Asymptotic normality of the time-domain generalized least squares estimator for linear regression models

TL;DR

This paper proves that the time-domain generalized least squares estimator is asymptotically normal for a broad class of error dependence models in linear regression, extending previous results limited to specific structures.

Contribution

It establishes the asymptotic normality of the time-domain IGLS estimator under general dependence models, broadening the scope beyond prior frequency domain results.

Findings

Asymptotic normality of time-domain IGLS estimator proven for general dependence models

Extends previous results limited to specific error structures

Provides theoretical foundation for inference with misspecified GLS in time domain

Abstract

In linear models, the generalized least squares (GLS) estimator is applicable when the structure of the error dependence is known. When it is unknown, such structure must be approximated and estimated in a manner that may lead to misspecification. The large-sample analysis of incorrectly-specified GLS (IGLS) estimators requires careful asymptotic manipulations. When performing estimation in the frequency domain, the asymptotic normality of the IGLS estimator, under the so-called Grenander assumptions, has been proved for a broad class of error dependence models. Under the same assumptions, asymptotic normality results for the time-domain IGLS estimator are only available for a limited class of error structures. We prove that the time-domain IGLS estimator is asymptotically normal for a general class of dependence models.

Equations

$$Y_t={\mathbf{x}}_t^{\top }\mathbf{\beta }+U_t\text{,}$$

$${\mathbf{y}}_T={\mathbf{X}}_T\mathbf{\beta }+{\mathbf{u}}_T\text{,}$$

$$U_t=\sum _{i=-\infty }^{\infty }{\theta }_i{E}_{t-i}\text{,}$$

$${\tilde{\mathbf{\beta }}}_T\left({\mathbf{\Sigma }}_T\right)={\left({\mathbf{X}}_T^{\top }{\text{Σ}}_T^{-1}{\mathbf{X}}_T\right)}^{-1}{\mathbf{X}}_T^{\top }{\mathbf{\Sigma }}_T^{-1}{\mathbf{y}}_T\text{,}$$

$$T^{1/2}\left[{\tilde{\mathbf{\beta }}}_T\left({\mathbf{\Sigma }}_T\right)-\mathbf{\beta }\right]\stackrel{\mathcal{L}}{⟶}\text{N}\left(0,{\mathbf{C}}_u\right)\text{,}$$

$${\tilde{\mathbf{\beta }}}_T\left({\mathbf{\Lambda }}_T\right)={\left({\mathbf{X}}_T^{\top }{\text{Λ}}_T^{-1}{\mathbf{X}}_T\right)}^{-1}{\mathbf{X}}_T^{\top }{\mathbf{\Lambda }}_T^{-1}{\mathbf{y}}_T\text{.}$$

$$U_t=\sum _{i=1}^{\infty }{\alpha }_i{U}_{t-i}+E_t\text{,}$$

$$V_t=\sum _{i=1}^N{\kappa }_i{V}_{t-i}+E_t\text{,}$$

$${\mathbf{S}}_T\left[{\tilde{\mathbf{\beta }}}_T\left({\mathbf{\Lambda }}_T\right)-\mathbf{\beta }\right]\stackrel{\mathcal{L}}{⟶}\text{N}\left(0,{\mathbf{C}}_v\right)\text{,}$$

$${\mathbf{C}}_v=\underset{T\to \infty }{\lim }{\left({\mathbf{Z}}_T^{\top }{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T\right)}^{-1}{\mathbf{Z}}_T^{\top }{\mathbf{\Lambda }}_T^{-1}{\mathbf{\Sigma }}_T{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T{\left({\mathbf{Z}}_T^{\top }{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T\right)}^{-1}\text{.}$$

$${\mathbf{C}}_v=2\pi {\left[{\int }_{-\pi }^{\pi }\frac{1}{f_v\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right)\right]}^{-1}{\int }_{-\pi }^{\pi }\frac{f_u\left(\omega \right)}{f_v^2\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right){\left[{\int }_{-\pi }^{\pi }\frac{1}{f_v\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right)\right]}^{-1}\text{.}$$

$$f_v\left(\omega \right)={\left(2\pi \right)}^{-1}\sum _{i=-\infty }^{\infty }{\eta }_i{e}^{-\iota i\omega }\text{,}$$

$$\mathbb{E}\left(\mathbf{v}_{T}\mathbf{v}_{T}^{\top}\right)=\bm{\Lambda}_{T}=\left[\begin{array}[]{ccccc}\eta_{0}&\eta_{1}&\eta_{2}&\cdots&\eta_{T-1}\\ \eta_{1}&\eta_{0}&\eta_{1}\\ \eta_{2}&\eta_{1}&\eta_{0}&&\vdots\\ \vdots&&&\ddots\\ \eta_{T-1}&&\cdots&&\eta_{0}\end{array}\right]\text{.}$$

$${\mathbf{J}}_{\mathbf{xx}}\left(\omega \right)=\frac{1}{2\pi T}\left(\sum _{t=1}^T{\mathbf{x}}_t{e}^{\iota t\omega }\right)\left(\sum _{t=1}^T{\mathbf{x}}_t^{\top }{e}^{-\iota t\omega }\right)$$

$${\mathbf{J}}_{\mathbf{x}Y}\left(\omega \right)=\frac{1}{2\pi T}\left(\sum _{t=1}^T{\mathbf{x}}_t{e}^{\iota t\omega }\right)\left(\sum _{t=1}^T{Y}_t{e}^{-\iota t\omega }\right)$$

$$\bar{\mathbf{\beta }}\left(f_v\right)={\left[\sum _{t=1}^T{f}_v^{-1}\left(\frac{2\pi t}{T}\right){\mathbf{J}}_{\mathbf{xx}}\left(\frac{2\pi t}{T}\right)\right]}^{-1}\sum _{t=1}^T{f}_v^{-1}\left(\frac{2\pi t}{T}\right){\mathbf{J}}_{\mathbf{x}Y}\left(\frac{2\pi t}{T}\right)\text{.}$$

$$\mathbb{E}\left(E_t|{\mathcal{F}}_{t-1}\right)=\mathbb{E}\left(E_t^2-\mathbb{E}\left(E_t^2\right)|{\mathcal{F}}_{t-1}\right)=0$$

$$\underset{\delta \to \infty }{\lim }\underset{t\in \mathbb{Z}}{\sup }{\int }_{|e|>\delta }{e}^2\text{d}{F}_{E_t}\left(e\right)=0\text{.}$$

$${\mathbf{S}}_T\left[\bar{\mathbf{\beta }}\left(f_v\right)-\mathbf{\beta }\right]\stackrel{\mathcal{L}}{⟶}\text{N}\left(0,{\mathbf{C}}_v\right)\text{,}$$

$${\Vert \mathbf{A}\Vert }_{\text{op}}=\sup \left\{{\Vert \mathbf{A}\mathbf{x}\Vert }_2/{\Vert \mathbf{x}\Vert }_2:\mathbf{x}\in {\mathbb{R}}^n\backslash \left\{0\right\}\right\}$$

$${\Vert \mathbf{A}\Vert }_1=\underset{j\in \left[n\right]}{\max }\sum _{i=1}^m\left|a_{ij}\right|\text{ and }{\Vert \mathbf{A}\Vert }_{\infty }=\underset{i\in \left[m\right]}{\max }\sum _{j=1}^n\left|a_{ij}\right|\text{,}$$

$$U_t=\sum _{i=-\infty }^{\infty }{\theta }_i{E}_{t-i}\text{,}$$

$$\mathbb{E}\left(E_t\right)=0\text{, }\text{var}\left(E_t\right)={\sigma }^2<\infty \text{, }\sum _{i=-\infty }^{\infty }\left|{\theta }_i\right|<\infty \text{, and }0<\left|\sum _{i=-\infty }^{\infty }{\theta }_i{e}^{-\iota i\omega }\right|\text{.}$$

$$\underset{\delta \to \infty }{\lim }\underset{t\in \mathbb{Z}}{\sup }{\int }_{|e|>\delta }{e}^2\text{d}{F}_{E_t}\left(e\right)=0\text{.}$$

$$\sum _{i=1}^{\infty }i\left|{\eta }_i\right|<\infty \text{.}$$

$${\mathbf{C}}_v$$

$$=2\pi {\left[{\int }_{-\pi }^{\pi }\frac{1}{f_v\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right)\right]}^{-1}{\int }_{-\pi }^{\pi }\frac{f_u\left(\omega \right)}{f_v^2\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right){\left[{\int }_{-\pi }^{\pi }\frac{1}{f_v\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right)\right]}^{-1}\text{,}$$

$${\mathbf{S}}_T\left[{\tilde{\mathbf{\beta }}}_T\left(f_v\right)-\mathbf{\beta }\right]={\left({\mathbf{Z}}_T{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T\right)}^{-1}{\mathbf{Z}}_T^{\top }{\mathbf{\Lambda }}_T^{-1}{\mathbf{u}}_T\text{,}$$

$$\underset{T\to \infty }{\lim }{\mathbf{Z}}_T^{\top }{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\frac{1}{f_v\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right)\text{,}$$

$${\mathbf{S}}_T\text{var}\left[{\tilde{\mathbf{\beta }}}_T\left(f_v\right)\right]{\mathbf{S}}_T={\left({\mathbf{Z}}_T{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T\right)}^{-1}{\mathbf{Z}}_T^{\top }{\mathbf{\Lambda }}_T^{-1}{\mathbf{\Sigma }}_T{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T{\left({\mathbf{Z}}_T{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T\right)}^{-1}\text{,}$$

$${\left(\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\frac{1}{f_v\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right)\right)}^{-1}\left[\underset{T\to \infty }{\lim }{\mathbf{Z}}_T^{\top }{\mathbf{\Lambda }}_T^{-1}{\mathbf{\Sigma }}_T{\mathbf{\Lambda }}_T^{-1}{\mathbf{Z}}_T\right]{\left(\frac{1}{2\pi }{\int }_{-\pi }^{\pi }\frac{1}{f_v\left(\omega \right)}\text{d}\mathbf{H}\left(\omega \right)\right)}^{-1}\text{.}$$

Full text

Keywords: asymptotic normality; autoregressive models; generalized least squares; misspecification; time-series analysis

1 Introduction

Let $\left\{\mathbf{x}_{t}\right\}_{t=1}^{T}$ be a non-stochastic sequence of vectors, such that $\mathbf{x}_{t}^{\top}=\left(x_{1t},\dots x_{dt}\right)\in\mathbb{R}^{1\times d}$, where $t\in\left[T\right]=\left\{1,\dots,T\right\}$, $d,T\in\mathbb{N}$, and $\left(\cdot\right)^{\top}$ is the matrix transposition operator. Let $\left\{U_{t}\right\}_{t=1}^{T}$ be a sequence of random errors. We are interested in the sequence $\left\{Y_{t}\right\}_{t=1}^{T}$ that is generated by the relationship

[TABLE]

where $\bm{\beta}^{\top}=\left(\beta_{1},\dots,\beta_{p}\right)\in\mathbb{R}^{1\times p}$ is a vector of non-stochastic regression coefficients. We can write relationship (1) in the matrix form:

[TABLE]

simultaneously for all $t\in\left[T\right]$, where $\mathbf{X}_{T}\in\mathbb{R}^{T\times d}$ has rows $\mathbf{x}_{t}^{\top}$, $\mathbf{y}_{T}^{\top}=\left(Y_{1},\dots,Y_{T}\right)\in\mathbb{R}^{1\times T}$, and $\mathbf{u}_{T}^{\top}=\left(U_{1},\dots,U_{T}\right)\in\mathbb{R}^{1\times T}$.

Relationships that are described by (1) are generally referred to as multiple linear regression models and are ubiquitous in the study of engineering, natural science, and social science phenomena (see e.g., Weisberg,, 2005). For general treatments of the topic of linear regression modeling, we refer the interested reader to the the manuscripts of Gross, (2003), Seber & Lee, (2003), and Yan & Su, (2009).

In this article, we consider the scenario where the sequence of errors $\left\{U_{t}\right\}_{t=1}^{T}$ is a finite segment of the stationary sequence $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$, such that

[TABLE]

for each $t\in\left[T\right]$, where $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$ is an independent sequence of random variables, such that $\mathbb{E}\left(E_{t}\right)=0$ and $\text{var}\left(E_{t}\right)=\sigma^{2}<\infty$. If the sequence of coefficients $\left\{\theta_{i}\right\}_{i=-\infty}^{\infty}$ is known, then one can write the covariance matrix of $\mathbf{u}_{T}$: $\bm{\Sigma}_{T}=\mathbb{E}\left(\mathbf{u}_{T}\mathbf{u}_{T}^{\top}\right)$ and use it to construct, the so-called generalized least squares (GLS) estimator

[TABLE]

which is known to be the best linear unbiased estimator (BLUE) of $\bm{\beta}$ (cf. Amemiya,, 1985, Sec. 6.1.3). Furthermore, Theorem 1 of Baltagi, (2002, Ch. 9) states that under the condition that $\mathbf{C}_{u}=\lim_{T\rightarrow\infty}T^{-1}\mathbf{X}_{T}^{\top}\bm{\Sigma}_{T}^{-1}\mathbf{X}_{T}^{\top}$ exists, we also have the fact that $\tilde{\bm{\beta}}_{T}\left(\bm{\Sigma}_{T}\right)$ is asymptotically normal in the sense that

[TABLE]

where we denote convergence in law by $\overset{\mathcal{L}}{\longrightarrow}$ and $\text{N}\left(\mathbf{\bm{\mu}},\bm{\Sigma}\right)$ denotes a normal distribution with mean vector $\mathbf{0}$ and covariance matrix $\bm{\Sigma}$.

In applications, the coefficients $\left\{\theta_{i}\right\}_{i=-\infty}^{\infty}$ are rarely if ever known. Thus, the DGP of the error sequence $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ is also unknown. In order to proceed to make inference, one generally assumes a hypothetical DGP for $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ that is equivalent to that of some error sequence $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$.

Let $\left\{V_{t}\right\}_{t=1}^{T}$ be a finite segment of $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$, and let $\mathbf{v}_{T}^{\top}=\left(V_{1},\dots,V_{T}\right)\in\mathbb{R}^{1\times T}$. Furthermore, write the covariance matrix $\mathbf{v}_{T}$ as $\bm{\Lambda}_{T}=\mathbb{E}\left(\mathbf{v}_{T}\mathbf{v}_{T}^{\top}\right)$. Then, by replacing $\bm{\Sigma}_{T}$ in (2) by $\bm{\Lambda}_{T}$, we obtain the so-called incorrectly-specified GLS (IGLS; Koreisha & Fang,, 2001) estimator

[TABLE]

The finite-sample properties of (3) were studied comprehensively in Koreisha & Fang, (2001) and Kariya & Kurata, (2004). General asymptotic results for the IGLS estimator are more difficult to establish, and thus only a small number of results are available in the literature. For example, Rothenberg, (1984) considered asymptotic normality of the IGLS estimator, when $\bm{\Lambda}_{T}$ has first-order autoregressive (AR) form. Some consistency results regarding the IGLS estimator appear in Samarov, (1987) and Koreisha & Fang, (2001). To date, the most general set of asymptotic theorems regarding the IGLS estimator are those reported in Amemiya, (1973).

Make the so-called Grenander regularity conditions (Grenander,, 1954):

• [Gren1]

$\lim_{T\rightarrow\infty}s_{iT}^{2}=\infty$ ($i\in\left[p\right]$), where $s_{iT}^{2}=\sum_{t=1}^{T}x_{it}^{2}$,

• [Gren2]

$\lim_{T\rightarrow\infty}\left(x_{iT}^{2}/s_{iT}^{2}\right)=0$ ($i\in\left[p\right]$),

• [Gren3]

$\lim_{T\rightarrow\infty}\left[\left(s_{iT}s_{jT}\right)^{-1}\sum_{t=1}^{T-k}x_{it}x_{j,t+k}\right]=\rho_{ij}\left(k\right)$ exists for each $i,k\in\left[p\right]$, and

• [Gren4]

the matrix $\mathbf{R}\left(0\right)$ is non-singular, where $\mathbf{R}\left(k\right)$ has element $\rho_{ij}\left(k\right)$ in the $i\text{th}$ row and $j\text{th}$ column.

Furthermore, make the following additional assumptions.

• [Amem1]

The $t\text{th}$ elements of $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ have the form

[TABLE]

where $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$ is a sequence of independent random variables such that $\mathbb{E}\left(E_{t}\right)=0$ and $\text{var}\left(E_{t}\right)=\sigma^{2}<\infty$, and $\left\{\alpha_{i}\right\}_{i=1}^{\infty}$ is such that $\left|1-\sum_{i=1}^{\infty}\alpha_{i}e^{\iota i\omega}\right|^{2}>0$ and $\sum_{i=1}^{N}\left|\alpha_{N+i}\right|=O\left(c^{N}\right)$, where $c\in\left[0,1\right)$ and $\omega\in\left[-\pi,\pi\right]$.

• [Amem2]

The sequence $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ is hypothesized to be equivalent to the stationary autoregressive (AR) process $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$, with $t\text{th}$ term of the form

[TABLE]

where $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$ is a sequence of independent random variables such that $\mathbb{E}\left(E_{t}\right)=0$ and $\text{var}\left(E_{t}\right)=\sigma^{2}<\infty$, and $\left\{\kappa_{i}\right\}_{i=1}^{N}$ are such that the roots of $\sum_{i=1}^{N}\kappa_{i}\zeta^{i}=1$ (with respect to $\zeta$) are all outside of the unit circle.

Here $\iota=\sqrt{-1}$ denotes the imaginary unit. See Amemiya, (1985, Ch. 5.2) for more details regarding AR processes.

Let $\mathbf{S}_{T}$ be the diagonal matrix with $i\text{th}$ element $s_{iT}$, for $i\in\left[p\right]$. Further, let $\mathbf{Z}_{T}=\mathbf{X}_{T}\mathbf{S}_{T}^{-1}$ and define $\mathbf{H}\left(\omega\right)$ to be a Hermitian matrix function with positive semidefinite increments such that $\mathbf{R}\left(k\right)=\int_{-\pi}^{\pi}e^{\iota k\omega}\text{d}\mathbf{H}\left(\omega\right)$. Under [Gren1]–[Gren4], [Amem1], and [Amem2], Amemiya, (1973) proved that

[TABLE]

where

[TABLE]

Furthermore, Amemiya, (1973) showed that if we denote the spectral density functions (SDFs) of the processes $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ and $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$ by $f_{u}\left(\omega\right)$ and $f_{v}\left(\omega\right)$, respectively, then we may write $\mathbf{C}_{v}$ in the spectral form

[TABLE]

It is remarkable that the IGLS is a time-domain estimator that can be proved to have a covariance matrix with simple spectral form.

Since the SDF of $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$ can be written as

[TABLE]

where $\eta_{i}=\text{cov}\left(V_{t},V_{t+i}\right)$ and $\eta_{i}=\eta_{-i}$ for each $i\in\mathbb{Z}$, we can write

[TABLE]

Thus, since the auto-covariance $\bm{\Lambda}_{T}$ is determined by (5), we may interchange the notation $\tilde{\bm{\beta}}_{T}\left(\bm{\Lambda}_{T}\right)$ with $\tilde{\bm{\beta}}_{T}\left(f_{v}\right)$. We shall refer to $\tilde{\bm{\beta}}_{T}\left(f_{v}\right)$ as the time-domain IGLS estimator in order to differentiate it from the frequency-domain IGLS estimator that is introduced in the sequel.

Let

[TABLE]

and

[TABLE]

to be the periodogram of $\left\{\mathbf{x}_{t}\right\}_{t=1}^{T}$ and the cross-spectra periodogram between $\left\{\mathbf{x}_{t}\right\}_{t=1}^{T}$ and $\left\{Y_{t}\right\}_{t=1}^{T}$, respectively (see, e.g., Brockwell & Davis,, 2006, Sec. 11.7). Using (7) and (8), Hannan, (1973) proposed to estimate $\bm{\beta}$ by the frequency-domain IGLS estimator

[TABLE]

Let $\mathcal{F}_{t-1}$ is the $\sigma\text{-algebra}$ that is generated by the sequence $\left\{E_{t-i}\right\}_{i=1}^{\infty}$ and make the following assumptions.

• [Hann1]

The random sequence $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ has the form $U_{t}=\sum_{i=-\infty}^{\infty}\theta_{i}E_{t-i}$, satisfying

[TABLE]

almost surely, and $\sum_{i=-\infty}^{\infty}\left|\theta_{i}\right|<\infty$.

• [Hann2]

The random sequence $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$ has distribution $F_{E_{t}}\left(e\right)$, which satisfies

[TABLE]

• [Hann3]

The SDF $f_{v}\left(\omega\right)$ is real, positive, continuous, and even over $\omega\in\left[-\pi,\pi\right]$.

Under [Gren1]–[Gren4] and [Hann1]–[Hann3], Hannan, (1973) proved that

[TABLE]

where $\mathbf{C}_{v}$ has form (4) (see also Robinson & Velasco,, 1997). That is, (3) and (9) have the same asymptotic distributions when both [Amem1] and [Amem2], or [Hann1]–[Hann3] are satisfied, in addition to [Gren1]–[Gren4].

It is notable, however, that [Hann1]–[Hann3] are more general assumptions that [Amem1] and [Amem2]. Thus, an obvious question to ask is whether the equivalence in asymptotic distributions between the time-domain estimator (3) and frequency-domain estimator (9) remains when one replaces [Amem1] and [Amem2] by assumptions that are more general and closer in spirit to [Hann1]–[Hann3]. In this article, we shall provide an affirmative answer to this question. Before presenting our main result, we wish to provide a review of the relevant literature.

The asymptotic covariance form (9) was used by Engle, 1974b and Nicholls & Pagan, (1977) to explore the efficiencies of the OLS estimator, the BLUE, and the IGLS, under various choices of $f_{u}$ and $f_{v}$, when $d=1$. Some finite sample properties of the frequency-domain estimator were established in Engle, 1974a and Engle & Gardner, (1976).

Under [Gren1]–[Gren4], the IGLS asymptotic covariance (4) was obtained via spectral methods in Kholevo, (1969), Rozanov & Kozlov, (1969), Kholevo, 1971a , and Ibragimov & Rozanov, (1978, Sec. 7.4), under general conditions (see Lemma 5 in the Appendix). In the cited papers, the IGLS was studied under the name of pseudo-best estimators. Unfortunately, no asymptotic normality result of the desired kind were established. It is notable, that Kholevo, 1971b obtained an asymptotic normality result for the continuous-time least squares problem that is hypothesized to be transferable to the pseudo-best estimator case. However, no such result was provided, nor a result regarding the discrete time case.

Hybrid time and frequency-domain IGLS estimators have also been considered, as well as extensions upon the frequency-domain estimator theme. Examples of hybrid estimators include Samarov, (1987) and Hambaba, (1992).

Extensions of the results of Hannan, (1973) to account for long-range dependence appear in Robinson & Hidalgo, (1997) and Hidalgo & Robinson, (2002). A non-linear frequency-domain estimator appears in Hannan, (1971). A broad generalization of the frequency-domain estimation approach to semi-parametric and non-parametric modeling is considered by Robinson, (1991).

Closely related to our article is the report of Aguero et al., (2010), which establishes the asymptotic equivalence between time and frequency-domain estimators for linear dynamic system identification problems. See Hannan & Deistler, (2012) regarding linear dynamic systems.

Using the Cholesky covariance matrix factorization method of Wu & Pourahmadi, (2003), Yang, (2012) constructed an IGLS estimator that is asymptotically efficient. Furthermore, they obtain an asymptotic normality result, under the [Gren1]–[Gren4], using a proof technique that is adapted from those of Anderson, (1971, Thm. 10.2.7) and Fuller, (1996, Thm. 9.1.2) (see Lemma 2 in the Appendix). A model averaging method akin to the construction of Yang, (2012) was studied in Cheng et al., (2015), and a long memory GLS estimator of the same form was considered by Ing et al., (2016).

Also related to our article is the work of Kapetanios & Psaradakis, (2016), which proposed to extend the results of Amemiya, (1973) in a different direction. Here, the [Gren1]–[Gren4] are replaced by various stochastic assumptions on the sequences $\left\{\mathbf{x}_{t}\right\}_{t=1}^{T}$ and $\left\{U_{t}\right\}_{t=1}^{T}$ that make use of mixing and stochastic approximation concepts, and higher moment bound (see Potscher & Prucha, 1997, Ch. 6 regarding mixing and approximation concepts). Compared to our work, the work of Kapetanios & Psaradakis, (2016) can be seen as a complementary and parallel direction of generalization of the results of Amemiya, (1973). Whereas we propose to relax [Amem1] and [Amem2], Kapetanios & Psaradakis, (2016) replaces [Gren1]–[Gren4], instead.

The remainder of the manuscript proceeds as follows. In Section 2, we state and prove our main result. Discussions and remarks are provided in Section 3. Here, we provide results regarding the practical case, where $f_{v}$ is both hypothesized and estimated from the data. Necessary lemmas and technical results are presented in the Appendix.

2 Main result

We retain all notation from the introduction. Furthermore for matrices $\mathbf{A}\in\mathbb{R}^{m\times n}$, let

[TABLE]

denote the operator norm, and let

[TABLE]

denote the $l_{1}$ and $l_{\infty}$ induced norms, respectively. For vectors $\bm{a}\in\mathbb{R}^{m}$, we denote the Euclidean norm of $\bm{a}$ by $\left\|\bm{a}\right\|_{2}$.

Make the following assumptions.

• [Main1]

The $t\text{th}$ element of the error sequence $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ has form

[TABLE]

and $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$ is an independent sequence, where

[TABLE]

• [Main2]

The random sequence $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$ has distribution $F_{E_{t}}\left(e\right)$, which satisfies

[TABLE]

• [Main3]

The SDF $f_{v}\left(\omega\right)$ is real, positive, continuous, and even over $\omega\in\left[-\pi,\pi\right]$.

• [Main4]

The covariance expansion (5) of $f_{v}\left(\omega\right)$ satisfies

[TABLE]

Lemma 1**.**

Under [Gren1]–[Gren4] and [Main1]–[Main4], $\mathbf{S}_{T}\text{var}\left[\tilde{\bm{\beta}}_{T}\left(f_{v}\right)\right]\mathbf{S}_{T}$ approaches

[TABLE]

as $T\rightarrow\infty$.

Proof.

Following from Amemiya, (1973), we write

[TABLE]

and let $\mathbf{w}_{T}=\mathbf{Z}_{T}^{\top}\bm{\Lambda}^{-1}\mathbf{u}_{T}$. Under [Gren1]–[Gren4] and [Main3],

[TABLE]

by Lemma 2. By Lemma 3, $\bm{\Lambda}_{T}$ is invertible and thus, for any $T$, $\bm{\Lambda}_{T}^{-1}$ exists. Thus, we have

[TABLE]

which has the limit, as $T\rightarrow\infty$,

[TABLE]

Assumption [Main1] implies that $f_{u}$ is real and positive since $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ is an absolutely summable linear filter of the independent finite variance sequence $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$ (cf. Theorems 2.11 and 2.12 of Fan & Yao,, 2003). Since $f_{v}$ is positive and continuous by [Main3] and $f_{u}$ is real, positive and continuous by [Main1], we can apply Lemma 5 to obtain

[TABLE]

Upon substitution of (14) into the left-hand side (LHS) of (15) and rearrangement, we obtain

[TABLE]

and have thus verified (12). ∎

Theorem 1**.**

Under [Gren1]–[Gren4] and [Main1]–[Main4],

[TABLE]

where $\mathbf{C}_{v}$ has the form (12).

Proof.

It suffices to show that $\mathbf{w}_{T}$ is asymptotically normal with mean $\mathbf{0}$ and covariance matrix equal to the LHS of (16). First, write $\mathbf{w}_{T,N}=\mathbf{Z}_{T}^{-1}\bm{\Lambda}_{T}^{-1}\bm{\upsilon}_{T,N}$, where

[TABLE]

and $N=N\left(T\right)$ is a positive and increasing integer function of $T$, such that $\lim_{T\rightarrow\infty}N\left(T\right)=\infty$. Let $\bm{\upsilon}_{T,N}=\bm{\Upsilon}_{T,N}\mathbf{e}_{T,N}$, where

[TABLE]

is a $T\times\left(T+2N-1\right)$ matrix and

[TABLE]

is a $\left(T+2N-1\right)\times 1$ vector.

To apply Lemma 6, we must show that for each $N$, $\mathbf{w}_{T,N}$ converges in law to some $\bm{w}_{N}$, as $T\rightarrow\infty$, where $\bm{\omega}_{N}$ is asymptotically normal with mean $\mathbf{0}$ and covariance matrix (16), as $N\rightarrow\infty$. Then, we must verify that

[TABLE]

for each $\varepsilon>0$.

For the purpose of applying the Cramer-Wold device, define $\bm{\alpha}^{\top}=\left(\alpha_{1},\dots,\alpha_{d}\right)$, where $\left\|\bm{\alpha}\right\|_{2}=1$. Let $\bm{\nu}_{k}$ denote the $k\text{th}$ column of $\mathbf{X}_{T}^{\top}\bm{\Lambda}_{T}^{-1}\bm{\Upsilon}_{T,N}$, for $k\in\left\{1-N,\dots,T+N\right\}$. That is,

[TABLE]

Therefore,

[TABLE]

where

[TABLE]

By [Main1] $\left\|\bm{\Upsilon}_{T,N}\right\|_{\text{op}}$ is bounded, and by [Gren4], $\left\|\mathbf{X}_{T}\mathbf{S}^{-1}\right\|_{\text{op}}$ is bounded. Further, by [Main3] and [Main4], we have the boundedness of $\left\|\bm{\Lambda}_{T}^{-1}\right\|_{\text{op}}$. Thus, we obtain the inequalities

[TABLE]

The last fact follows from an application of Lemma 3 and all of the bounds are independent of $T$ and $N$.

Observe that $\left\{W_{t}\right\}_{t=1-N}^{T+N}$ is a sequence of independent random variables with expectation \text{\mathbb{E}}\left(W_{t}\right)=0 and $\sum_{t=1-N}^{T+N}\text{var}\left(W_{t}\right)=1$. Let $F_{W_{t}}\left(w\right)$ be the distribution function of $W_{t}$, for each $t$. Then, for any $\delta>0$, we have the bound:

[TABLE]

where

[TABLE]

By the bound in (20) and [Main2], we must show that

[TABLE]

to prove that (21) converges to zero, as $T\rightarrow\infty$. Write the $i\text{th}$ row and $j\text{th}$ column element of $\bm{\Upsilon}_{T,N}$ and $\bm{\Lambda}_{T}^{-1}$ as $\Upsilon_{ij}$ and $\lambda_{ij}=\lambda_{\left|i-j\right|}$, respectively. Upon expansion we can obtain the following inequalities for the LHS of (22):

[TABLE]

where $C>0$ is some finite constant.

By [Main1], $\left\|\bm{\Upsilon}_{T,N}\right\|_{1}$ and $\left\|\bm{\Upsilon}_{T,N}\right\|_{\infty}$ are bounded, independently of $T$ and $N$.Therefore, for any $T$ and $N$, $\sup_{t}\sum_{j=1}^{T}\left|\Upsilon_{jt}\right|<\infty$. Similarly, by Assumptions [Main3] and [Main4], we apply Corollary 1 to show that $\left\|\bm{\Lambda}_{T}^{-1}\right\|_{1},\left\|\bm{\Lambda}_{T}^{-1}\right\|_{\infty}<\infty$, independently of $T$, and thus $\sum_{k=0}^{\infty}\left|\lambda_{k}\right|<\infty$. Lastly,

[TABLE]

by [Gren1], [Gren2], and Anderson, (1971, Lem. 2.6.1). Thus (22) is proved.

Next, (22) is sufficient to guarantee that (21) approaches zero as $T$ approaches infinity. We can apply the Lindeberg-Feller central limit theorem (DasGupta,, 2008, Thm. 5.1) to obtain $\sum_{t=1-N}^{T+N}W_{t}\overset{\mathcal{L}}{\longrightarrow}\text{N}\left(0,1\right)$.

Via (19), $\bm{\alpha}^{\top}\bm{w}_{T,N}$ is asymptotically equal in distribution to $\bm{\alpha}^{\top}\bm{\omega}_{N}$ (as $T\rightarrow\infty$), where, for any choice of $\bm{\alpha}$, $\bm{\alpha}^{\top}\bm{w}_{N}$ is normal with mean zero and variance

[TABLE]

Via the Cramer-Wold device (cf. DasGupta,, 2008, Th. 1.16), $\bm{w}_{T,N}$ is asymptotically normal with mean vector $\mathbf{0}$ and covariance

[TABLE]

using Lemma 5, where $f_{u,N}$ is the SDF corresponding to $\bm{\Sigma}_{T,N}=\mathbb{E}\left(\bm{\upsilon}_{T,N}\bm{\upsilon}_{T,N}^{\top}\right)$. In other words, $\bm{w}_{N}$ is normally distributed with zero mean vector and covariance matrix (23).

By [Main1], $f_{u}\left(\omega\right)=\left(\sigma^{2}/2\pi\right)\left|\sum_{i=-\infty}^{\infty}\theta_{i}e^{-\iota i\omega}\right|^{2}$ and $f_{u,N}\left(\omega\right)=\left(\sigma^{2}/2\pi\right)\left|\sum_{i=-N}^{N}\theta_{i}e^{-\iota i\omega}\right|^{2}$, via the power transfer formula (cf. Lemma 4). Furthermore, since $\sum_{i=-\infty}^{\infty}\left|\theta_{i}\right|<\infty$ and by preservation of uniform convergence under continuous composition (Bartle & Joichi,, 1961), $f_{u,N}$ converges uniformly to $f_{u}$, as $N$ approaches infinity (cf. Gray,, 2006, Sec. 4.1). Via the Cramer-Wold device, $\bm{w}_{N}$ converges in law to $\bm{w}$ ($N\rightarrow\infty$), where $\bm{\alpha}^{\top}\bm{w}$ has mean vector $\mathbf{0}$ and covariance matrix

[TABLE]

which is equal to (16).

Finally, we must verify (18). We use Chebyshev’s inequality, which states that for any $\varepsilon>0$,

[TABLE]

where we write the numerator of the right-hand side of (25) as

[TABLE]

which reduces to

[TABLE]

By (16), (23), and (24), we have

[TABLE]

Thus, by (25), condition (18) is verified. This completes the proof. ∎

3 Discussions and remarks

3.1 Notes regarding the assumptions of Theorem 1

We can directly compare [Main1] to [Hann1]. It is notable that [Hann1] is more general than [Main1] since it allows $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ to be a linear filter over a martingale sequence $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$, satisfying $\mathbb{E}\left(E_{t}|\mathcal{F}_{t-1}\right)=\mathbb{E}\left(E_{t}^{2}-\mathbb{E}\left(E_{t}^{2}\right)|\mathcal{F}_{t-1}\right)=0$. Further, the condition $0<\left|\sum_{i=-\infty}^{\infty}\theta_{i}e^{-\iota i\omega}\right|$ is necessitated so that Lemma 5 can be applied. It is remarked in Amemiya, (1973), however, that [Main1] and [Main2] are more general than [Amem1].

The addition [Main4] is the key that facilitates the proof. This assumption is necessary for bounding $\left\|\bm{\Lambda}_{T}^{-1}\right\|_{1}$ and $\left\|\bm{\Lambda}_{T}^{-1}\right\|_{\infty}$, which is required to prove (17). It must be remarked that [Main4] is a common condition in the literature, and has been made in similar proof methods, such as those of Cheng et al., (2015). The assumption is not restrictive, since a broad class of short-memory processes satisfy [Main4]. For example, any stationary autoregressive moving average (ARMA) process will satisfy [Main4] (cf. Fan & Yao,, 2003, Sec. 2.5). The assumption is also commonly used in the analysis of unit root processes (see, e.g., Hamilton,, 1994, 17.5).

3.2 Feasible generalized least squares

Generally the SDF $f_{v}$ is unknown and must be estimated from data. Suppose that $\hat{f}_{v,T}$ is an estimator of $f_{v}$, which is indexed by the sample size $T$. Denote the FGLS estimator of $\bm{\beta}$ by $\tilde{\bm{\beta}}_{T}\left(\hat{f}_{v,T}\right)$. For the FGLS to be of use, we require that the FGLS has the same asymptotic distribution as $\tilde{\bm{\beta}}_{T}\left(f_{v}\right)$. To this end, it is sufficient to show that

[TABLE]

where $\overset{\mathbb{P}}{\longrightarrow}$ denotes convergence in probability. Denote the $T\times T$ auto-covariance matrix corresponding to the estimator $\hat{f}_{v,T}$ as $\hat{\bm{\Lambda}}_{T}$. Then, we may write $\tilde{\bm{\beta}}_{T}\left(\hat{\bm{\Lambda}}_{T}\right)=\tilde{\bm{\beta}}_{T}\left(\hat{f}_{v,T}\right)$.

Under [Gren1]–[Gren4] and [Amem1], Amemiya, (1973, Thm. 2) proved that if $f_{v}$ is the SDP of an AR process of order $N\in\mathbb{N}$, that satisfies [Amem2], then (26) holds, when $\hat{\bm{\Lambda}}_{T}$ is obtained via the OLS estimator for the AR model coefficients (cf. Amemiya,, 1985, Sec. 5.4). The argument from Amemiya, (1973, Thm. 2) would hold whenever $\hat{\bm{\Lambda}}_{T}$ is obtained via any consistent estimator of the AR model coefficients.

The proof of the theorem also remains the same upon replacing [Amem1] by [Main1] and [Main2] and noting that [Amem2] is implied by [Main3] and [Main4]. Thus, under the hypothesis of Theorem 1, if $f_{v}$ is hypothesized to be the SDF of a stationary AR process of order $N\in\mathbb{N}$ (i.e., satisfying [Amem2]), then (26) holds, where $\hat{\bm{\Lambda}}_{T}$ is obtained via any consistent estimator of the AR coefficients.

It is notable that proving that (26) holds, under [Amem2] is permissive due to the fact that the inverse auto-covariance matrix $\bm{\Lambda}_{T}^{-1}$ has a banded Toeplitz form (cf. Verbyla,, 1985). We conjecture that it is possible to obtain similar results using the same techniques as those from Amemiya, (1973), when $f_{v}$ is any parametric family of SDFs with banded Toeplitz inverse auto-covariance matrices $\bm{\Lambda}_{T}^{-1}$. However, the proof of such a result is beyond the scope of the current paper.

3.3 Further comments regarding the frequency-domain IGLS estimator

We note that Hannan, (1973) proved a more general result than that which we reported in Section 1. Consider the following conditions.

• [Hann4]

The sequence $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ is stationary and $\alpha\text{-mixing}$ with $\sum_{t=1}^{\infty}\alpha_{t}^{2/\left(2+\delta\right)}<\infty$, where $\alpha_{t}$ is the $t\text{th}$ $\alpha\text{-mixing}$ number and $\mathbb{E}\left|U_{t}\right|^{2+\delta}<\infty$ for some $\delta>0$.

• [Hann5]

The sequence $\left\{\mathbf{x}_{t}\right\}_{t=1}^{\infty}$ is a finite segment of $\left\{\mathbf{x}_{t}\right\}_{t=-\infty}^{\infty}$, where $\left\{\mathbf{x}_{t}\right\}_{t=-\infty}^{\infty}$, is strictly stationary, ergodic, and independent of $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$.

From Hannan, (1973), it was proved that (10) could be obtained by assuming either [Gren1]–[Gren4] or [Hann5], and either [Hann4], or [Hann1] and [Hann2], together with [Hann3]. In the case where [Hann5] is assumed, the matrix $\mathbf{R}\left(k\right)$ in [Gren4] is replaced by $\bar{\mathbf{R}}\left(k\right)$, with elements $\bar{\rho}_{ij}\left(k\right)=\mathbb{E}\left(X_{it}X_{j,t+k}\right)/\left[\mathbb{E}\left(X_{it}^{2}\right)\mathbb{E}\left(X_{j,t}^{2}\right)\right]^{1/2}$, where $\mathbf{x}_{t}^{\top}=\left(X_{1t},\dots X_{pt}\right)$ is now stochastic.

Assumptions [Hann4] and [Hann5] are similar to the mixing and stochastic assumptions considered in the proofs of Kapetanios & Psaradakis, (2016). A proof of the main result under these conditions could be thus adapted from the techniques of Kapetanios & Psaradakis, (2016). We believe that this is an interesting direction of research. However, it falls outside the aim of our paper, which was to directly improve upon the IGLS results obtained in Amemiya, (1973).

Appendix

The following results are required in our main proofs. Sources for all unproved results are provided at the end of the section.

Lemma 2**.**

Let $\mathbf{v}_{T}^{\top}=\left(V_{1},\dots,V_{T}\right)$ have covariance matrix $\mathbb{E}\left(\mathbf{v}_{T}\mathbf{v}_{T}^{\top}\right)=\bm{\Lambda}_{T}$, where $\left\{V_{t}\right\}_{t=1}^{T}$ is a finite segment of the random sequence $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$ with SDF $f_{v}$. Under [Gren1]–[Gren4], and [Main3],

[TABLE]

Lemma 3**.**

Let $\mathbf{v}_{T}^{\top}=\left(V_{1},\dots,V_{T}\right)$ have covariance matrix $\mathbb{E}\left(\mathbf{v}_{T}\mathbf{v}_{T}^{\top}\right)=\bm{\Lambda}_{T}$, where $\left\{V_{t}\right\}_{t=1}^{T}$ is a finite segment of the random sequence $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$ with SDF $f_{v}$. If $f_{v}\left(\omega\right)\geq c>0$ for all $\omega\in\left[-\pi,\pi\right]$, then $\bm{\Lambda}_{T}$ is nonsingular and $\left\|\bm{\Lambda}_{T}^{-1}\right\|_{\text{op}}<\infty$.

Lemma 4**.**

Let the random sequence $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ satisfy [Main1] and have SDF $f_{u}$. Then $f_{u}\left(\omega\right)$ is real, positive, and continuous.

Proof.

Form (11) implies that $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ is a linear filter of $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$. We may write the so-called transfer function of this filtering as $\Gamma\left(\omega\right)=\sum_{i=-\infty}^{\infty}\theta_{i}e^{-\iota i\omega}$. Since $\sum_{i=-\infty}^{\infty}\left|\theta_{i}\right|<\infty$, we have the fact that $\Gamma\left(\omega\right)$ is continuous (cf. Rudin,, 1987, Sec. 9.4). By the same fact, $\Gamma\left(\omega\right)$ is also bounded in modulus (i.e. $\left|\Gamma\right|<\infty$). Next, using Theorem 2.12 of Fan & Yao, (2003), we may write

[TABLE]

for each $\omega\in\left[-\pi,\pi\right]$, by the fact that $\left\{E_{t}\right\}_{t=-\infty}^{\infty}$ is an independent sequence with $\mathbb{E}\left(E_{t}\right)=0$ and $\text{var}\left(E_{t}\right)=\sigma^{2}$. Since the squared modulus is real and positive, by our assumptions, we have the desired result. ∎

Lemma 5**.**

Under [Gren1]–[Gren4], if the sequences $\left\{U_{t}\right\}_{t=-\infty}^{\infty}$ and $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$ have positive and continuous SDFs $f_{u}$ and $f_{v}$, respectively, and if $f_{u}$ is bounded, then

[TABLE]

Lemma 6**.**

For each $N$, suppose that $\bm{X}_{T,N}\overset{\mathcal{L}}{\longrightarrow}\bm{X}_{N}$, as $T\rightarrow\infty$, and that $\bm{X}_{N}\overset{\mathcal{L}}{\longrightarrow}\bm{X}$, as $N\rightarrow\infty$. Furthermore, assume that

[TABLE]

for each $\epsilon>0$ and some $\left\{\bm{Y}_{T}\right\}$. If each of the random variables involved have a common separable domain, and if $\left\|\cdot\right\|$ is some appropriate norm on said domain, then $\bm{Y}_{T}\overset{\mathcal{L}}{\longrightarrow}\bm{X}$, as $T\rightarrow\infty$.

Lemma 7**.**

Let $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$ be a real-valued, where $\mathbb{E}\left(V_{t}\right)=0$ ($t\in\mathbb{Z}$), $\eta_{i}=\text{cov}\left(V_{t},V_{t+i}\right)$ ($i\in\mathbb{Z}$; $\eta_{-i}=\eta_{i}$), and SDF $f_{v}\left(\omega\right)=\left(2\pi\right)^{-1}\sum_{i=-\infty}^{\infty}\eta_{i}e^{-\iota i\omega}$. If $f_{v}$ is positive and $\sum_{i=-\infty}^{\infty}i\left|\eta_{i}\right|<\infty$, then, $V_{t}=\sum_{i=0}^{\infty}b_{i}E_{t-i}$, where $E_{t}=\sum_{i=0}^{\infty}a_{i}V_{t-i}$ is a random process, with $\mathbb{E}\left(E_{t}\right)=0$ and $\text{var}\left(E_{t}\right)=\varsigma^{2}>0$ ($a_{0}=1$ and $b_{0}=1$). Furthermore, $\sum_{i=0}^{\infty}i\left|a_{i}\right|<\infty$, $\sum_{i=0}^{\infty}i\left|b_{i}\right|<\infty$, and

[TABLE]

where $a\left(\omega\right)=\sum_{i=0}^{\infty}a_{i}e^{-\iota i\omega}$ and $b\left(\omega\right)=\sum_{i=0}^{\infty}b_{i}e^{-\iota i\omega}$.

Let $\bm{\Lambda}_{T}$ be a covariance matrix that satisfies Lemma 7. Using the Cholesky decomposition of Akaike, (1969), we may write the inverse of $\bm{\Lambda}_{T}$ in the form $\bm{\Lambda}_{T}^{-1}=\mathbf{L}_{T}^{\top}\bm{\Delta}_{T}^{-1}\mathbf{L}_{T}$, where

[TABLE]

and $\bm{\Delta}_{T}=\text{diag}\left(\varsigma_{0}^{2},\varsigma_{1}^{2},\dots,\varsigma_{T-1}^{2}\right)$. Here, $a_{ij}$ and $\varsigma_{k}^{2}$ ($i,j,k\in\left[T-1\right]$) are obtained by solving the following problems: for each $k$

[TABLE]

and $\varsigma_{k}^{2}>0$ is the minimal value of the problem. When $k=0$, set $\varsigma_{0}^{2}=\varsigma^{2}$. We have the following result.

Lemma 8**.**

Let $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$ be the time series from Lemma 7. Then, there exists a $J$ such that for any $j\geq J$, $\sum_{i=1}^{j}\left|a_{ij}-a_{i}\right|\leq C\sum_{i=j+1}^{\infty}\left|a_{i}\right|$, where $C>0$ is a finite constant that only depends on $f_{v}$, and $a_{ij}$ is the $i\text{th}$ row and $j\text{th}$ column element of the matrix $\mathbf{L}_{T}$.

Corollary 1**.**

Let $\left\{V_{t}\right\}_{t=-\infty}^{\infty}$ be the time series from Lemma 7. Then, $\left\|\bm{\Lambda}_{T}^{-1}\right\|_{1},\left\|\bm{\Lambda}_{T}^{-1}\right\|_{\infty}<\infty$, independently of $T$.

Proof.

Write $\bm{\Lambda}_{T}^{-1}=\mathbf{L}_{T}^{\top}\bm{\Delta}_{T}^{-1}\mathbf{L}_{T}$ and apply Seber, (2008, Thm. 4.6.5) to obtain

[TABLE]

Note that $\left\|\mathbf{L}_{T}^{\top}\right\|_{1}=\left\|\mathbf{L}_{T}\right\|_{\infty}$, $\left\|\mathbf{L}_{T}^{\top}\right\|_{\infty}=\left\|\mathbf{L}_{T}\right\|_{1}$, and $\left\|\bm{\Delta}_{T}^{-1}\right\|_{1}=\left\|\bm{\Delta}_{T}^{-1}\right\|_{\infty}$. Thus we require only three computations.

Let $a_{0k}=1$ and $a_{0}=1$ and write

[TABLE]

By the triangle inequality, we obtain

[TABLE]

which, for sufficiently large $T$, allows for the application of Lemma 8 in order to obtain

[TABLE]

where $C>0$ is a finite constant that is independent of $T$. Since $\sum_{i=0}^{\infty}i\left|a_{i}\right|<\infty$, we have the fact that $\sum_{i=0}^{\infty}i\left|a_{i}\right|<\infty$ and $\sum_{i=0}^{j}\left|a_{ij}\right|<\infty$, for any $j\in\left[T-1\right]$ . Thus, $\left\|\mathbf{L}_{T}\right\|_{\infty}$ is bounded, independently of $T$.

Next, we write $\left\|\mathbf{L}_{T}\right\|_{1}$ as

[TABLE]

Using Lemma 8, we can bound $\left|a_{1,T-1}\right|$ by $\sum_{i=1}^{T-1}\left|a_{i,T-1}-a_{i}\right|\leq C\sum_{i=T}^{\infty}\left|a_{i}\right|$, which convergences, since $\sum_{i=0}^{\infty}\left|a_{i}\right|<\infty$. Similarly, we can use Lemma 8 to bound $\left|a_{2,T-1}\right|+\left|a_{1,T-2}\right|$ by

[TABLE]

This converges, since $\sum_{i=0}^{\infty}\left|a_{i}\right|<\infty$. Continuing the pattern, we arrive at the final expression

[TABLE]

which can be bounded by

[TABLE]

Applying Lemma 8, we bound (28) by

[TABLE]

The expression $C\sum_{i=1}^{\infty}i\left|a_{i}\right|$ converges by Lemma 7. Furthermore, $1+C\sum_{i=1}^{\infty}i\left|a_{i}\right|$ bounds every term of (27), and hence bounds the maximum term. Therefore, $\left\|\mathbf{L}_{T}\right\|_{1}$ is bounded, independently of $T$.

Lastly, must bound $\left\|\bm{\Delta}_{T}^{-1}\right\|_{1}$. Since $\bm{\Delta}_{T}^{-1}$ is a diagonal matrix and each $0<\varsigma_{k}^{2}<\infty$,

[TABLE]

is bounded since each column contains a single finite and positive term. The proof is completed by combining the three bounds. ∎

Lemma 2 appears as Theorem 10.2.7, in Anderson, (1971). Lemma 3 is implied by Theorem 5.2 in Gray, (2006). Lemma 5 appears as Theorem 8 in Ibragimov & Rozanov, (1978, Ch. VII). Lemma 6 appears as Theorem 4.2 in Billingsley, (1968). Lemma 6 is obtained by taking the $l=1$ case (in the notation of the source material) of Brillinger, (2001, Thm. 3.8.4). Lemma 8 can be derived from Lemma 4 of Berk, (1974).

Acknowledgements

The author is indebted to Paul Kabaila, for his interest in this project and for his fruitful comments. The research is funded under Australian Research Council grants DE170101134 and DP180101192.