Uniform Quasi ML based inference for the panel AR(1) model

TL;DR

This paper develops robust ML-based inference methods for panel AR(1) models that are valid under various conditions, including heteroskedasticity and weak identification, providing reliable tests and confidence sets.

Contribution

It introduces identification-robust ML inference techniques for panel AR(1) models using Quasi LM tests and confidence sets based on the expected Hessian, applicable under heteroskedasticity and misspecification.

Findings

Quasi LM tests have correct asymptotic size and coverage.

The proposed tests are robust to heteroskedasticity and distribution misspecification.

Power of the tests matches the theoretical maximum in worst-case scenarios.

Abstract

Maximum Likelihood (ML) offers attractive alternatives to Generalized Method of Moments (GMM) estimators for dynamic panel data models. However, to date no identification-robust inference methods exist that can be used in conjunction with the ML estimators for these models. In this paper we propose ML based inference methods for panel AR(1) models with arbitrary initial conditions and heteroskedasticity that are robust to the strength of identification. We show that (Quasi) Lagrange Multiplier (LM) tests and confidence sets (CSs) that use the expected Hessian rather than the observed Hessian of the log-likelihood function have correct asymptotic size and coverage probability in a uniform sense, respectively. Such Quasi LM tests and CSs are also robust to misspecification of the distribution of the data and to heterogeneity, including heteroskedasticity. We derive the power envelope of a Fixed Effects version of such an LM test for hypotheses involving the autoregressive parameter when the average information matrix is estimated by a centered OPG estimator and the model is only second-order identified, and show that it coincides with the maximal attainable power curve in the worst-case setting. We also study the empirical size and power properties of these (Quasi) LM tests and find that the hypothesis that the (Quasi) LM test has correct size cannot be rejected.

Tables

Table 1. Table 1: Empirical size of Quasi LM test based on RE Likelihood; Nominal size is 0.05; T=4; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
$\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.20	.0532	.0504	.0552	.0500	.0508	.0520
0.50	.0504	.0468	.0488	.0540	.0492	.0580
0.80	.0452	.0448	.0540	.0488	.0520	.0448
0.90	.0528	.0492	.0452	.0492	.0568	.0548
0.95	.0520	.0556	.0460	.0476	.0488	.0628
0.98	.0552	.0444	.0392	.0452	.0512	.0536
0.99	.0504	.0468	.0400	.0440	.0380	.0508

Table 2. Table 4: Empirical size of Quasi LM test based on FE Likelihood; Nominal size is 0.05; T=9; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
$\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.20	.0520	.0576	.0484	.0588	.0432	.0508
0.50	.0500	.0532	.0496	.0464	.0572	.0444
0.80	.0524	.0496	.0516	.0556	.0576	.0456
0.90	.0576	.0504	.0524	.0572	.0492	.0464
0.95	.0592	.0496	.0516	.0564	.0480	.0524
0.98	.0596	.0436	.0536	.0468	.0608	.0568
0.99	.0520	.0456	.0524	.0532	.0460	.0524

Table 3. Table 5: Empirical power of Quasi LM test based on RE Likelihood; H 0 : ρ = 0.8 H_{0}:\rho=0.8 ; Nominal size is 0.05; T=4; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
true $\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.50	0.357	0.567	0.381	0.556	0.876	0.977
0.60	0.210	0.374	0.221	0.396	0.445	0.484
0.70	0.090	0.154	0.107	0.174	0.092	0.104
0.90	0.050	0.079	0.067	0.090	0.048	0.067
0.95	0.069	0.121	0.070	0.125	0.066	0.101
0.99	0.093	0.200	0.080	0.123	0.094	0.189

Table 4. Table 6: Empirical power of Quasi LM test based on FE Likelihood; H 0 : ρ = 0.8 H_{0}:\rho=0.8 ; Nominal size is 0.05; T=4; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
true $\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.50	0.191	0.339	0.231	0.398	0.865	0.975
0.60	0.114	0.220	0.145	0.276	0.437	0.479
0.70	0.072	0.095	0.074	0.122	0.086	0.103
0.90	0.052	0.074	0.060	0.065	0.047	0.065
0.95	0.070	0.122	0.061	0.094	0.065	0.102
0.99	0.096	0.200	0.076	0.113	0.097	0.192

Table 5. Table 7: Empirical power of Quasi LM test based on RE Likelihood; H 0 : ρ = 0.8 H_{0}:\rho=0.8 ; Nominal size is 0.05; T=9; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
true $\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.50	0.998	1.000	0.995	1.000	0.996	1.000
0.60	0.944	1.000	0.943	1.000	0.914	0.999
0.70	0.456	0.810	0.523	0.874	0.419	0.747
0.90	0.215	0.553	0.299	0.658	0.198	0.488
0.95	0.456	0.888	0.476	0.832	0.426	0.864
0.99	0.730	0.992	0.561	0.907	0.730	0.992

Table 6. Table 8: Empirical power of Quasi LM test based on FE Likelihood; H 0 : ρ = 0.8 H_{0}:\rho=0.8 ; Nominal size is 0.05; T=9; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
true $\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.50	0.999	1.000	0.992	1.000	0.996	1.000
0.60	0.939	0.999	0.939	0.999	0.915	0.999
0.70	0.426	0.772	0.475	0.826	0.425	0.749
0.90	0.210	0.541	0.257	0.569	0.198	0.488
0.95	0.453	0.887	0.438	0.792	0.427	0.864
0.99	0.730	0.992	0.553	0.899	0.731	0.992

Table 7. Table 9: Empirical size of Quasi LM test based on RE Likelihood; Nominal size is 0.05; T=4; σ μ 2 = 25 ; \sigma_{\mu}^{2}=25; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
$\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.20	.0472	.0580	.0472	.0504	.0500	.0512
0.50	.0524	.0504	.0460	.0468	.0436	.0500
0.80	.0564	.0492	.0596	.0564	.0540	.0428
0.90	.0528	.0492	.0416	.0496	.0524	.0520
0.95	.0540	.0508	.0448	.0504	.0472	.0556
0.98	.0460	.0520	.0440	.0448	.0588	.0476
0.99	.0460	.0524	.0476	.0504	.0472	.0484

Table 8. Table 10: Empirical size of Quasi LM test based on RE Likelihood; Nominal size is 0.05; T=9; σ μ 2 = 25 ; \sigma_{\mu}^{2}=25; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
$\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.20	.0452	.0508	.0480	.0476	.0524	.0472
0.50	.0596	.0508	.0512	.0504	.0548	.0488
0.80	.0576	.0564	.0564	.0524	.0588	.0584
0.90	.0480	.0644	.0576	.0512	.0552	.0504
0.95	.0540	.0512	.0624	.0508	.0552	.0548
0.98	.0504	.0512	.0504	.0580	.0584	.0664
0.99	.0540	.0544	.0532	.0480	.0632	.0456

Table 9. Table 11: Empirical power of Quasi LM test based on RE Likelihood; H 0 : ρ = 0.8 H_{0}:\rho=0.8 ; Nominal size is 0.05; T=4; σ μ 2 = 25 ; \sigma_{\mu}^{2}=25; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
$\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.50	.205	.348	.238	.408	.883	.982
0.60	.129	.238	.136	.300	.462	.501
0.70	.078	.110	.076	.128	.098	.102
0.90	.052	.078	.055	.066	.048	.075
0.95	.066	.115	.060	.098	.059	.123
0.99	.088	.197	.076	.127	.093	.203

Table 10. Table 12: Empirical power of Quasi LM test based on RE Likelihood; H 0 : ρ = 0.8 H_{0}:\rho=0.8 ; Nominal size is 0.05; T=9; σ μ 2 = 25 ; \sigma_{\mu}^{2}=25; 2500 replications.

model	S-Normal		S-${\chi }^2$		NS-Normal
$\rho$	$N=100$	$N=250$	$N=100$	$N=250$	$N=100$	$N=250$
0.50	.999	1.000	.993	1.000	.997	1.000
0.60	.919	1.000	.930	.999	.914	1.000
0.70	.430	.784	.502	.818	.398	.756
0.90	.211	.544	.260	.582	.188	.502
0.95	.456	.872	.452	.813	.438	.860
0.99	.731	.992	.570	.898	.730	.992

Equations

$$y_{i,t}=\rho y_{i,t-1}+{\eta }_i+{\epsilon }_{i,t},\text{ where }{\eta }_i=(1-\rho ){\mu }_i,\text{vskip}-13.00806pt$$

$$E({\epsilon }_{i,t})=0\text{ and }E{\left|{\epsilon }_{i,t}\right|}^{k+\varsigma }<\infty \text{ for }i=1,...,N\text{ and }t=2,...,T,$$

$$(y_{i,1}{\eta }_i{)}^{\prime },i=1,...,N,\text{ are i.i.d. with }{\sigma }_y^2=Var(y_{i,1}),E{\left|y_{i,1}\right|}^{k+\varsigma }<\infty ,\text{ and}$$

$$E({\mu }_i)=0,{\sigma }_{\mu }^2=E({\mu }_i^2),E{\left|{\mu }_i\right|}^{k+\varsigma }<\infty \text{ and }E({\mu }_i{y}_{i,1})={\sigma }_{\mu y}\text{ when }\left|\rho \right|<1.\text{vskip}-8.67204pt$$

$$v_{i,1},i=1,...,N,\text{ are i.i.d. with }{\sigma }_{v_1}^2=Var(v_{i,1})\text{ and }$$

$$E{\left|v_{i,1}\right|}^{k+\varsigma }<\infty \text{ when }\left|\rho \right|<1.$$

$${\epsilon }_{i,s}\perp {\epsilon }_{i,t}\text{ for }i=1,...,N\text{ and }t\ne s.\text{vskip}-7.22743pt$$

$${\epsilon }_{i,t}\perp y_{i,1}\text{ and }{\epsilon }_{i,t}\perp {\eta }_i\text{ for }i=1,...,N\text{ and }t=2,...,T,$$

$${\epsilon }_{i,t}\perp v_{i,1}\text{ for }i=1,...,N\text{ and }t=2,...,T\text{ when }\left|\rho \right|<1.$$

$$E(y_{i,1}-{\mu }_i)=0\text{and}E({\mu }_i(y_{i,1}-{\mu }_i))=0.$$

$$E({\epsilon }_{i,t}^2)={\lambda }_t^2<\infty ,\text{ for }i=1,...,N\text{ and }t=2,...,T.$$

$$E({\epsilon }_{i,t}^2)={\sigma }^2<\infty ,\text{ for }i=1,...,N\text{ and }t=2,...,T\text{.}$$

$${\eta }_i=\pi (1-\rho )y_{i,1}+(1-\rho )v_i,i=1,...,N,$$

$$y_i=\rho y_{i,-1}+\pi (1-\rho )y_{i,1}\iota +u_i,\text{where}$$

$$u_i=(1-\rho )v_i\iota +{\epsilon }_i\text{with}E({\epsilon }_i{\epsilon }_i^{\prime })=\Psi (\zeta )=diag({\lambda }_2^2,{\lambda }_3^2,...,{\lambda }_T^2).$$

$$y_i=\rho y_{i,-1}+(1-\rho )y_{i,1}\iota +u_i\text{,where }$$

$$u_i=(1-\rho )v_i\iota +{\epsilon }_i\text{with}E({\epsilon }_i{\epsilon }_i^{\prime })={\sigma }_i^2\Psi (\zeta ),$$

$$S_{i,1}$$

$$\mathcal{I}$$

$${\mathcal{H}}_{1,1}$$

$${\mathcal{H}}_{2,2}$$

$${\mathcal{H}}^{-1}$$

$$C_{1,1}$$

$$C_{2,2}$$

$$U_{1,N}$$

$$U_N$$

$${\breve{Z}}_N$$

$$\breve{Z}$$

$$\breve{B}\sim Bernoulli\text{ with }\mathrm{Pr}(\breve{B}=0)=\mathrm{Pr}(U^{\prime }\breve{Z}>0).$$

$$QLM({\tilde{\theta }}_n)$$

$$\tilde{S}({\tilde{\theta }}_n)$$

$$S_i$$

$${\tilde{\mathcal{H}}}_{1,1}$$

$${\tilde{\mathcal{H}}}_{2,2}$$

$$QL{M}^c({\tilde{\theta }}_n)$$

$$GMM-AR(\rho )=Nm(\rho {)}^{\prime }[{\hat{V}}_{mm}(\rho ){]}^{-1}m(\rho )\text{vskip}-8.67204pt$$

$$m(r)=N^{-1}{\sum }_{i=1}^N\left[P\times vech\left(D_r(\Delta y_i(\Delta y_i{)}^{\prime })D_r^{\prime }\right)\right]$$

Full text

Uniform Quasi ML based inference for the panel AR(1) model.

Hugo Kruiniger

Durham University Address: hugo.kruiniger@durham.ac.uk; Department of Economics, 1 Mill Hill Lane, Durham DH1 3LB, England. I thank N. Peyerimhoff for helpful discussions.

(This version: 12 December 2025)

JEL classification: C12, C13, C23.

Keywords: dynamic panel data model, identification robust inference, quasi Lagrange multiplier test, score test, second-order identification, singular information matrix, uniform inference, Wald test.

Abstract

Maximum Likelihood (ML) offers attractive alternatives to Generalized Method of Moments (GMM) estimators for dynamic panel data models. However, to date noidentification robust inference methods exist that can be used in conjunction with the ML estimators for these models. In this paper we propose ML based inference methods for panel AR(1) models with arbitrary initial conditions and heteroskedasticity that arerobust to the strength of identification. We show that (Quasi) Lagrange Multiplier (LM) tests and confidence sets (CSs) that use the expected Hessian rather than the observed Hessian of the log-likelihood function have correct asymptotic size and coverageprobability in a uniform sense, respectively. Such Quasi LM tests and CSs are also robust to misspecification of the distribution of the data and to heterogeneity, including heteroskedasticity. We derive the power envelope of a Fixed Effects version of such an LM test for hypotheses involving the autoregressive parameter when the average information matrix is estimated by a centered OPG estimator and the model is only second-order identified, and show that it coincides with the maximal attainable power curve in the worst case setting. We also study the empirical size and power properties of these (Quasi) LM tests and find that the hypothesis that the (Quasi) LM test has correct size cannot be rejected.

1 Introduction

This paper proposes new inference methods for panel AR models with arbitrary initial conditions and heteroskedasticity and possibly additional regressors that are robust to the strength of identification. Specifically, we consider several Maximum Likelihood based methods of constructing tests and confidence sets (CSs) and show that (Quasi) LM tests and CSs that use the expected Hessian rather than the observed Hessian of the log-likelihood have correct asymptotic size in a uniform sense.

There exists a vast literature devoted to estimation of versions of the panel AR(1) model or, more generally, dynamic panel data models. The estimators in this literature can roughly be classified into two groups: Instrumental Variables/Generalized Method of Moments (IV/GMM) type estimators and Maximum Likelihood (ML) type estimators. Seminal contributions to the first group include Anderson and Hsiao (1981, 1982), Holtz-Eakin et al. (1988), Arellano and Bond (AB, 1991), Arellano and Bover (ABov, 1995) and Ahn and Schmidt (AS, 1995), while seminal contributions to the second group include Chamberlain (1980), Anderson and Hsiao (1981, 1982), MaCurdy (1982) and Hsiao et al. (2002). The last two papers considered Fixed Effects (FE) ML estimators, that is, estimators that are based on data in first differences and are consistent under minimal assumptions, whereas the other two papers in the second group considered (correlated) Random Effects (RE) ML estimators.111The RE approach assumes finite second moments of the data in levels while the FE approach only assumes finite second moments of the data in differences, cf. Kruiniger (2001, 2022).222The Transformed MLE of Hsiao et al. (2002) and the FEMLE in Kruiniger (2001, 2013, 2022) are the same estimator and are the FE counterpart of the REMLE.

Kruiniger (2001) has shown that when the data are i.i.d. and normal, then the RE and FE MLEs of the autoregressive parameter $\rho$ in a panel AR(1) model with homoskedastic errors and $\left|\rho\right|<1$ are asymptotically equivalent to optimal AS-type GMM estimators,which in this case also exploit moment conditions that rely on time-series homoskedasticity, and that when $\rho=1,$ the information matrix associated with the FEMLE for the panel AR(1) model with homoskedastic errors is singular. Kruiniger (2013) and Alvarez and Arellano (2022) have developed RE and FE MLEs for the panel AR(1) model that allows for time-series heteroskedasticity. These estimators of $\rho$ remain large $N$, fixed $T$ consistent when the data are non-normal or i.h.d., where $N$ and $T$ are the dimensions of the panel. Kruiniger (2013) and Ahn and Thomas (2023) have shown that when $\rho=1$ and the errors bar possibly the error of the last period are homoskedastic over time, then RE and FE MLEs of $\rho$ that allow for time-series heteroskedasticity are $N^{1/4}$-consistent and have non-normal limiting distributions. The slower rate of convergence is related to the fact that in this case the information matrix is singular so that $\rho$ is only second-order locally identified, cf. Sargan (1983) and Rotnitzky et al. (2000).

Monte Carlo studies in Hsiao et al. (2002), Kruiniger (2013) and Hayakawa and Pesaran (2015) have found that the (Quasi) MLEs for the panel AR(1) and ARX(1) models have very good finite sample properties but that when $\rho$ is close to one, Wald tests for hypotheses about $\rho$ are oversized and asymptotic confidence intervals based on the QMLEs of $\rho$ are too narrow because $\rho$ is weakly identified. However, to date no iden-tification robust inference methods have been proposed that are related to these QMLEs.

Weak identification including second- rather than first-order local identification affect the rate of convergence and the limiting distributions of the estimators and pose a challenge to conducting inference both in the case of GMM and the ML method. For reliable inference it is important that tests and/or confidence sets (CSs) have correct asymptotic size in a uniform sense. As Andrews et al. (2020) explain, pointwise asymptotics often provide very poor approximations to the finite-sample size when the test statistic of interest has a discontinuous pointwise asymptotic distribution. For this reason Wald-type tests and CSs for (hypotheses/parameter vectors that include) $\rho$ will generally not have correct asymptotic size in a uniform sense. Also likelihood based (Quasi) LR-type tests and CSs for (hypotheses/parameter vectors that include) $\rho$ will in many cases not have correct (locally) uniform asymptotic size, cf. Rotnitzky et al. (2000) and Bottai (2003). In this paper we will instead propose tests and CSs for (hypotheses/parameter vectors that include) $\rho$ that are based on (Quasi) LM test statistics that use the expected Hessian of a RE or FE log-likelihood and show that they have correct uniform asymptotic size.

Our approach generalizes that of Bottai (2003), who considered CSs based on similar LM test statistics in the context of identifiable one-dimensional parametric models with a smooth likelihood function and information equal to zero at a critical point, in at least two ways. Firstly, we consider multi-parameter models and hypotheses. To show that our (Quasi) LM tests and CSs have correct uniform asymptotic size, we make use of the fact that in suitably reparametrized versions of the panel AR(1) model, the parameters other than $\rho$ are still strongly identified when $\rho=1.$ Secondly, by using Quasi LM test statistics, we also allow for misspecification of the distribution of the data and heterogeneity.

Bottai (2003) explains for the one-parameter case why an LM test that is based on the observed Hessian rather than the expected Hessian will not have correct uniform asymptotic size when the parameter, say $\theta$, is second-order identified at a critical point $\theta^{\ast}$. In this case the LM test statistic does not converge in distribution to a $\chi^{2}(1)$ random variable under the sequence $\theta_{n}=\theta^{\ast}-cn^{-1/4}$ as the sample size $n$ tends to infinity although it does converge to a $\chi^{2}(1)$ random variable under sequences $\theta_{n}=\theta^{\ast}-cn^{-b}$ with $b>1/4.$ The same situation arises in the multi-parameter case.

We will now discuss alternative, GMM-based methods. When the autoregressive parameter $\rho$ is local to unity, then the AB GMM estimator has poor finite sample properties and is inconsistent due to weak instruments, cf. ABov, Blundell and Bond (BB, 1998) and Kruiniger (K, 2009). If in addition the individual time series are covariance stationary or the number of pre-sample realizations is large, then the System estimator of Abov and BB may have a non-normal limiting distribution and its rate of convergence may depend on the choice of the weight matrix, cf. K2009. Furthermore, when mean-stationarity holds but the ratio of the variance of the individual effects to the variance of the idiosyncratic errors is large, then the AB GMM and the System estimator can also suffer from a weak instruments problem and be biased, cf. Hayakawa (2007), Bun and Windmeijer (2010) and Kruiniger (2001, 2022). Incidentally, Bun and Kleibergen (2022) and Kruiniger (2022) have shown that provided that $T$ is not too small, $\rho$ is identified, albeit possibly only second-order identified when $\rho=1$, by a set of linear and quadratic AS-type moment conditions that only depend on a lack of serial correlation assumption for the errors and on differenced data and hence do not require mean-stationarity to hold.

GMM-based tests (and CSs) for $\rho$ that have correct uniform asymptotic size when $\rho=1$ include (CSs based on) the Newey and West (1987) type GMM LM test(-statistic)s that exploit ABov and SYS moment conditions, respectively, see Madsen (2003) and K2009, and identification-robust test(-statistics)s such as the GMM AR test of Stock and Wright (2000) and the KLM and GMM-CLR tests of Kleibergen (2005) that exploit AB, ABov, SYS and AS moments conditions, cf. Bun and Kleibergen (2014).333These GMM LM tests have correct uniform asymptotic size when $\rho=1$ because the scaled sample moment conditions they depend on (evaluated at $\rho=1$) and their first derivates are asymptotically independent. As a result these GMM LM test statistics have the same limiting distribution as they would have under strong identification, i.e., $N(0,1)$ (or $\chi^{2}(1)$ when squared), see Kruiniger (2009). However, in general GMM LM tests are not robust to lack of identification. For instance, if $\rho=1,$ the GMM LM test that only exploits the moment conditions of AB does not have correct uniform asymptotic size and shouldn’t be used, cf. Bond and Windmeijer (2005). Results in Newey and Windmeijer (NW, 2009) imply that most of these identification robust test statistics also have correct asymptotic size under many weak moment conditions asymptotics with a restriction on the relative rate at which the number of moment conditions and $N$ tend to infinity. The only possible exceptions are the tests that exploit the non-linear AS moment conditions when NW’s second assumption, in particular global identification, does not hold. However, this only happens when $T\geq 3$ and the variances of the errors change over time at a constant rate, cf. Alvarez and Arellano (2022). Andrews and Guggenberger (2017) have shown that under rather general conditions the KLM and GMM-CLR tests for nonlinear moment condition models have correct asymptotic size as $N\rightarrow\infty$ although they note that for GMM-CLR tests this result depends in the multi-parameter case on how the conditioning statistic, upon which the GMM-CLR test depends, is weighted.

Bun and Kleibergen (2014) have shown for $T\geq 4$ that in a worst case scenario, where the variance of the initial observations goes to infinity, the true value of $\rho$ is equal to one and the errors are homoskedastic so that $\rho$ is only second-order locally identified, the power envelope of the KLM test based on the AS moment conditions and a centered optimal weight matrix, viz. the AS KLM test, coincides with the maximal attainable power curve for testing $H_{0}:\rho=1-cN^{-1/4},$ and that in such a scenario the AS GMM AR and the AS GMM LM tests have less power than the AS KLM test when $T\geq 5$.444In the worst case scenario effectively only moment conditions based on second moments of differences of the data are exploited, which do not require mean-stationarity when $\left|\rho\right|<1$. The analysis of local power of these tests that is provided in Bun and Kleibergen (2014) is different from that in Dovonon et al. (2020). The latter assumes that the true value of $\rho$ equals $1-cN^{-1/4}$ and tests $H_{0}:\rho=1$, and finds in a simulation study that the AS GMM AR test has better power properties than the AS KLM and AS GMM-CLR tests.

Andrews et al. (2019) list some limitations and weaknesses of inference procedures that are based on identification robust test statistics. Firstly, the KLM and GMM-CLR tests for subvectors of $\theta$ generally no longer have correct asymptotic size when the nuisance parameters are not strongly identified, although the GMM AR test for subvectors of $\theta$ still has correct asymptotic size in this case when the errors are homoskedastic, cf. Guggenberger et al. (2012, 2019) and Kleibergen (2021). Secondly, a large number of inference procedures have been proposed for overidentified models with nonhomoskedastic errors, some of which are based on analogs or generalizations of the GMM-CLR test, but there is no consensus on what procedures to use in practice, beyond the recommendation to use methods that are efficient when the instruments are strong. Recently, Andrews (2018) introduced a widely applicable approach to detecting weak identification and constructing two-step confidence sets that controls coverage distortions under weak identification. In cases where the model is well identified, his method indicates this and reports nonrobust confidence sets with probability tending to 1. Finally, it has been found that increases in the number of moment conditions can lead to size distortions of the identification robust tests in finite samples, cf. Kleibergen and Mavroeidis (2009), Kleibergen (2019) and Bun et al. (2020). The GMM LM test is also susceptible to this problem. In their Monte Carlo studies based on various panel AR(1) and panel ARX(1) models with parameter values that correspond to different degrees of identification strength both Hayakawa and Pesaran (2015) and Bun and Poldermans (2015) find evidence for size distortions for the GMM LM, GMM AR, KLM and GMM-CLR tests that exploit different sets of moment conditions (e.g. AB, AS, SYS and subsets and collapsed versions of these sets) and that the distortions get worse as $T$ or the number of moment conditions increases. The size distortions become smaller when $N$ gets larger. Bun et al. (2020) show that when the number of moment conditions is moderately large, then the GMM AR test that is based on a weighting matrix that uses centered moment conditions is oversized, whereas the uncentered version of the GMM AR test is conservative. Bun et al. (2020) also propose a degrees-of-freedom corrected centered GMM AR test that has good size properties.

The paper proceeds as follows. In section 2 we present the panel AR(1) model, the assumptions, the (Q)ML estimators and their asymptotic properties including their local-to-unit root limiting distributions. In section 3 we discuss the asymptotic size properties of various ML based tests including Quasi LM tests that use the expected Hessian of the RE or the FE log-likelihood function. We also derive the power envelope of the FE version of such an QLM test for $H_{0}:\rho=1-cN^{-1/4}$ when the average information matrix is estimated by a centered OPG estimator and the model is only second-order identified, and the maximal attainable power curve for testing $H_{0}:\rho=1-cN^{-1/4}$ in the worst case scenario. In section 4 we conduct a Monte Carlo study into the empirical size and power properties of the QLM tests. Section 5 concludes and the appendix contains the proofs.

2 The panel AR(1) model

Consider the panel AR(1) model with individual effects:

[TABLE]

for $i=1,...,N$ and $t=2,...,T.$ When deriving the asymptotics results, we let $N$ become large while the number of observations per ‘individual’, $T,$ remains fixed. The analysis below can be extended straightforwardly for models that also include strictly exogenous covariates; we have omitted them from the model to keep the presentation simple.

We assume that $-1<\rho\leq 1.$ Note that the model can be rewritten as $y_{i,t}-\mu_{i}=\rho(y_{i,t-1}-\mu_{i})+\varepsilon_{i,t}$ and that the $\eta_{i}$ disappear (the $\mu_{i}$ drop out) when $\rho=1$. The parame- trization $\eta_{i}=(1-\rho)\mu_{i}$ prevents the individual effects from turning into individual trends at $\rho=1$ and thereby avoids a discontinuity in the data generating process at $\rho=1$.

The vectors of idiosyncratic errors $\varepsilon_{i}=(\varepsilon_{i,2}$ $...$ $\varepsilon_{i,T})^{\prime}$ are independently distributed across individuals and satisfy the following Standard Assumptions, SA$k$, for $k=2$ or $k=4$:

[TABLE]

where $\varsigma\geq 0$ is an arbitrarily small constant. In the sequel SA2 is denoted by SA.

The individual effects can often be treated as random effects. In this case we make the following Random Effects Assumptions, REA$k$, for $k=2$ or $k=4$:

[TABLE]

In addition, we let $\sigma_{\eta}^{2}=E(\eta_{i}^{2})$ and $\sigma_{\eta y}=E(\eta_{i}y_{i,1}).$ The i.i.d. assumption in (3) is only made for presentational convenience. It can easily be relaxed. The assumption $E\left|\mu_{i}\right|^{k+\varsigma}<\infty$ (for $k=2$ or $k=4$) ensures that under covariance stationarity the means of the data, i.e., $\eta_{i}/(1-\rho)=\mu_{i},$ $i=1,...,N$, are drawn from a distribution with a finite variance rather than a variance that tends to infinity when $\rho$ approaches one.

Unlike the RE ML estimators, the FE ML estimators only exploit data in first differences. This reflects the fact that the FE approach entails making minimal assumptions about the $\mu_{i}$ and the $y_{i,1}.$ In the FE case we assume that $v_{i,1}\equiv y_{i,1}-\mu_{i},$ $i=1,...,N$, satisfy a Fixed Effects Assumption, FEA$k$, for $k=2$ or $k=4$, cf. Kruiniger (2013):

[TABLE]

The i.i.d. assumption in (4) is made for presentational convenience. It can be relaxed.

Suppose that $y_{i,1}$ depends on $\mu_{i}$ in a linear fashion, i.e. $y_{i,1}=E(y_{i,1})+\alpha_{1}\mu_{i}+\varepsilon_{i,1}$ with $E(\varepsilon_{i,1})=0,$ and $\mu_{i}\perp\varepsilon_{i,1}$. In the important case that $\alpha_{1}=1,$ we have $\mu_{i}\perp v_{i,1}$ and FEA$k$ does not impose any restrictions on $\mu_{i}$ and $y_{i,1}$ other than those on $y_{i,1}-\mu_{i}$. However when $\alpha_{1}\neq 1,$ FEA$k$ implies restrictions on the $\mu_{i}$ themselves.

In the sequel REA2 and FEA2 are denoted by REA and FEA, respectively.

We add assumption B (Basic), which in the RE case amounts to

[TABLE]

and

[TABLE]

and in the FE case amounts to (5) and

[TABLE]

We sometimes use an augmented version of assumption B, i.e., assumption B*′* that in addi-tion to (5) and (6) assumes that $\varepsilon_{i,l}\varepsilon_{i,s}\varepsilon_{i,t}\perp y_{i,1}$ for all $l,s,t\in\{2,...,T\}$ and $i=1,...,N$.

In the paper we will also make reference to the assumption of mean-stationarity:

[TABLE]

We can allow for heteroskedasticity of the $\varepsilon_{i,t}$ in the time-series dimension:

[TABLE]

In some cases we make the stronger assumption of Time Series Homoskedasticity (TSH):

[TABLE]

Let $\overline{\sigma}_{s}^{2}=N^{-1}\sum_{i=1}^{N}E(\varepsilon_{i,s}^{2})$, then we say that assumption TSH${}_{t}^{\ast}$ holds if and only if $\overline{\sigma}_{s}^{2}=\overline{\sigma}_{2}^{2},$ for $s=3,...,t.$ We will often make a ‘homoskedasticity’ assumption that is weaker than TSH, namely TSH*∗*, which is short for TSH${}_{T}^{\ast}.$ Similarly we can use a weaker version of time-series heteroskedasticity, namely $\overline{\sigma}_{t}^{2}=\lambda_{t}^{2}<\infty,$ for $t=2,...,T.$

2.1 ML estimators

Direct application of the Maximum Likelihood method to the nonstationary panel AR(1) model with RE will generally yield an inconsistent estimator for $\rho$ due to correlation between the individual effects ($\eta_{i}$) and the regressors ($y_{i,t-1},$ $t=2,...,T$). However, a consistent ML estimator for $\rho$ can be obtained after reformulating the model. Following Chamberlain (1980) we can decompose the $\eta_{i}$ into a term that depends on the initial observation, $y_{i,1},$ and a term that does not: 555For the sake of a simple exposition we assume that $E(y_{i,1})=0$. A situation where $E(y_{i,1})\neq 0$ can be handled by including an intercept term in (11) and in (12).

[TABLE]

where $v_{i}$ is a new individual effect with $E(v_{i})=0$, $\pi(1-\rho)=$ plim${}_{N\rightarrow\infty}\sum_{i=1}^{N}(\eta_{i}y_{i,1})/$ $\sum_{i=1}^{N}y_{i,1}^{2}$ and plim${}_{N\rightarrow\infty}N^{-1}\mathop{\textstyle\sum}\nolimits_{i=1}^{N}(y_{i,1}v_{i})=0.$

Let $y_{i}=(y_{i,2}$ $...$ $y_{i,T})^{\prime}$ and $y_{i,-1}=(y_{i,1}$ $...$ $y_{i,T-1})^{\prime}$ and let $\iota$ denote a vector of ones. Then using the decomposition of the ‘correlated effects’ $\eta_{i}$ given in (11), we can rewrite the panel AR(1) model with RE as

[TABLE]

Let plim${}_{N\rightarrow\infty}N^{-1}\mathop{\textstyle\sum}\nolimits_{i=1}^{N}v_{i}^{2}=\sigma_{v}^{2}$ and $\Phi=(1-\rho)^{2}\sigma_{v}^{2}\iota\iota^{\prime}+\Psi(\zeta)$. Then it is easily verified that plim${}_{N\rightarrow\infty}N^{-1}\mathop{\textstyle\sum}\nolimits_{i=1}^{N}(y_{i,0}\iota^{\prime}\Phi^{-1}u_{i})=0$ and plim${}_{N\rightarrow\infty}N^{-1}\mathop{\textstyle\sum}\nolimits_{i=1}^{N}(y_{i,-1}^{\prime}\Phi^{-1}u_{i})=0,$ cf. similar results under homoskedasticity in Blundell and Bond (1998). After imposing the assumption that the error components are i.i.d. and normal, i.e., $u_{i}\sim i.i.d.$ $N(0,\Phi)$, application of the ML method to (12) yields the RE ML estimator of $\rho$, $\pi$, $\sigma_{v}^{2}$ and $\zeta=(\lambda_{2}^{2}$ $\lambda_{3}^{2}$ $...$ $\lambda_{T}^{2})^{\prime}$. This estimator will still be consistent when the $\varepsilon_{i,t}$ are heteroskedastic across both dimensions of the panel.

When calculating the REMLE it is convenient to use the reparameterization $\tilde{\pi}=\pi(1-\rho)$ and $\widetilde{\sigma}_{v}^{2}=(1-\rho)^{2}\sigma_{v}^{2}.$ Let $\theta_{0}=(\rho,\tilde{\pi},\widetilde{\sigma}_{v}^{2},\zeta^{\prime})^{\prime}.$ Then the log-likelihood function for the above model will be denoted by $l_{RE}(\theta)=\mathop{\textstyle\sum}\nolimits_{i=1}^{N}l_{RE,i}(\theta)$ where $\theta=(r,\widetilde{p},\widetilde{s}_{v}^{2},z^{\prime})^{\prime}$ and $l_{RE,i}(\theta)$ is the contribution to $l_{RE}(\theta)$ from ‘individual’ $i$. The REMLE of $\theta_{0}$ will be denoted by $\widehat{\theta}_{RE}$ or simply by $\widehat{\theta}.$

Hsiao et al. (2002) has proposed the Transformed MLE for the panel AR(1) model, which can be viewed as the FE counterpart of the REMLE, cf. Kruiniger (2001). One can obtain the FEMLE for the model that allows for heteroskedasticity by replacing $\mu_{i}$ in (1) by $y_{i,1}+v_{i},$ and imposing that the $(v_{i}$ $\varepsilon_{i}^{\prime})$ are i.i.d. and normal with $v_{i}\perp\varepsilon_{i}$ when $\left|\rho\right|<1$.666For the sake of a simple exposition we assume that $E(v_{i})=0$. A situation where $E(v_{i})\neq 0$ can be handled by including an intercept term in (13). This amounts to imposing the restriction $\pi=1$ on the model in (12) and leads to the following formulation of the nonstationary panel AR(1) model with FE:

[TABLE]

and where $v_{i}=-v_{i,1}$ satisfy assumption (7). After imposing $u_{i}\sim i.i.d.$ $N(0,\Phi)$ with $\Phi=\widetilde{\sigma}_{v}^{2}\iota\iota^{\prime}+\Psi(\zeta)$, application of the ML method to (13) yields the FE MLE of $\rho$, $\widetilde{\sigma}_{v}^{2}$, $\zeta=(\lambda_{2}^{2}$ $\lambda_{3}^{2}$ $...$ $\lambda_{T}^{2})^{\prime}$. This estimator will also be consistent when the $\varepsilon_{i,t}$ are heteroskedastic across both dimensions of the panel.  The log-likelihood function for the above model will be denoted by $l_{FE}(\theta)$ where in this case $\theta=(r,\widetilde{s}_{v}^{2},z^{\prime})^{\prime}$.

2.2 QML estimation

In the previous subsection we allowed for cross-sectional heteroskedasticity but otherwise assumed homogeneity of the distributions of the (standardized) idiosyncratic errors. However, such strong distributional assumptions with respect to the errors are almost never satisfied by panel data. K2013 has shown that when the data exhibit heterogeneity the ML method still yields consistent RE and FE Quasi ML estimators for $\rho$ if SA, REA (or FEA) and B hold, and $T\geq 3$ when $\left|\rho\right|<1$ and $T\geq 4$ when $\rho=1$ (cf. Theorem 2 in K2013). Under stronger conditions on the $\varepsilon_{i},$ $y_{i,1}$ and $\mu_{i}$ (or $v_{i,1}$), namely under SA4, REA4 (or FEA4), B and an appropriate Lindeberg condition, K2013 has also shown that the first-order fixed parameter asymptotic distributions of the RE and FE QMLEs of $\theta_{0}$ are normal when either $\rho=1$ and the average information matrix and the Expected Hessian of the log-likelihood function are nonsingular or when $\left|\rho\right|<1$, that is, $\sqrt{N}(\widehat{\theta}-\theta_{0})\overset{d}{\rightarrow}N(0,H(\theta_{0})^{-1}G(\theta_{0})H(\theta_{0})^{-1})$ in these cases, where $H(\theta_{0})$ is the asymptotic Hessian of the log-likelihood function and $G(\theta_{0})$ is the asymptotic information matrix (cf. Theorem 3 in K2013).

2.2.1 Asymptotic properties of the QMLEs when $\rho=1$ and

$TSH^{\ast}$ holds

K2001 has shown that if $\rho=1,$ the $\varepsilon_{i}$ are i.i.d. and normal, and assumption TSH holds and has been imposed on the likelihood function, then the Expected Hessian of $N^{-1}l_{FE}(\theta)$ (and thus also the information matrix) is singular. Ahn and Thomas (2023) and Bond et al. (2005) have established an analogous result for the Expected Hessian of $N^{-1}l_{RE}(\theta).$ In these cases standard ML theory cannot be used to derive the limiting distribution of the MLE. Applying the asymptotic theory recently developed by Rotnitzky, Cox, Bottai and Robins (2000, henceforth RCBR) for cases with an information matrix of rank one less than full, Ahn and Thomas (2023) have derived the non-normal limiting distribution of the REMLE. The theory of RCBR is based on using suitable Taylor expansions of the possibly reparametrized log-likelihood and score functions around the value of the parameter vector $\theta$ at which the information matrix is singular, viz. $\theta_{\ast}.$

Unsurprisingly, when $\rho=1$ and TSH*∗* holds but TSH has not been imposed, the Expected Hessians of $N^{-1}l_{RE}(\theta)$ and $N^{-1}l_{FE}(\theta)$ are singular as well. By extending the theory of RCBR to allow for i.h.d. data, K2013 has generalized the result of Ahn and Thomas (2023) as follows.

Application of the asymptotic results of RCBR or their extension for i.h.d. data to the (Q)MLEs requires that one reparametrizes the RE and FE models described in section 2.1 so that three conditions, viz. (B1)-(B3), which are stated just above Theorem 1, are satisfied. They have been verified in K2013 and are similar to conditions given in RCBR. The reparametrizations are such that the scores at $\theta_{\ast}$ for all the new parameters but one are linearly independent, the QMLEs of these parameters converge at rate $O_{p}(N^{-1/2})$ and the score at $\theta_{\ast}$ for the remaining new parameter, which is $\rho$, is equal to zero w.p.1.

In the case of the RE(Q)MLE we need to use the following new parametrization (indicated by the subscript $n$): $\theta_{0,n}=(\rho_{n},\widetilde{\sigma}_{v,n}^{2},\zeta_{n}^{\prime},\tilde{\pi}_{n})^{\prime}$ where $\rho_{n}=\rho,$ $\widetilde{\sigma}_{v,n}^{2}=\widetilde{\sigma}_{v}^{2}/\sigma_{2}^{2}-(1-\rho),$ $\zeta_{n}=(\sigma_{2,n}^{2},\sigma_{3,n}^{2},...,\sigma_{T,n}^{2})^{\prime}$ with $\sigma_{t,n}^{2}=\sigma_{t}^{2}/\rho,$ $t=2,...,T,$ so that $\zeta_{n}=\zeta/\rho$, and $\tilde{\pi}_{n}=\tilde{\pi}-(1-\rho).$ Under this reparametrization the regression equation becomes $y_{i}-y_{i,1}\iota=\rho_{n}(y_{i,-1}-y_{i,1}\iota)+\tilde{\pi}_{n}y_{i,1}\iota+u_{i}.$ Noting that we can express the elements of $\theta$ as functions of the elements of $\theta_{n}=(r_{n},\widetilde{s}_{v,n}^{2},z_{n}^{\prime},\widetilde{p}_{n})^{\prime},$ viz. $\theta(\theta_{n})=(r_{n},r_{n}s_{2,n}^{2}(\widetilde{s}_{v,n}^{2}+(1-r_{n})),r_{n}z_{n}^{\prime},\widetilde{p}_{n}+(1-r_{n}))^{\prime},$ the reparameterized log-likelihood function is given by $l_{n,RE}(\theta_{n})=\sum_{i=1}^{N}l_{n,RE,i}(\theta_{n})$ where $l_{n,RE,i}(\theta_{n})=l_{RE,i}(\theta(\theta_{n})).\,$To obtain results for the FE(Q)MLE one leaves out the last elements of $\theta_{0},$ $\theta,$ $\theta_{0,n}$ and $\theta_{n},$ i.e., the elements that correpond to $\widetilde{\pi},$ $\widetilde{p},$ $\widetilde{\pi}_{n}$ and $\widetilde{p}_{n}$, respectively. Hence it suffices to focus on the RE(Q)MLE.

Note that if $\rho=1$ and TSH*∗* holds, then $\theta_{0}=\theta_{0,n}=(1,0,\sigma^{2}\iota^{\prime},0)^{\prime}$ for some $\sigma^{2}$. Thus $\theta_{\ast}=(1,0,\sigma^{2}\iota^{\prime},0)^{\prime}$ for some $\sigma^{2}$. We may partition $\theta_{0}$ as $\theta_{0}=(\rho,\delta^{\prime})^{\prime}$, partition $\theta_{\ast}$ as $\theta_{\ast}=(\rho_{\ast},\delta_{\ast}^{\prime})^{\prime}=(1,\delta_{\ast}^{\prime})^{\prime}$, partition $\theta_{0,n}$ as $\theta_{0,n}=(\rho_{n},\delta_{n}^{\prime})^{\prime}$ and partition $\theta_{n}$ as $\theta_{n}=(r_{n},d_{n}^{\prime})^{\prime}.$ We will use $l_{i},$ $l_{n}$ and $l_{n,i}$ as short for $l_{RE,i}(\theta),$ $l_{n,RE}(\theta_{n})$ and $l_{n,RE,i}(\theta_{n}),$ respectively. We also define:

[TABLE]

[TABLE]

Note that $U_{2,N}=(N^{-1/2}\mathop{\textstyle\sum}\nolimits_{i=1}^{N}S_{i,1},$ $N^{-1/2}\mathop{\textstyle\sum}\nolimits_{i=1}^{N}S_{i,2}^{\prime})^{\prime}$ and that condition (B2), which requires that for all $K\in\mathbb{R}^{T+1}$ $\frac{\partial^{2}l_{n}}{\partial\rho_{n}^{2}}|_{\theta_{\ast}}\neq K^{\prime}\frac{\partial l_{n}}{\partial\delta_{n}}|_{\theta_{\ast}}$ with positive probability, implies that $\mathcal{I}$ is nonsingular.

Condition (B3), which requires that for all $\overline{K}=(k$ $K^{\prime})^{\prime}\in\mathbb{R}^{T+2}$ $\frac{\partial^{3}l_{n}}{\partial\rho_{n}^{3}}|_{\theta_{\ast}}\neq k\frac{\partial^{2}l_{n}}{\partial\rho_{n}^{2}}|_{\theta_{\ast}}+K^{\prime}\frac{\partial l_{n}}{\partial\delta_{n}}|_{\theta_{\ast}}$ with positive probability, ensures that $U_{N}$ is not identically equal to zero.

Condition (B1) requires that $\frac{\partial l_{n}}{\partial\rho_{n}}|_{\theta_{\ast}}=0$ w.p.1.

In order to obtain the parametrization for which conditions (B1)-(B3) are metK2013 has used the formula $\theta_{n}=\theta+[0,\widetilde{K}^{\prime}]^{\prime}(r-1),$ where $\widetilde{K}=\mathop{\textstyle\sum}_{i}E((\frac{\partial l_{i}}{\partial\rho}|_{\theta_{\ast}})(\frac{\partial l_{i}}{\partial\delta}|_{\theta_{\ast}}))\times$ $[\mathop{\textstyle\sum}_{i}E((\frac{\partial l_{i}}{\partial\delta}|_{\theta_{\ast}})(\frac{\partial l_{i}}{\partial\delta}|_{\theta_{\ast}})^{\prime})]^{-1}$, cf. RCBR p. 264.

Let $\mathbf{1}\{\breve{Z}_{1}>0\}=1$ if $\breve{Z}_{1}>0,$ let $\mathbf{1}\{\breve{Z}_{1}>0\}=0$ if $\breve{Z}_{1}\leq 0,$ let $\mathbf{1}\{\breve{Z}_{1}\leq 0\}=1-\mathbf{1}\{\breve{Z}_{1}>0\},$ and let $\theta_{a,n}=(\rho_{n},\widetilde{\sigma}_{v,n}^{2},\zeta_{n}^{\prime})^{\prime}.$ Then K2013 obtained the following result when the $\varepsilon_{i}$ (or, equivalently, the data) are i.h.d.:

Theorem 1** (K2013)**

Let assumptions SA4, REA4 (FEA4) and B hold and let $T\geq 4$. Let $\rho=1$ and TSH∗ hold, that is, let $\theta_{0,n}=\theta_{\ast}$ for some $\sigma^{2}.$ Finally, let $\widehat{\theta}_{n}\equiv(\widehat{\rho}_{n},\widehat{\delta}_{n})^{\prime}$ be the REQMLE (FEQMLE) that maximizes the value of $l_{n}(\theta_{n})=l_{n,RE}(\theta_{n})$ $(l_{n,FE}(\theta_{n}))$. If the regularity conditions (A1)-(A7) given in K2013 hold and $\mathcal{H}$ is nonsingular, then

a) $\frac{\partial l_{n}}{\partial r_{n}}|_{\theta_{\ast}}=0$ and $\frac{\partial l_{n}}{\partial d_{n}}|_{\theta_{\ast}}\neq 0$ a.s. Furthermore, $rank(I_{\delta_{n}\delta_{n}})=\dim(d_{n})$ a.s.

b) \left[\begin{array}[]{c}N^{1/4}(\widehat{\rho}_{n}-1)\\ N^{1/2}(\widehat{\delta}_{n}-\delta_{n})\end{array}\right]\overset{d}{\rightarrow}\left[\begin{array}[]{c}(-1)^{\breve{B}}\breve{Z}_{1}^{1/2}\\ \breve{Z}_{2}\end{array}\right]\mathbf{1}\{\breve{Z}_{1}>0\}+\left[\begin{array}[]{c}0\\ \breve{Z}_{2}-(\mathcal{H}^{2,1}/\mathcal{H}^{1,1})\breve{Z}_{1}\end{array}\right]\mathbf{1}\{\breve{Z}_{1}\leq 0\}

$U_{N}\overset{d}{\rightarrow}U$* and $\breve{Z}_{N}\overset{d}{\rightarrow}\breve{Z},$ and if in addition assumption B′ holds, then

c1) $\mathcal{I}_{RE}=diag(\mathcal{I}_{RE,\theta_{a,n}\theta_{a,n}},\mathcal{I}_{RE,\tilde{\pi}_{n}\tilde{\pi}_{n}})$ where $\dim(\mathcal{I}_{RE,\tilde{\pi}_{n}\tilde{\pi}_{n}})=1;$ $\mathcal{I}_{RE,\theta_{a,n}\theta_{a,n}}=\mathcal{I}_{FE}.\vskip 3.0pt plus 1.0pt minus 1.0pt$

c2) $\mathcal{H}_{RE}=diag(\mathcal{H}_{RE,\theta_{a,n}\theta_{a,n}},\mathcal{H}_{RE,\tilde{\pi}_{n}\tilde{\pi}_{n}})$ where $\dim(\mathcal{H}_{RE,\tilde{\pi}_{n}\tilde{\pi}_{n}})=1;$ $\mathcal{H}_{RE,\theta_{a,n}\theta_{a,n}}=\mathcal{H}_{FE}.\vskip 3.0pt plus 1.0pt minus 1.0pt\newline$d) in general $\widehat{\rho}_{RE}\overset{asy}{\nsim}\widehat{\rho}_{FE}$ but if the $\varepsilon_{i}$ are i.i.d. and normal, then $\widehat{\rho}_{RE}\overset{asy}{\sim}\widehat{\rho}_{FE}.$*

The QMLEs of $\theta$ are given by $\widehat{\theta}=\theta(\widehat{\theta}_{n})$. The parameter $\rho$ is second-order identified (cf. Sargan, 1983) because $\frac{\partial l_{n}}{\partial r_{n}}|_{\theta_{\ast}}=0,$ while $\frac{\partial^{2}l_{n,i}}{\partial r_{n}^{2}}|_{\theta_{\ast}}\neq 0$, and because for all $K\in\mathbb{R}^{T+1}$ $\frac{\partial^{2}l_{n}}{\partial r_{n}^{2}}|_{\theta_{\ast}}\neq K^{\prime}\frac{\partial l_{n}}{\partial d_{n}}|_{\theta_{\ast}}$ with positive probability, that is, (B.2) holds. As a result the expansion of the log-likelihood involves $N^{1/4}(\widetilde{\rho}-1)$ rather than $N^{1/2}(\widetilde{\rho}-1)$ and the QMLEs of $\left|\rho-1\right|$ converge at rate $O_{p}(N^{-1/4}).$ Furthermore, when $\theta_{0,n}=\theta_{\ast}$ and $\breve{Z}_{1,N}>0,$ $l_{n}(\theta_{n})$ is bimodal. In this case the QMLE of $\rho$ is determined by higher-order terms in the expansion of $l_{n}(\theta_{n})$ around $\theta_{n}=\theta_{\ast}$, i.e., $U_{N}^{\prime}\breve{Z}_{N}$, cf. K2013. Although $\frac{\partial l_{n}}{\partial d_{n}}|_{\theta_{\ast}}\neq 0$ a.s. and $\widehat{\delta}_{n}-\delta_{n}=O_{p}(N^{-1/2}),$ whereas $\frac{\partial l_{n}}{\partial r_{n}}|_{\theta_{\ast}}=0$ a.s. and $\widehat{\rho}_{n}-1=O_{p}(N^{-1/4}),$ all elements of $\widehat{\theta}_{a,n}$ have a non-normal limiting distribution. All these findings would still hold if the model also included exogenous regressors.

Theorem 1 holds for i.h.d. data. In the special case where the $\varepsilon_{i}$ are i.i.d. and normal, $\mathcal{H}=-\mathcal{I}$, $\breve{Z}\sim N(0,\mathcal{I}^{-1}),$ $U\perp\breve{Z}$ and hence $\Pr(\breve{B}=0|\breve{Z}_{1})=1/2,$ $\widehat{\rho}$ has a symmetric limiting distribution, and $\widehat{\rho}_{RE}$ is asymptotically equivalent to $\widehat{\rho}_{FE}$. However, in general (i.e., except for some special cases) if the $\varepsilon_{i}$ are i.i.d and non-normal or i.h.d., then $\mathcal{H}\neq-\mathcal{I}$, $E(U\breve{Z}^{\prime})\neq 0$ and hence $\Pr(\breve{B}=0|\breve{Z}_{1})\neq\Pr(\breve{B}=0)$, and $\widehat{\rho}$ has an asymmetric limiting distribution. Moreover, $\widehat{\rho}_{RE}$ is not asymptotically equivalent to $\widehat{\rho}_{FE}$.

2.3 The distributional properties of QML estimators when $\rho$ is close to unity

It is well-known that (Quasi) ML estimators can be reinterpreted as GMM estimators, cf. e.g. Newey and McFadden (NMcF, 1994). The underlying moment conditions can be obtained by setting the expected score vector equal to zero. It follows that the Expected Hessian of the (quasi) log-likelihood function equals the first derivative of the vector of moment conditions exploited by the (Q)MLE with respect to the parameters. Therefore when the Expected Hessian of the (quasi) log-likelihood function is almost singular, the (Q)MLE suffers from a ‘weak moment conditions problem.’ K2013 gives necessary and sufficient conditions for this situation to arise when $Var(y_{i,1}-\mu_{i})\propto(1-\rho)^{0}=1$ $\forall i\in\{1,2,...,N\}$:

Theorem 2** (K2013)**

Let assumptions SA, REA (or FEA) and B hold and let$Var(y_{i,1}-\mu_{i})\propto(1-\rho)^{0}=1$ $\forall i\in\{1,2,...,N\}$. Furthermore let $\rho$ be local to unity. Then the Expected Hessian of $l_{RE}(\theta)$ $(l_{FE}(\theta))$ is almost singular if and only if either assumption TSH∗ holds, assumption TSH has been imposed on $l_{RE}(\theta)$ $(l_{FE}(\theta))$ and $T\geq 3,$ or assumption TSH${}_{T-1}^{\ast}$ (almost) holds and $T\geq 4$.

If the Expected Hessian of the (quasi) log-likelihood function is almost singular, one can obtain a better approximation to the finite sample distribution of the (Q)MLE than the usual approximation by using local asymptotics. K2013 obtained the following result:

Theorem 3** (K2013)**

*Let assumptions SA4, FEA4 (or REA4) and B hold, let $T\geq 4$, let $\delta_{\ast}$ be such that $(1,\delta_{\ast}^{\prime})^{\prime}=\theta_{\ast},$ and let $\{\theta_{n,N}\}$ be such that $N^{1/4}(\rho_{n,N}-1)=o(1)$ and $N^{1/2}(\delta_{n,N}-\delta_{\ast})=o(1)$ for some $\sigma^{2}.$ Let $\{\theta_{i,n,N}\},$ $i=1,...,N,$ be such that $\rho_{i,n,N}=\rho_{n,N},$ $N^{-1}\sum_{i=1}^{N}\widetilde{\sigma}_{v,i,n,N}^{2}=\widetilde{\sigma}_{v,n,N}^{2},$ $N^{-1}\sum_{i=1}^{N}\zeta_{i,n,N}=\zeta_{n,N}$ $($and $\tilde{\pi}_{i,n,N}=\tilde{\pi}_{n,N}).$ Let $\widehat{\theta}_{n}=(\widehat{\rho}_{n},\widehat{\delta}_{n})^{\prime}$ be the FEQMLE (REQMLE) that maximizes the value of l_{n}(\theta_{n})=l_{n,FE}(\theta_{n})$$(l_{n,RE}(\theta_{n})). Let $\mathcal{H},$ $\breve{B}$ and $\breve{Z}=(\breve{Z}_{1},\breve{Z}_{2}^{\prime})^{\prime}$ satisfy the definitions given just above theorem

1. If $\mathcal{H}$ is nonsingular and $(y_{i,1}$ $y_{i}^{\prime})^{\prime}$ is generated under $\theta_{i,n,N},$ $i=1,...,N,$ then*

a) $\widehat{\theta}_{n}-\theta_{\ast}=o_{p}(1).$

b) \left[\begin{array}[]{c}N^{1/4}(\widehat{\rho}_{n}-1)\\ N^{1/2}(\widehat{\delta}_{n}-\delta_{\ast})\end{array}\right]\overset{d}{\rightarrow}\left[\begin{array}[]{c}(-1)^{\breve{B}}\breve{Z}_{1}^{1/2}\\ \breve{Z}_{2}\end{array}\right]\mathbf{1}\{\breve{Z}_{1}>0\}+\left[\begin{array}[]{c}0\\ \breve{Z}_{2}-(\mathcal{H}^{2,1}/\mathcal{H}^{1,1})\breve{Z}_{1}\end{array}\right]\mathbf{1}\{\breve{Z}_{1}\leq 0\}

c) in general $\widehat{\rho}_{RE}\overset{asy}{\nsim}\widehat{\rho}_{FE}$ but if assumption B′ also holds and the $u_{i}$ are i.i.d. and normal, then $\widehat{\rho}_{RE}\overset{asy}{\sim}\widehat{\rho}_{FE}.$

Note that since $\widetilde{\sigma}_{v,n}^{2}=\widetilde{\sigma}_{v}^{2}/\sigma_{2}^{2}-(1-\rho)=(1-\rho)((1-\rho)\sigma_{v}^{2}/\sigma_{2}^{2}-1)$ and $\tilde{\pi}_{n}=(\pi-1)(1-\rho)$, both condition $N^{1/2}(\widetilde{\sigma}_{v,n,N}^{2}-0)=o(1)$ and condition $N^{1/2}(\tilde{\pi}_{n,N}-0)=o(1)$ normally require that $N^{1/2}(\rho_{n,N}-1)=o(1)$ rather than $N^{1/4}(\rho_{n,N}-1)=o(1).$ However, when for instance $\pi=1$ and $\sigma_{v}^{2}=2\sigma_{2}^{2}/(1-\rho^{2}),$ then the results of theorem 3 hold under any $\{\rho_{n,N}\}$ such that $N^{1/4}(\rho_{n,N}-1)=o(1).$ In any case, when both $N^{1/4}(\rho_{n,N}-1)=o(1)$ and $N^{1/2}(\delta_{n,N}-\delta_{\ast})=o(1)$ for some $\sigma_{2}^{2},$ the RE- and the FEQMLE of $\rho$ are $N^{1/4}-$consistent.

Results in RCBR suggest that in principle under $(\rho_{n,N}-1)=-\lambda N^{-b}$ different local asymptotic distributions and rates of convergence are obtained for the QMLE of $\rho$ when $b=1/4,$ $1/6<b<1/4$, $b=1/6$ or $0<b<1/6,$ respectively.

3 Likelihood based tests

Wald test statistics, some versions of (Quasi) LM test statistics, and (Quasi) LR test statistics that are used for testing an hypothesis about $\rho$ and are based on the reparametrized RE or FE likelihood do not uniformly converge to their fixed parameter first-order limiting distributions near the singularity point, cf. Bottai (2003). As a consequence these tests do not have correct asymptotic size in a uniform sense. (Q)LM test statistics that are standardised by (using a sandwich formula involving) the expected rather than the observed average Hessian are exceptions to this rule. Bottai (2003) explains why this is the case in the context of models with a single parameter. As we will show in the proof of theorem 4, when testing a hypothesis that includes a restriction on the parameter that has zero score at the singularity point, these test statistics still converge uniformly to their fixed parameter first-order limiting distribution, which is a central $\chi^{2}$-distribution, near the singularity point and these tests have correct asymptotic size in a uniform sense. Crucially, the expected average Hessian of $l_{n}(\theta_{n})$, i.e., $\overline{H}(\mathcal{\breve{\theta}}_{n})\equiv E_{\mathcal{\breve{\theta}}_{n}}(\frac{1}{N}\frac{\partial^{2}l_{n}(\theta_{n})}{\partial\theta_{n}\partial\theta_{n}^{\prime}}|_{\mathcal{\breve{\theta}}_{n}})$, isnegative definite and hence nonsingular for any value of $\mathcal{\breve{\theta}}_{n}$ that differs from the singu- larity point $\theta_{\ast}$. Note that the values of the elements of $\overline{H}(\mathcal{\breve{\theta}}_{n})$ do not depend on the true distribution of the data. Alternative testing approaches that are based on a QLR test statistic but use a so-called type 2 robust critical value, which is determined by a statistic that indicates the closeness to the singularity point, or a least favourable critical value (cf. Andrews and Cheng, 2013) either do not work due to non-monotonicity of the limiting distributions of the QLR test statistic near the singularity point or are conservative and have low power.

We will now introduce the QLM test statistic $QLM(\widetilde{\mathcal{\theta}}_{n})$ for testing $H_{0}:$ $A\mathcal{\theta}_{0,n}=a$, which includes a restriction on $\rho$, where $\widetilde{\mathcal{\theta}}_{n}$ is a restricted QML estimate of $\mathcal{\theta}_{0,n}$ such that $A\widetilde{\mathcal{\theta}}_{n}=a,$ $A$ is a $J\times\dim(\mathcal{\theta})$ constant matrix of rank $J,$ and a\is a constant vector. Define the average information matrix $\overline{\mathcal{J}}\mathcal{(\mathcal{\breve{\theta}}}_{n}\mathcal{)}=N^{-1}\mathop{\textstyle\sum}_{i=1}^{N}\mathcal{J}_{i}\mathcal{(\mathcal{\breve{\theta}}}_{n}\mathcal{)}$ with $\mathcal{J}_{i}\mathcal{(\breve{\theta}}_{n}\mathcal{)}=\left(\frac{\partial l_{n,i}(\theta_{n})}{\partial\theta_{n}}|_{\mathcal{\breve{\theta}}_{n}}\right)\left(\frac{\partial l_{n,i}(\theta_{n})}{\partial\theta_{n}^{\prime}}|_{\mathcal{\breve{\theta}}_{n}}\right),$ where $l_{n,i}(\theta_{n})$ is the contribution to the reparametrized log-likelihood function, $l_{n}(\theta_{n})$, by individual $i$. If $\widetilde{\mathcal{\theta}}_{n}\neq\theta_{\ast}$ for all $\sigma^{2}>0$, then $QLM(\widetilde{\mathcal{\theta}}_{n})$ is given by, cf. White (1994, p. 173):

[TABLE]

If $H_{0}$ is true and $\mathcal{\theta}_{0,n}\neq\theta_{\ast}$ (for all $\sigma^{2}>0$), then $QLM(\widetilde{\mathcal{\theta}}_{n})\overset{d}{\rightarrow}\chi^{2}(J).$ To test $H_{0}:$ $\rho=a$ when $\widetilde{\mathcal{\theta}}_{n}\neq\theta_{\ast}$ (for all $\sigma^{2}>0$,) for some known value of $a\in(-1,1]$, one can use $QLM(\widetilde{\mathcal{\theta}}_{n})$ in (3) with $A=(1$ $\mathbf{0}^{\prime})$ and $\frac{\partial l_{n}(\widetilde{\mathcal{\theta}}_{n})}{\partial\theta_{n}}=A^{\prime}\frac{\partial l_{n}(\widetilde{\mathcal{\theta}}_{n})}{\partial\rho}$. Note also that, unlike the values of the(Quasi) Wald and Hausman test-statistics that use $\overline{H}^{-1}(\widehat{\mathcal{\theta}}_{n})$ (cf. White, 1994, p. 173), the value of $QLM(\widetilde{\mathcal{\theta}}_{n})$ remains the same when $\overline{H}^{-1}(\mathcal{\breve{\theta}}_{n})$ for some $\mathcal{\breve{\theta}}_{n}$ is replaced by $adj(\overline{H}(\mathcal{\breve{\theta}}_{n}))$. Furthermore, if $\widetilde{\mathcal{\theta}}_{n}\neq\theta_{\ast}$, then $\overline{\mathcal{J}}(\widetilde{\mathcal{\theta}}_{n})$ and p$\lim_{N\rightarrow\infty}\overline{\mathcal{J}}(\widetilde{\mathcal{\theta}}_{n})$ are positive definite.

If $\widetilde{\mathcal{\theta}}_{n}=\theta_{\ast}$ for some $\sigma^{2}>0,$ $rk(\overline{H}(\widetilde{\mathcal{\theta}}_{n}))=\dim(\overline{H}(\widetilde{\mathcal{\theta}}_{n}))-1$ and hence $rk(adj(\overline{H}(\widetilde{\mathcal{\theta}}_{n})))=1.$ Furthermore, $\overline{H}(\widetilde{\mathcal{\theta}}_{n})_{i,j}=0$ iff $i=1$ and/or $j=1,$ $adj(\overline{H}(\widetilde{\mathcal{\theta}}_{n}))_{i,j}\neq 0$ iff $i=j=1,$ and $adj(\overline{H}(\widetilde{\mathcal{\theta}}_{n}))(1$ $\mathbf{0}^{\prime})^{\prime}\propto(1$ $\mathbf{0}^{\prime})^{\prime}.$ Hence if $\widetilde{\mathcal{\theta}}_{n}=\theta_{\ast}$ (for some $\sigma^{2}>0$), the QLM test statistic in (3) for $H_{0}:$ $\rho=a$ would be equal to $(\frac{\partial l_{n}(\mathcal{\theta}_{n})}{\partial r}|_{\theta_{\ast}})^{2}/(\mathop{\textstyle\sum}_{i=1}^{N}(\frac{\partial l_{n,i}(\mathcal{\theta}_{n})}{\partial r}|_{\theta_{\ast}})^{2})$. However, this test statistic cannot be used, because $\frac{\partial l_{n,i}(\mathcal{\theta}_{n})}{\partial r}|_{\theta_{\ast}}=0$ a.s. and p$\lim_{N\rightarrow\infty}N^{-1}\mathop{\textstyle\sum}_{i=1}^{N}(\frac{\partial l_{n,i}(\mathcal{\theta}_{n})}{\partial r}|_{\theta_{\ast}})^{2}=0$. More generally, $N^{-1/2}A\,adj(\overline{H}(\theta_{\ast}))\frac{\partial l_{n}(\mathcal{\theta}_{n})}{\partial\mathcal{\theta}_{n}}|_{\theta_{\ast}}=0$ a.s. and p$\lim_{N\rightarrow\infty}[A\,adj(\overline{H}(\theta_{\ast}))\times$ $\overline{\mathcal{J}}(\theta_{\ast})adj(\overline{H}(\theta_{\ast}))A^{\prime}]=\mathbf{0.}$ To test $H_{0}:\nolinebreak$ $\nolinebreak A\mathcal{\theta}_{0,n}=a$ when $\widetilde{\mathcal{\theta}}_{n}=\theta_{\ast}$ (for some $\sigma^{2}>0$), one can use the following QLM test statistic, cf. Bottai (2003):

[TABLE]

with

[TABLE]

and

[TABLE]

where we have used $l_{n}$ and $l_{n,i}$ as short for $l_{n}(\theta_{n})$ and $l_{n,i}(\theta_{n})$. In contrast, the Wald and Hausman test-statistics that use $\overline{H}^{-1}(\widehat{\mathcal{\theta}}_{n})$ when $\widehat{\mathcal{\theta}}_{n}\neq\theta_{\ast}$ cannot be defined when $\widehat{\mathcal{\theta}}_{n}=\theta_{\ast}.$ If $H_{0}$ is true and $\mathcal{\theta}_{0,n}=\theta_{\ast}$ for some $\sigma^{2}>0$, then $QLM(\theta_{\ast})\overset{d}{\rightarrow}\chi^{2}(J).$ To test $H_{0}:$ $\rho=a$ when $\widetilde{\mathcal{\theta}}_{n}=\theta_{\ast}$, one can use $QLM(\widetilde{\mathcal{\theta}}_{n})$ in (3) with $A=(1$ $\mathbf{0}^{\prime})$ and $\widetilde{S}(\widetilde{\mathcal{\theta}}_{n})=A^{\prime}(\sum\nolimits_{i=1}^{N}S_{i,1})$. It can easily be shown that the test statistic $QLM(\widetilde{\mathcal{\theta}}_{n})$ for testing $H_{0}:$ $\rho=a$ given by (3) and (3) is continuous at $\widetilde{\mathcal{\theta}}_{n}=\theta_{\ast}$ (for any $\sigma^{2}>0$) by using de l’Hôpital’s rule twice.

Theorem 4

Under regularity conditions (A1)-(A7) given in the appendix, the Quasi LM test for $H_{0}:$ $A\mathcal{\theta}_{0,n}=a,$ where $A_{1,.}=(1$ $\mathbf{0}^{\prime}),$ that is based on (3) if $\widetilde{\mathcal{\theta}}_{n}\neq\theta_{\ast}$ for all $\sigma^{2}>0,$ and on (3) if $\widetilde{\mathcal{\theta}}_{n}=\theta_{\ast}$ for some $\sigma^{2}>0,$ has correct asymptotic size in a uniform sense.

Confidence Sets (CSs) that have correct uniform asymptotic size can be obtained by ”inverting” the QLM test, i.e., $QLM(\widetilde{\mathcal{\theta}}_{n})$ given by (3) and (3), or any of the other aforementioned tests that have correct uniform asymptotic size. For instance, a CS for $\rho$ of level $(1-\alpha)\ast 100\%$ constructed in this way is the set of points $a\in{\mathbb{R}}$ for which the test fails to reject $H_{0}:\rho=a$ at significance level $\alpha.$

When the data are i.i.d. and normal and $\rho$ is not local or equal to one, the (Q)LM test that uses the expected Hessian has optimal local power properties. Next we derive the power envelope of a centered version of $QLM(\widetilde{\mathcal{\theta}}_{n})$ for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$, where $\kappa$ is a constant, viz. $QLM^{c}(\widetilde{\mathcal{\theta}}_{n}),$ when the data are i.i.d. and normal and $\theta_{0}=\theta_{\ast}.$ Let $\overline{\mathcal{J}}^{c}\mathcal{(\theta}_{n}\mathcal{)}=N^{-1}\mathop{\textstyle\sum}_{i=1}^{N}\mathcal{J}_{i}^{c}\mathcal{(\theta}_{n}\mathcal{)}$ with $\mathcal{J}_{i}^{c}\mathcal{(\mathcal{\mathcal{\breve{\theta}}}}_{n}\mathcal{)}=(\frac{\partial l_{n,i}(\theta_{n})}{\partial\theta_{n}}|_{\mathcal{\breve{\theta}}_{n}}-N^{-1}\mathop{\textstyle\sum}_{i=1}^{N}(\frac{\partial l_{n,i}(\theta_{n})}{\partial\theta_{n}}|_{\mathcal{\breve{\theta}}_{n}}))(\frac{\partial l_{n,i}(\theta_{n})}{\partial\theta_{n}^{\prime}}|_{\mathcal{\breve{\theta}}_{n}}-N^{-1}\mathop{\textstyle\sum}_{i=1}^{N}(\frac{\partial l_{n,i}(\theta_{n})}{\partial\theta_{n}^{\prime}}|_{\mathcal{\breve{\theta}}_{n}}))$. If $\widetilde{\mathcal{\theta}}_{n}\neq\theta_{\ast},$ we have

[TABLE]

where $A=(1$ $\mathbf{0}^{\prime}).$ If $\widetilde{\mathcal{\theta}}_{n}=\theta_{\ast},$ $QLM^{c}(\widetilde{\mathcal{\theta}}_{n})$ is similar to (3) with $\widetilde{S}(\widetilde{\mathcal{\theta}}_{n})=A^{\prime}(\sum\nolimits_{i=1}^{N}S_{i,1})$ and $\widetilde{\mathcal{J}}(\widetilde{\mathcal{\theta}}_{n})$ replaced by $\widetilde{\mathcal{J}}^{c}(\widetilde{\mathcal{\theta}}_{n})=N^{-1}\sum\nolimits_{i=1}^{N}(S_{i}-\overline{S})(S_{i}-\overline{S})^{\prime},$ where $\overline{S}=N^{-1}\sum\nolimits_{i=1}^{N}S_{i}$.

If mean stationarity (8) holds when $\left|\rho\right|<1$ and $Var(\Delta y_{i,t})/Var(y_{i,1})=o(N^{-1}),$ $i=1,2,...,N,$ then only the full set of moment conditions based on differenceddata matters asymptotically for identification of $\rho$ and there is no loss in efficiencywhen exploiting only the latter for estimation or testing purposes; additional moment conditions involving levels of the data are redundant (cf. Kruiniger, 2022, and Bun and Kleibergen, 2022). Note that $Var(\Delta y_{i,t})=O(1)$ for any $\rho\in(-1,1].$ We will investigate the power of the $QLM^{c}(\widetilde{\mathcal{\theta}}_{n})$ tests under a worst case scenario where $\theta_{0}=\theta_{\ast}$ and $1/Var(y_{i,1})=o(N^{-1})$ $i=1,2,...,N$ and consider a sequence of null hypotheses that is local-to-unity: $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ (cf. Bun and Kleibergen, 2022). In this scenario it is sufficient to focus on the FE version of $QLM^{c}(\widetilde{\mathcal{\theta}}_{n}),$ viz. $QLM_{FE}^{c}(\widetilde{\mathcal{\theta}}_{n}).$

Theorem 5

When the data are i.i.d. and normal and $\theta_{0}=\theta_{\ast},$ (so that TSH holds,) then the large sample distribution of $QLM_{FE}^{c}(\widetilde{\mathcal{\theta}}_{n})$ for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ is given by $\chi^{2}($p$\lim_{N\rightarrow\infty}c_{1}^{\prime}S^{-1}\overline{H}^{-1}A^{\prime}(A\overline{H}^{-1}\overline{\mathcal{J}}^{c}\overline{H}^{-1}A^{\prime})^{-1}A\overline{H}^{-1}S^{-1}c_{1}\kappa^{4},1)$ where $c_{1}$ and $S$ are defined in the proof in the appendix.

The quartic root rate in the sequence of hypotheses is related to the fact that $\rho$ is only second-order identified when $\theta_{0}=\theta_{\ast}$. When we impose TSH on $l_{n}(\theta_{n}),$ we obtain the following result:

Theorem 6

When the data are i.i.d. and normal, $\theta_{0}=\theta_{\ast}$ and TSH has been imposed on $l_{n}(\theta_{n})$, then the large sample distribution of $QLM_{FE}^{c}(\widetilde{\mathcal{\theta}}_{n})$ for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ is given by $\chi^{2}(\frac{(2T-3)T(T-1)(T-2)}{72}\kappa^{4},1).$

We will obtain the maximal attainable power (MAP) curve for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ by first considering the asymptotic distribution of an GMM-Anderson-Rubin statistic which tests $H_{0}$ by using all the moment conditions based on first-differences of the data whilst (the true value of) $\rho=1.$ This GMM-AR statistic is given by

[TABLE]

with

[TABLE]

where $D_{r}$ is a $(T-1)\times(T-1)$ band matrix with $(D_{r})_{i,i}=1$ and $(D_{r})_{i+1,i}=-r$ for $i=1,2,...,T-2,($and $T-1)$ and $(D_{r})_{i,j}=0$ elsewhere; $P=(0$ $g$ $I_{p})$ is a $p\times\frac{1}{2}(T-1)T$ matrix where $p=\frac{1}{2}T(T-1)-2$ and $g$ is a $p$-vector such that $(1,-1,$ $g^{\prime})^{\prime}=vech(D_{r}D_{r}^{\prime})$; and $\widehat{V}_{mm}(\rho)$ is the Eicker-White estimator of the covariance matrix of $m(\rho).$

Theorem 7

When the data are i.i.d. and normal, $\theta_{0}=\theta_{\ast}$ and TSH is exploited by the testing procedure, then the large sample distribution of $GMM-AR(\rho)$ for testing$H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ is given by $\chi^{2}(\frac{(2T-3)T(T-1)(T-2)}{72}\kappa^{4},\frac{1}{2}T(T-1)-2).$

We obtain the MAP curve for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ by deriving the asymptotic distribution of an GMM-Anderson-Rubin statistic which tests $H_{0}$ by using the (infeasible) weighted average of the moment conditions in $E(m(\rho))=0_{p}$ (with $m(r)$ given in (19)) that leads to the largest value of the non-centrality parameter of this distribution whilst (the true value of) $\rho=1$.

Theorem 8

When the data are i.i.d. and normal, $\theta_{0}=\theta_{\ast}$ and TSH is exploited by the testing procedure, then the maximal attainable power curve for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ is given by $\chi^{2}(\frac{(2T-3)T(T-1)(T-2)}{72}\kappa^{4},1).$

Corollary 1

When the data are i.i.d. and normal, $\theta_{0}=\theta_{\ast}$ and TSH has been imposed on $l_{n}(\theta_{n})$, then the large sample distribution of $QLM_{FE}^{c}(\widetilde{\mathcal{\theta}}_{n})$ for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ attains the maximal attainable power curve for testing $H_{0}.$

This result implies that, like the KLM-statistic, the centered LM-statistic $QLM^{c}(\widetilde{\mathcal{\theta}}_{n})$ is efficient both when $\rho$ is less than one and when $\rho$ is equal to one.

To test $H_{0}:$ $\rho=1$ one could also use a Wald test based on $\sqrt{N}(\widehat{\rho}-1)^{2}$ where $\widehat{\rho}$ is the REQMLE or FEQMLE of $\rho.$ Under H_{0}\we have $\sqrt{N}(\widehat{\rho}-1)^{2}\overset{d}{\rightarrow}\breve{Z}_{1}\mathbf{1}\{\breve{Z}_{1}>0\},$ cf. Theorem 1. Recall that $\breve{Z}_{N}=(\breve{Z}_{1,N},\breve{Z}_{2,N}^{\prime})^{\prime}=-\mathcal{H}^{-1}U_{2,N}\overset{d}{\rightarrow}\breve{Z}$ with $U_{2,N}=(\frac{1}{2!}N^{-1/2}\frac{\partial^{2}l_{n}}{\partial r_{n}^{2}}|_{\theta_{\ast}},N^{-1/2}\frac{\partial l_{n}}{\partial d_{n}^{\prime}}|_{\theta_{\ast}})^{\prime}$ and $\breve{Z}=(\breve{Z}_{1},\breve{Z}_{2}^{\prime})^{\prime}\sim N(0,\mathcal{H}^{-1}\mathcal{IH}^{-1}).$ When the data are i.i.d. non-normal or i.h.d., one can bootstrap the distribution of $U_{2,N}$. To do this, one can make use of the fact that under $H_{0}$, $\varepsilon_{i}=y_{i}-y_{i,-1}$ for $i=1,...,N.$

4 The finite sample performance of the QLM tests

In this section we investigate through Monte Carlo simulations the empirical size and power properties of QLM-tests for testing a simple hypothesis of the type $H_{0}:$ $\rho=a$, namely $QLM(\mathcal{\rho}),$ that are based on the RE and FE likelihood functions for various panel AR(1) models without covariates. The data were generated using the panel AR(1) model given in (1). We study how the properties of these tests are affected if we change (1) the value of $\rho,$ (2) the distributions of the initial conditions $v_{i,1}=y_{i,1}-\mu_{i}$, (3) the distributions of the idiosyncratic errors (the $\varepsilon_{i,t}$) and/or (4) the ratio of the variances of the error components. We conducted the simulation experiments for $(T,N)=(4,100),$ $(9,100),$ $(4,250)$ or $(9,250).$ The nominal size of the tests was $0.05$ and for each scenario the number of replications was 2500.

We calculated the empirical size of the tests for $\rho=0.2,$ $0.5,$ $0.8,$ $0.9,$ $0.95,$ $0.98$ or $0.99$ and we calculated the empirical power of the tests for $H_{0}:\rho=0.8$ when $\rho=0.5,$ $0.6,$ $0.7,$ $0.9,$ $0.95$ or $0.99.$

In all simulation experiments the individual effects, the $\mu_{i}$, were i.i.d. $N(0,\sigma_{\mu}^{2})$ with $\sigma_{\mu}^{2}=1$ or $25.$ The errors, the $\varepsilon_{i,t}$, were either i.i.d. $N(0,1)$ or i.i.d. $(\chi^{2}(1)-1)/\sqrt{2}.$ Note that in both cases $Var(\varepsilon_{i,t})=1$ for $i=1,...,N$ and $t=2,...,T$.

In order to assess how the assumptions with respect to $y_{i,1}-\mu_{i}$, $i=1,...,N,$ affect the finite sample properties of the tests, we conducted three different kinds of experiments: in one set, labeled NS-Normal, the initial observations are non-stationary, i.e., $y_{i,1}-\mu_{i}=0$, $i=1,...,N,$ and the $\varepsilon_{i,t}\sim N(0,1)$, whereas in the other two sets, labeled S-Normal and S-ChiSq., respectively, the initial observations are drawn from stationary distributions, i.e., either $(y_{i,1}-\mu_{i})\sim N(0,1/(1-\rho^{2}))$ when the $\varepsilon_{i,t}\sim N(0,1)$ or $(y_{i,1}-\mu_{i})\sim(\chi^{2}(1)-1)/\sqrt{2(1-\rho^{2})}$ when the $\varepsilon_{i,t}\sim(\chi^{2}(1)-1)/\sqrt{2}$. Note that in design NS-Normal the data are still mean stationary, i.e., $E(y_{i,1}-\mu_{i})=0$ and $E(\mu_{i}(y_{i,1}-\mu_{i}))=0$, and that in all designs $E(y_{i,t}-y_{i,t-1})=0$.

In the cases of the RE and FE models, $(1-\rho)\mu_{i}+\varepsilon_{i}$ is decomposed as $(1-\rho)\pi y_{i,1}+(1-\rho)v_{i}+\varepsilon_{i}=(1-\rho)\pi y_{i,1}+u_{i}$ with $\pi=1$ for the FE case. In the experiments we imposed homoskedasticity on the likelihood functions and added the restrictions $\sigma^{2}>0$ and $\widetilde{\sigma}_{v}^{2}\geq 0$ or the restrictions $\sigma^{2}>0$ and $(T-1)\widetilde{\sigma}_{v}^{2}+\sigma^{2}>0$ in case the restriction $\widetilde{\sigma}_{v}^{2}\geq 0$ was binding to ensure that the estimates of $E(u_{i}u_{i}^{\prime})$ were positive definite.

We allowed for time effects by subtracting cross-sectional averages from the data.

Note that the QML estimators suffer from a weak moment conditions problem when $\rho$ is close to one.

Tables 1-12 report the simulation results in terms of the relative rejection frequenties of the tests. Tables 1-4, 9, 10 report results concerning the empirical size of the tests whereas tables 5-8, 11, 12 report results concerning the empirical power of the tests. The tables also differ with respect to the the value of $\sigma_{\mu}^{2}$: tables 1-8 correspond to $\sigma_{\mu}^{2}=1$ and report results for both RE and FE versions of the QLM tests whereas tables 9-12 correspond to $\sigma_{\mu}^{2}=25$ and report results for the RE version of the QLM test only because changes in the value of $\sigma_{\mu}^{2}$ do not affect the FE affects version of the QLM test by construction. If the tests have correct size (i.e., 0.05), then the standard errors of the estimates of the empirical size are $\sqrt{0.05\ast(1-0.05)}/\sqrt{2500}\approx 0.0044.$

Inspection of the results in tables 1-4, 9 and 10 leads to the following conclusions regarding the empirical size of the QLM tests:

Using a 5% significance level, we would not be able to reject the hypothesis that the tests have correct size for the various scenarios with $\sigma_{\mu}^{2}=1.$ Only 8 out of 168 estimates of the empirical size lie outside the acceptance region $(0.0412,0.0588).$ The most extreme estimates of the empirical sizes are $0.0628$ and $\ 0.0380.$ If we restrict attention to $N=250$ and $\rho=0.95,$ $0.98$ or $0.99,$ then 2 out of 36 estimates of the empirical size lie outside the acceptance region $(0.0412,0.0588).$ 2. 2.

Using a 5% significance level, we would not be able to reject the hypothesis that the tests have correct size for the various scenarios with $\sigma_{\mu}^{2}=25.$ Only 5 out of 84 estimates of the empirical size lie outside the acceptance region $(0.0412,0.0588).$ The most extreme estimates of the empirical sizes are $0.0664$ and $\ 0.0644.$ If we restrict attention to $N=250$ and $\rho=0.95,$ $0.98$ or $0.99,$ then 1 out of 18 estimates of the empirical size lies outside the acceptance region $(0.0412,0.0588).$

Inspection of the results in tables 5-8, 11 and 12 leads to the following conclusions regarding the empirical power of the QLM tests:

The power of the tests increases in $N$ and $T.$ 2. 4.

The power of the RE version of the QLM test is higher than the power of the FE version of the QLM test unless the initial conditions $v_{i,1}=y_{i,1}-\mu_{i}$ are zero (NS) in which case the power is the same for both versions of the QLM tests. 3. 5.

The power curves for testing $H_{0}:\rho=0.8$ are asymmetric around $\rho=0.8$, i.e., the QLM tests have more power against an alternative below $\rho=0.8$ than against an equidistant alternative above $\rho=0.8.$

5 Concluding remarks

In this paper we proposed new ML based inference methods for panel AR models with arbitrary initial conditions and heteroskedasticity and possibly additional regressors that are robust to the strength of identification. Specifically, we showed that (Quasi) LM tests and CSs that use the expected Hessian rather than the observed Hessian of the RE or the FE log-likelihood function have correct asymptotic size in a uniform sense. We also derived the power envelope of a FE version of such an LM test for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ when the average information matrix is estimated by a centered OPG estimator and the model is only second-order identified, and showed that it coincides with the maximal attainable power curve for testing $H_{0}:$ $\rho=a=1-\kappa/\sqrt[4]{N}$ in the worst case setting. In a Monte Carlo study that included a variety of experiments we found that these (Quasi) LM tests have correct empirical size and good empirical power properties.

None of the existing ML based inference methods for dynamic panel data models have correct uniform asymptotic size close to the point in the parameter space at which such a model is only second-order identified and therefore the methods proposed in this paper will result in more reliable inference. The proposed methods can be adapted to generalizations of the panel AR(1) model that was considered in this paper including dynamic panel models with a factor structure and panel VAR models. Similarly, Quasi LM test that have correct uniform asymptotic size can also be developed for other multi-parameter models that are only second-order identified at some point in the parameter space, such as the examples mentioned in e.g. Rotnitzky et. al. (2000) and Dovonon and Hall (2018).

An issue that requires additional study is the effect of the choice of the estimator for the average information matrix on the power of the QLM tests.

Appendix A Proofs

Regularity conditions:

Let $f(y_{i}^{+};\theta)$ denote the density of the random vector $y_{i}^{+}$ so that $l_{i}(\theta)=\ln f(y_{i}^{+};\theta).$ Let $p=\dim(\theta).$ We can write $\theta$ as $(\theta_{1},...,\theta_{p})^{\prime}.$ For any $1\times p$ vector $a=(a_{1},...,a_{p}),$ let $l_{i}^{(a)}(\theta)$ denote $\partial^{a.}l_{i}(\theta)/\partial^{a_{1}}\theta_{1}\partial^{a_{2}}\theta_{2}...\partial^{a_{p}}\theta_{p}$ where $a.=\mathop{\textstyle\sum}\nolimits_{k=1}^{p}a_{k}.$ Let $l^{(a)}(\theta)=\mathop{\textstyle\sum}\nolimits_{i=1}^{N}l_{i}^{(a)}(\theta)$ and define $f^{(a)}(y_{i}^{+};\theta)$ similarly. Let w.p.1 denote ”with probability 1”. Let $S_{k}(\theta)=\partial l(\theta)/\partial\theta_{k},$ $1\leq k\leq p,$ and $S_{k}=S_{k}(\theta_{\ast}).$ Let $S_{1}^{(q+j)},$ $j=0,1,$ denote $\partial^{q+j}l(\theta)/\partial^{q+j}\theta_{1}|_{\theta_{\ast}}.$ Let $q\in\mathbb{N}$ be such that conditions (B1*′)-(B3′*) given in K2013 hold, i.e., $\theta_{1}$ is $q$-th order identified. Then the regularity conditions for the case of i.h.d. data are given by (cf. Bottai, 2003):

(A1′)**

$\theta_{\ast}\in\Theta$, a compact subset of $\mathbb{R}^{p}$ that contains an open neighbourhood $\mathcal{N}$ of $\theta_{\ast}$.

(A2′)**

Distinct values of $\theta$ in $\Theta$ correspond to distinct probability distributions.

(A3′)**

W.p.1 (under $\theta_{\ast}$), $f(y_{i}^{+};\theta)>0$ for all $\theta\in\mathcal{N}$.

(A4′)**

There exist $B_{i}(y_{i}^{+})$ such that $\left|l_{i}(\theta)\right|\leq B_{i}(y_{i}^{+})$ for all $\theta\in\Theta$ and $E\left|B_{i}(y_{i}^{+})\right|^{1+\xi}<\infty$ for some $\xi>0$ and all $i\in\{1,...,N\}.$

(A5′)**

For all $\theta$ in $\mathcal{N}$ and all $a$ with $1\leq a.\leq 2q+1$, the derivatives $f^{(a)}(y_{i}^{+};\theta)$ and $l_{i}^{(a)}(\theta)$ exist w.p.1 and there exist $B_{i}(y_{i}^{+})$ such that $\left|l_{i}^{(a)}(\theta)\right|\leq B_{i}(y_{i}^{+})$ for all $\theta\in\mathcal{N}$ and $E\left|B_{i}(y_{i}^{+})\right|^{1+\xi}<\infty$ for some $\xi$ and all $i\in\{1,...,N\}.$ Furthermore (for all $a$ with $1\leq a.\leq 2q+1$), $\int\sup_{\theta\in\mathcal{N}}\left|f^{(a)}(y_{i}^{+};\theta)\right|dy_{i}<\infty$ , $\int\sup_{\theta\in\mathcal{N}}[\{f^{(a)}(y_{i}^{+};\theta)\}^{2}/f(y_{i}^{+};\theta)]dy_{i}<\infty,$ and $\int\sup_{\theta\in\mathcal{N}}\{\left|l_{i}^{(a)}(\theta)\right|^{j}f^{(a^{\prime})}(y_{i}^{+};\theta)\}dy_{i}<\infty$ for $j=1,2$ and all $a^{\prime}$ with $0\leq a^{\prime}.\leq 2q+1,$ where $dy_{i}$ is short for $dy_{i,1}dy_{i,2},...,dy_{i,T}$

(A6a′)**

For all $a$ with $1\leq a.\leq 2q+1,$ there exist $\xi(a)>2$ such that $\sup_{\theta\in\mathcal{N}}E_{\theta}\left|l_{i}^{(a)}(\theta^{\prime})\right|^{\xi(a)}$ $<\infty$ for all $\theta^{\prime}\in\mathcal{N}$ and all $i\in\{1,...,N\}.$

(A6b′)**

When $a.=2q+1$ there exists $\varpi>0$ and some function $h(.)$ satisfying $E\left|h(y_{i}^{+})\right|^{\xi}$ $<\infty$ for some $\xi>2$ and all $i\in\{1,...,N\}$, such that for any $\theta$ and $\theta^{\prime}$ in $\mathcal{N}$, w.p.1, $\left|l^{(a)}(\theta)-l^{(a)}(\theta^{\prime})\right|\leq\left\|\theta-\theta^{\prime}\right\|^{\varpi}h(y_{i}^{+}),$ where $\left\|\theta\right\|=\left(\mathop{\textstyle\sum}\nolimits_{k=1}^{p}\theta_{k}^{2}\right)^{1/2}.$

(A7′)**

Conditions (B1*′)-(B3′*) given in K2013 hold and w.p.1, $S_{2},...,S_{p}$ are linearly independent. Furthermore, for each $\theta\neq\theta_{\ast}$, $S_{1}(\theta)\neq 0$ with positive probability.

Conditions (A1)-(A7) alluded to in our theorem 4 are equal to conditions (A1*′)-(A7′), respectively, with $p=T+2$, $q=2$ and $\theta_{\ast}=(\rho_{\ast},\delta_{\ast}^{\prime})^{\prime}=(1,\delta_{\ast}^{\prime})^{\prime}$ and with $\theta$ and $l_{i}(\theta)$ replaced by $\theta_{n}=(r_{n},d_{n}^{\prime})^{\prime}$ and $l_{n,i}(\theta_{n}),$ respectively. In this case (B1′)-(B3′*) given in K2013 are equal to conditions (B1)-(B3) given above theorem 1. Conditions (A1)-(A7) are satisfied when the data are i.i.d. and normal but in general have to be checked.

Proof of theorem 4:

We first prove that if (A1)-(A7) hold, the restricted QMLE $\widetilde{\mathcal{\theta}}_{n}=\widetilde{\mathcal{\theta}}_{n,N}$ that satisfies $A\widetilde{\mathcal{\theta}}_{n,N}=a_{N},$ where $A_{1,.}=(1$ $\mathbf{0}^{\prime}),$ is root-$N$ consistent under the parameter sequence $\mathcal{\theta}_{0,n,N}\linebreak$with $\mathcal{\theta}_{0,n,N}\rightarrow\mathcal{\theta}_{\ast}$ and $A\mathcal{\theta}_{0,n,N}=a_{N}$ (so that $A\mathcal{\theta}_{\ast}=a=\lim_{N\rightarrow\infty}a_{N}$). Consistency of $\widetilde{\mathcal{\theta}}_{n,N}$ follows from Theorem 2.1 in NMcF. The proof is similar to that of consistency of $\widehat{\theta}_{n}$ given in K2013wp. Let $\mathcal{\ddot{\theta}}_{n,N}=(a_{N},\mathcal{\ddot{\delta}}_{n,N}^{\prime})^{\prime}$ be the QMLE that maximizes $l_{n}(\theta_{n,N})$ subject to $r_{n,N}=a_{N}$. One can show that $\mathcal{\ddot{\delta}}_{n,N}-\mathcal{\delta}_{n,N}=O_{p}(N^{-1/2})$ by expanding the likeli-hood equations for $\delta$ corresponding to $l_{n}(\theta_{n,N})$ with $r_{n,N}=a_{N}$. Following Davidson and MacKinnon (1993, pp. 276-277), one can similarly show that $\widetilde{\mathcal{\theta}}_{n,N}-\mathcal{\theta}_{0,n,N}=O_{p}(N^{-1/2})$.

Let $\overline{S}_{N,i}(\widetilde{\mathcal{\theta}}_{n,N};\mathcal{\theta}_{0,n,N})=(A\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n,N})\overline{\mathcal{J}}(\widetilde{\mathcal{\theta}}_{n,N})\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n,N})A^{\prime})^{-1/2}A\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n,N})\frac{\partial\widetilde{l}_{n,i}(\widetilde{\mathcal{\theta}}_{n,N})}{\partial\theta_{n}}\linebreak$when $\widetilde{\mathcal{\theta}}_{n,N}\neq\theta_{\ast},$ $\mathcal{\theta}_{0,n,N}\neq\theta_{\ast}$ (for all$\,\sigma^{2}>0$), $A\mathcal{\theta}_{0,n,N}=a_{N},$ and $\mathcal{\theta}_{0,n,N}\in\Theta.$ For any $\{\widetilde{\mathcal{\theta}}_{n,N}\}$ such that $\widetilde{\mathcal{\theta}}_{n,N}\rightarrow\mathcal{\theta}_{\ast},$ the limit of $\overline{QLM}(\widetilde{\mathcal{\theta}}_{n,N};\mathcal{\theta}_{0,n,N})\equiv(N^{-1/2}\mathop{\textstyle\sum}_{i=1}^{N}\overline{S}_{N,i}(\widetilde{\mathcal{\theta}}_{n,N};\mathcal{\theta}_{0,n,N}))^{\prime}\times$ $(N^{-1/2}\mathop{\textstyle\sum}_{i=1}^{N}\overline{S}_{N,i}(\widetilde{\mathcal{\theta}}_{n,N};\mathcal{\theta}_{0,n,N}))$ when $\widetilde{\mathcal{\theta}}_{n,N}(=\widetilde{\mathcal{\theta}}_{n})\rightarrow\mathcal{\theta}_{\ast}$ (for any $\sigma^{2}>0$) and $\mathcal{\theta}_{0,n,N}\rightarrow\mathcal{\theta}_{\ast}$ (for any $\sigma^{2}>0$), while $A\mathcal{\theta}_{0,n,N}=a_{N}$, can be obtained by applying de l’Hôpital’s rule.

Following Davidson and MacKinnon (1993, pp. 276-277), we obtain $N^{1/2}(\widetilde{\mathcal{\theta}}_{n}-\mathcal{\theta}_{0,n,N})\overset{asy}{=}\linebreak-\overline{H}^{-1}(I-A^{\prime}(A\overline{H}^{-1}A^{\prime})^{-1}A\overline{H}^{-1})N^{-1/2}\frac{\partial l_{n}(\mathcal{\theta}_{0,n,N})}{\partial\theta_{n}}$ where $\overline{H}\mathcal{=}\overline{H}(\mathcal{\theta}_{0,n,N})$ and $\mathcal{\theta}_{0,n,N}\neq\theta_{\ast}.$ We also have $N^{-1/2}\frac{\partial l_{n}(\widetilde{\mathcal{\theta}}_{n})}{\partial\theta_{n}}\overset{asy}{=}N^{-1/2}\frac{\partial l_{n}(\mathcal{\theta}_{0,n,N})}{\partial\theta_{n}}+\overline{H}N^{1/2}(\widetilde{\mathcal{\theta}}_{n}-\mathcal{\theta}_{0,n,N}).$ Hence $N^{-1/2}\frac{\partial l_{n}(\widetilde{\mathcal{\theta}}_{n})}{\partial\theta_{n}}\overset{asy}{=}\linebreak A^{\prime}(A\overline{H}^{-1}A^{\prime})^{-1}A\overline{H}^{-1}N^{-1/2}\frac{\partial l_{n}(\mathcal{\theta}_{0,n,N})}{\partial\theta_{n}}$ and (A\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n})\overline{\mathcal{J}}(\widetilde{\mathcal{\theta}}_{n})\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n})A^{\prime})^{-1/2}A\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n})\times$$N^{-1/2}\frac{\partial l_{n}(\widetilde{\mathcal{\theta}}_{n})}{\partial\theta_{n}}\overset{asy}{=}(A\overline{H}^{-1}(\mathcal{\theta}_{0,n,N})E_{\mathcal{\theta}_{0,n,N}}\left(\overline{\mathcal{J}}(\mathcal{\theta}_{0,n,N})\right)\overline{H}^{-1}(\mathcal{\theta}_{0,n,N})A^{\prime})^{-1/2}A\overline{H}^{-1}(\mathcal{\theta}_{0,n,N})\times\linebreak N^{-1/2}\frac{\partial l_{n}(\mathcal{\theta}_{0,n,N})}{\partial\theta_{n}} under the parameter sequence $\mathcal{\theta}_{0,n,N}$ with $\mathcal{\theta}_{0,n,N}\rightarrow\mathcal{\theta}_{\ast}$, $A\mathcal{\theta}_{0,n,N}=a_{N}$ and $\mathcal{\theta}_{0,n,N}\neq\theta_{\ast},$ where we have used that p$\lim_{N\rightarrow\infty}[\overline{\mathcal{J}}(\widetilde{\mathcal{\theta}}_{n})-E_{\mathcal{\theta}_{0,n,N}}\left(\overline{\mathcal{J}}(\mathcal{\theta}_{0,n,N})\right)]=0$ under $\mathcal{\theta}_{0,n,N}\rightarrow\mathcal{\theta}_{\ast}$ and the fact that $\overline{\mathcal{J}}(\widetilde{\mathcal{\theta}}_{n})$ and $E_{\mathcal{\theta}_{0,n,N}}\left(\overline{\mathcal{J}}(\mathcal{\theta}_{0,n,N})\right)$ are positive definite when $\widetilde{\mathcal{\theta}}_{n,N}\neq\theta_{\ast}$ and $\mathcal{\theta}_{0,n,N}\neq\theta_{\ast}$.

Let $S_{N,i}(\mathcal{\theta}_{n})=A\overline{H}^{-1}(\mathcal{\theta}_{n})\frac{\partial l_{n,i}(\mathcal{\theta}_{n})}{\partial\theta_{n}}$, $S_{N,i}=S_{N,i}(\mathcal{\theta}_{0,n,N})$ and $\mathcal{\theta}_{0,n,N}\rightarrow\mathcal{\theta}_{\ast}\in\Theta$ with $A\mathcal{\theta}_{0,n,N}=a_{N}$ and $\mathcal{\theta}_{0,n,N}\neq\theta_{\ast}.$ Under conditions (A5)-(A6) and similarly to Bottai (2003), $E_{\mathcal{\theta}_{0,n,N}}(S_{N,i})=0,$ $Var_{\mathcal{\theta}_{0,n,N}}(S_{N,i})=A\overline{H}^{-1}(\mathcal{\theta}_{0,n,N})E_{\mathcal{\theta}_{0,n,N}}(\mathcal{J}_{i}(\mathcal{\theta}_{0,n,N}))\overline{H}^{-1}(\mathcal{\theta}_{0,n,N})A^{\prime}$ and$\linebreak\sup_{i}\sup_{\mathcal{\theta}_{n}\in\mathcal{N}}E_{\mathcal{\theta}_{0,n,N}}(\left|\lambda^{\prime}S_{N,i}\right|^{\varsigma})<\infty$ for some $\varsigma>2$ and for all $\lambda\in\mathbb{R}^{J}$ where $\mathcal{N}\subset\Theta$ is an open neighbourhood around $\mathcal{\theta}_{\ast}$. We also have $N^{-1}\mathop{\textstyle\sum}_{i=1}^{N}Var_{\mathcal{\theta}_{0,n,N}}(\lambda^{\prime}S_{N,i})>0$ uniformly in $N$ for all $\lambda\in\mathbb{R}^{J}\backslash\{\mathbf{0}\}.$ Thus the Lyapunov conditions are satisfied and by (a multivariate version of) Lindeberg’s CLT for triangular arrays, $(\mathop{\textstyle\sum}_{i=1}^{N}Var_{\mathcal{\theta}_{0,n,N}}(S_{N,i}))^{-1/2}\times\linebreak\mathop{\textstyle\sum}_{i=1}^{N}S_{N,i}$ converges under the parameter sequence $\mathcal{\theta}_{0,n,N}$ to $N(0,I_{J}).$ It follows that $(A\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n})\overline{\mathcal{J}}(\widetilde{\mathcal{\theta}}_{n})\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n})A^{\prime})^{-1/2}A\overline{H}^{-1}(\widetilde{\mathcal{\theta}}_{n})N^{-1/2}\frac{\partial l_{n}(\widetilde{\mathcal{\theta}}_{n})}{\partial\theta_{n}}\overset{d}{\rightarrow}N(0,I_{J})$ and $QLM(\widetilde{\theta}_{n})\overset{d}{\rightarrow}\chi^{2}(J)$ under the parameter sequence $\mathcal{\theta}_{0,n,N}$ with $\mathcal{\theta}_{0,n,N}\rightarrow\mathcal{\theta}_{\ast}$, $A\mathcal{\theta}_{0,n,N}=a_{N}$ and $\mathcal{\theta}_{0,n,N}\neq\theta_{\ast}.$

Next let $\widetilde{S}_{N,i}(\mathcal{\theta}_{n})=A\widetilde{\mathcal{H}}^{-1}(\mathcal{\theta}_{n})\widetilde{S}(\mathcal{\theta}_{n})$ and $\widetilde{S}_{N,i}=\widetilde{S}_{N,i}(\mathcal{\theta}_{\ast}).$ Under condition (A7) and similarly to Bottai (2003), we can also show that $(\mathop{\textstyle\sum}_{i=1}^{N}Var_{\mathcal{\theta}_{0,n,N}}(\widetilde{S}_{N,i}))^{-1/2}\mathop{\textstyle\sum}_{i=1}^{N}\widetilde{S}_{N,i}\overset{d}{\rightarrow}N(0,I_{J})$ and $QLM(\mathcal{\theta}_{\ast})\overset{d}{\rightarrow}\chi^{2}(J)$ when $\mathcal{\theta}_{0,n,N}=\mathcal{\theta}_{\ast}.$ We conclude that$\linebreak\lim_{N\rightarrow\infty}\sup_{\theta_{0}\in\mathcal{N}}\left|\Pr_{\theta_{0}}\{QLM(\widetilde{\theta}_{n})>\chi_{J,\alpha}^{2}\}-\alpha\right|=0.$ $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof of theorem 5:

We first derive $\frac{\partial l}{\partial\theta_{0}},$ where $l=l_{FE}(\theta_{0})$ with $\theta_{0}=(\rho,\widetilde{\sigma}_{v}^{2},\zeta^{\prime})^{\prime}$, by following the working paper version of K2013.

Let $g(y_{i}|y_{i,1},\theta_{0})$ denote the normal p.d.f. of $y_{i}$ given $y_{i,1},$ let $D$ denote a $(T-2)\times(T-1)$ first-difference matrix, let $\widetilde{y}_{i}=y_{i}-y_{i,1}\iota,$ $\widetilde{y}_{i,-1}=y_{i,-1}-y_{i,1}\iota,$ $w_{i}=\widetilde{y}_{i}-\rho\widetilde{y}_{i,-1}$ and $d=\Psi^{-1}\iota/\iota^{\prime}\Psi^{-1}\iota.$ Note that $g(y_{i}|y_{i,1},\theta_{0})=g(w_{i}|y_{i,1},\theta_{0}),$ $E(Dw_{i}w_{i}^{\prime}d|y_{i,1})=0$ and $d^{\prime}\iota=1.$ To derive $\frac{\partial l}{\partial\rho}$, it is useful to factorise $g(w_{i}|y_{i,1},\theta_{0})=g(Dw_{i}$ $d^{\prime}w_{i}|y_{i,1},\theta_{0})$ as

[TABLE]

Now let $\sigma_{u}^{2}=Var(d^{\prime}w_{i}|y_{i,1})=Var(d^{\prime}u_{i})=1/\iota^{\prime}\Psi^{-1}\iota+\widetilde{\sigma}_{v}^{2}.$ Then we can parametrize$\ln g(d^{\prime}w_{i}|y_{i,1},\theta_{0})$ more parsimoniously as

[TABLE]

Finally, given that $l=l_{FE}(\theta_{0})=\sum_{i}\ln g(Dw_{i}|y_{i,1},\rho,\zeta)+\sum_{i}\ln g(d^{\prime}w_{i}|y_{i,1},\theta_{0}),$ we obtain

[TABLE]

Using the reparametrization given above theorem 1, i.e., $r=r_{n},$ $s_{t}^{2}=rs_{t,n}^{2},$ $t=2,...,T$, so that $z=\rho z_{n},$ and $\widetilde{s}_{v}^{2}/s_{2}^{2}=\widetilde{s}_{v,n}^{2}+(1-r),$ we find that $\sigma_{u}^{2}=\sigma_{u}^{2}(\theta_{n})=r/(\iota^{\prime}\Psi_{n}^{-1}\iota)+s_{2,n}^{2}r(\widetilde{s}_{v,n}^{2}+(1-r))$ where $\Psi_{n}=diag(s_{2,n}^{2},...,s_{T,n}^{2}).\vskip 3.0pt plus 1.0pt minus 1.0pt$

Let $l_{n}=l_{n}(\theta_{n}).$ Then from $l_{n}=l(\theta(\theta_{n}))$ and $\frac{\partial l}{\partial\rho}$ and $\frac{\partial l}{\partial\sigma_{u}^{2}}$ in (23) we obtain that

[TABLE]

where $\frac{\partial\sigma_{u}^{2}}{\partial r}=\frac{\partial\sigma_{u}^{2}(\theta_{n})}{\partial r}.\vskip 3.0pt plus 1.0pt minus 1.0pt$

We have assumed that $\mathcal{\theta}_{0}=\mathcal{\theta}_{\ast}$ and TSH. Letting $\widetilde{\sigma}_{u}^{2}=r(1+(T-1)(1-r))$ and $\theta_{0,n,N}=(r,\widetilde{\sigma}_{v,n}^{2},\zeta_{n}^{\prime})^{\prime}=(a_{N},$ $0,$ $\sigma^{2}\iota^{\prime})^{\prime}$, replacing $s_{2,n}^{2},...,s_{T,n}^{2}$ and $\widetilde{s}_{v,n}^{2}$ by elements of $\widetilde{\theta}_{n},$ which are $\sqrt{N}$-consistent estimators, noting that $D^{\prime}(DD^{\prime})^{-1}D=I_{T-1}-\iota\iota^{\prime}/(T-1)\equiv Q,$ and using a Taylor expansion of $\frac{\partial l_{n}(\theta_{n})}{\partial r}|_{\widetilde{\theta}_{n}}$, we obtain

[TABLE]

where $\overline{H}_{1,.}N^{1/2}(\widetilde{\theta}_{n}-\theta_{0,n,N})=O_{p}(1)$ and

[TABLE]

with $\frac{\partial\widetilde{\sigma}_{u}^{2}}{\partial r}=1+(T-1)(2(1-r)-1).$

We can write $\widetilde{y}_{i}-r\widetilde{y}_{i,-1}=\varepsilon_{i}+(1-r)\widetilde{y}_{i,-1}$ and $\widetilde{y}_{i,-1}=B\varepsilon_{i}$ where $B$ is a $(T-1)\times(T-1)$ matrix with $B_{i,j}=\rho^{i-j-1}=1$ for $i>j$ and $B_{i,j}=0$ for $i\leq j.$ Let $\phi=1-r.$ From $\frac{\partial l_{n}}{\partial\rho_{n}}|_{\theta_{\ast}}=0$ (cf. Theorem 1) we deduce that the RHS of (A) contains the common factor $\phi$ at least once. Note that $tr(B^{\prime}QB)=\frac{1}{6}(T-2)T,$ $tr(QB)=-\frac{1}{2}(T-2),\ \iota^{\prime}B^{\prime}B\iota=\frac{1}{6}(T-2)(T-1)(2T-3)$ and $\iota^{\prime}B\iota=\frac{1}{2}(T-2)(T-1).$ Then we obtain

[TABLE]

Moreover,

[TABLE]

Thus $\frac{1}{N}\frac{\partial l_{n}(\theta_{n})}{\partial r}|_{\theta_{0,n,N}}$ contains the common factor $\phi$ exactly once and we can divide $\frac{2\widetilde{\sigma}_{u}^{4}}{N}\frac{\partial l_{n}}{\partial r}|_{\theta_{0,n,N}}$ by $\phi.$

Let $S$ be a $(T-1)\times(T-1)$ diagonal matrix with $S_{1,1}=\phi^{-1}$ and $S_{i,i}=1$ for $i\geq 2$. Using reasoning similar to that used in Davidson and MacKinnon (1993), cf. the proof of theorem 4, we obtain that

[TABLE]

Note that all elements of the vector $N^{-1/2}S\frac{\partial l_{n}(\theta_{n})}{\partial\theta_{n}}|_{\widetilde{\theta}_{n}}$ apart from its first element are zero.

Using (26) and $\phi=1-r=\kappa/\sqrt[4]{N}$, we also have

[TABLE]

Next we consider $E(\frac{\widetilde{\sigma}_{u}^{4}}{N}\frac{\partial l_{n}}{\partial\widetilde{s}_{v,n}^{2}}|_{\theta_{0,n,N}}).$

[TABLE]

Hence

[TABLE]

Finally, we consider $E(\frac{\partial l_{n}}{\partial z_{n}}|_{\theta_{0,n,N}}).$ From the working paper version of K2013 we have

[TABLE]

and

[TABLE]

Note that

[TABLE]

and

[TABLE]

It follows that

[TABLE]

Simplifying yields

[TABLE]

and for $k=2,...,T-1,$

[TABLE]

Hence $\sqrt{N}E(\frac{\sigma^{2}\widetilde{\sigma}_{u}^{4}}{rN}\frac{\partial l_{n}}{\partial s_{2,n}^{2}}|_{\theta_{0,n,N}})=\frac{1}{12}\kappa^{2}\left(12T-18\right)+O(N^{-1/4})$ and for k=2,...,T-1,$$\sqrt{N}E(\frac{\sigma^{2}\widetilde{\sigma}_{u}^{4}}{rN}\frac{\partial l_{n}}{\partial s_{k+1,n}^{2}}|_{\theta_{0,n,N}})=\frac{1}{12}\kappa^{2}\left(12T-18k\right)+O(N^{-1/4}).

Let $c_{1}$ be a ($T-1$)-vector such that $c_{1}\kappa^{2}=\lim_{N\rightarrow\infty}N^{-1/2}E(S\frac{\partial l_{n}(\theta_{n})}{\partial\theta_{n}}|_{\theta_{0,n,N}}).$

Then we have $SN^{-1/2}\frac{\partial l_{n}(\theta_{n})}{\partial\theta_{n}}|_{\theta_{0,n,N}}\overset{d}{\rightarrow}N(c_{1}\kappa^{2},$ p$\lim_{N\rightarrow\infty}S\overline{\mathcal{J}}^{c}S)$ and QLM_{FE}^{c}(\widetilde{\mathcal{\theta}}_{n})\overset{d}{\rightarrow}$$\chi^{2}(p$\lim_{N\rightarrow\infty}c_{1}^{\prime}S^{-1}\overline{H}^{-1}A^{\prime}(A\overline{H}^{-1}\overline{\mathcal{J}}^{c}\overline{H}^{-1}A^{\prime})^{-1}A\overline{H}^{-1}S^{-1}c_{1}\kappa^{4},1).$ $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof of theorem 6:

Note that

[TABLE]

Using results in the proof of Theorem 5, it follows that

[TABLE]

Similarly to the proof of Theorem 5, we also have

[TABLE]

and

[TABLE]

Let $\widetilde{S}$ be a $3\times 3$ diagonal matrix with $\widetilde{S}_{1,1}=\phi^{-1},$ $\widetilde{S}_{2,2}=1$ and $\widetilde{S}_{3,3}=\sigma^{2}.$ Then weobtain c_{3}\equiv\lim_{N\rightarrow\infty}N^{-1/2}E(\widetilde{S}\frac{\partial l_{n}(\theta_{n})}{\partial\theta_{n}}|_{\theta_{0,n,N}})=\left[\begin{array}[]{c}\frac{1}{6}T\left(T-1\right)\left(T^{2}-T+1\right)\\ \frac{1}{12}T\left(2T-1\right)\left(T-1\right)\\ \frac{1}{4}T\left(T-1\right)\end{array}\right]\kappa^{2}. We alsohave p\lim_{N\rightarrow\infty}\widetilde{S}\overline{H}\widetilde{S}=\left[\begin{array}[]{ccc}\frac{1}{2}T\left(T-1\right)\left(-T+T^{2}+1\right)&\frac{1}{6}T\left(2T^{2}-3T+1\right)&\frac{1}{2}T\left(T-1\right)\\ \frac{1}{6}T\left(2T^{2}-3T+1\right)&\frac{1}{2}(T-1)^{2}&\frac{1}{2}(T-1)\\ \frac{1}{2}T\left(T-1\right)&\frac{1}{2}(T-1)&\frac{1}{2}(T-1)\end{array}\right].\vskip 6.0pt plus 2.0pt minus 2.0pt

Using $tr(B^{\prime}B)=\frac{1}{2}(T-2)(T-1),$ $tr(B^{\prime}BB^{\prime}B)=\frac{1}{6}(T-2)(T-1)(T^{2}-3T+3),$ $tr(QBQB)=-\frac{1}{12}(T-2)(T-6)$ and $tr(B^{\prime}BQB)=-\frac{1}{24}(T-2)T(T+1),$ we obtain

p\lim_{N\rightarrow\infty}\widetilde{S}\overline{\mathcal{J}}^{c}\widetilde{S}=\left[\begin{array}[]{ccc}\frac{1}{3}T\left(T-1\right)\left(-T+T^{2}+1\right)&\frac{1}{6}T\left(2T^{2}-3T+1\right)&\frac{1}{2}T\left(T-1\right)\\ \frac{1}{6}T\left(2T^{2}-3T+1\right)&\frac{1}{2}(T-1)^{2}&\frac{1}{2}(T-1)\\ \frac{1}{2}T\left(T-1\right)&\frac{1}{2}(T-1)&\frac{1}{2}(T-1)\end{array}\right].\vskip 6.0pt plus 2.0pt minus 2.0pt Finally, we have $\widetilde{S}N^{-1/2}\frac{\partial l_{n}(\theta_{n})}{\partial\theta_{n}}|_{\theta_{0,n,N}}\overset{d}{\rightarrow}N(c_{3},$ p$\lim_{N\rightarrow\infty}\widetilde{S}\overline{\mathcal{J}}^{c}\widetilde{S})$ and $QLM_{FE}^{c}(\widetilde{\mathcal{\theta}}_{n})\overset{d}{\rightarrow}\chi^{2}(\gamma,1)$ with $\gamma=$ p$\lim_{N\rightarrow\infty}c_{3}^{\prime}\widetilde{S}^{-1}\overline{H}^{-1}A^{\prime}(A\overline{H}^{-1}\overline{\mathcal{J}}^{c}\overline{H}^{-1}A^{\prime})^{-1}A\overline{H}^{-1}\widetilde{S}^{-1}c_{3}=\frac{(2T-3)T(T-1)(T-2)}{72}\kappa^{4}.$ $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

To prove theorem 7 we will make use of some intermediate results. It will also be useful to introduce some additional notation. For $n\in{\mathbb{N}}$ and $a,b,c\in{\mathbb{R}}$, $\Delta_{n}(a,b,c)$ denotes the $(n\times n)$ tridiagonal matrix of the following form

[TABLE]

Lemma 1

Let $T\geq 2$ and $G_{T}=\Delta_{T}(1,2,-1)$. Then the inverse $G_{T}^{-1}$ has the following simple structure:

[TABLE]

**Proof of lemma 1: **It is straightforward to verify that $G_{T}G_{T}^{-1}=I_{T}$. $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Let $D_{u,T}$ be the $(T^{2}\times\frac{T(T+1)}{2})$ matrix such that $\mathrm{vec}(A)=D_{u,T}\mathrm{vech}(A)$ for any $(T\times T)$ matrix $A.$ $D_{u,T}$ is a so-called duplication matrix as defined in Ch. 3 of Magnus and Neudecker (2007). Let $D_{u,T}^{+}=(D_{u,T}^{\prime}D_{u,T})^{-1}D_{u,T}^{\prime}$ be the Penrose-Moore inverse of $D_{u,T}.$

Proposition 1

Let $T\geq 2$ and $M_{T}=2D_{u,T}^{+}(G_{T}\otimes G_{T})(D_{u,T}^{+})^{\prime}$. Then $M_{T}$ is invertible and its inverse $M_{T}^{-1}$ has the form

[TABLE]

**Proof of proposition 1: **This follows from application of equation (13) in Theorem 13 in Ch. 3 of Magnus and Neudecker (2007). $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 2

Let $T\geq 2$, $G_{T}=\Delta_{T}(1,2,-1),$ $H_{T}=\Delta_{T}(0,-2,1),$

[TABLE]

[TABLE]

where $e_{1}$ is the first standard column basis vector of ${\mathbb{R}}^{T}.$

Then we have $G_{T}^{-1}H_{T}=F_{T}-I_{T}$ and $-6G_{T}^{-1}A_{T}=2(T+1)F_{T}-6G_{T}^{-1}+(T+1)I_{T}.$

**Proof of lemma 2: **Verification of both claims is straightforward. $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Let $G_{T}=\Delta_{T}(1,2,-1).$ Let

[TABLE]

be a $p_{T}\times\frac{1}{2}T(T+1)$ matrix where $p_{T}=\frac{1}{2}T(T+1)-2$ and $g_{T}$ is a $p_{T}$-vector such that

[TABLE]

and let

[TABLE]

where $M_{T}^{-1}$ is given in (28) and

[TABLE]

Lemma 3

Let $T\geq 2,$ let $A_{T}=\Delta_{T}(-\frac{T-1}{2},-\frac{T-2}{3},\frac{T+1}{6})$ and let $\overline{P}_{T}$ be defined by (31), (32), (30), (29) and (28). Then we have

[TABLE]

where $e_{1}$ is the first standard column basis vector of ${\mathbb{R}}^{T(T+1)/2}$.

Proof of lemma 3:

First note that $P_{T}\mathrm{vech}(A_{T})=\frac{T+1}{6}g_{T}+(0_{p_{T},2}$ $I_{p_{T}})\mathrm{vech}(A_{T})=(0_{p_{T},2}$ $I_{p_{T}})\mathrm{vech}(I_{T}).$

Next we need to show that $\begin{pmatrix}1&0&0_{1,p_{T}}\end{pmatrix}M_{T}^{-1}\mathrm{vech}(A_{T})=0.$ For this it suffices to show that

[TABLE]

The LHS of (34) equals $(-\frac{T-2}{3})\frac{1}{6}T(T+1)(2T+1)+\frac{T-2}{3}T^{2}-\frac{T-1}{2}T^{2}+\frac{T+1}{6}(2T(T-1)+2(T-1)(T-2)+...+2\cdot 2\cdot 1)=-\frac{1}{18}(T-2)T(T+1)(2T+1)-\frac{T^{2}}{6}(T+1)+\frac{T+1}{3}\left(\sum_{t=1}^{T-1}t^{2}+\sum_{t=1}^{T-1}t\right)=-\frac{1}{18}(T-2)T(T+1)(2T+1)-\frac{T^{2}}{6}(T+1)+\frac{T+1}{3}(\frac{1}{6}(T-1)T(2T-1)+\frac{1}{2}(T-1)T)=0.$

Finally, we need to show that $\begin{pmatrix}0&1&-g_{T}^{\prime}\end{pmatrix}M_{T}^{-1}\mathrm{vech}(A_{T})=0.$ Note that $\begin{pmatrix}0&1&-g_{T}^{\prime}\end{pmatrix}^{\prime}=\mathrm{vech}(H_{T})$ where $H_{T}=\Delta_{T}(0,-2,1)$ and that $(\mathrm{vech}(H_{T}))^{\prime}M_{T}^{-1}\mathrm{vech}(A_{T})=\frac{1}{2}(\mathrm{vec}(H_{T}))^{\prime}\times$ $(G_{T}^{-1}\otimes G_{T}^{-1})\mathrm{vec}(A_{T})=\frac{1}{2}tr(G_{T}^{-1}A_{T}G_{T}^{-1}H_{T}).$ The last equality follows from application of Theorem 3 in Ch. 2 of Magnus and Neudecker (2007). From lemma 2 and additivity of the trace, it follows straightforwardly that $\frac{1}{2}tr(G_{T}^{-1}A_{T}G_{T}^{-1}H_{T})=0.$ $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 4

Let $T\geq 2,$ $G_{T}=\Delta_{T}(1,2,-1)$ and $A_{T}=\Delta_{T}(-\frac{T-1}{2},-\frac{T-2}{3},\frac{T+1}{6}).$ Then we have

[TABLE]

**Proof of lemma 4: **The claim in (35) follows straightforwardly from $(\mathrm{vec}(A_{T}))^{\prime}(G_{T}^{-1}\otimes G_{T}^{-1})\mathrm{vec}(A_{T})=tr(G_{T}^{-1}A_{T}G_{T}^{-1}A_{T})$, lemma 2 and additivity of the trace. $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Lemma 5

Let $T\geq 2$, $G_{T}=\Delta_{T}(1,2,-1)$ and $M_{T}=2D_{u,T}^{+}(G_{T}\otimes G_{T})(D_{u,T}^{+})^{\prime}.$ Let $P_{T}$ be defined by (29) and (30) and let $R_{T}=P_{T}M_{T}P_{T}^{\prime}$. Let $w_{T}$ be the column vector of size $\frac{T(T+1)}{2}-2$ defined by $(1,0,w_{T}^{\prime})^{\prime}=\mathrm{vech}(I_{T}).$ Then $R_{T}$ is invertible and we have

[TABLE]

**Proof of lemma 5: **First note that

[TABLE]

where $\overline{P}_{T}$ is defined by (31) and (32), $M_{0,T}$ is an $(2\times 2)$ matrix and $0_{p,q}$ denotes the zero matrix of size $(p\times q)$. From (37) and invertibility of $\overline{P}_{T}$ and $M_{T}$, we obtain

[TABLE]

and if we choose $\widehat{w}_{T}=(0,0,w_{T}^{\prime})^{\prime},$ we obtain

[TABLE]

which we are after. Next, by using the formula for $M_{T}^{-1}$ given in Proposition 1, we obtain

[TABLE]

with $z_{T}=D_{u,T}(\overline{P}_{T})^{-1}\widehat{w}_{T}$.

Lemma 3 implies $\mathrm{vech}(A_{T})=(\overline{P}_{T})^{-1}\widehat{w}_{T}$ where $A_{T}=\Delta_{T}(-\frac{T-1}{2},-\frac{T-2}{3},\frac{T+1}{6})$ and hence $\mathrm{vec}(A_{T})=D_{u,T}(\overline{P}_{T})^{-1}\widehat{w}_{T}=z_{T}.$ From this, equation (38) and lemma 4, it follows that $w_{T}^{\prime}R_{T}^{-1}w_{T}=\frac{1}{2}\mathrm{vec}(A_{T})^{\top}(G_{T}^{-1}\otimes G_{T}^{-1})\mathrm{vec}(A_{T})=\frac{(2T-1)(T+1)T(T-1)}{72}$. $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof of theorem 7:

When the data are i.i.d. and normal, $\theta_{0}=\theta_{\ast}$ and when $r=1-\kappa/\sqrt[4]{N}$,$Evech\left(D_{r}(\Delta y_{i}(\Delta y_{i})^{\prime})D_{r}^{\prime}\right)=\sigma^{2}vech\left(D_{r}D_{r}^{\prime}\right)$ and $\lim_{N\rightarrow\infty}N^{1/2}E(m(r))=\sigma^{2}\kappa^{2}\times\linebreak vech\left(diag(0,1,1,...,1,1\right))\equiv\sigma^{2}\kappa^{2}c_{1}.$

Let $G=D_{1}D_{1}^{\prime}.$ We have $E(vech(D_{1}\varepsilon_{i}\varepsilon_{i}^{\prime}D_{1}^{\prime}-\sigma^{2}G)vech(D_{1}\varepsilon_{i}\varepsilon_{i}^{\prime}D_{1}^{\prime}-\sigma^{2}G)^{\prime})=2\sigma^{4}D_{u}^{+}(G\otimes G)(D_{u}^{+})^{\prime}$ with $D_{u}^{+}=(D_{u}^{\prime}D_{u})^{-1}D_{u}^{\prime}$ where $D_{u}$ is the so-called duplication matrix such that $vec(A)=D_{u}vech(A)$ for any $(T-1)\times(T-1)$ matrix $A,$ see Magnus and Neudecker (2007). It follows that p$\lim_{N\rightarrow\infty}\widehat{V}_{mm}(\rho)=2\sigma^{4}PD_{u}^{+}(G\otimes G)(D_{u}^{+})^{\prime}P^{\prime}$ and $GMM-AR(\rho)\overset{d}{\rightarrow}\chi^{2}(\gamma,p)$ where $\gamma=c_{1}^{\prime}(2PD_{u}^{+}(G\otimes G)(D_{u}^{+})^{\prime}P^{\prime})^{-1}c_{1}\kappa^{4}$ and $p=\dim(m(r))=\frac{1}{2}T(T-1)-2.$

Finally, we show that $\gamma=\frac{(2T-3)T(T-1)(T-2)}{72}\kappa^{4}.$ This follows immediately fromlemma 5 upon replacing $T$ by $T-1$ (Note $G=G_{T-1},$ $D_{u}^{+}=D_{u,T-1}^{+},$ $P=P_{T-1}$ and $c_{1}=w_{T-1}$). $\square\vskip 6.0pt plus 2.0pt minus 2.0pt$

Proof of theorem 8:

Let $w$ be a $p$-vector with $w^{\prime}w=1.$ When the data are i.i.d. and normal, and $\theta_{0}=\theta_{\ast}$ (so that TSH holds), the large sample distribution of the GMM-AR test statistic that uses $w^{\prime}m(\rho)$ for testing $H_{0}:$ $\rho=1-\kappa/\sqrt[4]{N}$ is given by $\chi^{2}(\gamma_{w},1)$ where $\gamma_{w}=c_{1}^{\prime}w(2w^{\prime}PD_{u}^{+}(G\otimes G)(D_{u}^{+})^{\prime}P^{\prime}w)^{-1}w^{\prime}c_{1}\kappa^{4}$ with $c_{1}=vech\left(diag(0,1,1,...,1,1\right))$ (cf. the proof of Theorem 7). The combination $w$ that maximizes the local power of the GMM-AR test is the one that maximizes the value of $\gamma_{w}.$

The maximal value of $\gamma_{w}$ is the largest root of the following eigenvalue problem

[TABLE]

and the power maximizing combination $w$ is given by the eigenvector associated with this largest root. The largest root is equal to $\gamma=c_{1}^{\prime}(2PD_{u}^{+}(G\otimes G)(D_{u}^{+})^{\prime}P^{\prime})^{-1}c_{1}=\frac{(2T-3)T(T-1)(T-2)}{72}.$ $\square\vskip-7.22743pt$