\bm{\mathcal{X}_{i}}^{\prime}(\bm{\hat{\mathcal{S}}_{N}}^{-1}-\bm{\mathcal{H}^{-1}_{N}})\bm{\mathcal{X}_{i}}\right\|=o_{p}(1)$. 2. (ii)

$\left\|T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}(\bm{\hat{\mathcal{S}}_{N}}^{-1}-\bm{\mathcal{H}^{-1}_{N}})\bm{\epsilon_{i}}\right\|=o_{p}(1)$.

Proposition B.12

For any $N$:

1. (i)

$\left\|\bm{\mathcal{F}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\mathcal{F}}\right\|=O_{p}(1).$** 2. (ii)

$\left\|\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\mathcal{F}}\right\|=O_{p}\left(T^{-\nicefrac{{1}}{{2}}}\right).$** 3. (iii)

$\left\|T^{-1}\left(\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\mathcal{V}}_{\bm{i}}-{\bf{V}}_{\bm{i}}^{\prime}\bm{\mathrm{C}_{N}}^{-1}{\bf{V}}_{\bm{i}}\right)\right\|=o_{p}(1).$** 4. (iv)

$T^{-1}\left(\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\mathcal{V}}_{\bm{i}}\right)\xrightarrow{p}\bm{\Sigma}_{{\bf{V}}_{\bm{i}}^{\prime}\bm{\mathrm{C}_{N}}^{-1}{\bf{V}}_{\bm{i}}}>0.$**

Proposition B.13

For any $N$

1. (i)

$\left\|\bm{\Gamma_{i}}^{\prime}\bm{\mathcal{F}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\epsilon_{i}}\right\|=O_{p}\left(T^{-\nicefrac{{1}}{{2}}}\right).$** 2. (ii)

$\left\|\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\epsilon_{i}}-{\bf{V}}_{\bm{i}}^{\prime}\bm{\mathrm{C}_{N}}^{-1}\bm{\varepsilon_{i}}\right\|=O_{p}(1).$**

Proposition B.14

Let $\bm{\bar{C}_{N}}:=2\pi\bm{P}\bm{G}_{c_{N}}^{-1}\bm{P}^{\prime}$, with $\bm{P}$ defined in (E.3) and $\bm{G}_{c_{N}}={\rm diag}({\bm{g}}(c_{N},\omega))$ is defined as in display (E.2) with $g_{c_{N}}(\omega)=\sum_{h=-\infty}^{\infty}c_{N}(h)\cos(h\omega)$. The scalar $c_{N}(h)=c_{N,ts}$ is the $(t,s)-$entry of the matrix $\bm{\mathrm{C}_{N}}$ defined in (28). Then, $\forall N$

1. (i)

$\left\|{\bf{V}}_{\bm{i}}^{\prime}\bm{\mathrm{C}_{N}}^{-1}\bm{\varepsilon_{i}}-{\bf{V}}_{\bm{i}}^{\prime}\bm{\bar{C}_{N}}\bm{\varepsilon_{i}}\right\|=O_{p}(1)$. 2. (ii)

$\left\|\bm{\bar{C}_{N}}\right\|_{row}<\infty$.

Proposition B.15

For any $N$ and $T_{0}$ increasing suitably with $T$

[TABLE]

Proposition B.16

For any $N$, as $T\to\infty$

[TABLE]

Proposition B.17

For any $N$

[TABLE]

C The matrix $\SS$

By Equation (3), (15) and (19)

[TABLE]

with

[TABLE]

To define $\bm{\hat{A}}_{i}$, note that

[TABLE]

Hence, $\bm{\hat{A}_{i}}=\left(\bm{I}-\bm{\Gamma_{i}}\bm{\mathcal{X}_{i}}^{+}\bm{\mathcal{F}}\right)\bm{\mathrm{b}_{i}}\bm{\mathrm{b}_{i}}^{\prime}\left(\bm{I}-\bm{\Gamma_{i}}\bm{\mathcal{X}_{i}}^{+}\bm{\mathcal{F}}\right)^{\prime}+\left(\bm{\Gamma_{i}}\bm{\mathcal{X}_{i}}^{+}\right)\bm{\epsilon_{i}}\bm{\epsilon_{i}}^{\prime}\left(\bm{\Gamma_{i}}\bm{\mathcal{X}_{i}}^{+}\right)^{\prime}.$

Likewise, $\bm{\hat{\bm{\mathcal{C}}}_{1i}}=\left(\bm{I}-\bm{\mathcal{V}}_{\bm{i}}\bm{\mathcal{X}_{i}}^{+}\right)\bm{\epsilon_{i}}\bm{\epsilon_{i}}^{\prime}\left(\bm{I}-\bm{\mathcal{V}}_{\bm{i}}\bm{\mathcal{X}_{i}}^{+}\right)^{\prime},$ and $\bm{\hat{\bm{\mathcal{C}}}_{2i}}=\left(\bm{\mathcal{V}}_{\bm{i}}\bm{\mathcal{X}_{i}}^{+}\right)\bm{\mathcal{F}}\bm{\mathrm{b}_{i}}\bm{\mathrm{b}_{i}}^{\prime}\bm{\mathcal{F}}^{\prime}\left(\bm{\mathcal{V}}_{\bm{i}}\bm{\mathcal{X}_{i}}^{+}\right)^{\prime}.$ The term $\bm{\hat{\bm{\mathcal{C}}}_{3i}}$ is defined as $\bm{\hat{\bm{\mathcal{C}}}_{3i}}=\sum_{j=1}^{13}\left(\bm{\hat{\bm{\mathcal{C}}}_{3i,j}}+\bm{\hat{\bm{\mathcal{C}}}^{\prime}_{3i,j}}\right),$ where

[TABLE]

Next, define the matrices

[TABLE]

where $\bm{\Xi_{i}}$ and $\bm{A_{N}}$ are defined in Equations (2) and (9), and

[TABLE]

with $\bm{\Psi}_{i}$ defined as in (10)

Some properties of the matrices defined above are established Lemmas H.1 and H.4 in Appendix H.1. In particular, Lemma H.1 verifies that the matrix $\SS$ satisfies the assumptions in Lemma A.1, on which rely the proof of most of the results in Lemma H.2.

Supplementary Material to

“Robust Nearly-Efficient Estimation of Large Panels with Factor Structures”

**Appendices

**

This Supplement is made by seven appendixes: Appendix D contains some results of linear matrix algebra (Lemmas D.1 to D.6), Appendix E contains some results used to construct bounds on the inverse of various covariance matrices (Lemmas E.1 to E.3 and Corollary E.3), Appendix F contains the proofs to Lemmas A.1 to A.3 (stated in Appendix A of the manuscript), Appendix G contains the proofs to Propositions B.1 to B.17 (stated in Appendix B of the manuscript), Appendix H contains auxiliary results for the proof of Theorem 3.2 (Lemmas H.1 to H.19), Appendix I formalises the asymptotic properties of the estimators for the common observed regressors’ coefficient (Theorems I.1 to I.3) and, finally, Appendix J provides some technical details for the case when the regressors and the residuals have different, yet correlated, factor structures.

D Some results on matrix algebra

In this section, we report for reference some auxiliary results on matrix algebra. The first two Lemmas are reported without proof. In the following, all the matrices have real entries.

Lemma D.1** (Sherman-Morrison-Woodbury formula)**

For every matrices $\bm{A},\bm{B}$ and $\bm{C}$ of suitable dimension:

[TABLE]

if all the involved inverses exist.

Lemma D.2** (Bernstein (2009), Fact 6.4.10)**

Let $\bm{A}(n\times m)$ a matrix of rank $m$ and $\bm{B}(m\times m)$ a positive definite matrix. Then,

[TABLE]

Lemma D.3

Let $\bf{A}$ be defined as in Lemma D.2. Assume further that $\bf{A}^{\prime}\bf{A}=\bm{I_{m}}$. Let $\bm{C}(n\times n)$ a positive definite matrix, and define $\bm{E}:=\mathscr{P}_{\bm{A}}\bm{C}\mathscr{P}_{\bm{A}}$. Then,

[TABLE]

where $\bm{Q}(n\times n)$ satisfies $\bm{Q}^{\prime}\bm{Q}=\bm{C^{-1}}$ and the diagonal matrix $\bm{L}$ is obtained setting equal to zero the last $n-m$ diagonal elements of the matrix $\bm{I}_{n}$.

**Proof ** By simple manipulation and Lemma D.2

[TABLE]

with $\bm{\mathcal{A}}=\bm{C^{\frac{1}{2}}}\bf{A}$. Rewriting $\mathscr{M}_{\mathcal{A}}=\bm{PLP^{\prime}}$, with $\bm{P^{\prime}P}=\bm{I}_{n}$, the proof is completed setting $\bm{Q=C^{\frac{1}{2}}P}$. $\Box$

Lemma D.4

2. (i)

Let $\bm{A}(m\times m)$ positive semidefinite, and $\bm{B}(n\times m)$. Then

[TABLE] 3. (ii)

Let $\mathscr{P}_{B}$ a projection matrix, with $\bm{B}(n\times m)$, $n>m$, and $\bm{A}(m\times m)$. Then,

[TABLE] 4. (iii)

For any symmetric matrix $\bm{A}(n\times n)$ and positive semidefinite matrix $\bm{B}(n\times n)$

[TABLE]

and

[TABLE] 5. (iv)

Let $\bm{A}(n\times n)$ be a positive semidefinite matrix, then

[TABLE]

.

**Proof ** The first inequality of part (i) can be found in (Magnus & Neudecker 1988, p. 237). About the second inequality, first note that ${\rm tr}(\bm{B^{\prime}}\bm{A}\bm{B})={\rm tr}(\bm{B}\bm{B}^{\prime}\bm{A})$. The proof is concluded using the quasilinear representation of the extremal eigenvalues (Magnus & Neudecker 1988, p. 204). For part (ii), by Reayleigh quotient we have

[TABLE]

For part (iii) see (Magnus & Neudecker 1988, p. 204). Note that for the second inequality to hold, only requires $\bm{B}$ to be symmetric. Part (iv) follows from (Abadir & Magnus 2005, Exercise 8.7) and (Lütkepohl 1996, Section 9.13.4(4)). $\Box$

Lemma D.5

Let $\bm{A}(m\times m)$ a matrix satisfying $\|\bm{A}\|_{col}<\infty$ and $\|\bm{A}\|_{row}<\infty$, and $\bm{p}$ and $\bm{q}$ two conformable vectors. Then,

[TABLE]

**Proof ** Follows from (20) and (14), on pages 110 and 112 in Lütkepohl (1996), respectively. $\Box$

Lemma D.6

Let $\bm{A}(m\times m)$ positive semi-definite matrix with $\lambda_{m}(\bm{A})<\infty$, and denote with $\bm{a_{i}}$ its $i$-th row. Then, $\|\bm{a_{i}}\|<\lambda_{m}(\bm{A})$.

**Proof ** Let $\bm{B}=\bm{A}\bm{A}=[b_{ij}]_{i,j=1}^{m}$. By the second inequality in Lemma D.4(iv), $\|\bm{a_{i}}\|^{2}=b_{ii}\leq\lambda_{m}(\bm{B})=\lambda_{m}^{2}(\bm{A})$. $\Box$

E The inverse of a covariance matrix

Lemma E.1 reports a result well known in time series analysis. We prefer to report it anyway with proof, both for completeness and to introduce some further notation. Approximating the matrix $\bm{\mathfrak{C}}$ with the circulant symmetric matrix $\bm{\mathfrak{C}}^{(s)}$, is convenient for deriving the properties of the inverse of $\bm{\mathfrak{C}}$. Diagonalizing $\bm{\mathfrak{C}}^{(s)}$ we obtain the matrix $\Lambda$ of easy interpretation (see E.6). Lemma E.2 highlights the importance of Lemma A.3. The latter implies that the inverse of the matrix $\bm{\mathfrak{C}}^{(s)}$ has bounded row-norm. The same property is inherited by the matrix $\bm{P^{\prime}G^{-1}_{\varsigma}\bm{P}}$, that is more convenient to work with. The main advantage is that the nonzero entries of the diagonal matrix $\bm{G^{-1}_{\varsigma}}$ do not depend on $T$. Corollary E.3 adapt the result derived for time series analysis to a panel framework.

Lemma E.1

Let $\varsigma(\cdot)$ a real function defined on the integers. Assume that $\varsigma(\cdot)$ is symmetric, non-negative definite and $\sum_{h=1}^{\infty}h^{\delta}\left|\varsigma(h)\right|<\infty$, with $\delta\geq 1$. Let

[TABLE]

and ${\bm{G}}_{\varsigma}:={\rm diag}\left({\bm{g}}(\varsigma,\omega)\right)$ be the $T\times T$ matrix, with

[TABLE]

where $\omega_{j}=2\pi j/T$, $j=0,\dots,[T/2]$ and $[x]$ denotes the integer part of $x$. Define the real orthogonal $T\times T$ matrix ${\bm{P}}$ by

[TABLE]

where

[TABLE]

and the matrix $\bm{\mathfrak{C}}=\left[\varsigma(i-j)\right]_{i,j=1}^{T}$. Denote by $\Delta_{pq}^{(T)}$ is the $p,q$ component of the matrix ${\bm{P}}\bm{\mathfrak{C}}{\bm{P}}^{\prime}-2\pi\bm{G}_{\varsigma},$ then

[TABLE]

**Proof ** The proof follows closely (Brockwell & Davis 1991, Proposition 4.5.2). Define the $T\times T$ circulant symmetric matrix

[TABLE]

For brevity’s sake we only consider the case of $T$ odd131313For the case when $T$ is even the reader is referred to Brockwell & Davis (1991), p. 135. Following (Brockwell & Davis 1991, Section 4.5), the above matrix can be diagonalized as $\bm{P}\bm{\mathfrak{C}}^{(s)}\bm{P}^{\prime}=\bm{\Lambda}$,

[TABLE]

with

[TABLE]

Let $\bm{p_{i}}=\left[p_{i1},p_{i2},\dots,p_{iT}\right]$ and $\bm{e_{i}}$ denote the $i^{th}$ row of the matrices $\bm{P}$, $\bm{I_{T}}$, respectively. Hence we have to show that

[TABLE]

The first term on the right hand side of the above inequality is bounded in absolute value by

[TABLE]

For the second term we have

[TABLE]

Since both (E.9),(E.10) terms are independent of $i$ and $j$, the proof is completed. $\Box$

Lemma E.2

Let $\bm{P}$, $\bm{G_{\varsigma}}$ and $\bm{\Lambda}$ be as in (E.2), (E.3) and (E.6), respectively. Let $\lambda(\omega)=\sum_{|h|\leq[T/2]}\varsigma(h)\exp(ih\omega)$, with $\lambda(\omega_{j})=\lambda_{j}$. Suppose that $\inf_{\omega}\lambda(\omega)>0$, $\inf_{\omega}g_{\varsigma}(\omega)>0$, and the conditions assumptions of Lemma E.1 are satisfied with $\delta\geq 2$. Then, $\left\|\bm{P}^{\prime}\bm{G_{\varsigma}}^{-1}\bm{P}\right\|_{row}<\infty$,

**Proof ** For sake of brevity, we consider only the case of $T$ odd, as in the proof of Lemma E.1. Write

[TABLE]

where $\bm{\mathfrak{C}}^{(s)}$ has been defined in (E.5).

The inverse of $\bm{\mathfrak{C}}^{(s)}$ is circulant and symmetric, with eigenvalues $1/\lambda_{j}$, $j=0,\dots,\left[T/2\right]$, where $\lambda_{j}$ is defined as (E.7). Because

[TABLE]

is continuous in $\omega$, Lemma A.3 implies that $\varsigma(h)$ is absolutely summable, and $\left\|\left(\bm{\mathfrak{C}}^{(s)}\right)^{-1}\right\|_{row}<\infty$.

By the norm inequalities (6) and (12) in (Lütkepohl 1996, 8.5.12, (12))

[TABLE]

where the latter equality follows from(E.9). $\Box$

Corollary E.3

Let $\varsigma_{i}(\cdot)$, $i=1,\dots,N$ a set of real functions defined on the integers. Assume that,

1. (a)

$\varsigma_{i}(\cdot)$* is symmetric, non-negative definite and $\sup_{i}\sum_{h=1}^{\infty}h^{\delta}\left|\varsigma_{i}(h)\right|<\infty$, with $\delta\geq 2$ , $\forall\;i$;* 2. (b)

$m_{g}:=\inf_{i}\inf_{\omega}g_{i}(\omega)>0$, and $M_{g}:=\sup_{i}\sup_{\omega}g_{i}(\omega)<\infty$;

where $g_{\varsigma_{i}}(\omega)=(2\pi)^{-1}\sum_{h=-\infty}^{\infty}\varsigma_{i}(h)\cos(h\omega)$, $-\pi\leq\omega\leq\pi$ and $m_{g},M_{g}$ are positive constants. Let $\bm{\mathcal{C}_{N}}=\left[\varsigma_{N}(i-j)\right]_{\ell,j=1}^{T}$, with $\varsigma_{N}(h)=N^{-1}\sum_{i=1}^{N}\varsigma_{i}(h)$. Define the function

[TABLE]

and $T\times T$ diagonal matrix $G_{\varsigma_{N}}$ as in display(E.2). Denote by $\Delta_{pq}^{(NT)}$ is the $p,q$ component of the matrix ${\bm{P}}\bm{\mathcal{C}_{N}}{\bm{P}}^{\prime}-2\pi\bm{G_{\varsigma_{N}}},$ with $\bm{P}$ defined in (E.3). Then,

1. (i)

$\sup_{1\leq\ell,j\leq T}|\Delta_{\ell,j}^{(NT)}|=O(1/T),\quad\textrm{as}\quad N,T\to\infty,\qquad\ell,j=1,\dots,T.$** 2. (ii)

$\left\|\bm{P^{\prime}}\bm{G}^{-1}_{\varsigma_{N}}\bm{P}\right\|_{row}<\infty$.

**Proof ** Tonelli’s lemma imply that, under Assumption $(a)$, $\sum_{h}\varsigma_{N}(h)=N^{-1}\sum_{i}\sum_{h}\varsigma_{i}(h)$. The proof of part $(i)$ then follows very closely equations (E.8)-(E.10).

Part $(ii)$ of the lemma is proved following the proof of Lemma E.2. A slight modification of the calculations in (E.11) show that $d^{2}\lambda^{-1}_{N}(\omega)/d\omega^{2}$ is continuous in $\omega$, with $\lambda_{N}(\omega)=\sum_{|h|\leq[T/2]}\varsigma_{N}(h)\exp(ih\omega)$ . $\Box$

F Proof of the lemmas in Appendix A

Proof Lemma A.1 Part (i) By the identity in equation (F.1) we have

[TABLE]

Combining the above equality with Lemma D.1, we get

[TABLE]

Hence, by Assumptions $(a)-(d)$

[TABLE]

Parts (ii) and (iii): The proof follows straightforwardly from the inequality

[TABLE]

$\Box$

Proof Lemma A.2 By 3.5.2(5) in Lütkepohl (1996)

[TABLE]

Plugging-in the RHS of (F.2) in $\bm{A}^{-1}$ in RHS of (F.1) proves Part $(ii)$ of the Lemma. Similarly, Part $(ii)$ of the Lemma is proved replacing $r$ times the RHS of (F.2) for $\bm{A}^{-1}$ in the RHS of (F.1). $\Box$

Proof Lemma A.3 For a proof of case $r=1$ see (Zygmund 1959, Chapter II, Theorem 4.7, p.48). Consider case $r=2$ and, without loss of generality, set $h>0$. Replace $\omega$ with $\omega-\pi/h$; using integration by substitution, the Fourier coefficient $\varsigma_{h}$ can be re-written as

[TABLE]

Similarly, substituting $\omega$ with $\omega+\pi/h$, we get

[TABLE]

Hence, for some for some $0\leq s_{1}=s_{1}(\omega),s_{2}=s_{2}(\omega),s_{3}=s_{3}(\omega)\leq 1$

[TABLE]

The first equality follows from expressions (F.3) and(F.4). The second and third equalities follows by applying twice the mean-value theorem.

Since $|s_{1}+s_{2}|\leq 2$ and, by assumption $|g^{(2)}(\omega)|\leq\kappa$, we have

[TABLE]

proving the theorem for $r=2$. The general case follows along the same lines. $\Box$

G Proof of the propositions in Appendix B

G.1 Proof of the propositions in Appendix B.1

Proof of Proposition B.1 Part $(i)$ follows from Remark 2.6. Combining Remark 2.4, Assumptions 2.2 and 2.6 and Lemma D.4(i)

[TABLE]

The proof of Part $(ii)$ is completed noting that

[TABLE]

Using similar arguments, for Part $(iii)$ we have

[TABLE]

noting that $\mathbb{E}{\rm tr}(\bm{\mathrm{F}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}/T)\leq\mathbb{E}(\bm{\mathrm{F}}^{\prime}\bm{\mathrm{F}}/T)<\infty$ by Assumption 2.3. The result in Part $(iv)$ follows along the same lines, showing

[TABLE]

Finally, Part $(v)$ follows from Lemma D.4(iii) and Remark 2.4. $\Box$

Proof Proposition B.2

[TABLE]

which tends to zero by Assumption 2.1 as $T\to\infty$, for a suitable choice $\tau_{0}$ increasing with $T$. $\Box$

Proof Proposition B.3 Convergence in distribution in (B.11) is implied if the convergence holds conditional on $\operatorname{\mathscr{F}}(\bm{\mathrm{Z}})$, and we establish the latter. For any vector $\bm{h}$ satisfying $\left\|\bm{h}\right\|=1$ and any $T,\tau_{0}$, define ${d}_{ju}:=T^{-{\nicefrac{{1}}{{2}}}}\bm{h}^{\prime}\bm{W}_{j0}^{-{1\over 2}}\bm{s}_{ju}$. Then, following Robinson & Hidalgo (1997), Proposition 2 (see also Scott (1973), Theorem 1, Conditions (B)), it is sufficient to show that

[TABLE]

and

[TABLE]

where $\bm{1}(\cdot)$ denotes the indicator function. By construction $\sum_{-T_{0}}^{T}d^{2}_{ju}=1/(\mathbb{E}\eta^{2}_{j0})$, hence (G.3) follows trivially. For some $\delta_{2}>0$, (G.4) is bounded by

[TABLE]

The first term is made arbitrarily small by choosing $\delta_{2}$ suitably small and noting that $\eta_{ju}^{2}$ is uniformly integrable by Assumption (2.1) and in (Davidson 1994, Theorem 12.10). Concerning the second term in (G.5), Propositions (B.2) and (B.4) below imply that $\|\bm{W}_{j0}^{-1}\|=O_{p}(1)$, so we can consider the set $\parallel\bm{W}_{j0}^{-{1\over 2}}\parallel\leq\kappa$ on which, for any $\epsilon>0$

[TABLE]

for $L=L(\epsilon)$ satisfying $\sum_{u\geq L}\phi_{ju}^{2}<\epsilon$. By Assumption 2.3, $\mathbb{E}\left\|\bm{\mathrm{z}}_{t}\right\|^{2}<\infty$ and $\max_{1\leq t\leq T}\left\|\bm{\mathrm{z}}_{t}\right\|\leq\kappa\left(\sum_{t=1}^{T}\left\|\bm{\mathrm{z}}_{t}\right\|^{4}\right)^{\nicefrac{{1}}{{4}}}=O_{p}(T^{\nicefrac{{1}}{{4}}})$. The proof of (G.4) is concluded by Markov inequality. $\Box$

Proof Proposition B.4 Note that

[TABLE]

Remark 2.3 implies that the matrix $\bm{\Phi}_{j}$ is positive definite for a sufficiently large $T$. The proof follows letting $T$ diverge to infinity. $\Box$

Proof of proposition B.5 Part $(i)$ follows from Lemma A.1(ii). To verify that $\left\|T^{-1}\bm{\mathcal{F}}^{\prime}\left(\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\Xi^{-1}_{N}}\bm{\mathrm{D}}_{\bot}\right)^{-1}\bm{\mathcal{F}}\right\|=O_{p}(1)$, by Lemma D.4(i) it is enough to prove that $\lambda_{1}\left(\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\Xi_{N}}\bm{\mathrm{D}}_{\bot}\right)>\kappa$. Define $\bar{\bm{\mathrm{D}}}:=\bm{\mathrm{D}}\left(\bm{\mathrm{D}}^{\prime}\bm{\mathrm{D}}\right)^{-\nicefrac{{1}}{{2}}}$, then $\lambda_{i}\left(\left[\bm{\mathrm{D}}_{\bot}\;\bar{\bm{\mathrm{D}}}\right]^{\prime}\bm{\Xi_{N}}\left[\bm{\mathrm{D}}_{\bot}\;\bar{\bm{\mathrm{D}}}\right]\right)$ $=\lambda_{i}\left(\bm{\Xi_{N}}\right),$ for $i=1,\dots,T$. The inclusion principle (Lütkepohl 1996, p. 160) implies that

[TABLE]

Remark 2.3, and Lemma D.4(iii) imply that

[TABLE]

proving the result. Assumption $(a)-(d)$ of the Lemma are satisfied because of Assumptions 2.3 and 2.5, completing the proof. Similarly, Part $(ii)$ follows from Lemma A.1, part $(iii)$ if we show that $\left\|T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{V}}_{\bm{i}}^{\prime}\left(\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\Xi_{N}}\bm{\mathrm{D}}_{\bot}\right)^{-1}\bm{\mathcal{F}}\right\|=O_{p}(1)$. The latter is proved noting that, by Assumptions 2.6, Lemma  D.2 and Lemma D.4.(i)

[TABLE]

The last bound follows noting that $\left\|\bm{\mathrm{D}}_{\bot}\left(\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\Xi_{N}}\bm{\mathrm{D}}_{\bot}\right)^{-1}\bm{\mathrm{D}}_{\bot}^{\prime}\right\|=\left\|\left(\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\Xi_{N}}\bm{\mathrm{D}}_{\bot}\right)^{-1}\right\|$, display (G.7), Remarks 2.3 and 2.4, Assumption 2.3.

To prove part $(iii)$, we first show that $T^{-1}\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\mathcal{S}}^{-1}_{N}\bm{\mathcal{V}}_{\bm{i}}\approx$ $T^{-1}$ $\bm{\mathcal{V}}_{\bm{i}}^{\prime}\left(\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\Xi_{N}}\bm{\mathrm{D}}_{\bot}\right)^{-1}\bm{\mathcal{V}}_{\bm{i}}$. By Lemma D.1,

[TABLE]

where the bound follow from part $(ii)$, Assumption 2.3, and Remark 2.3. Next we show that $T^{-1}\bm{\mathcal{V}}_{\bm{i}}^{\prime}\left(\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\Xi_{N}}\bm{\mathrm{D}}_{\bot}\right)^{-1}\bm{\mathcal{V}}_{\bm{i}}\approx$ $T^{-1}$ ${\bf{V}}_{\bm{i}}^{\prime}\bm{\Xi^{-1}_{N}}{\bf{V}}_{\bm{i}}$. Lemma D.3 and Lemma D.4(i),(iii) entail that

[TABLE]

The desired result follows from Remark 2.4. $\Box$

Proof of proposition B.6 The proof of parts $(i)$ and $(ii)$ are analogous to the proofs of Proposition B.5), parts $(i)$ and $(iii)$, respectively, and hence omitted. $\Box$

Proof of proposition B.7 Note that $\xi_{i}(h)=\sum_{j=1}^{N}r^{2}_{ij}{\rm cov}(a_{jt},a_{jt+h})$, therefore by Assumption 2.1 satisfies the condition of Corollary E.3, implying that ${\rm tr}\left(\bm{P}\bm{\Xi_{N}}\bm{P}^{\prime}-2\pi\bm{G}_{\xi_{N}}\right)^{2}=O(1)$. Hence, by Lemma D.4(i),

[TABLE]

The proof of Part (i) follows by Markov’s inequality, whereas Part (ii) follows along the lines of Lemma E.2’s proof. $\Box$

Proof of proposition B.8 Following (Robinson & Hidalgo 1997, Propositon 1) and Proposition B.3

[TABLE]

which tends to zero as $T\to\infty$ for a suitable choice $\tau_{0}$ increasing with $T$, by Assumption 2.1. $\Box$

Proof of proposition B.9 Following (Robinson & Hidalgo 1997, Propositon 2) and Proposition B.4 convergence in distribution in (B.19) is implied if the convergence holds conditional on $\operatorname{\mathscr{F}}({\bf{V}}_{\bm{i}})$, the sigma algebra generated by $\{\bm{v_{it}}\}_{t=1}^{T}$, and we establish the latter. For any vector $\bm{h}$ satisfying $\parallel\bm{h}\parallel=1$ and any $T,\tau_{0}$, define ${d}_{iju}=T^{-{\nicefrac{{1}}{{2}}}}\bm{h}^{\prime}\left(\bm{W}^{(\bar{\xi})}_{ij0}\right)^{-{1\over 2}}\bm{s}_{iju}$. The proof of the following results

[TABLE]

and

[TABLE]

follow closely the proof of (G.3) and (G.4). For conciseness we will limit ourselves to prove that the second term of (G.13)

[TABLE]

converges in probability to zero, for some $\delta_{2}>0$. Consider the set $\left\|\left(\bm{W}^{(\bar{\xi})}_{ij0}\right)^{-{1\over 2}}\right\|\leq\kappa$ on which, for any $\kappa_{2}>0$,

[TABLE]

for $L=L(\kappa_{2})$ satisfying $\sum_{u\geq L}\phi_{ju}^{2}<\epsilon$. The proof of (G.12) is completed noting that, by Assumption 2.2 and Proposition B.8.(ii)

[TABLE]

and

[TABLE]

$\Box$

Proof B.10 Proceeding as for the proof of Proposition B.4, define

[TABLE]

with $\bm{\Phi_{j}}$ is positive definite for large $T$. The proof is concluded if we prove that

[TABLE]

The bound follows noting that $\mathbb{E}\left\|T^{-\nicefrac{{1}}{{2}}}\bm{\Xi^{-1}_{N}}{\bf{V}}_{\bm{i}}\right\|=O(1)$ and, analogously to the Lemma B.7’ proof,

[TABLE]

$\Box$

G.2 Proof of the propositions in Appendix B.2

Proof of proposition B.11 The results follows from Lemmata 2 (Section H.3) and the triangular inequality. $\Box$

Proof of proposition B.12 The proof is similar to that of Proposition B.5, and hence details are omitted. We will only check that $\bm{\mathcal{H}_{N}}$ satisfies the assumption of Lemma A.1. It will be enough to establish that the eigenvalues of $\bar{c}_{N}$ are bounded, and uniformly bounded away from zero. Denote by $\theta_{its}$ the $t,s$-element of $\bm{\Theta_{i}}$.

[TABLE]

where $\mathcal{q}_{ik\ell}$ is the $k,\ell-$element of the symmetric $(K\times K)$ matrix $\bm{\mathscr{Q}}_{i}$

[TABLE]

Assumptions 2.3 and 2.5, Lemma D.4(iii)(iv) and Proposition B.1(v) imply that $\sup_{i,k,\ell}|\mathcal{q}_{ik\ell}|<\infty$. Assumption 2.2 implies that $\sup_{it}\sum_{s=1}^{T}|\theta_{its}|<\infty$, and hence $\lambda_{T}(\bm{\Theta_{i}})<\infty$. Using again Lemma D.4(iv) and display (G.8) we find that

[TABLE]

proving the result. $\Box$

**Proof of proposition B.13 ** Analogous to Proposition B.6, and hence omitted. $\Box$

**Proof of proposition B.14 ** Recall that $\bm{\Theta_{i}}=[\theta_{its}]$ and $\bm{\Xi_{i}}=[\xi_{its}]$. For $h=t-s$, Assumptions 2.1 and 2.2, and the uniform boundness of $\mathcal{q}_{ik\ell}$ (see the proof of Lemma B.12) entail that $\sup_{i}\sum_{h=1}^{\infty}h^{2}\left(|\xi_{i}(h)|+|\theta_{i}(h)|\right)<\infty$. Hence,

[TABLE]

is even and positive definite and the conditions in Lemmas E.1 and A.3 and Corollary E.3 are satisfied. The proof is completed along the line of Proposition B.7. $\Box$

**Proof of proposition B.15 ** The proof makes use of Corollary E.3 and proceed as for Lemma B.8 $\Box$

**Proof of proposition B.16 ** See the Proof of Proposition (B.9) $\Box$

**Proof of proposition B.17 ** Equation (B.25) implies that

[TABLE]

For sake of brevity, we only consider the first norm on the RHS of the inequality, that is bounded by

[TABLE]

The results follows from Remark 2.3, Assumption 2.2, inequality (G.17) and noting that, using similar arguments as in the proof of Proposition B.7

[TABLE]

The remaining terms can be bounded similarly. $\Box$

H Auxiliary results for proof of Theorem 3.2

As mentioned at the end of Appendix C, Lemmata I derive some properties of the matrix $\SS$ and results frequently used in the following proofs. Lemmata II and Lemmata III develop key results for the proof of Theorem 3.2 (see Proposition B.11). Following the discussion in Section 5, we consider the double weighted convergence of the difference $\bm{\hat{\mathcal{S}}_{N}}^{-1}-\bm{\mathcal{H}^{-1}_{N}}$, that by the identity in (36), can be rewritten as

[TABLE]

H.1 Lemmata I

Lemma H.1

Under Assumptions of Theorem 3.2

1. (i)

$\left\|\bm{\Psi_{i}}\right\|_{sp}=O_{p}\left(1\right)$* and $\left\|\bm{\Psi_{i}}^{-1}\right\|_{sp}<\infty$, $\forall i$.* 2. (ii)

$\lambda_{T}\left(\bm{\breve{\Theta}_{i}}\right)=O_{p}(1)$, $\forall i$. 3. (iii)

$\lambda_{T-K}\left(\bm{\Omega_{N}}^{-1}\right)<\infty.$** 4. (iv)

$0<\lambda_{i}\left(\bm{I}_{M}-\bm{\Gamma_{i}}\bm{\Psi_{i}}^{-1}\bm{\Gamma_{i}}^{\prime}\frac{\bm{\mathcal{F}}^{\prime}\bm{\mathcal{F}}}{T}\right)\leq 1$* for $i=1,\dots,M$.*

Lemma H.2

Under Assumptions of Theorem 3.2

1. (i)

$\left\|\bm{\Omega_{N}}^{-1}\bm{\mathcal{F}}\right\|=O_{p}\left(\frac{1}{\sqrt{T}}\right).$** 2. (ii)

$\left\|\bm{\mathcal{F}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\mathcal{F}}\right\|=O_{p}\left(1\right).$** 3. (iii)

$\left\|\bm{\mathcal{V}_{j}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\mathcal{F}}\right\|=O_{p}\left(\frac{1}{\sqrt{T}}\right).$** 4. (iv)

$\left\|\bm{\mathcal{F}}\bm{\Omega_{N}}^{-1}\bm{\epsilon_{i}}\right\|=O_{p}\left(\frac{1}{\sqrt{T}}\right).$** 5. (v)

$\left\|\bm{\mathcal{V}}_{\bm{i}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\epsilon_{i}}\right\|=O_{p}(\sqrt{T}).$** 6. (vi)

$\left\|\bm{\mathcal{V}_{j}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\mathcal{V}}_{\bm{i}}\right\|=O_{p}(T)$* for any $i,j$.*

Lemma H.3

Under Assumptions of Theorem 3.2

1. (i)

$\left\|\bm{\varepsilon_{i}}^{\prime}\bm{\mathrm{D}}\right\|=O_{p}\left(\sqrt{T}\right).$** 2. (ii)

$\left\|{\bf{V}}_{\bm{i}}^{\prime}\bm{\mathrm{D}}\right\|=O_{p}\left(\sqrt{T}\right).$** 3. (iii)

$\left\|\bm{\epsilon_{i}}^{\prime}\bm{\mathcal{X}_{i}}\right\|=\left\|\bm{\varepsilon_{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{X}_{i}}\right\|=O_{p}\left(\sqrt{T}\right).$** 4. (iv)

$\left\|\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{F}}\right\|=\left\|\bm{\mathrm{X}_{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}\right\|=O_{p}\left(T\right).$**

Lemma H.4

Under Assumptions 2.2-2.6, for $j=1,2,\dots$ $J$,

[TABLE]

with

[TABLE]

and

[TABLE]

Remark H.1

Lemma H.4 implies that, for $J=0$

[TABLE]

H.2 Proof Lemmata I

Proof Lemma H.1

1. (i)

The proof of the first part follows from Remark 2.4 and Assumption 2.3. For the second part, it is enough to show that $\lambda_{1}\left(\bm{\Psi_{i}}\right)>0$ as $T\to\infty$. By Lemma D.4(iii) and Remark 2.4

[TABLE] 2. (ii)

The proof is similar that of Proposition B.12, and hence some details are omitted. Similarly to Equation (G.14), let $\breve{\theta}_{its}={\rm tr}\left(\bm{\breve{\mathscr{Q}}}_{i}\mathbb{E}\left(\bm{\mathrm{v}}_{it}\bm{\mathrm{v}}_{is}^{\prime}\right)\right)$ where

[TABLE]

with $k,\ell-$ entry equal to $\breve{\mathcal{q}}_{ik\ell}$. Because $\sup_{i,k,\ell}|\breve{q}_{ik\ell}|=O_{p}(1)$, $\sup_{it}\sum_{s=1}^{T}|\breve{\theta}_{its}|=O_{p}(1)$, and hence $\lambda_{T}(\bm{\breve{\Theta}_{i}})=O_{p}(1)$. 3. (iii)

From Lemma D.4, part (i) $\lambda_{T-k}(\bm{\Omega_{N}}^{-1})\leq\lambda_{T}(\bm{\mathrm{C}_{N}}^{-1})\lambda_{T}(\mathscr{M}_{\bm{\mathrm{D}}})$. Part (iii) of the same lemma implies that $\lambda_{T}(\bm{\mathrm{C}_{N}}^{-1})\leq\lambda_{1}^{-1}(\bm{\Xi_{N}})<\infty$, where the last inequality follows from Remark 2.3. 4. (iv)

To simplify the exposition, we assume that $K\geq M$ (the case $K<M$ follows similarly). Because

[TABLE]

are equal to $\lambda_{i}\left(T^{-1}\bm{\Psi_{i}}^{-\nicefrac{{1}}{{2}}}\bm{\Gamma_{i}}^{\prime}\bm{\mathcal{F}}^{\prime}\bm{\mathcal{F}}\bm{\Gamma_{i}}\bm{\Psi_{i}}^{-\nicefrac{{1}}{{2}}}\right)$, $i=K-M+1,\dots,K$, and hence the eigenvalues in (H.5) are non-negative. Moreover

[TABLE]

It follows that the eigenvalues in (H.5) are non-negative and smaller than one, proving the theorem.

$\Box$

Proof Lemma H.2 In view of LemmaH.1, Part (i) follows from Lemma A.1(a). The proofs of Part(ii) is similar to B.5(i), whereas that of Part(iii) and (iv) are similar to B.5(ii), and hence omitted. Finally, for Part (v) we have

[TABLE]

$\Box$

Proof Lemma H.3 For part (ii), by Lemma D.4, Assumptions 2.6 and 2.3 and Remark 2.3, we have

[TABLE]

The proofs of parts (i) and (ii) use similar arguments, and hence are omitted. For the latter part, $\left\|\bm{\mathrm{X}_{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}\right\|\leq\left\|\bm{\Gamma_{i}}\right\|\left\|\bm{\mathrm{F}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}\right\|+\left\|{\bf{V}}_{\bm{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}\right\|=O_{p}(T)+O_{p}(\sqrt{T})$ in view of Assumption 2.3 and noting that $\mathbb{E}\left\|{\bf{V}}_{\bm{i}}^{\prime}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}\right\|^{2}\leq\mathbb{E}\|\bm{\mathrm{F}}\|^{2}\lambda_{T}(\mathbb{E}{\bf{V}}_{\bm{i}}{\bf{V}}_{\bm{i}}^{\prime})=O_{p}(T)$. $\Box$

Proof Lemma H.4 The equality in displays (H.1)-(H.2) follows from Lemma A.2. Concerning the bound in (H.3), by triangular inequality

[TABLE]

For the first term on the RHS of the inequality, Assumption 2.2 imply that

[TABLE]

By Lemma D.4(i)

[TABLE]

implying that $\left\|\frac{{\bf{V}}_{\bm{i}}^{\prime}\mathscr{P}_{\bm{\mathrm{D}}}{\bf{V}}_{\bm{i}}}{T}\right\|\leq\frac{1}{T}\left\|{\bf{V}}_{\bm{i}}^{\prime}\mathscr{P}_{\bm{\mathrm{D}}}\right\|^{2}=O_{p}\left(\frac{1}{T}\right)$. Similarly, for the last two terms we have

[TABLE]

concluding the proof. $\Box$

H.3 Lemmata II

Under the Assumption of Theorem 3.2, the following results holds:

Lemma H.5

2. (i)

$\left\|\frac{1}{\sqrt{T}}\bm{\mathcal{X}_{j}}^{\prime}\left(\bm{\hat{\mathcal{S}}_{N}}^{-1}-\bm{\Omega_{N}}^{-1}\right)\bm{\mathcal{u}_{j}}\right\|=o_{p}(1).$** 3. (ii)

$\left\|\frac{1}{T}\bm{\mathcal{X}_{j}}^{\prime}\left(\bm{\hat{\mathcal{S}}_{N}}^{-1}-\bm{\Omega_{N}}^{-1}\right)\bm{\mathcal{X}_{j}}\right\|=o_{p}(1).$**

Lemma H.6

2. (i)

$\left\|\frac{1}{\sqrt{T}}\bm{\mathcal{X}_{j}}^{\prime}\left(\bm{\Omega_{N}}^{-1}-\bm{\mathcal{H}^{-1}_{N}}\right)\bm{\mathcal{u}_{j}}\right\|=o_{p}(1).$** 3. (ii)

$\left\|\frac{1}{T}\bm{\mathcal{X}_{j}}^{\prime}\left(\bm{\Omega_{N}}^{-1}-\bm{\mathcal{H}^{-1}_{N}}\right)\bm{\mathcal{X}_{j}}\right\|=o_{p}(1).$**

H.4 Proofs Lemmata II

Proof Lemma H.5 The proof rely on the results stated in the next section, Lemmata III.

1. missingfnum@PartniPart (i)Part (i)

By Lemma A.2(i) and the triangular inequality

[TABLE]

For the term (H.7) we have

[TABLE]

The term (H.11) is bounded in Lemma H.7. Lemma H.8, Lemma H.2(i), and Assumption 2.5 entail that the term (H.11) can be bound as

[TABLE]

Similarly, but using Lemma H.9 instead of Lemma H.8, we find that the term (H.11) is $o_{p}(1)$.

Next, for the term (H.11), Assumption 2.5, Lemma H.2(i) and Lemma H.10 imply

[TABLE]

Hence, we conclude that the term (H.7) is $o_{p}(1)$.

As noted in Remark 2.8, Assumption 2.4 implies that $\|\bm{\hat{\mathcal{S}}_{N}}^{-1}\|<\infty$. Hence, the term (H.7) is bounded by

[TABLE]

Taking similar steps as for the term in (H.7), Lemmas H.8, H.9 and H.10 entail that (H.7) is of order $\max\left\{O_{p}(1),O_{p}(T/\sqrt{N}),O_{p}(T^{2}/N)\right\}$, concluding the proof. 2. missingfnum@PartniPart (ii)Part (ii)

The proof follows closely that of Part (i), and hence is omitted.

$\Box$

Proof Lemma H.6 Similar to the proof of Lemma H.5 $\Box$

H.5 Lemmata III

Lemma H.7

[TABLE]

Lemma H.8

[TABLE]

Lemma H.9

[TABLE]

Lemma H.10

[TABLE]

Lemma H.11

[TABLE]

Lemma H.12

[TABLE]

H.6 Proof Lemmata III

The proofs exploits results on moments and cumulants. A summary of the main results on cumulants used in the proof are provided in Section H.6.1, that also serves to introduce further notation.

H.6.1 Combinatorial expressions of cumulants and moments

In the presentation we follows Peccati & Taqqu (2008). Let $a$ denotes a finite non-empty set, and $\mathcal{P}(a)$ the set of partitions of $a$. By definition, an element $\pi\in\mathcal{P}(a)$ is a collection of non-empty and disjoint subset of $a$ (called blocks), such that their union equals $a$. To clarify the notation used in the following, if $\pi=\{a_{1},\dots,a_{j}\}\in\mathcal{P}(a)$, then $\cap_{i=1}^{j}a_{i}=\emptyset$, $\cup_{i=1}^{j}a_{i}=a$, where $\emptyset$ is the empty set. The symbol $|\pi|$ indicates the number of blocks (or the size) of the partition $\pi$.

For every $\sigma,\pi\in\mathcal{P}(a)$, we write $\sigma\leq\pi$ if, and only if, each block of $\sigma$ is contained in a block of $\pi$. Whenever $\sigma\leq\pi$, one has $|\sigma|\geq|\pi|$. The partition $\sigma\wedge\pi$, meet of $\sigma,\pi\in\mathcal{P}(a)$, is the element of $\mathcal{P}(a)$ whose blocks are constructed by taking the non-empty intersection of the blocks of $\sigma$ and $\pi$. The partition $\sigma\vee\pi$, join of $\sigma,\pi\in\mathcal{P}(a)$, is the element of $\mathcal{P}(a)$ whose blocks are constructed by taking the non-disjoint unions of the blocks of $\sigma$ and $\pi$, that is, by taking the union of those blocks that have at least one element in common. In particular, $|\sigma\wedge\pi|\geq|\sigma\vee\pi|$. The partitions $\sigma\wedge\pi$ and $\sigma\vee\pi$ are the greatest lower bound and the least upper bounds associated with the pair $(\sigma,\delta)$. The maximal element of $\mathcal{P}(a)$ is the trivial partition $\hat{1}=\left\{a\right\}$. The minimal element of $\mathcal{P}(a)$ is the partition $\hat{0}$, such that each block of $\hat{0}$ contains exactly one element of $a$. Observe that $|\hat{1}|=1$ and $|\hat{0}|=|a|$.

Example H.13

Take $a=\{1,2,3,4,5\}$. If $\pi=\{\{1,2,3\},\{4,5\}\}$, $\sigma=\{\{1,2\},\{3\},\{4,5\}\}$. Then, $\sigma\leq\pi$ and $\delta\wedge\pi=\delta$, $\delta\vee\pi=\pi$.

Let $[n]=\left\{1,2,\dots,n\right\}$. For $n\geq 1$ we consider a vector-values random variables $\bm{X^{,[n]}}=(X_{1},\dots,X_{n})$ such that $\mathbb{E}|X_{j}|^{n}\leq\infty$. For every subset $a=\left\{j_{1},\dots,j_{k}\right\}\subseteq[n]$, we write

[TABLE]

For instance, $\forall m\leq n$,

[TABLE]

The next result contains three crucial relations, linking the cumulants and the moments associated with a random vector $\bf{X^{,[m]}}$.

Proposition H.14** (Proposition 3.1. Peccati & Taqqu (2008))**

For every $a\subset[n]$,

[TABLE] 2. 2.

[TABLE] 3. 3.

$\forall\pi=\{a_{1},\dots,a_{k}\}\in\mathcal{P}(a)$,

[TABLE]

Part 1 and 2 are due to Leonov & Shiryaev (1959), part 3 to Malyshev (1980).

Example H.15

As an illustration, consider the cumulant ${\rm cum}(X_{1}X_{2},X_{3})$, in which case one has $\pi={a_{1},a_{2}}$, with $a_{1}=\{1,2\}$ and $a_{2}=\{3\}$. There are three partition $\sigma\in\mathcal{P}\left([3]\right)$ such that $\sigma\vee\pi=\hat{1}=\{1,2,3\}$, namely

[TABLE]

It follows that

[TABLE]

Similarly

[TABLE]

Part of the notions presented in the above sections can be translated into the language of diagrams, which are often used to compute higher order cumulants.

Consider a finite set $a$. A diagram is a graphical representation of a pair of partitions $(\pi,\sigma)\subseteq\mathcal{P}(a)$ such that $\pi=\{a_{1},\dots,a_{k}\}$ and $\sigma=\{p_{1},\dots,p_{s}\}$. It is obtained as follows

Order the elements of each block $a_{i}$, for $i=1,\dots,k;$ 2. 2.

Associate with each block $a_{i}\in\pi$ a row of each $|a_{i}|$ vertices (represented as dots), in such a way that the $j$th vertex of the $i$th row corresponds to the $j$th element of the block $a_{i}$; 3. 3.

For every $\ell=1,\dots,s$, draw a closed curve around the vertices corresponding to the elements of the block $p_{l}\in\sigma$. We will denote by $\mathscr{D}(\pi,\sigma)$ the diagram of a pair of partitions $(\pi,\sigma).$

The diagram $\mathscr{D}(\pi,\sigma)$ associated with two partitions $(\pi,\sigma)$ is said to be connected is $\delta\vee\pi=\hat{1}$, that is if the only partition $\rho$ such that $\pi\leq\rho$ and $\sigma\leq\rho$ is the maximal partition $\hat{1}$. In other words, $\mathscr{D}(\pi,\sigma)$ is connected if and only if the rows of the diagram (the blocks of $\pi$), cannot be divided into two subsets, each defined a separate diagram.

The Malyshev formula (H.16) can be reformulated as follows. For every finite set $a$ and every $\pi=\{a_{1},\dots,a_{k}\}\in\mathcal{P}(a)$

[TABLE]

where $\mathcal{D}^{c}(\pi,\sigma)$ denotes the class of connected diagrams for the pair of partitions $(\pi,\omega)\subset\mathcal{P}(a)$, that is

[TABLE]

Example H.16

Consider again the Example H.13. Recall that in this case $a=[3]$ and that the relevant partition is $\pi=\{\{1,2\},\{3\}\}$. There are only three partitions $\sigma_{1},\sigma_{2},\sigma_{3}\in\mathcal{P}([3])$ such that $\mathscr{D}(\pi,\sigma_{1})$, $\mathscr{D}(\pi,\sigma_{2})$, and $\mathscr{D}(\pi,\sigma_{3})$ are connected, namely

[TABLE]

The cumulants ${\rm cum}(X_{1},X_{2},X_{3})$,${\rm cum}(X_{1},X_{3}){\rm cum}_{1}(X_{2})$, and ${\rm cum}(X_{1}){\rm cum}(X_{2},X_{3})$ are associated with the diagrams $\mathscr{D}(\pi,\sigma_{1})$, $\mathscr{D}(\pi,\sigma_{2})$ and $\mathscr{D}(\pi,\sigma_{3})$, respectively. Taking the sums over the partitions associated with the connected diagram, the formula (H.17) is obtained.

H.6.2 Proof Lemma H.7

Recalling the definitions in Appendix C; for $j=1,2,3$ let $\bm{\hat{C}_{jN}}:=N^{-1}\sum_{i=1}^{N}\bm{\hat{C}_{ji}}$, with $\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\hat{C}}_{ji}\bm{\mathrm{D}}_{\bot}=\bm{\hat{\mathcal{C}}}_{ji}$, and $\bm{\breve{\Theta}}_{N}=N^{-1}\sum_{i=1}^{N}\bm{\breve{\Theta}_{i}}$.

To save notation, define the $T\times T$ matrix $\bm{\bar{\Omega}}=[\bar{\omega}_{ts}]$, with

[TABLE]

From the bounds in displays (H.21), (H.6), (H.57) and (H.66) we have

[TABLE]

The proof of most of the terms is quite tedious. For the terms that cannot be easily bounded using matrix norm inequality, we apply Markov inequality and bound the expectation. It is useful to note that the conditional expectation (with respect $\operatorname{\mathscr{F}}(\bm{\mathrm{Z}})$) of $\bm{\bar{\Omega}}$ and $\bm{\Psi_{i}}^{-1}$ have uniformly bounded entries, so they can be treated as bounded constants. The save space, the conditioning will be not made explicit, because of Assumption 2.6. For illustrative purposed, a quite detailed proof for $Part(a)$ of Term II will be provided. More concise proof will be provided for the remaining parts and terms. Below, we report as remarks few facts, following from the combination of the Assumptions and previous lemmas, that will be repeatedly used later.

Remark H.2

Property (4) in (Lütkepohl 1996, Section 5.2) and Lemma H.1(iii) imply that $\lambda_{T-K}(\bm{\bar{\Omega}})=\lambda_{T-K}(\bm{\Omega_{N}}^{-1})$. By Lemma D.4(iv), $\sup_{t,s}|\bar{\omega}_{ts}|<\infty$ and Lemma D.6 implies that $\sup_{t}\sum_{s}|\bar{\omega}_{ts}|^{2}<\infty$. Hence, by Lemma D.5, Remark 2.2 and Assumptions 2.2 imply that,

[TABLE]

Similarly, $\sup_{p}\left|\sum_{t,s}\bar{\omega}_{pt}{\rm cov}\left(v_{ikt},v_{jhs}\right)\right|$ and $\sup_{p}\sup_{j}\left|\sum_{i}\sum_{t,s}|\bar{\omega}_{pt}{\rm cov}\left(\varepsilon_{it},\varepsilon_{js}\right)\right|$ are both $O(\sqrt{T})$.

The same bounds can be obtained replacing ${\rm cov}(v_{ikt},v_{jhs})$ with ${\rm cov}(\varepsilon_{it},\varepsilon_{js})$.

Remark H.3

Let $\psi^{-}_{ikh}$ denote the $k,h$-entry of the $K\times K$ matrix $\bm{\Psi_{i}}^{-1}$. Lemma H.1(i), Lemma D.4(iv) and Remark 2.4 imply that $\sup_{i}|\psi^{-}_{ikh}|<\infty$.

Bound Term I

Displays (B.2) and (B.3), Assumption 2.5 and Remark H.1, entail that

[TABLE]

Hence, $\left\|\bm{\hat{A}_{N}}-\bm{A_{N}}\right\|=O_{p}(T^{-\nicefrac{{1}}{{2}}})$. Using Lemma H.2(iii),(iv) we conclude that

[TABLE]

Bound Term II

We split the proof of the latter equality in four parts (Parts (a)-(d)), one for each term on the RHS of the inequality below.

[TABLE]

1. missingfnum@PartiPart (a)Part (a)

To bound the term (H.22) show that

[TABLE]

The bound follows from (H.32) and (H.34) below.

Consider the first cumulant in (H.30). Using the notation introduced in Section H.6.1

[TABLE]

be the partition defining the vertices of the diagram. Then,

[TABLE]

and

[TABLE]

For $l=1,2$, we have

[TABLE]

By Remark H.2, the term in (H.30)-(H.30) can be bounded as follows

[TABLE]

Next, we consider the cumulant in (H.30):

[TABLE]

for $l=1,2,3.$ We can distinguish four different sets of diagrams, defined by the number of blocks in the partition ($l$) and the size of the blocks. We label these sets as $\sigma^{(a)},\dots,\sigma^{(d)}$. For example, the first partition $\sigma^{(a)}=\left\{p_{1},p_{2},p_{3}:|p_{1}|=|p_{2}|=|p_{3}|=2\right\}$ includes all the partitions made of three blocks, $p_{1},p_{2}$ and $p_{3}$, and each block is of size two ($|p_{i}|=2$, for $i=1,2,3$). In a diagram, we would have three rows, each with two vertices.

• $\sigma^{(a)}$

$:=\left\{p_{1},p_{2},p_{3}:|p_{1}|=|p_{2}|=|p_{3}|=2\right\}$

[TABLE]

• $\sigma^{(b)}$

$:=\left\{p_{1},p_{2}:|p_{1}|=2,|p_{2}|=4\right\}$

[TABLE]

• $\sigma^{(c)}$

$:=\left\{p_{1},p_{2}:|p_{1}|=3,|p_{2}|=3\right\}$

[TABLE]

• $\sigma^{(d)}$

$:=\left\{p_{1},|p_{1}|=6\right\}$

[TABLE]

Using the equality in (H.33), for the terms in (H.30)-(H.30) we find

[TABLE]

The latter bound results from the bounds in displays (H.35)-(H.38), which are in turn derived for each of the four sets of diagrams, $\sigma^{(a)},\dots,\sigma^{(d)},$. To save space, for each set, computations are shown for only one diagram. The remaining terms can be shown to be at most of the same order, using similar arguments.

Consider first $\sigma_{1}^{(a)}\in\sigma^{(a)}$

[TABLE]

Then, by Remark H.2

[TABLE]

Similarly, for

[TABLE]

[TABLE]

For

[TABLE]

we have

[TABLE]

Finally, for $\sigma^{(d)}$ we have

[TABLE] 2. 3. missingfnum@PartiPart (b)Part (b)

Setting $J=0$ in Lemma H.4, the term (H.23) can be rewritten as

[TABLE]

The latter bound in (H.41) follows from the results in Lemmata I:

[TABLE]

Recalling that $\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{X}_{i}}=\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}\bm{\Gamma_{i}}+{\bf{V}}_{\bm{i}}-\mathscr{P}_{\bm{\mathrm{D}}}{\bf{V}}_{\bm{i}}$, for the term in (H.40) we have

[TABLE]

The term (H.44) is bounded using again the results in Lemmata I,

[TABLE]

For the term(H.44) we have

[TABLE]

The upper bound follows noting that $\sum_{k,h}\sup_{i}|\psi^{-}_{ikh}|\leq\infty$ by Remark H.3 and the bounds in(H.49) and (H.50) below.

The term (H.48) can be written as

[TABLE]

for $l=1,2,3$, and

[TABLE]

First we consider the partition $\pi_{1}\in\pi$ where each block has the same cross-sectional index:

[TABLE]

leading to

[TABLE]

Using similar arguments, it can be shown that the remaining partitions the bound is $O(T^{2}N)$. For illustrative purposes, we consider the partition

[TABLE]

for which, again by Remark H.2

[TABLE]

The proof of the result below,

[TABLE]

follows along the same lines, and hence details are omitted.

Next we consider the second term (H.44). Define the $T\times K$ matrix $\bm{\chi_{i}}:=T^{-\nicefrac{{1}}{{2}}}\mathscr{M}_{\bm{\mathrm{D}}}\bm{\mathrm{F}}\bm{\Gamma_{i}}\bm{\Psi_{i}}^{-1}$ with entry $\chi_{itk}$. Noting that

[TABLE]

Hence, by Lemma H.1(i), $\sup_{i}\left\|\bm{\chi_{i}}\right\|$ is uniformly bounded. Then,

[TABLE]

To prove that latter bound, we first consider the sum in Equation (H.53). By Remark H.2,

[TABLE]

Next, for the term in (H.52), we show that, for $l=1,2,3$,

[TABLE]

For the summation over $\pi$, we consider only two partitions for illustrative purposes, namely

[TABLE]

The partition $\pi_{1}$ is formed by blocks with the same cross-sectional indices. By Lemma D.5 and the result in display (H.51), we have

[TABLE]

whereas, for $\pi_{2}$

[TABLE] 4. 5. missingfnum@PartiPart (c)Part (c)

Similarly to Part (b) we first consider the inequality

[TABLE]

Using Lemma H.4 with $J=2$, the squared norm in (H.55) is bounded by

[TABLE]

To show how the latter bound is obtained, we analyse each term separately. Let $\text{\calligra m}_{ts}$ denote $t,s$-entry of $\mathscr{M}_{\bm{\mathrm{D}}}$. We first consider the expectation of the term (H.58):

[TABLE]

where $k,l=1,2,3$.

Consider first the partitions where all the blocks have the same cross-sectional indexes

[TABLE]

We only show how to derive the bound for the term in (H.63). By inequality 4.(e) p. 111 in Lütkepohl (1996), is it bounded by $T^{\nicefrac{{3}}{{2}}}\left\|\mathscr{M}_{\bm{\mathrm{D}}}\bm{\Xi_{i}}\left[\bm{\bar{\Omega}}\bm{\Xi_{j}}\bm{\bar{\Omega}}\right]\bm{\Xi_{\ell}}\mathscr{M}_{\bm{\mathrm{D}}}\right\|_{sp}=O_{p}\left(T^{\nicefrac{{3}}{{2}}}\right)$.

We consider two further sets of partitions, characterized by different upper bounds.

[TABLE]

For the partitions $\pi_{1},\sigma_{1}$ we have

[TABLE]

Finally, for the partitions $\pi_{2},\sigma_{2}$

[TABLE]

Using the same techniques illustrated above, it can be shown that the terms in (H.59) and (H.60) are $\max\left(O_{p}(1),O_{p}\left(T^{\nicefrac{{3}}{{2}}}/N\right)\right)$. The proof is tedious, due the the high order of the cumulants involving the $v_{ikt}$s, and hence omitted to save space. It might be worth to note that bound of the latter term requires the high order moment assumptions on the process $\{v_{itk}\}$ in Assumption 2.2.

Concerning (H.61) we have

[TABLE]

completing the proof of (H.62).

For the term (H.56) we have (see equation (H.51))

[TABLE]

The latter bound proceeding as for terms (H.44) and (H.55); the proof is omitted. 6. missingfnum@PartiPart (d)Part (d)

Setting $J=1$ in the expansion in Lemma H.4, we get

[TABLE]

The proof for the latter bound follows closely that of part $(c)$, and hence is omitted.

Bound Term III

Using the expansion in Lemma H.4 for $J=2$,

[TABLE]

To prove the bound, we start considering the norm in equation (H.72). The latter can be further bounded by

[TABLE]

We first consider the expectation of the term (H.74).

[TABLE]

About term (H.75)

[TABLE]

where $\mathrm{f}_{ts}$ and $f_{ts}$ denote the elements for the matrices $\bm{\mathrm{F}}$ and $\bm{\mathcal{F}}$, respectively. The expectation in (H.6) can be rewritten as $\sum_{\pi=\{a_{1},\dots a_{l}\}\in\mathcal{P}(a)}{\rm cum}(\bm{v^{,a_{1}}},\dots,\bm{v^{,a_{l}}})$, for $l=1,2,3,4$, and

[TABLE]

We focus on the partition $\pi$, where the blocks have the same cross-sectional indexes:

[TABLE]

that leads to bound the following expectation

[TABLE]

because by Lemma D.5, $\left|\sum_{t_{1},t_{2},t_{3}}{\rm cum}\left(v_{k_{1}it_{1}},v_{k_{2}it_{2}},v_{k_{3}it_{3}}\right)\sum_{t_{4}}\text{\calligra m}_{t_{3}t_{4}}f_{mt_{4}}\right|\leq\kappa\sqrt{T}\|\bm{\mathcal{F}}\|$ For the remaining partitions $\mathcal{P}(a)\backslash\pi$, suing the same techniques one can show that

[TABLE]

Proceeding in a similar way, terms (H.76)-(H.77) can be show to be $O_{p}(1)$, whereas the terms (H.72)-(H.72) are $O_{p}(1)$. Finally, the term in (H.6) is $O_{p}(1)$, using the results in Lemmas H.1, H.2 and H.3, Remark 2.11 and noting that $\left\|\bm{\mathfrak{R}}^{3}\right\|=O_{p}(T^{\nicefrac{{3}}{{2}}})$.

Bound Term IV

In the sequel we will show that

[TABLE]

We will split each norm in thirteen terms ($h=1,\dots,13$). To save notation, we write $\bm{\hat{C}_{3i,h^{\prime}}}$ to indicate $\bm{\hat{C}^{\prime}_{3i,h}}$. For example, $h,h^{\prime}=4$ mean that we are considering the terms $N^{-1}\sum_{i=1}^{N}{\bf{V}}_{\bm{j}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\hat{C}_{3i,4}}\bm{\Omega_{N}}^{-1}\bm{\varepsilon_{j}}$ and $N^{-1}\sum_{i=1}^{N}{\bf{V}}_{\bm{j}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\hat{C}^{\prime}_{3i,4}}\bm{\Omega_{N}}^{-1}\bm{\varepsilon_{j}}$, respectively. Due the high number of terms involved, and because the derivation of the bounds exploit similar techniques to the ones previously used, we will provide very concise proofs. Since most of the $\hat{C}_{3i,h}$s are function $\bm{\mathcal{X}_{i}}^{+}$, the bounds’ derivations will require the use of Lemma H.4. For each term, the order of the expansion will be specified.

• $\bm{h=1}:$

Setting $J=0$,

[TABLE]

• $\bm{h^{\prime}=1}:$

Setting $J=0$,

[TABLE]

• $\bm{h=2}:$

Similarly to the case $h=1$,

[TABLE]

• $\bm{h^{\prime}=2}:$

Similarly to the case $h^{\prime}=1$,

[TABLE]

• $\bm{h=3}:$

[TABLE]

• $\bm{h^{\prime}=3}:$

[TABLE]

• $\bm{h=4}:$

Setting $J=1$,

[TABLE]

• $\bm{h^{\prime}=4}$:

$\left\|N^{-1}\sum_{i=1}^{N}\bm{\mathcal{V}_{j}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\hat{C}^{\prime}_{3i,4}}\bm{\Omega_{N}}^{-1}\bm{\epsilon_{j}}\right\|=\max\left\{O_{p}(1),O_{p}\left(\frac{T}{\sqrt{N}}\right)\right\}.$

• $\bm{h=5}$:

[TABLE]

• $\bm{h^{\prime}=5}$:

Setting $J=1$

[TABLE]

• $\bm{h=6}$:

[TABLE]

• $\bm{h^{\prime}=6}$:

[TABLE]

• $\bm{h=7}$:

Setting $J=0$,

[TABLE]

• $\bm{h^{\prime}=7}$:

[TABLE]

The remaining terms ($h,h^{\prime}=8,9,\dots,13$) can be shown to be $O_{p}(1)$ using similar arguments to those used for the term labelled $h^{\prime}=7$.

H.6.3 Proof of Lemma H.8

The proof is similar to the proof of Lemma H.7, and hence omitted.

H.6.4 Proof Lemma H.9

The proof is similar to the proof of Lemma H.7 and hence omitted.

H.6.5 Proof Lemma H.10

Proceeding as in the proof of Lemma H.7, and using Lemma H.2

[TABLE]

The Term $I^{*}$ is bounded by $\left\|\bm{\mathcal{F}}\right\|\left\|\bm{\hat{A}_{N}}-\bm{A_{N}}\right\|\left\|\bm{\mathcal{F}}^{\prime}\bm{\Omega_{N}}^{-1}\bm{\mathcal{F}}\right\|\leq O_{p}(\sqrt{T})O_{p}\left(T^{-\nicefrac{{1}}{{2}}}\right)O_{p}\left(1\right)$. Concerning the Term $II^{*}$, he have

[TABLE]

and $\left\|\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\Xi_{N}}\bm{\mathrm{D}}_{\bot}\bm{\Omega_{N}}^{-1}\bm{\mathcal{F}}\right\|\leq\left\|\bm{\Xi_{N}}\right\|_{sp}\left\|\bm{\Omega_{N}}^{-1}\bm{\mathcal{F}}\right\|=O_{p}\left(T^{-\nicefrac{{1}}{{2}}}\right)$. Using similar argument, it can be shown that the Term $III^{*}$ is $O_{p}(1)$. For the last term ($IV^{*}$), we will show that

[TABLE]

We will consider only the terms for $h=1$. By Lemma A.2 we find

[TABLE]

Because

[TABLE]

we conclude that $\left\|\bm{\hat{C}}_{3i,1}\bm{\Omega_{N}}^{-1}\bm{\mathcal{F}}\right\|=O_{p}(1)$. For $h^{\prime}=1$ we have

[TABLE]

Using the same arguments for $h,h^{\prime}=2,\dots,13$ we prove the results.

H.6.6 Proof Lemma H.11

Let $\bm{\bar{\mathcal{H}}}:=\bm{\mathrm{D}}_{\bot}\bm{\mathcal{H}^{-1}_{N}}\bm{\mathrm{D}}_{\bot}^{\prime}$, and denote by $\bar{\mathcal{h}}_{ts}$ its $(t,s)$-entry, for $s,t=1,\dots,(T-S).$ Then, equations (B.22) and (C.1), (G.15) and (H.4) entail that

[TABLE]

where $\bar{\mathcal{q}}_{ik\ell}:=\breve{\mathcal{q}}_{ik\ell}-\mathcal{q}_{ik\ell}$ and ${\bf{v}}_{ik}$ denotes the $k-$th column of the matrix ${\bf{V}}_{\bm{i}}$. Note that $\bar{\mathcal{q}}_{ik\ell}\xrightarrow{p}0$. From arguments similar to those used in proof of Lemmata III, if follow that $\left\|{\bf{V}}_{\bm{j}}^{\prime}\bm{\bar{\Omega}}\mathbb{E}\left({\bf{v}}_{ik}{\bf{v}}^{\prime}_{i\ell}\right)\bm{\bar{\mathcal{H}}}\bm{\varepsilon_{j}}\right\|=O_{p}(\sqrt{T})$, concluding the proof.

I Common regressors

In this Section we provide some abbreviated proofs for the theorems of Section 4. Further details are available upon request.

Theorem I.1

Under Assumption 2.4:

[TABLE]

**Proof ** Using Equation (1) in (Lütkepohl 1996, Section 3.5.3) for the partitioned inverse of $\bf{Z}_{i}^{\prime}\bf{Z}_{i}$, with $\bf{Z}_{i}=\left[\bm{\mathrm{D}},\bm{\mathrm{X}_{i}}\right]$, $\mathscr{M}_{\bf{Z}_{i}}$ can be rewritten as

[TABLE]

Equation (29) entail that $\bm{\hat{\mathrm{u}}_{i}}=\mathscr{M}_{\bf{Z}_{i}}\bm{\mathrm{u}_{i}}=\bm{\mathrm{D}}_{\bot}\mathscr{M}_{\bm{\mathcal{X}_{i}}}\bm{\mathcal{u}_{i}}$; hence $\bm{\tilde{S}^{+}}=\bm{\mathrm{D}}_{\bot}\left(\bm{\hat{\mathcal{S}}_{N}}\right)^{-1}\bm{\mathrm{D}}_{\bot}^{\prime}$, by Lemma D.2, with $\bm{\hat{\mathcal{S}}_{N}}$ defined in (19). It follows that

[TABLE]

completing the proof. $\Box$

Theorem I.2

Under the assumptions of Theorem 3.2, if

[TABLE]

then,

[TABLE]

where

[TABLE]

**Proof ** Rewrite

[TABLE]

where we used $\bm{\mathrm{D}}^{\prime}\bm{\mathrm{u}_{i}}=\bm{\mathrm{D}}^{\prime}\bm{\varepsilon_{i}}$. Then apply Theorem 3.2 together with:

[TABLE]

The asymptotic covariance matrix follows by taking into account the covariance between $T^{\nicefrac{{1}}{{2}}}(\hat{\bm{\beta}}_{i}^{GLS}-\bm{\beta}_{i0})$ and $T^{-\nicefrac{{1}}{{2}}}\bm{\mathrm{D}}^{\prime}\bm{\varepsilon_{i}}$. $\Box$

Theorem I.3

Under the assumptions of Theorem 3.2, if

[TABLE]

then,

[TABLE]

where

[TABLE]

**Proof ** For the matrix $\breve{\bm{S}}_{N}$ defined in (32), it is easy to verify that (see Property 15 in (Lütkepohl 1996, 3.6.2)

[TABLE]

yielding

[TABLE]

with $\bf{A}_{i}:=\bm{\mathrm{X}_{i}}^{\prime}\bm{\mathrm{D}}_{\bot}\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\mathrm{D}}_{\bot}^{\prime}\bm{\mathrm{X}_{i}}+\bm{\mathrm{X}_{i}}^{\prime}\mathscr{P}_{\bm{\mathrm{D}}}\bm{\mathrm{X}_{i}}$ and ${\bf E_{i}}:=(\bm{\mathrm{D}}^{\prime}({\bf I}-\bm{\mathrm{X}_{i}}\bf{A}^{-1}\bm{\mathrm{X}_{i}}^{\prime})\bm{\mathrm{D}})^{-1}$. Similarly,

[TABLE]

where, using the identification assumption (I.4).

[TABLE]

and, from the proof of Theorem 3.2, $T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\bm{\hat{\mathcal{S}}_{N}}^{-1}\bm{\mathcal{u}_{i}}\approx T^{-\nicefrac{{1}}{{2}}}\bm{\mathcal{X}_{i}}^{\prime}\bm{\mathcal{H}^{-1}_{N}}\bm{\epsilon_{i}}$. $\Box$

J Different factor structure

Following up the discussion in Section 6.1, we now consider the feasible GLS. In this section we allow for $\bm{\mathrm{D}}\neq 0$, that is we consider the model

[TABLE]

The aim of this section is to provide an heuristic proof that the results of Theorem 3.2 apply to model (J.1). Proceeding along the lines of Appendix C, we define

[TABLE]

The matrices $\bm{\mathcal{G}}$ and $\hat{\bf{A}}_{i}$ are defined below. Noting that

[TABLE]

the matrix

[TABLE]

can be rewritten as $\bm{\mathcal{G}}\bm{\hat{A}_{N}}^{(\bm{\mathcal{G}})}\bm{\mathcal{G}}^{\prime}$, with $\bm{\mathcal{G}}:=\left[\bm{\mathcal{F}}_{2},\;\mathscr{M}_{\bm{\mathcal{F}}_{2}}\bm{\mathcal{F}}_{1}\right]$ of dimension $T\times(M_{1}+M_{2})$. Write

[TABLE]

Then,

[TABLE]

Likewise

[TABLE]

The terms $\bm{\hat{C}^{(\bm{\mathcal{G}})}_{3i}}$ is defined as $\bm{\hat{C}^{(\bm{\mathcal{G}})}_{3i}}=\sum_{j=1}^{13}\left(\bm{\hat{C}^{(\bm{\mathcal{G}})}_{3i,j}}+\bm{\hat{C}^{(\bm{\mathcal{G}})^{\prime}}_{3i,j}}\right),$ where

[TABLE]

Next define the matrices

[TABLE]

and

[TABLE]

with $\bm{\mathcal{F}}_{2}^{+}=\left(\bm{\mathcal{F}}_{2}^{\prime}\bm{\mathcal{F}}_{2}\right)^{-1}\bm{\mathcal{F}}_{2}$, and

[TABLE]

The key difference with respect to what discuss in Section 6.1 lies in the fact that the inverse of $\SS^{(\bm{\mathcal{G}})}$ is orthogonal both to $\bm{\mathcal{F}}_{1}$ and $\bm{\mathcal{F}}_{2}$.