Estimating Linear Dynamic Panels with Recentered Moments

Abstract

This paper proposes estimating linear dynamic panels by explicitly exploiting the endogeneity of lagged dependent variables and expressing the crossmoments between the endogenous lagged dependent variables and disturbances in terms of model parameters. These moments, when recentered, form the basis for model estimation. The resulting estimator’s asymptotic properties are derived under different asymptotic regimes (large number of cross-sectional units or long time spans), stable conditions (with or without a unit root), and error characteristics (homoskedasticity or heteroskedasticity of different forms). Monte Carlo experiments show that it has very good finite-sample performance.

1. Introduction

It is well known that the standard fixed-effects or within-group (WG) estimator for linear dynamic panel (DP) models suffers from the issue of bias (Nickell 1981) when the time series dimension T is small. Alternative estimators include those based on instrumental variables (IVs) (Anderson and Hsiao 1981), the generalized method of moments (GMM) (e.g., Arellano and Bond 1991; Holtz-Eakin et al. 1988), the maximum likelihood (ML) approach (e.g., Alvarez and Arellano 2003, 2022; Blundell and Smith 1991; Hsiao et al. 2002; Lancaster 2002), those that correct the score function in the ML framework (e.g., Alvarez and Arellano 2022; Breitung et al. 2022; Dhaene and Jochmans 2016), and those that directly correct the bias of the WG estimator (e.g., Bao and Yu 2023; Bun and Carree 2005; Dhaene and Jochmans 2015; Everaert and Pozzi 2007; Gouriéroux et al. 2010; Hahn and Kuersteiner 2002; Kiviet 1995).1

This paper proposes estimating linear dynamic panels by recentering the cross moments between the lagged dependent variables, which are endogenous, and the error term in the model by their non-zero expectations.2 The resulting estimator is named the recentered method of moments (RMM) estimator accordingly. These recentered moments are functions of model parameters and, together with moment conditions from other exogenous regressors, if any, form the basis for model estimation. Essentially, it is based on the idea that the best “instrument” for any of the endogenous lagged dependent variables is itself. As such, one does not need to search for IVs and can avoid issues of weak instruments and many instruments that exist in the GMM framework. It is closely related to the bias-correction literature, but there is no correction procedure involved. In particular, Appendix B illustrates that the estimator proposed in this paper is numerically equivalent to the indirect inference (II) estimator, as introduced by Bao and Yu (2023). However, they are motivated very differently. The strategy of Bao and Yu (2023) starts with a biased estimator, but this paper does not have a biased estimator to begin with and designs recentered moment conditions directly by expressing the cross moments between the endogenous lagged dependent variables and disturbances in terms of model parameters.3 Furthermore, there are three major contributions in this paper that are absent elsewhere.

First, it allows for more general assumptions regarding the data-generating process (DGP). Specifically, the most general form of error heteroskedasticity, arising from cross-sectional units, time, or both, is considered. Breitung et al. (2022) focus primarily on the situation of cross-sectional heteroskedasticity.4 Alvarez and Arellano (2022) consider only time-series heteroskedasticity. Bao and Yu (2023) discuss a robust version of their II estimator that is valid under both forms of heteroskedasticity, though falling short of deriving its asymptotic distribution. They focus instead on the case when there is only time-series heteroskedasticity. Bun and Carree (2006) derive the asymptotic bias of the WG estimator when both forms of heteroskedasticity are present and propose bias correcting the WG estimator when T is fixed for the first-order DP (DP(1)) model. Juodis (2013) points out that the bias-correction procedure of Bun and Carree (2006) is, in fact, inconsistent and designs a consistent one for a higher-order DP.5 Note that both Bun and Carree (2006) and Juodis (2013) estimate the temporal heteroskedasticity parameters so that they can be plugged into the bias expression of the WG estimator for the purpose of bias correction. However, they do not provide insights on how to conduct asymptotic inference on the resulting bias-corrected estimator. In contrast, Alvarez and Arellano (2022) jointly estimate the temporal heteroskedasticity parameters and model parameters and also derive their joint asymptotic distribution.

Second, this paper explicitly includes the case when there is a unit root. If the time span is short, this may not matter, as the inference procedure is under the large-N asymptotic regime, where N is the number of cross-sectional units. Bun and Carree (2006), Juodis (2013), Alvarez and Arellano (2022), and Bao and Yu (2023) all consider short panels. But for long panels, the issue of unit root cannot be simply ignored. Dhaene and Jochmans (2015) do not consider the unit-root case, and their jackknife method is developed under the rectangular-array asymptotic regime (namely, both N and T are large). Breitung et al. (2022) also present some results under the rectangular-array regime, but do not explicitly consider the unit-root case.

Third, asymptotic distribution results are derived that, in general, do not require both N and T to be large. The asymptotic distribution of the proposed RMM estimator under large T resembles the familiar ordinary least squares (OLS) result in traditional regression analysis, and its asymptotic variance achieves the efficiency bound under homoskedasticity. Similar to time series literature, the convergence rate of the estimator of the autoregressive parameters is different when there is a unit root under large T, but the standard t-test procedure carries through when one is conducting hypothesis testing. Under homoskedasticity, Han and Phillips (2010) report a unified asymptotic distribution result for their first difference least squares (FDLS) estimator for DP(1) when the unit-root case is allowed. In their setup, the fixed effects disappear under the unit-root case. Hayakawa (2009) derives the asymptotic properties of the IV estimator for higher-order DP models, with neither heteroskedasticity nor exogenous regressors present, when the panel is dynamically stable under large N and large T, though he remarks that it should also hold under other asymptotic regimes.

The plan of this paper is as follows. The next section introduces the model and notation. Section 3 presents the RMM estimator under the baseline set-up of homoskedasticity, though it is further shown that the estimator is robust to cross-sectional heteroskedasticity as well. An important message here is that when T is large, there is no asymptotic bias, standing in contrast to the consistent WG estimator that may possess an asymptotic bias. Section 4 introduces a robust estimator under cross-sectional and temporal heteroskedasticity. Section 5 illustrates the good finite-sample performance of the proposed estimator by Monte Carlo experiments. The last section concludes and discusses possible future research. Technical details, including lemmas that are used for the proofs of the main results in this paper, together with some extended discussions and additional simulation results, are provided in the appendices. Throughout, matrix/vector dimension subscripts are typically omitted unless confusion may arise. A subscript 0 is used to signify the true value of a parameter that is to be estimated.

2. Model, Notation, and Assumptions

The p-th order linear dynamic panel model, DP(p) for short, is

$$\begin{array}{cc}\hfill y_{it}& =\sum _{\ell =1}^p{\varphi }_{\ell }{y}_{i,t-\ell }+{\mathit{x}}_{it}^{\prime }\mathit{\beta }+{\alpha }_i+u_{it}={\mathit{w}}_{it}^{\prime }\mathit{\theta }+{\alpha }_i+u_{it},\hfill \end{array}$$

(1)

$i=1,\cdots ,N,t=1,\cdots ,T$, where the dependent variable $y_{it}$ is related to its lagged values, up to order p, fixed effects ${\alpha }_i$, the $k\times 1$ vector of exogenous variables ${\mathit{x}}_{it}={(x_{it,1},\cdots ,x_{it,k})}^{\prime }$, and an idiosyncratic disturbance term $u_{it}$. The vector ${\mathit{w}}_{it}={(y_{i,t-1},\cdots ,y_{i,t-p},{\mathit{x}}_{it}^{\prime })}^{\prime }$ collects all the right-hand side variables and $\mathit{\theta }={({\varphi }^{\prime },{\mathit{\beta }}^{\prime })}^{\prime }$, where $\mathit{\varphi }={({\varphi }_1,\cdots ,{\varphi }_p)}^{\prime }$. For each i, let ${\mathit{y}}_i={(y_{i1},\cdots ,y_{iT})}^{\prime }$, ${\mathit{y}}_{i,(-\ell )}={(y_{i,1-\ell },\cdots ,y_{i,T-\ell })}^{\prime }$, ${\mathit{Y}}_i=({\mathit{y}}_{i,(-1)},\cdots ,{\mathit{y}}_{i,(-p)})$, ${\mathit{X}}_i={({\mathit{x}}_{i1},\cdots ,{\mathit{x}}_{iT})}^{\prime }$, ${\mathit{W}}_i=({\mathit{Y}}_i,{\mathit{X}}_i)={({\mathit{w}}_{i1},\cdots ,{\mathit{w}}_{iT})}^{\prime }$, and ${\mathit{u}}_i={(u_{i1},\cdots ,u_{iT})}^{\prime }$. Stacking over i, one has $\alpha ={({\alpha }_1,\cdots ,{\alpha }_N)}^{\prime }$, $\mathit{y}={({\mathit{y}}_1^{\prime },\cdots ,{\mathit{y}}_N^{\prime })}^{\prime }$, ${\mathit{y}}_{(-\ell )}={({\mathit{y}}_{1,(-\ell )}^{\prime },\cdots ,{\mathit{y}}_{N,(-\ell )}^{\prime })}^{\prime }$, $\mathit{Y}={({\mathit{Y}}_1^{\prime },\cdots ,{\mathit{Y}}_N^{\prime })}^{\prime }=({\mathit{y}}_{(-1)},\cdots ,{\mathit{y}}_{(-p)})$, $\mathit{X}={({\mathit{X}}_1^{\prime },\cdots ,{\mathit{X}}_N^{\prime })}^{\prime }$, $\mathit{W}=(\mathit{Y},\mathit{X})={({\mathit{W}}_1^{\prime },\cdots ,{\mathit{W}}_N^{\prime })}^{\prime }$, and $\mathit{u}={({\mathit{u}}_1^{\prime },\cdots ,{\mathit{u}}_N^{\prime })}^{\prime }$. Let ⊗ denote the matrix Kronecker product operator and $\mathbf{1}={\mathbf{1}}_T$ be a $T\times 1$ column vector of ones. Then, in matrix notation, (1) can be written as

$$\mathit{y}=\mathit{W}\mathit{\theta }+\alpha \otimes \mathbf{1}+\mathit{u}.$$

(2)

Suppose there exists a matrix $\mathit{A}$ such that when pre-multiplying it to (2), the fixed effects are wiped out, namely,

$$\mathit{A}\mathit{y}=\mathit{A}\mathit{W}\mathit{\theta }+\mathit{A}\mathit{u}.$$

(3)

The $NT\times NT$ matrix ${\mathit{I}}_N\otimes {\mathit{M}}_T$ comes as a natural choice for $\mathit{A}$, where ${\mathit{M}}_T={\mathit{I}}_T-T^{-1}\mathbf{1}{\mathbf{1}}^{\prime }\equiv \mathit{M}$, and ${\mathit{I}}_N$ (${\mathit{I}}_T$) denotes the identity matrix of size N (T). In this case, applying the OLS procedure to (3) yields the WG estimator. For a given $\mathit{A}$, the regression model (3) contains endogenous lagged dependent variables ${\mathit{y}}_{(-1)}$, ⋯, ${\mathit{y}}_{(-p)}$ as regressors and $\mathrm{E}\left({\mathit{W}}^{\prime }{\mathit{A}}^{\prime }\mathit{A}\mathit{u}\right)\ne \mathbf{0}$. The IV/GMM literature focuses on using various instruments for ${\mathit{y}}_{(-\ell )}$ (or its various differences). The best “instrument” for ${\mathit{y}}_{(-\ell )}$ is of course itself, though it violates the definition of instrument. Nevertheless, if one can explicitly analyze how ${\mathit{y}}_{(-\ell )}$ is correlated with $\mathit{u}$, subject to the transformation induced by $\mathit{A}$, then this piece of information can be used to estimate model parameters. This is essentially the idea of this paper. In the sequel, $\mathit{A}={\mathit{I}}_N\otimes \mathit{M}$ is used (and thus, ${\mathit{A}}^{\prime }\mathit{A}=\mathit{A}$), and the following assumptions are made. Different assumptions about the idiosyncratic error term are deferred into the next two sections when different forms of heteroskedasticity are discussed.

Assumption 1.

The series of fixed effects ${\alpha }_i$, $i=1,\cdots ,N$, is i.i.d. across individuals with finite moments up to the fourth order.

Assumption 2.

The error terms $u_{it}$ and fixed effects ${\alpha }_i$ are independent of each other for any $i=1,\cdots ,N$, $t=1,\cdots ,T$.

Assumption 3.

The regressors $\mathit{X}$, if present, are either fixed or random and ${\left(NT\right)}^{-1}{\mathit{X}}^{\prime }\mathit{A}\mathit{X}$ converges (in probability) to a nonsingular matrix as $NT\to \infty$. When they are fixed, ${\mathit{x}}_{it}=O\left(1\right)$. When they are random: (i) they are strictly exogenous with respect to error terms; (ii) ${\mathit{x}}_{it}=O_p\left(1\right)$ with finite moments up to the fourth order; (ii) $\mathrm{E}\left({\alpha }_i^{r_1}{x}_{it,s}^{r_2}\right)=O\left(1\right)$ and $\mathrm{Cov}({\alpha }_i^{r_1}{x}_{it,s_i}^{r_2},{\alpha }_j^{r_1}{x}_{jt,s_j}^{r_2})=0$, $r_1+r_2\le 4$, $r_1\ge 0$, $r_2\ge 0$, $i\ne j$, $i,j=1,\cdots ,N$, $s,s_i,s_j=1,\cdots ,k$.

Assumption 4.

The initial values $y_{i,-s}$, $i=1,\cdots ,N$, $s=0,1,\cdots ,p$, are either fixed or random. When they are fixed, $y_{i,-s}=O\left(1\right)$. When they are random: (i) $y_{i,-s}=O_p\left(1\right)$ with finite moments up to the fourth order; (ii) $\mathrm{E}\left({\alpha }_i^{r_1}{y}_{i,-s}^{r_2}\right)=O\left(1\right)$ and $\mathrm{Cov}({\alpha }_i^{r_1}{y}_{i,-s}^{r_2},{\alpha }_j^{r_1}{y}_{j,-s}^{r_2})=0$, $r_1+r_2\le 4$, $r_1\ge 0$, $r_2\ge 0$, $i\ne j$, $i,j=1,\cdots ,N$, $s=0,\cdots ,p$; (iii) $\mathrm{Cov}(y_{i,-s},u_{it})=0$, $s=0,\cdots ,p$, $t=1,\cdots ,T$, $i=1,\cdots ,N$.

Without loss of generality, in what follows, the analysis in this paper is conditioning on $\mathit{X}$ for ease of presentation. No distribution assumption is made, so long as some moment conditions hold. Assumption 3 does not rule out the possible correlation between $\mathit{X}$ and the fixed effects, but rules out correlation among the products of them. For simplicity, the strictly exogenous regressors contained in $\mathit{X}$ are bounded (in probability). For the estimation strategy to be introduced in the next section, the inclusion of explosive exogenous regressors can affect the convergence rate of the associated parameter estimators, but does not affect the convergence rate of those associated with the lagged dependent variables. Assumption 4 does not spell out how the initial values are generated, except that there are restrictions on how they may be correlated with the fixed effects and idiosyncratic errors. The panel under consideration is not restricted to be dynamically stable. The fixed effects in Assumption 1 are treated as randomly generated from some distribution. This assumption itself is not needed for the purpose of estimation, since the recentered moment conditions to be presented wipe out the fixed effects. However, it is used, together with Assumptions 2–4 and Assumption 5 or 6, to ensure that the (scaled) moments and the associated gradient have properly defined probability limits (see, for example, Lemma A5 in the Appendix C), which in turn are used to establish the asymptotic distribution of the RMM estimator.6 When there is a unit root, the asymptotic behavior of the RMM estimator to be introduced depends on the fixed effects, so for the sake of tractability, they are assumed to be deterministic by following Hahn and Kuersteiner (2002).7

In the next two sections, the asymptotic properties are derived of the RMM estimator under homoskedasticity and heteroskedasticity, without or with a unit root. Under large N and finite T, the asymptotic distribution is of the same form, whereas under large T, subject to the appropriate scaling factors, the asymptotic distribution resembles the OLS result in traditional regression analysis.

Note that, as established in Bao and Yu (2023), the vectors of lagged observations from model (1) (at the true parameter vector ${\mathit{\theta }}_0$) for each cross-sectional unit can be written as

$${\mathit{y}}_{i,(-\ell )}={\mathbf{\Phi }}_p^{-1}[{\mathit{L}}^{\ell }({\mathit{u}}_i+{\alpha }_i\mathbf{1}+{\mathit{X}}_i{\mathit{\beta }}_0)+\sum _{s=0}^{\ell -1}{\mathbf{\Phi }}_s{\mathit{L}}^{\ell -1-s}{\mathit{e}}_1{y}_{i,-s}+\sum _{s=\ell }^{p-1}{\mathbf{\Phi }}_{(s-p)}{\mathit{L}}^{\ell -1-s}{\mathit{e}}_1{y}_{i,-s}],$$

(4)

$\ell =1,\cdots ,p$, where $\mathit{L}$ is a $T\times T$ strict lower triangular matrix with 1’s on the first sub-diagonals and 0’s elsewhere, ${\mathit{e}}_1={(1,0,\cdots ,0)}^{\prime }$ is a $T\times 1$ vector, ${\mathbf{\Phi }}_p={\mathbf{\Phi }}_p\left({\varphi }_0\right)$, ${\mathbf{\Phi }}_p\left(\varphi \right)=\mathit{I}-{\varphi }_1\mathit{L}-\cdots -{\varphi }_p{\mathit{L}}^p$, and ${\mathbf{\Phi }}_{(s-p)}={\mathbf{\Phi }}_s\left({\varphi }_0\right)-{\mathbf{\Phi }}_p\left({\varphi }_0\right)$.8 Then, stacking over index i, one has, for $\ell =1,\cdots ,p$,

$$\begin{array}{cc}\hfill {\mathit{y}}_{(-\ell )}& =({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{u}+({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{X}{\mathit{\beta }}_0+({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\mathbf{1})\alpha \hfill \\ \hfill & +\sum _{s=0}^{\ell -1}({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathbf{\Phi }}_s{\mathit{L}}^{\ell -1-s}{\mathit{e}}_1){\mathbf{y}}_{(-s)}+\sum _{s=\ell }^{p-1}({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathbf{\Phi }}_{(s-p)}{\mathit{L}}^{\ell -1-s}{\mathit{e}}_1){\mathbf{y}}_{(-s)},\hfill \end{array}$$

(5)

where ${\mathbf{y}}_{(-s)}={(y_{1,-s},\cdots ,y_{N,-s})}^{\prime }$ is an $N\times 1$ vector collecting initial cross-sectional observations at time $-s=0,\cdots ,1-p$. This representation of ${\mathit{y}}_{(-\ell )}$ is in terms of $\mathit{u}$, $\mathit{X}$, $\alpha$, and initial conditions, so ${\mathit{W}}^{\prime }\mathit{Au}$ essentially boils down to linear and quadratic forms in the random vector $\mathit{u}$. This facilitates the derivation of the expectation of ${\mathit{W}}^{\prime }\mathit{Au}$, which forms the basis of the moment conditions in this paper for estimating model parameters. When $u_{i,t}$ is independent and identically distributed (i.i.d.) across i and t, moments of quadratic forms in $\mathit{u}$ can be found in Bao and Ullah (2010), and in the presence of heteroskedasticity, results are provided in Appendix A.7 of Ullah (2004).

3. The Baseline Set-Up

This section presents the estimator under the framework when the idiosyncratic errors are homoskedastic across i and t.

Assumption 5.

$u_{it}$, $i=1,\cdots ,N$, $t=1,\cdots ,T$, is i.i.d. across i and t, $\mathrm{E}\left(u_{it}\right)=0$, $\mathrm{Var}\left(u_{it}\right)={\sigma }^2$, and has finite moments up to the fourth order.

3.1. Estimation

Given Assumptions 1–5 and using the representation (5), one has, at ${\mathit{\theta }}_0$,

$$\mathrm{E}\left({\mathit{W}}^{\prime }\mathit{Au}\right)=\left(\begin{array}{c}N{\sigma }^2\mathrm{tr}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^1\right)\\ ⋮\\ N{\sigma }^2\mathrm{tr}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^p\right)\\ {\mathbf{0}}_k\end{array}\right)=-\frac{N{\sigma }^2}{T}\left(\begin{array}{c}{\mathbf{1}}^{\prime }{\mathbf{\Phi }}_p^{-1}\mathit{L}\mathbf{1}\\ ⋮\\ {\mathbf{1}}^{\prime }{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^p\mathbf{1}\\ {\mathbf{0}}_k\end{array}\right),$$

(6)

where $\mathrm{tr}(·)$ is the matrix trace operator. The first equality in (6) follows when one substitutes (5) into $\mathit{W}=({\mathit{y}}_{(-1)},\cdots ,{\mathit{y}}_{(-p)},\mathit{X})$ and takes expectation of quadratic forms in $\mathit{u}$ and the second equality follows from $\mathrm{tr}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)=\mathrm{tr}\left({\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)-T^{-1}{\mathbf{1}}^{\prime }{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\mathbf{1}$, where, since ${\mathit{L}}^{\ell }{\mathbf{\Phi }}_p^{-1}$ is strictly lower triangular, $\mathrm{tr}\left({\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)=0$.

This set of moment conditions additionally involves the variance parameter ${\sigma }^2$. Since $\mathrm{E}\left({\mathit{u}}^{\prime }\mathit{A}\mathit{u}\right)=N(T-1){\sigma }^2$, one can define the following

$$\begin{array}{cc}\hfill {\mathit{g}}_{NT}\left(\mathit{\theta }\right)& =\frac{1}{NT}{\mathit{W}}^{\prime }\mathit{A}(\mathit{y}-\mathit{W}\mathit{\theta })+\frac{1}{NT}{(\mathit{y}-\mathit{W}\mathit{\theta })}^{\prime }\mathit{A}(\mathit{y}-\mathit{W}\mathit{\theta })\mathit{h}\left(\mathit{\theta }\right),\hfill \end{array}$$

(7)

where $\mathit{h}\left(\mathit{\theta }\right)={\left[T(T-1)\right]}^{-1}{({\mathbf{1}}^{\prime }{\mathbf{\Phi }}_p^{-1}\left(\varphi \right)\mathit{L}\mathbf{1},\cdots ,{\mathbf{1}}^{\prime }{\mathbf{\Phi }}_p^{-1}\left(\varphi \right){\mathit{L}}^p\mathbf{1},{\mathbf{0}}_k^{\prime })}^{\prime }$, such that $\mathrm{E}\left({\mathit{g}}_{NT}\left({\mathit{\theta }}_0\right)\right)=\mathbf{0}$. Thus, an estimator can be defined as $\widehat{\mathit{\theta }}={arg}_{\mathit{\theta }}\{{\mathit{g}}_{NT}\left(\mathit{\theta }\right)=\mathbf{0}\}$. Appendix B discusses several closely related estimators that are motivated very differently. The traditional method of moments (MM) or GMM is based on moment conditions that have expectations (or the probability limits) equal to zero exactly, where these moments are usually defined as cross moments of exogenous regressors or instruments and disturbances in the model under consideration. Here, the cross moments pertaining to the endogenous lagged dependent variables have non-zero expectations and the moment conditions ${\mathit{g}}_{NT}\left(\mathit{\theta }\right)=\mathbf{0}$ for estimation are designed by recentering the cross moments by their non-zero expectation parts. For this reason, $\widehat{\mathit{\theta }}$ is named the RMM estimator in this paper.

Note that one can equivalently write the set of moment conditions (7) as

$$\begin{array}{cc}\hfill {\mathit{g}}_{NT}\left(\mathit{\theta }\right)& =\frac{1}{NT}\sum _{i=1}^N\sum _{t=1}^T\left[({\mathit{w}}_{it}-{\overline{\mathit{w}}}_i){ϵ}_{it}\left(\mathit{\theta }\right)+({ϵ}_{it}\left(\mathit{\theta }\right)-{\overline{ϵ}}_i\left(\mathit{\theta }\right)){ϵ}_{it}\left(\mathit{\theta }\right)\mathit{h}\left(\mathit{\theta }\right)\right]\hfill \\ & \equiv \frac{1}{NT}\sum _{i=1}^N\sum _{t=1}^T\left[({\mathit{z}}_{it}\left(\mathit{\theta }\right)-{\overline{\mathit{z}}}_i\left(\mathit{\theta }\right)){ϵ}_{it}\left(\mathit{\theta }\right)\right],\hfill \end{array}$$

where ${ϵ}_{it}\left(\mathit{\theta }\right)=y_{it}-{\mathit{w}}_{it}^{\prime }\mathit{\theta }$, ${\mathit{z}}_{it}\left(\mathit{\theta }\right)={\mathit{w}}_{it}+\mathit{h}\left(\mathit{\theta }\right){ϵ}_{it}\left(\mathit{\theta }\right)$, ${\overline{\mathit{w}}}_i=T^{-1}{\sum }_{t=1}^T{\mathit{w}}_{it}$, and ${\overline{ϵ}}_i\left(\mathit{\theta }\right)$ and ${\overline{\mathit{z}}}_i\left(\mathit{\theta }\right)$ are defined similarly. It turns out that $T^{-1}{\sum }_{t=1}^T\left[({\mathit{z}}_{it}\left(\mathit{\theta }\right)-{\overline{\mathit{z}}}_i\left(\mathit{\theta }\right)){ϵ}_{it}\left(\mathit{\theta }\right)\right]$ is the same as ${\mathit{m}}_{Ti}\left(\mathit{\theta }\right)$ in Breitung et al. (2022).9 They show that ${\mathit{m}}_{Ti}\left(\mathit{\theta }\right)$ has the property of $\mathrm{E}\left({\mathit{m}}_{Ti}\left({\mathit{\theta }}_0\right)\right)=\mathbf{0}$ under cross-sectional heteroskedasticity, namely, $\mathrm{E}\left(u_{it}^2\right)={\sigma }_i^2$. Essentially, this is because, in this case, ${\widehat{\sigma }}_{Ti}^2\left({\mathit{\theta }}_0\right)={(T-1)}^{-1}{\sum }_{t=1}^T({ϵ}_{it}\left({\mathit{\theta }}_0\right)-{\overline{ϵ}}_i\left({\mathit{\theta }}_0\right)){ϵ}_{it}\left({\mathit{\theta }}_0\right)$ is an unbiased estimator of ${\sigma }_i^2$ and thus ${(T-1)}^{-1}{\sum }_{i=1}^N{\sum }_{t=1}^T({ϵ}_{it}\left({\mathit{\theta }}_0\right)-{\overline{ϵ}}_i\left({\mathit{\theta }}_0\right)){ϵ}_{it}\left({\mathit{\theta }}_0\right)={(T-1)}^{-1}{\mathit{u}}^{\prime }\mathit{A}\mathit{u}$ is in fact an unbiased estimator of ${\sum }_{i=1}^N{\sigma }_i^2$. In other words, if Assumption 5 is modified such that $\mathrm{E}\left(u_{it}^2\right)={\sigma }_i^2$, then it is still the case that $\mathrm{E}\left({\mathit{g}}_{NT}\left({\mathit{\theta }}_0\right)\right)=\mathbf{0}$ and the estimation procedure does not change. This is invalid, however, when there is time-series heteroskedasticity ($\mathrm{E}\left(u_{it}^2\right)={\sigma }_t^2$) or both forms of heteroskedasticity ($\mathrm{E}\left(u_{it}^2\right)={\sigma }_{it}^2$). Remark 2 of Breitung et al. (2022) suggests that ${\mathit{m}}_{Ti}\left(\mathit{\theta }\right)$ may still be considered a valid set of moment conditions when both forms of heteroskedasticity are present, in the sense that the limit of ${\mathit{m}}_{Ti}\left(\mathit{\theta }\right)$ (as $T\to \infty$) goes to zero, provided that: (i) there is no unit root and (ii) T is large. In the next section, a robust estimator is introduced such that it is based on a set of moment conditions having exact expectation zero under the most general form of heteroskedasticity without imposing the two conditions. To fix ideas, various results in this section are presented by assuming homoskedasticity.

Obviously,

$$\begin{array}{cc}\hfill {\mathit{G}}_{NT}\left(\mathit{\theta }\right)& =\frac{\partial {\mathit{g}}_{NT}\left(\mathit{\theta }\right)}{\partial {\mathit{\theta }}^{\prime }}\hfill \\ \hfill & =\frac{1}{NT}[{(\mathit{y}-\mathit{W}\mathit{\theta })}^{\prime }\mathit{A}(\mathit{y}-\mathit{W}\mathit{\theta })\mathit{H}\left(\mathit{\theta }\right)-{\mathit{W}}^{\prime }\mathit{A}\mathit{W}-2\mathit{h}\left(\mathit{\theta }\right){(\mathit{y}-\mathit{W}\mathit{\theta })}^{\prime }\mathit{A}\mathit{W}],\hfill \end{array}$$

(8)

where

$$\mathit{H}\left(\mathit{\theta }\right)=\frac{\partial \mathit{h}\left(\mathit{\theta }\right)}{\partial {\mathit{\theta }}^{\prime }}=\frac{1}{T(T-1)}\left(\begin{array}{cc}{\mathit{R}}_p\left(\varphi \right)& {\mathbf{O}}_{p\times k}\\ {\mathbf{O}}_{k\times p}& {\mathbf{O}}_k\end{array}\right),$$

(9)

and ${\mathit{R}}_p\left(\varphi \right)$ is a $p\times p$ matrix with ${\mathbf{1}}^{\prime }{\mathbf{\Phi }}_p^{-1}\left(\varphi \right){\mathit{L}}^s{\mathbf{\Phi }}_p^{-1}\left(\varphi \right){\mathit{L}}^{\ell }\mathbf{1}$ in its $(\ell ,s)$-th position, $\ell ,s=1,\cdots ,p$. In light of (5), one notices that both ${\mathit{g}}_{NT}={\mathit{g}}_{NT}\left({\mathit{\theta }}_0\right)={\left(NT\right)}^{-1}[{\mathit{W}}^{\prime }\mathit{A}\mathit{u}+{\mathit{u}}^{\prime }\mathit{A}\mathit{u}\mathit{h}]$ and ${\mathit{G}}_{NT}={\mathit{G}}_{NT}\left({\mathit{\theta }}_0\right)={\left(NT\right)}^{-1}[{\mathit{u}}^{\prime }\mathit{A}\mathit{u}\mathit{H}-{\mathit{W}}^{\prime }\mathit{A}\mathit{W}-2\mathit{h}{\mathit{u}}^{\prime }\mathit{A}\mathit{W}]$ are in terms of linear and quadratic forms in $\mathit{u}$, whereas $\mathit{h}=\mathit{h}\left({\varphi }_0\right)$ and $\mathit{H}=\mathit{H}\left({\varphi }_0\right)$ are purely functions of ${\varphi }_0$. From Lemma A1 in Appendix C, regardless of N and T, under homoskedasticity,

$$\mathrm{Var}\left(\sqrt{NT}{\mathit{g}}_{NT}\right)=\frac{1}{NT}\mathrm{Var}\left({\mathit{W}}^{\prime }\mathit{A}\mathit{u}\right)-{\sigma }^4\left[\frac{2(T-1)}{T}+{\gamma }_2\frac{{(T-1)}^2}{T^2}\right]\mathit{h}{\mathit{h}}^{\prime }\equiv {\mathbf{\Omega }}_{NT},$$

(10)

where ${\gamma }_2$ is the excess kurtosis of $u_{it}$.10

At the true parameter vector ${\mathit{\theta }}_0$, when there is no unit root, $-{\left(NT\right)}^{-1}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}=O_p\left(1\right)$, $T\mathit{h}=O\left(1\right)$, $T\mathit{H}=O\left(1\right)$, $N^{-1}{\mathit{u}}^{\prime }\mathit{A}\mathit{W}=O\left(1\right)+O_p\left(N^{-1/2}{T}^{1/2}\right)$, and ${\left(NT\right)}^{-1}{\mathit{u}}^{\prime }\mathit{A}\mathit{u}=O_p\left(1\right)$, so ${\mathit{G}}_{NT}=O_p\left(1\right)+[O\left(T^{-2}\right)+O_p\left(N^{-1/2}{T}^{-3/2}\right)]+O_p\left(T^{-1}\right)$. When T is large, $\mathit{G}$ is obviously dominated by the $O_p\left(1\right)$ term, namely, $-{\left(NT\right)}^{-1}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}$. (When there is a unit root, the properly scaled ${\mathit{G}}_{NT}$ has the scaled $-{\mathit{W}}^{\prime }\mathit{A}\mathit{W}$ as its leading term.) From Lemma A6 in Appendix C (or Lemma A14 when there is a unit root), the estimator is locally identified in large samples. When T is finite but N is large, it is still very likely that $-{\left(NT\right)}^{-1}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}$ is relatively large compared with the other two terms of finite order in ${\mathit{G}}_{NT}$. In finite samples, one can always check numerically whether ${\mathit{G}}_{NT}\left(\mathit{\theta }\right)$ is singular against a grid of values of $\mathit{\theta }$. This is similar to the approach adopted by Gospodinov et al. (2017) and Bao and Yu (2023). In this section and the one to follow, it is implicitly assumed that the estimator is (first-order) locally identified. A general statement about sufficient conditions for identification would be desirable but extremely difficult.11

3.2. Inference under Large N

If $N\to \infty$ and T is finite, by writing ${\mathit{g}}_{NT}={\left(NT\right)}^{-1}{\sum }_{i=1}^N({\mathit{W}}_i^{\prime }\mathit{M}{\mathit{u}}_i+{\mathit{u}}_i^{\prime }\mathit{M}{\mathit{u}}_i\mathit{h})\equiv {\left(NT\right)}^{-1}{\sum }_{i=1}^N{\mathit{g}}_i$, where ${\mathit{g}}_i$ is independent (across i) with mean $\mathbf{0}$ and variance $O\left(1\right)$ under Assumptions 1–5, one can apply Lyapunov’s central limit theorem to $N^{-1/2}{\mathit{g}}_{NT}$. This straightforwardly gives the following result.

Theorem 1.

Under Assumptions 1–5, if T is finite and ${\mathit{G}}_T={\mathrm{plim}}_{N\to \infty }{\mathit{G}}_{NT}$ and ${\mathbf{\Omega }}_T={lim}_{N\to \infty }{\mathbf{\Omega }}_{NT}$ exist and are nonsingular, then as $N\to \infty$,

$$\sqrt{N}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\mathit{G}}_T^{-1}{\mathbf{\Omega }}_T{\mathit{G}}_T^{-1\prime }\right).$$

(11)

For practical inference, while ${\mathit{G}}_T$ may be consistently estimated by ${\widehat{\mathit{G}}}_T={\left(NT\right)}^{-1}[{(\mathit{y}-\mathit{W}\widehat{\mathit{\theta }})}^{\prime }\mathit{A}(\mathit{y}-\mathit{W}\widehat{\mathit{\theta }})\widehat{\mathit{H}}-{\mathit{W}}^{\prime }\mathit{A}\mathit{W}-2\widehat{\mathit{h}}{(\mathit{y}-\mathit{W}\widehat{\mathit{\theta }})}^{\prime }\mathit{A}\mathit{W}]$ with $\widehat{\mathit{h}}=\mathit{h}\left(\widehat{\mathit{\theta }}\right)$ and $\widehat{\mathit{H}}=\mathit{H}\left(\widehat{\mathit{\theta }}\right)$, it is not advisable to use a plug-in approach to estimate ${\mathbf{\Omega }}_T$. Recall ${\mathbf{\Omega }}_{NT}$ (see (10)) contains $\mathrm{Var}\left({\mathit{W}}^{\prime }\mathit{A}\mathit{u}\right)$, which, as shown in Bao and Yu (2023), has a very complicated expression involving further the skewness and kurtosis of $u_{it}$, fixed effects, and possible interactions between the fixed effects and initial conditions. Given these complications and the fact that ${\mathit{W}}_i^{\prime }\mathit{M}{\mathit{u}}_i+{\mathit{u}}_i^{\prime }\mathit{M}{\mathit{u}}_i\mathit{h}$ is independent (across i), a White-type (White 1980) estimator is natural for estimating ${\mathbf{\Omega }}_T$ in this asymptotic regime of large N and finite T. In consequence, a consistent variance estimator of the asymptotic variance of $\sqrt{N}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)$ is

$$\widehat{\mathit{V}}\left(\widehat{\mathit{\theta }}\right)={\widehat{\mathit{G}}}_T^{-1}\left(\frac{1}{N}\sum _{i=1}^N{\widehat{\mathit{g}}}_{Ti}{\widehat{\mathit{g}}}_{Ti}^{\prime }\right){\widehat{\mathit{G}}}_T^{-1\prime },$$

(12)

where ${\widehat{\mathit{g}}}_{Ti}=T^{-1}({\mathit{W}}_i^{\prime }\mathit{M}{\widehat{\mathit{u}}}_i+{\widehat{\mathit{u}}}_i^{\prime }\mathit{M}{\widehat{\mathit{u}}}_i\widehat{\mathit{h}})$ with $\mathit{M}{\widehat{\mathit{u}}}_i=\mathit{M}({\mathit{y}}_i-{\mathit{W}}_i\widehat{\mathit{\theta }})$.

One can show that ${\mathit{G}}_T$ is equal to ${\mathrm{plim}}_{N\to \infty }\nabla {\mathit{m}}_{NT}\left({\mathit{\theta }}_0\right)$ in Breitung et al. (2022), which, in turn, is equal to ${\lim }_{N\to \infty }{N}^{-1}{\sum }_{i=1}^N\mathrm{E}[\nabla {\mathit{m}}_{Ti}\left({\mathit{\theta }}_0\right)]$, where $\nabla {\mathit{m}}_{NT}\left(\mathit{\theta }\right)$ ($=\partial {\mathit{m}}_{NT}\left(\mathit{\theta }\right)/\partial {\mathit{\theta }}^{\prime }$) and $\mathrm{E}[\nabla {\mathit{m}}_{Ti}\left(\mathit{\theta }\right)]$ are given in Appendix A of Breitung et al. (2022). Further, ${\mathbf{\Omega }}_T$ is the same as ${\mathrm{plim}}_{N\to \infty }{N}^{-1}{\sum }_{i=1}^N{\mathit{m}}_{Ti}\left({\mathit{\theta }}_0\right){\mathit{m}}_{Ti}{\left({\mathit{\theta }}_0\right)}^{\prime }$. Thus, $\widehat{\mathit{V}}\left(\widehat{\mathit{\theta }}\right)$ is asymptotically the same as the fixed-T consistent variance matrix estimator of Breitung et al. (2022) (see their Equation (6)). Again, despite the discussion in this section being under the set-up of homoskedastic errors, it also applies to the case of cross-sectional heteroskedasticity.

3.3. Inference under Large T

If $T\to \infty$, regardless of N, upon substituting (5), one can check that the term ${\mathit{y}}_{(-\ell )}^{\prime }\mathit{A}\mathit{u}$, $\ell =1,\cdots ,p$, in ${\mathit{W}}^{\prime }\mathit{A}\mathit{u}$ will be dominated by ${\mathit{u}}^{\prime }\mathit{A}({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{u}+{\mathit{u}}^{\prime }\mathit{A}({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{X}{\mathit{\beta }}_0$. When there is a unit root, ${\mathit{u}}^{\prime }\mathit{A}({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\mathbf{1})\alpha$ is also a dominating term. In the following, the stable and unit-root cases are discussed separately, since the leading terms in ${\mathit{y}}_{(-\ell )}^{\prime }\mathit{A}\mathit{u}$ are of different orders. The relevant matrices in the quadratic forms are uniformly (in T) bounded in row and column sums under the stable case, but not under the unit-root case. These differences create different distribution results for $\widehat{\mathit{\theta }}$ that are mainly in terms of the scaling factors. In what follows, let ${\lambda }_r$, $r=1,\cdots ,p$, denote the inverse of roots of $1-{\varphi }_{1}z-\cdots -{\varphi }_p{z}^p=0$ when $\varphi ={\varphi }_0$. The stable case refers to the situation if $|{\lambda }_r| < 1$, $r=1,\cdots ,p$, and without loss of generality, the unit-root case refers to the scenario when ${\lambda }_1=1$ and $|{\lambda }_r| < 1$, $r=2,\cdots ,p$.

3.3.1. Stable Case

In this case, the matrix $\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }$ in ${\mathit{u}}^{\prime }\mathit{A}({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{u}+{\mathit{u}}^{\prime }\mathit{A}({\mathit{I}}_N\otimes {\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{X}{\mathit{\beta }}_0={\mathit{u}}^{\prime }({\mathit{I}}_N\otimes \mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{u}+{\mathit{u}}^{\prime }({\mathit{I}}_N\otimes \mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{X}{\mathit{\beta }}_0$ is uniformly (with respect to T) bounded in row and column sums. This boundedness property also holds for ${\mathit{I}}_N\otimes \mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }$ and thus the central limit theorem of Kelejian and Prucha (2010) on linear and quadratic forms can be invoked.12 That is, as $T\to \infty$, ${\mathbf{\Omega }}_{1,NT}^{-1/2}\sqrt{NT}{\mathit{g}}_{NT}\stackrel{d}{\to }\mathrm{N}(\mathbf{0},\mathit{I})$ (see Lemma A7), in which

$${\mathbf{\Omega }}_{1,NT}=\frac{1}{NT}\left(\begin{array}{cc}{\sigma }^{4}N{\mathbf{\Gamma }}_1+{\sigma }^2{\mathbf{\Gamma }}_2& {\sigma }^2{\mathit{F}}_1^{\prime }\\ {\sigma }^2{\mathit{F}}_1& {\sigma }^2{\mathit{X}}^{\prime }\mathit{AX}\end{array}\right),$$

(13)

where ${\mathbf{\Gamma }}_1$ and ${\mathbf{\Gamma }}_2$ are $p\times p$ matrices, consisting respectively of elements $\mathrm{tr}\left[{\left({\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)}^{\prime }\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^s\right]$ and ${\mathit{\beta }}_0^{\prime }{\mathit{X}}^{\prime }[{\mathit{I}}_N\otimes {\left({\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)}^{\prime }\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^s]\mathit{X}{\mathit{\beta }}_0$, $\ell ,s=1,\cdots ,p$, in their $(\ell ,s)$-positions, and ${\mathit{F}}_1$ is $k\times p$ with ${\mathit{X}}^{\prime }({\mathit{I}}_N\otimes \mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell })\mathit{X}{\mathit{\beta }}_0$, $\ell =1,\cdots ,p$, as its ℓ-th column. Note that ${\mathbf{\Omega }}_{1,NT}$ is not the exact variance of $\sqrt{NT}{\mathit{g}}_{NT}$, which is ${\mathbf{\Omega }}_{NT}$ as defined by (10). It is better interpreted as an approximation, namely, ${\mathbf{\Omega }}_{1,NT}={\mathbf{\Omega }}_{NT}+o\left(1\right)$. It follows that ${\mathbf{\Omega }}_{NT}^{-1/2}\sqrt{NT}{\mathit{g}}_{NT}={\mathbf{\Omega }}_{1,NT}^{-1/2}\sqrt{NT}{\mathit{g}}_{NT}+o_p\left(1\right)$, regardless of N. In contrast to the exact variance matrix ${\mathbf{\Omega }}_{NT}$, the variance matrix ${\mathbf{\Omega }}_{1,NT}$ involves ${\mathit{\theta }}_0$, $\mathit{X}$, and ${\sigma }^2$, but not the fixed effects, initial conditions, or the skewness and kurtosis of the error distribution.

Theorem 2.

Under Assumptions 1–5 and that DP$\left(p\right)$ is dynamically stable, if ${\mathrm{plim}}_{T\to \infty }{\mathit{G}}_{NT}$ and ${lim}_{T\to \infty }{\mathbf{\Omega }}_{1,NT}$ exist and are nonsingular, then as $T\to \infty$,

$$\sqrt{NT}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\left(\underset{T\to \infty }{\mathrm{plim}}{\mathit{G}}_{NT}\right)}^{-1}\left(\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}\right){\left(\underset{T\to \infty }{\mathrm{plim}}{\mathit{G}}_{NT}\right)}^{-1\prime }\right).$$

(14)

If N is finite, one could have replaced the convergence rate $\sqrt{NT}$ by $\sqrt{T}$ and denoted ${\mathrm{plim}}_{T\to \infty }{\mathit{G}}_{NT}$ by ${\mathit{G}}_N$ and ${lim}_{T\to \infty }{\mathbf{\Omega }}_{1,NT}$ by ${\mathbf{\Omega }}_{1,N}$. But since there is no restriction in N (except that the probability limit of ${\mathit{G}}_{NT}$ and the limit of ${\mathbf{\Omega }}_{1,NT}$ are well defined when $T\to \infty$), which may be finite or also diverge as $T\to \infty$, no attempt is made to implement such replacements here (and also in the next (sub)sections).13 Lemma A6 gives ${\mathrm{plim}}_{T\to \infty }{\left(NT\right)}^{-1}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}={lim}_{T\to \infty }{\sigma }^{-2}{\mathbf{\Omega }}_{1,NT}$. Combining all these results, one has the following.

Corollary 1.

Under the conditions in Theorem 2, as $T\to \infty$,

$$\begin{array}{cc}\hfill & \sqrt{NT}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\left(\underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}\left(\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}\right){\left(\underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}\right)\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\sigma }^4{\left(\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}\right)}^{-1}\right)\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\sigma }^2{\left(\underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}\right).\hfill \end{array}$$

(17)

The asymptotic variances in (15)–(17) resemble the variance formula of a consistent OLS estimator. Recall that the WG estimator, which is also named the least squares dummy variable (LSDV) estimator, is essentially an OLS estimator and is consistent under large T, though it may still possess an asymptotic bias, depending on the divergence rates of N and T. The asymptotic distribution result (17) is the same as Theorem 1(ii) of Breitung et al. (2022) that is derived under cross-sectional heteroskedasticity.14 In view of Theorem 3 in Hahn and Kuersteiner (2002), the asymptotic variance (17) achieves the efficiency bound.

In practice, the asymptotic variance may be estimated by ${\widehat{\mathit{G}}}_{NT}^{-1}{\widehat{\mathbf{\Omega }}}_{1,NT}{\widehat{\mathit{G}}}_{NT}^{-1\prime }$, where ${\widehat{\mathbf{\Omega }}}_{1,NT}={\mathbf{\Omega }}_{1,NT}\left(\widehat{\mathit{\theta }}\right)$ and ${\widehat{\mathit{G}}}_{NT}={\mathit{G}}_{NT}\left(\widehat{\mathit{\theta }}\right)$, with $\widehat{\mathit{\theta }}$ replacing ${\mathit{\theta }}_0$ in ${\mathbf{\Omega }}_{1,NT}$ and ${\mathit{G}}_{NT}$, respectively, or ${({\mathit{W}}^{\prime }\mathit{A}\mathit{W}/NT)}^{-1}{\widehat{\mathbf{\Omega }}}_1{({\mathit{W}}^{\prime }\mathit{A}\mathit{W}/NT)}^{-1}$, or ${\widehat{\sigma }}^4{\widehat{\mathbf{\Omega }}}_{1,NT}^{-1}$, or ${\widehat{\sigma }}^2{({\mathit{W}}^{\prime }\mathit{A}\mathit{W}/NT)}^{-1}$, where ${\widehat{\sigma }}^2={(\mathit{y}-\mathit{W}\widehat{\mathit{\theta }})}^{\prime }\mathit{A}(\mathit{y}-\mathit{W}\widehat{\mathit{\theta }})/\left[N(T-1)\right]$. The last choice is perhaps the easiest one to use.

When $k=0$ and $p=1$, ${\mathbf{\Omega }}_{1,NT}/{\sigma }^4=\mathrm{tr}\left[{\left({\mathbf{\Phi }}_1^{-1}\mathit{L}\right)}^{\prime }\mathit{M}{\mathbf{\Phi }}_1^{-1}\mathit{L}\right]/T=1/(1-{\varphi }_0^2)+O\left(T^{-1}\right)$, so

$$\sqrt{NT}(\widehat{\varphi }-{\varphi }_0)\stackrel{d}{\to }\mathrm{N}(0,1-{\varphi }_0^2),$$

(18)

as $T\to \infty$. This is also the asymptotic distribution of the bias-corrected WG estimator in Hahn and Kuersteiner (2002).

3.3.2. Unit-Root Case

When there is a unit root, the leading terms in ${\mathit{W}}^{\prime }\mathit{A}\mathit{u}$ are linear forms in $\mathit{u}$. The matrices in quadratic forms in $\mathit{u}$ that appear in ${\mathit{W}}^{\prime }\mathit{A}\mathit{u}$ are no longer uniformly (in T) bounded in row and column sums, but they are of lower order compared with the leading linear forms. Thus, define the scaling matrix

$$\mathbf{{\rm Y}}=\left(\begin{array}{cc}N{T}^3{\mathit{I}}_p& {\mathit{O}}_{p\times k}\\ {\mathit{O}}_{k\times p}& NT{\mathit{I}}_k\end{array}\right).$$

Correspondingly, define ${\mathit{g}}_{NT}^{★}={\mathbf{{\rm Y}}}^{-1}[{\mathit{W}}^{\prime }\mathit{A}\mathit{u}-\mathrm{E}\left({\mathit{W}}^{\prime }\mathit{Au}\right)]$ and

$${\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)={\mathbf{{\rm Y}}}^{-1/2}{\sigma }^2\left(\begin{array}{cc}{\mathbf{\Gamma }}_2+{\mathbf{\Gamma }}_3\left(\alpha \right)& {\mathit{F}}_1^{\prime }+{\mathit{F}}_2^{\prime }\left(\alpha \right)\\ {\mathit{F}}_1+{\mathit{F}}_2\left(\alpha \right)& {\mathit{X}}^{\prime }\mathit{AX}\end{array}\right){\mathbf{{\rm Y}}}^{-1/2},$$

(19)

where ${\mathbf{\Gamma }}_2$ and ${\mathit{F}}_1$ are defined in (13), ${\mathbf{\Gamma }}_3\left(\alpha \right)$ has ${\mathbf{1}}^{\prime }{\left({\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)}^{\prime }\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^s\mathbf{1}{\alpha }^{\prime }\alpha$ in its ($\ell ,s$)-position, $\ell ,s=1,\cdots ,p$, and ${\mathit{F}}_2\left(\alpha \right)$ has ${\mathit{X}}^{\prime }({\mathit{I}}_N\otimes \mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\mathbf{1})\alpha$ as its ℓ-th column, $t=1,\cdots ,p$. Note that the matrix ${\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)$ involves the fixed effects. If the fixed effects are deterministic and of finite magnitudes, then ${\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)$ can be interpreted as the approximate variance of ${\mathbf{{\rm Y}}}^{1/2}{\mathit{g}}_{NT}^{★}$, namely, ${\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)=\mathrm{Var}\left({\mathbf{{\rm Y}}}^{1/2}{\mathit{g}}_{NT}^{★}\right)+o\left(1\right)$. Lemma A15 in Appendix C shows that ${\mathbf{\Omega }}_{1,NT}^{★-1/2}\left(\alpha \right){\mathbf{{\rm Y}}}^{1/2}{\mathit{g}}_{NT}^{★}\stackrel{d}{\to }\mathrm{N}(\mathbf{0},\mathit{I})$ as $T\to \infty$. Denote ${\mathit{G}}_{NT}^{★}={\mathit{G}}_{NT}^{★}\left({\mathit{\theta }}_0\right)$ with ${\mathit{G}}_{NT}^{★}\left(\mathit{\theta }\right)={\mathbf{{\rm Y}}}^{-1/2}\{\partial [{\mathit{W}}^{\prime }\mathit{A}(\mathit{y}-\mathit{W}\mathit{\theta })-\mathrm{E}\left({\mathit{W}}^{\prime }\mathit{A}\left((\mathit{y}-\mathit{W}\mathit{\theta })\right)\right]/\partial {\mathit{\theta }}^{\prime }\}{\mathbf{{\rm Y}}}^{-1/2}$.

Theorem 3.

Under Assumptions 3–5 and that DP$\left(p\right)$ has a unit root, assume that the fixed effects are deterministic and of finite magnitudes. If ${\mathrm{plim}}_{T\to \infty }{\mathit{G}}_{NT}^{★}$ and ${lim}_{T\to \infty }{\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)$ exist and are nonsingular, then as $T\to \infty$,

$${\mathbf{{\rm Y}}}^{1/2}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\left(\underset{T\to \infty }{\mathrm{plim}}{\mathit{G}}_{NT}^{★}\right)}^{-1}\left(\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)\right){\left(\underset{T\to \infty }{\mathrm{plim}}{\mathit{G}}_{NT}^{★}\right)}^{-1\prime }\right).$$

(20)

This, together with Lemmas A13 and A14, implies the following.

Corollary 2.

Under the conditions in Theorem 3, as $T\to \infty$,

$$\begin{array}{cc}\hfill & {\mathbf{{\rm Y}}}^{1/2}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\left(\underset{T\to \infty }{\mathrm{plim}}{\mathbf{{\rm Y}}}^{-1/2}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}{\mathbf{{\rm Y}}}^{-1/2}\right)}^{-1}\left(\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)\right){\left(\underset{T\to \infty }{\mathrm{plim}}{\mathbf{{\rm Y}}}^{-1/2}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}{\mathbf{{\rm Y}}}^{-1/2}\right)}^{-1}\right)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\sigma }^2{\left(\underset{T\to \infty }{\mathrm{plim}}{\mathbf{{\rm Y}}}^{-1/2}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}{\mathbf{{\rm Y}}}^{-1/2}\right)}^{-1}\right).\hfill \end{array}$$

(22)

When there are no fixed effects ($\alpha =\mathbf{0}$), but there are still relevant exogenous regressors contained in $\mathit{X}$, the asymptotic distribution result still holds as $T\to \infty$. When there is no $\mathit{X}$ and $p=1$, the leading term in ${\mathit{W}}^{\prime }\mathit{A}\mathit{W}={\sum }_{i=1}^N{\mathit{y}}_{i,(-1)}^{\prime }\mathit{M}{\mathit{y}}_{i,(-1)}$ is ${\sum }_{i=1}^N{\alpha }_i^2{\mathbf{1}}^{\prime }\mathit{L}{\mathbf{\Phi }}_1^{-1\prime }\mathit{M}{\mathbf{\Phi }}_1^{-1}\mathit{L}\mathbf{1}$. As $T\to \infty$, $T^{-3}{\mathbf{1}}^{\prime }\mathit{L}{\mathbf{\Phi }}_1^{-1\prime }\mathit{M}{\mathbf{\Phi }}_1^{-1}\mathit{L}\mathbf{1}\to 1/12$ and $T^{-3}\mathrm{Var}\left({\mathit{W}}^{\prime }\mathit{A}\mathit{u}\right)\to {\sigma }^2{\sum }_{i=1}^N{\alpha }_i^2/12$.

Corollary 3.

Under Assumptions 4 and 5, if further $k=0$, $p=1$, ${\varphi }_0=1$, the fixed effects are deterministic and of finite magnitudes, and $\alpha \ne \mathbf{0}$, then as $T\to \infty$,

$$\sqrt{N{T}^3}(\widehat{\varphi }-{\varphi }_0)\stackrel{d}{\to }\mathrm{N}\left(0,\frac{12{\sigma }^2}{N^{-1}{\alpha }^{\prime }\alpha }\right).$$

(23)

This echoes Theorem 5 of Hahn and Kuersteiner (2002). If $N\to \infty$ also, then the asymptotic variance in (23) is replaced accordingly with $12{\sigma }^2/\left({lim}_{N\to \infty }{N}^{-1}{\alpha }^{\prime }\alpha \right)$ (assuming that the limit exists).15

It should be pointed out that unit-root situation considered here is different from those in Han and Phillips (2010) and Kruiniger (2018), both focusing on DP(1) and ruling out trends due to the fixed effects and/or exogenous regressors. The specification in Han and Phillips (2010) is $y_{it}=\varphi y_{i,t-1}+(1-\varphi ){\alpha }_i+u_{it}$. Their FDLS estimator has a unified convergence rate of $\sqrt{N(T-1)}$ for ${\varphi }_0\in (-1,1]$ and the limiting distribution is Gaussian for any $N/T$ ratio as long as $N(T-1)\to \infty$. The model in Kruiniger (2018) is $y_{it}=\varphi y_{i,t-1}+{\mathit{x}}_{it}^{\prime }(\mathit{\beta }-\varphi \mathit{\beta })+(1-\varphi ){\alpha }_i+u_{it}$. He shows that the modified ML approach (Lancaster 2002) is undesirable in the unit-root case since ${\varphi }_0$ is first-order unidentified and is only second-order identified under the large-N-finite-T asymptotic regime. His generalized modified ML estimator, on the other hand, exists in the large-N-finite-T asymptotic regime with probability approaching 1; it is uniquely identified with probability 1 if it exists, and the asymptotic distribution of the estimated autoregressive parameter is non-Gaussian with a convergence rate of $N^{1/4}$ in the unit-root case. In this paper, it is assumed that either there are always fixed effects or when the fixed effects do not exist, there are relevant exogenous regressors. In this framework, the convergence rate of the RMM estimator is different only under large T for the unit-root case and the asymptotic distribution is still normal.

4. Heteroskedasticity

This section considers the situation of heteroskedastic idiosyncratic errors. Assumption 5 is modified accordingly such that both forms of heteroskedasticity are allowed.

Assumption 6.

The series of error terms $u_{it}$, $i=1,\cdots ,N$, $t=1,\cdots ,T$, is independent across i and t, $\mathrm{E}\left(u_{it}\right)=0$, $\mathrm{Var}\left(u_{it}\right)={\sigma }_{it}^2$, and has finite $(4+\eta )$-th moments, $\eta >0$.

Let $\mathrm{Var}\left({\mathit{u}}_i\right)={\mathbf{\Sigma }}_i=\mathrm{Dg}({\sigma }_{i1}^2,\cdots ,{\sigma }_{iT}^2)$, where $\mathrm{Dg}(·)$ denotes a diagonal matrix with its arguments in order on the diagonal. (When $\mathrm{Dg}(·)$ has a matrix argument, it collects, in order, the diagonal elements of the matrix and forms a diagonal matrix.) It is obvious that

$$\mathrm{E}\left({\mathit{W}}^{\prime }\mathit{Au}\right)=\sum _{i=1}^N\left(\begin{array}{c}\mathrm{tr}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^1{\mathbf{\Sigma }}_i\right)\\ ⋮\\ \mathrm{tr}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^p{\mathbf{\Sigma }}_i\right)\\ {\mathbf{0}}_k\end{array}\right).$$

(24)

Following Bao and Yu (2023), define

$${\mathbf{\Psi }}_{\ell }\left(\varphi \right)=\frac{T}{T-2}\mathrm{Dg}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}\left(\varphi \right){\mathit{L}}^{\ell }\right)-\frac{\mathrm{tr}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}\left(\varphi \right){\mathit{L}}^{\ell }\right)}{(T-1)(T-2)}\mathit{I}.$$

Then, using $\mathrm{tr}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }{\mathbf{\Sigma }}_i\right)=\mathrm{E}\left({\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{\ell }\mathit{M}{\mathit{u}}_i\right)$, where ${\mathbf{\Psi }}_{\ell }={\mathbf{\Psi }}_{\ell }\left({\varphi }_0\right)$, one may define the moment vector as

$$\begin{array}{cc}\hfill {\mathit{g}}_{NT}\left(\mathit{\theta }\right)& =\frac{1}{NT}\sum _{i=1}^N\left(\begin{array}{c}{\mathit{y}}_{i,(-1)}^{\prime }\mathit{M}({\mathit{y}}_i-{\mathit{W}}_i\mathit{\theta })-{({\mathit{y}}_i-{\mathit{W}}_i\mathit{\theta })}^{\prime }\mathit{M}{\mathbf{\Psi }}_1\left(\varphi \right)\mathit{M}({\mathit{y}}_i-{\mathit{W}}_i\mathit{\theta })\\ ⋮\\ {\mathit{y}}_{i,(-p)}^{\prime }\mathit{M}({\mathit{y}}_i-{\mathit{W}}_i\mathit{\theta })-{({\mathit{y}}_i-{\mathit{W}}_i\mathit{\theta })}^{\prime }\mathit{M}{\mathbf{\Psi }}_p\left(\varphi \right)\mathit{M}({\mathit{y}}_i-{\mathit{W}}_i\mathit{\theta })\\ {\mathit{X}}_i^{\prime }\mathit{M}({\mathit{y}}_i-{\mathit{W}}_i\mathit{\theta })\end{array}\right)\hfill \\ \hfill & \equiv \frac{1}{NT}\sum _{i=1}^N{\mathit{g}}_i\left(\mathit{\theta }\right)\hfill \end{array}$$

(25)

such that $\mathrm{E}\left({\mathit{g}}_{NT}\right)=\mathbf{0}$. Note that this set of moment conditions is still valid under Assumption 5. In the previous section under homoskedasticity, ${\sigma }^2$ is treated as a nuisance parameter, and it is replaced with ${\mathit{u}}^{\prime }\mathit{A}\mathit{u}/\left[N(T-1)\right]$ as if it were estimated. (It has also been emphasized that ${\mathit{u}}^{\prime }\mathit{A}\mathit{u}/\left[N(T-1)\right]$, in fact, is unbiased for estimating ${\sum }_{i=1}^N{\sigma }_i^2$ under cross-sectional heteroskedasticity.) Under the most general form of heteroskedasticity, there is no feasible way of estimating all the variance parameters unbiasedly. So this section does not seek to estimate ${\mathbf{\Sigma }}_i$. Instead, the set of moment conditions (25) is designed such that it is robust to heteroskedasticity.

Denote ${\mathbf{\Omega }}_{NT}=\mathrm{Var}\left(\sqrt{NT}{\mathit{g}}_{NT}\right)$ as before, but its expression is different, see (A9) in Appendix C. The gradient (evaluated at ${\mathit{\theta }}_0$) now becomes

$$\begin{array}{cc}& {\mathit{G}}_{NT}=\frac{1}{NT}{\displaystyle \sum _{i=1}^N}\left(\begin{array}{c}2{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_1\mathit{M}{\mathit{W}}_i-{\mathit{y}}_{i,(-1)}^{\prime }\mathit{M}{\mathit{W}}_i\\ ⋮\\ 2{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_p\mathit{M}{\mathit{W}}_i-{\mathit{y}}_{i,(-p)}^{\prime }\mathit{M}{\mathit{W}}_i\\ -{\mathit{X}}_i^{\prime }\mathit{M}{\mathit{W}}_i\end{array}\right)\hfill \\ \hfill & -\frac{1}{NT}{\displaystyle \sum _{i=1}^N}\left(\begin{array}{cccc}{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{11}\mathit{M}{\mathit{u}}_i& \cdots & {\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{1p}\mathit{M}{\mathit{u}}_i& {\mathbf{0}}_k^{\prime }\\ \\ {\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{p1}\mathit{M}{\mathit{u}}_i& \cdots & {\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{pp}\mathit{M}{\mathit{u}}_i& {\mathbf{0}}_k^{\prime }\\ {\mathbf{0}}_k& \cdots & {\mathbf{0}}_k& {\mathbf{O}}_k\end{array}\right),\hfill \end{array}$$

(26)

where

$${\mathbf{\Psi }}_{\ell s}=\frac{\partial {\mathbf{\Psi }}_{\ell }\left({\mathit{\theta }}_0\right)}{\partial {\varphi }_s}=\frac{T}{T-2}\mathrm{Dg}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^s{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)-\frac{\mathrm{tr}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^s{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)}{(T-1)(T-2)}\mathit{I}.$$

(27)

By substituting (4), one can see that ${\mathit{y}}_{i,(-\ell )}^{\prime }\mathit{M}{\mathit{u}}_i-{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{\ell }\mathit{M}{\mathit{u}}_i$ that populates ${\mathit{g}}_i\left({\mathit{\theta }}_0\right)$ is in terms of linear and quadratic forms in ${\mathit{u}}_i$ (see (A10) in the proof of Lemma A1 in Appendix C). This structure is to be exploited for one to study the asymptotic distribution of the resulting RMM estimator from the set of sample moment conditions (25).

4.1. Inference under Large N

Given any finite T, for each $\ell =1,\cdots ,p$, one can verify that the Lyapunov condition holds by checking $N^{-1}{\sum }_{i=1}^N\mathrm{Var}({\mathit{y}}_{i,(-\ell )}^{\prime }\mathit{M}{\mathit{u}}_i-{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{\ell }\mathit{M}{\mathit{u}}_i)=O\left(1\right)$ (see Lemma A1 in Appendix C for the expression of $\mathrm{Var}({\mathit{y}}_{i,(-\ell )}^{\prime }\mathit{M}{\mathit{u}}_i-{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{\ell }\mathit{M}{\mathit{u}}_i)$) and $\mathrm{E}\left(\right|{\mathit{y}}_{i,(-\ell )}^{\prime }\mathit{M}{\mathit{u}}_i-{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{\ell }\mathit{M}{\mathit{u}}_i{|}^{2+\delta })<\infty$ for some $\delta >0$. The Lyapunov condition also holds for ${\mathit{u}}_i^{\prime }\mathit{M}{\mathit{X}}_i$, since $N^{-1}{\sum }_{i=1}^N\mathrm{Var}\left({\mathit{u}}_i^{\prime }\mathit{M}{\mathit{X}}_i\right)=N^{-1}{\sum }_{i=1}^N\mathrm{tr}\left(\mathit{M}{\mathit{X}}_i{\mathit{X}}_i^{\prime }\mathit{M}{\mathbf{\Sigma }}_i\right)=O\left(1\right)$ and $\mathrm{E}\left(\right|{\mathit{u}}_i^{\prime }\mathit{M}{\mathit{X}}_i{|}^{2+\delta })<\infty$. Moreover, for any linear combination of ${\mathit{y}}_{i,(-1)}^{\prime }\mathit{M}{\mathit{u}}_i-{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_1\mathit{M}{\mathit{u}}_i$, ⋯, ${\mathit{y}}_{i,(-p)}^{\prime }\mathit{M}{\mathit{u}}_i-{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_p\mathit{M}{\mathit{u}}_i$, and ${\mathit{u}}_i^{\prime }\mathit{M}{\mathit{X}}_i$, the Lyapunov condition holds as well. Thus, with ${\mathbf{\Omega }}_{NT}$ and ${\mathit{G}}_{NT}$ updated, one has the same asymptotic distribution result as in Theorem 1, and to save space, it is not listed explicitly as a separate theorem.

4.2. Inference under Large T

As in the previous section, cases without and with a unit root are discussed separately, since the orders of various dominating terms in ${\mathit{G}}_{NT}$ and ${\mathbf{\Omega }}_{NT}$ are different.

4.2.1. Stable Case

Note that, for any $\ell =1,\cdots ,p$, ${\mathit{y}}_{i,(-\ell )}^{\prime }\mathit{M}{\mathit{u}}_i$ is dominated by ${\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }({\mathit{u}}_i+{\mathit{X}}_i{\mathit{\beta }}_0)$. $\mathrm{Dg}\left(\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\right)$ has diagonal elements of order $O\left(T^{-1}\right)$ (see Lemma A2 in Appendix C). Therefore, ${\mathit{y}}_{i,(-\ell )}^{\prime }\mathit{M}{\mathit{u}}_i-{\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Psi }}_{\ell }\mathit{M}{\mathit{u}}_i$ is dominated by ${\mathit{u}}_i^{\prime }\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }({\mathit{u}}_i+{\mathit{X}}_i{\mathit{\beta }}_0)$.

Theorem 4.

Under Assumptions 1–4 and 6 and that DP$\left(p\right)$ is dynamically stable, if ${\mathrm{plim}}_{T\to \infty }{\mathit{G}}_{NT}$ and ${lim}_{T\to \infty }{\mathbf{\Omega }}_{1,NT}$ exist and are nonsingular, then as $T\to \infty$,

$$\sqrt{NT}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\left(\underset{T\to \infty }{\mathrm{plim}}{\mathit{G}}_{NT}\right)}^{-1}\left(\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}\right){\left(\underset{T\to \infty }{\mathrm{plim}}{\mathit{G}}_{NT}\right)}^{-1\prime }\right),$$

(28)

where

$${\mathbf{\Omega }}_{1,NT}=\frac{1}{NT}\left(\begin{array}{cc}{\mathbf{\Gamma }}_1+{\mathbf{\Gamma }}_2& {\mathit{F}}_1^{\prime }\\ {\mathit{F}}_1& \sum _{i=1}^N{\mathit{X}}_i^{\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathit{X}}_i\end{array}\right)$$

(29)

in which ${\mathbf{\Gamma }}_1$ and ${\mathbf{\Gamma }}_2$ are $p\times p$ matrices with, respectively, ${\sum }_{i=1}^N\mathrm{tr}\left({\mathbf{\Sigma }}_i{\mathit{L}}^{\ell \prime }{\mathbf{\Phi }}_p^{-1\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^s\right)$ and ${\sum }_{i=1}^N{\mathit{\beta }}_0^{\prime }{\mathit{X}}_i^{\prime }{\mathit{L}}^{\ell \prime }{\mathbf{\Phi }}_p^{-1\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^s{\mathit{X}}_i{\mathit{\beta }}_0$ in their $(\ell ,s$)-th positions, $\ell ,s=1,\cdots ,p$, and ${\mathit{F}}_1$ is $k\times p$ with ${\mathit{X}}_i^{\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }{\mathit{X}}_i{\mathit{\beta }}_0$ as its ℓ-th column, $\ell =1,\cdots ,p$.

In view of Lemmas A9 and A10 in Appendix C, one also has the following corollary.

Corollary 4.

Under the assumptions in Theorem 4, as $T\to \infty$,

$$\begin{array}{cc}\hfill & \sqrt{NT}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\left(\underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}\left(\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}\right){\left(\underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}\right)\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\left(\underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}\left(\underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}\sum _{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathit{W}}_i\right){\left(\underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}\right).\hfill \end{array}$$

(31)

The asymptotic variance (31) resembles the OLS variance result under heteroskedasticity in cross-sectional regressions. In practice, for meaningful inference, one needs to estimate the meat part of the sandwich form of the asymptotic variance. From the proof of Lemma A8, ${\mathit{g}}_{NT}$ is dominated by ${\left(NT\right)}^{-1}{\sum }_{i=1}^N{({\mathit{y}}_{i,(-1)}^{\prime }\mathit{M}{\mathit{u}}_i,\cdots ,{\mathit{y}}_{i,(-p)}^{\prime }\mathit{M}{\mathit{u}}_i,{\mathit{u}}_i^{\prime }\mathit{M}{\mathit{X}}_i)}^{\prime }={\left(NT\right)}^{-1}{\sum }_{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{u}}_i$. Therefore, as $T\to \infty$,

$$\begin{array}{cc}\hfill \underset{T\to \infty }{\mathrm{plim}}\frac{1}{NT}\sum _{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{u}}_i{\mathit{u}}_i^{\prime }\mathit{M}{\mathit{W}}_i& =\underset{T\to \infty }{lim}\frac{1}{NT}\sum _{i=1}^N\mathrm{E}\left({\mathit{W}}_i^{\prime }\mathit{M}{\mathit{u}}_i{\mathit{u}}_i^{\prime }\mathit{M}{\mathit{W}}_i\right)\hfill \\ \hfill & =\underset{T\to \infty }{lim}\mathrm{Var}\left(\sqrt{NT}{\mathit{g}}_{NT}\right)\hfill \\ \hfill & =\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}.\hfill \end{array}$$

But still, ${\mathit{u}}_i$ is not observable. Given that $\mathit{M}{\mathit{u}}_i=\mathit{M}({\mathit{y}}_i-{\mathit{W}}_i{\mathit{\theta }}_0)\equiv \mathit{M}{\mathit{v}}_i$, one may be tempted to use $\mathit{M}{\widehat{\mathit{v}}}_i=\mathit{M}({\mathit{y}}_i-{\mathit{W}}_i\widehat{\mathit{\theta }})=\mathit{M}[{\mathit{u}}_i+{\mathit{W}}_i({\mathit{\theta }}_0-\widehat{\mathit{\theta }})]$. By substitution,

$$\begin{array}{cc}\hfill & \frac{1}{NT}\sum _{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\widehat{\mathit{v}}}_i{\widehat{\mathit{v}}}_i^{\prime }\mathit{M}{\mathit{W}}_i\hfill \\ \hfill & =\frac{1}{NT}\sum _{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{u}}_i{\mathit{u}}_i^{\prime }\mathit{M}{\mathit{W}}_i+\frac{1}{NT}\sum _{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{W}}_i(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0){(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)}^{\prime }{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{W}}_i\hfill \\ \hfill & -\frac{1}{NT}\sum _{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{W}}_i(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0){\mathit{u}}_i^{\prime }\mathit{M}{\mathit{W}}_i-\frac{1}{NT}\sum _{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{u}}_i{(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)}^{\prime }{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{W}}_i,\hfill \end{array}$$

where $\widehat{\mathit{\theta }}-{\mathit{\theta }}_0=O_p(1/\sqrt{NT})$, ${\mathit{W}}_i^{\prime }\mathit{M}{\mathit{W}}_i=O_p\left(T\right)$, and ${\mathit{W}}_i^{\prime }\mathit{M}{\mathit{u}}_i=O\left(1\right)+O_p\left(\sqrt{T/N}\right)$. If it is also the case that N diverges, then ${\left(NT\right)}^{-1}{\sum }_{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{W}}_i(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0){\mathit{u}}_i^{\prime }\mathit{M}{\mathit{W}}_i=o_p\left(1\right)$ and ${\left(NT\right)}^{-1}{\sum }_{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{W}}_i({\mathit{\theta }}_0-\widehat{\mathit{\theta }}){({\mathit{\theta }}_0-\widehat{\mathit{\theta }})}^{\prime }{\mathit{W}}_i^{\prime }\mathit{M}{\mathit{W}}_i=o_p\left(1\right)$.

Therefore, if $N,T\to \infty$, a feasible consistent estimator of the asymptotic variance is

$$\widehat{\mathit{V}}\left(\widehat{\mathit{\theta }}\right)=NT{\left({\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}\left(\sum _{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\widehat{\mathit{v}}}_i{\widehat{\mathit{v}}}_i^{\prime }\mathit{M}{\mathit{W}}_i\right){\left({\mathit{W}}^{\prime }\mathit{A}\mathit{W}\right)}^{-1}.$$

(32)

The asymptotic distribution result (31) with the the variance matrix estimated by (32) resembles Theorem 3 of Hansen (2007). Appendix D discusses the asymptotic distribution of ${\left(NT\right)}^{-1}{\sum }_{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\widehat{\mathit{v}}}_i{\widehat{\mathit{v}}}_i^{\prime }\mathit{M}{\mathit{W}}_i$ if N is fixed and its implication on asymptotic inference when one is using (32). In particular, the $\sqrt{N/(N-1)}{t}_{N-1}$ approximation proposed by Hansen (2007), where $t_{N-1}$ is the t distribution with $N-1$ degrees of freedom, may be used to approximate the asymptotic distribution of the t-statistic.

4.2.2. Unit-Root Case

In this case, the diagonal elements of $\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }{\mathbf{\Sigma }}_i={\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }{\mathbf{\Sigma }}_i-T^{-1}\mathbf{1}{\mathbf{1}}^{\prime }{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }{\mathbf{\Sigma }}_i$ are $O\left(1\right)$, since ${\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }{\mathbf{\Sigma }}_i$ is strictly lower triangular and $T^{-1}\mathbf{1}{\mathbf{1}}^{\prime }{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }$ has $O\left(1\right)$ elements. Further, ${\mathbf{\Psi }}_{\ell }$ is dominated by ${(T-2)}^{-1}T\mathrm{Dg}\left({\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }{\mathbf{\Sigma }}_i\right)$. As before, the scaling matrix $\mathbf{{\rm Y}}$ is used such that the set of moment conditions ${\mathit{g}}_{NT}^{★}\left(\mathit{\theta }\right)$ and gradient matrix ${\mathit{G}}_{NT}^{★}\left(\mathit{\theta }\right)$ are defined accordingly. Specifically, ${\mathit{g}}_{NT}^{★}\left(\mathit{\theta }\right)$ is ${\mathbf{{\rm Y}}}^{-1}\left[NT{\mathit{g}}_{NT}\left(\mathit{\theta }\right)\right]$ with ${\mathit{g}}_{NT}\left(\mathit{\theta }\right)$ given by (25) and ${\mathit{G}}_{NT}^{★}\left(\mathit{\theta }\right)$ is ${\mathbf{{\rm Y}}}^{-1/2}\left[NT{\mathit{G}}_{NT}\left(\mathit{\theta }\right){\mathbf{{\rm Y}}}^{-1/2}\right]$ with ${\mathit{G}}_{NT}\left(\mathit{\theta }\right)$ given by (26).

Theorem 5.

Under Assumptions 3, 4 and 6 and that DP$\left(p\right)$ has a unit root, assume the fixed effects are deterministic and of finite magnitudes. If ${\mathrm{plim}}_{T\to \infty }{\mathit{G}}_{NT}^{★}$ and ${lim}_{T\to \infty }{\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)$ exist and are nonsingular, then as $T\to \infty$,

$${\mathbf{{\rm Y}}}^{1/2}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\stackrel{d}{\to }\mathrm{N}\left(\mathbf{0},{\left(\underset{T\to \infty }{\mathrm{plim}}{\mathit{G}}^{★}\right)}^{-1}\left(\underset{T\to \infty }{lim}{\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)\right){\left(\underset{T\to \infty }{\mathrm{plim}}{\mathit{G}}^{★}\right)}^{-1\prime }\right),$$

(33)

where

$${\mathbf{\Omega }}_{1,NT}^{★}\left(\alpha \right)={\mathbf{{\rm Y}}}^{-1/2}\left(\begin{array}{cc}{\mathbf{\Gamma }}_2+{\mathbf{\Gamma }}_3\left(\alpha \right)& {\mathit{F}}_1^{\prime }+{\mathit{F}}_2^{\prime }\left(\alpha \right)\\ {\mathit{F}}_1+{\mathit{F}}_2\left(\alpha \right)& \sum _{i=1}^N{\mathit{X}}_i^{\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathit{X}}_i\end{array}\right){\mathbf{{\rm Y}}}^{-1/2},$$

(34)

in which ${\mathbf{\Gamma }}_2$ and ${\mathit{F}}_1$ are defined in (29), ${\mathbf{\Gamma }}_3\left(\alpha \right)$ is $p\times p$ with ${\sum }_{i=1}^N{\mathbf{1}}^{\prime }{\mathbf{\Sigma }}_i{\mathit{L}}^{\ell \prime }{\mathbf{\Phi }}_p^{-1\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^s\mathbf{1}{\alpha }_i^2$ in its $(\ell ,s)$-th position, $\ell ,s=1,\cdots ,p$, and ${\mathit{F}}_2\left(\alpha \right)$ is $k\times p$ with ${\sum }_{i=1}^N{\mathit{X}}_i^{\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathbf{\Phi }}_p^{-1}{\mathit{L}}^{\ell }\mathbf{1}{\alpha }_i$ as its ℓ-th column, $\ell =1,\cdots ,p$.

Based on Lemmas A17 and A18, one also has the following result.

Corollary 5.

Under the assumptions in Theorem 5, as $T\to \infty$,

$$\begin{array}{cc}\hfill {\mathbf{{\rm Y}}}^{1/2}(\widehat{\mathit{\theta }}-{\mathit{\theta }}_0)\stackrel{d}{\to }& \mathrm{N}\left(\mathbf{0},\mathit{V}\right),\hfill \end{array}$$

(35)

where $\mathit{V}={\left({\mathrm{plim}}_{T\to \infty }{\mathbf{{\rm Y}}}^{-1/2}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}{\mathbf{{\rm Y}}}^{-1/2}\right)}^{-1}{\mathrm{plim}}_{T\to \infty }{\mathbf{{\rm Y}}}^{-1/2}\left({\sum }_{i=1}^N{\mathit{W}}_i^{\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathit{W}}_i\right){\mathbf{{\rm Y}}}^{-1/2}$

${\left({\mathrm{plim}}_{T\to \infty }{\mathbf{{\rm Y}}}^{-1/2}{\mathit{W}}^{\prime }\mathit{A}\mathit{W}{\mathbf{{\rm Y}}}^{-1/2}\right)}^{-1}$.

Similar to (32), for practical inference, one needs $N,T\to \infty$ so that ${\mathit{W}}_i^{\prime }\mathit{M}{\mathbf{\Sigma }}_i\mathit{M}{\mathit{W}}_i$ can be replaced with ${\mathit{W}}_i^{\prime }\mathit{M}{\widehat{\mathit{v}}}_i{\widehat{\mathit{v}}}_i^{\prime }\mathit{M}{\mathit{W}}_i$ in (35) and the probability limits are replaced with the sample analogs to form an estimated asymptotic variance.

5. Monte Carlo Evidence

In this section, Monte Carlo simulations are conducted to assess the finite-sample performance of the proposed estimator. Bao and Yu (2023) provide simulation results for the first- and second-order models when the idiosyncratic errors are either homoskedastic or temporally heteroskedastic. Recall that under the baseline framework, the proposed estimator in this paper is also robust to cross-sectional heteroskedasticity. This section considers a third-order model, and for one to get a comprehensive spectrum of possible heteroskedasticity, four scenarios are included: homoskedasticity (across i and t), cross-sectional heteroskedasticity (across i only), temporal heteroskedasticity (over t only), and double (both cross-sectional and temporal) heteroskedasticity.

The following DP(3) is used:

$$\begin{array}{cc}\hfill y_{it}& ={\alpha }_i+{\varphi }_1{y}_{i,t-1}+{\varphi }_2{y}_{i,t-2}+{\varphi }_3{y}_{i,t-3}+{\mathit{\beta }}_1{x}_{1,it}+{\mathit{\beta }}_2{x}_{2,it}+u_{it},\hfill \\ \hfill x_{1,it}& ={\rho }_x{x}_{1,i,t-1}+{\xi }_{1,it},{\rho }_x=0.8,\hfill \\ \hfill x_{2,it}& ={\rho }_i{\alpha }_i+{\xi }_{2,it},\hfill \end{array}$$

(36)

where ${\alpha }_i$ is i.i.d. (across i) following a standard normal distribution, ${\rho }_i$ is i.i.d. (across i) uniformly on the interval $[0,1]$, ${\xi }_{1,it}$ is i.i.d. (across i and t) following a standard normal distribution, and ${\xi }_{2,it}$ is similarly defined. The initial value of $x_{1,it}$ is set to be ${\xi }_{0,i}/\sqrt{1-{\rho }_x^2}$, where ${\xi }_{0,i}$ is i.i.d. (across i) following a standard normal distribution. The initial observations on $y_{it}$ are simulated as ${\alpha }_i/(1-{\varphi }_{01}-{\varphi }_{02}-{\varphi }_{03})+{\mathit{x}}_{it}^{\prime }{\mathit{\beta }}_0+u_{it}\sqrt{\mathrm{Var}\left({\delta }_t\right)}$ if there is no unit root and ${\alpha }_i+{\mathit{x}}_{it}^{\prime }{\mathit{\beta }}_0+u_{it}$ otherwise, where ${\delta }_t$ follows a stationary zero-mean third-order autoregressive (AR(3)) process with coefficients ${\varphi }_{01}$, ${\varphi }_{02}$, and ${\varphi }_{03}$, and its shock term is a unit-variance white noise.

Let $e_{it}$ be i.i.d. (across i and t) with mean 0 and variance 1. The error term $u_{it}$ is simulated as follows in the four different scenarios (homoskedasticity, cross-sectional heteroskedasticity, temporal heteroskedasticity, and double heteroskedasticity):

$$u_{it}=\left\{\begin{array}{c}{e}_{it}\hfill \\ \sqrt{z_i}{e}_{it}\hfill & \hfill & z_i\sim \mathrm{U}[0.5,i]\mathrm{or}{\chi }_{10}^2\mathrm{if}\mathrm{U}[0.5,i]\ge 100\hfill \\ \sqrt{z_t}{e}_{it}\hfill & \hfill & z_t\sim \mathrm{U}[0.5,t^2]\mathrm{or}{\chi }_{10}^2\mathrm{if}\mathrm{U}[0.5,t^2]\ge 100\hfill \\ \sqrt{z_{it}}{e}_{it}\hfill & \hfill & z_{it}\sim \mathrm{U}[0.5,i]·\mathrm{U}[0.5,t^2]\mathrm{or}{\chi }_{10}^2\mathrm{if}\mathrm{U}[0.5,i]·\mathrm{U}[0.5,t^2]\ge 100\hfill \end{array}\right.$$

(37)

where, for example, $z_i\sim \mathrm{U}[0.5,i]\mathrm{or}{\chi }_{10}^2\mathrm{if}\mathrm{U}[0.5,i]\ge 100$ means that $z_i$ is independently (across i) drawn from a uniform distribution on the interval $[0.5,i]$, and if the realization is no smaller than 100, then it is redrawn from a chi-squared distribution with 10 degrees of freedom. $e_{it}$ is simulated from a normal distribution, but results under non-normal distributions are available upon request and the conclusions in this section are largely consistent across all distributions.16

In reality, data themselves rarely reveal clearly the lack or presence of heteroskedasticity, so estimation results from the estimator using the recentered moment conditions (7) and the robust one using (25), referred to as RMM and RMM${}_r$, respectively, are presented. When the empirical rejection rates of a $5\%$ two-sided t-test of the relevant parameter equal to its true value are reported, for WG and GMM, also included are their empirical sizes from t-ratios using the White-type robust standard errors (clustered at the individual level, referred to as WG(h) and GMM(h), respectively). There are different choices for the GMM estimator, depending on what and how many instruments are used. Breitung et al. (2022) show that the one-step estimator of Arellano and Bond (1991) is very comparable to other popular choices (Ahn and Schmidt 1995; Blundell and Bond 1998), so the GMM estimator used in this section is the one-step one.17 Different combinations of N and T are experimented with: [100 10; 50 20; 50, 50; 25 40; 20 50; 10 100]. Regardless of N and T, one can always use the estimator in this paper, but in reality, in a situation like $N=25$, $T=40$, to conduct inference, there is typically no convincing evidence for one to favor one asymptotic regime over the other. So in what follows, in addition to the bias and root mean squared error (RMSE), all out of 10,000 simulations, also reported are the empirical sizes using standard errors constructed under different asymptotics. In particular, RMM(N) and RMM${}_r$(N) denote the empirical sizes from t-ratios using the large-N standard errors (see Section 3.2 and Section 4.1) for RMM and RMM${}_r$, respectively. Likewise, RMM(T) refers to the empirical size from the t-ratio when the large-T standard errors from (17) (or (22)) are used for RMM. Furthermore, RMM${}_r$($NT$) means when the feasible variance (32) (or its version under unit root) is used for RMM${}_r$, which is valid if both N and T are large. On the other hand, RMM${}_r$(T) does not mean that a different t-ratio is used and instead, it signals that, for the RMM${}_r$ t-ratio when the feasible variance is used, the $\sqrt{N/(N-1)}{t}_{N-1}$ approximation developed by Hansen (2007) under large T is used for conducting hypothesis testing.

Included for comparison are the bias-corrected estimators of Bun and Carree (2006) (BC for short) and Juodis (2013) (BCJ for short).18 As Juodis (2013) points out, BC is consistent under homoskedasticity and cross-sectional heteroskedasticity, but invalid under temporal heteroskedasticity. BCJ, on the other hand, is robust to both forms of heteroskedasticity. Neither Bun and Carree (2006) nor Juodis (2013) derives the asymptotic distributions of their bias-corrected estimators, so only their bias and RMSE results are reported in this section. Included also is the half-panel jackknife (HPJ) estimator of Chudik et al. (2018), where errors can be heteroskedastic across both i and t.

In the experiments, ${\mathit{\beta }}_0={(1,1)}^{\prime }$ and three sets of parameter configurations of ${\varphi }_0$ are used: ${(0.3,0.3,0.2)}^{\prime }$, ${(0.3,-0.2,-0.1)}^{\prime }$, and ${(0.3,0.6,0.1)}^{\prime }$. The first configurations reflects a situation when the degree of time-series correlation is relatively strong in the sense that the cumulative partial effect of a past shock, measured by powers of ${\varphi }_{01}+{\varphi }_{02}+{\varphi }_{03}$, can be high. The second set corresponds to a case of zero past effect, and the last one is a unit-root case where the past effect never dies out. To save space, only results related to ${\varphi }_{01}+{\varphi }_{02}+{\varphi }_{03}$ under the first set of parameter configuration ${(0.3,0.3,0.2)}^{\prime }$ are reported, whereas results under the other two parameter configurations are contained in Appendix F.19

Table 1 reports the bias and RMSE, both multiplied by 100, and empirical rejection rate of the two-sided $5\%$t-test related to ${\varphi }_{01}+{\varphi }_{02}+{\varphi }_{03}$ when $u_{it}$ is homoskedastic. One sees clearly the superb performance of both RMM and RMM${}_r$, with the smallest bias and lowest RMSE. GMM performs reasonably well, but its bias and RMSE peak at $(N,T)=(50,20)$. The WG estimator performs the worst when T is small but improves as T gets relatively larger, but is still more biased even when $T=100$ compared with RMM or RMM${}_r$. BC and BCJ report larger bias and higher RMSE than GMM on many occasions, though there are also cases where they are much better than GMM. Note that BC and BCJ bias-correct the WG estimator, which is in fact consistent under large T. Table A1 in Appendix F reveals that when there is a unit root, BC and BCJ give a much larger bias than WG, even when T is relatively large. So, in this case, the action of bias-correction gives more biased estimates. HPJ is the most biased and possesses the highest RMSE among all the consistent estimators at $(N,T)=(100,10)$, namely, when the panel is relatively short. Its performance improves quickly as T increases, though still quite a bit below RMM/RMM${}_r$.

Now, consider the size performance of the associated two-sided $5\%$ test from the different estimators. From Table 1, one sees severe size distortions of the WG-based inference when T is relatively small, but its empirical size is close to the nominal size when $T=100$. The GMM-based inference performs really poorly. The t-test based on HPJ severely over-rejects when $(N,T)=(100,10)$, but its size distortion goes down quickly when the panel has longer spans. The large-N-based inferences from RMM and RMM${}_r$, namely, RMM(N) and RMM${}_r$(N), give empirical sizes close to $5\%$ when N is relatively large. On the other hand, RMM(T) delivers a reasonably good size performance even when T is relatively small. Results from RMM${}_r$($NT$) and RMM${}_r$(N) are mixed: when they work, they may perform slightly worse than RMM(N) and RMM(T), but their performances get worse when N is small and T is large. Finally, the size result from RMM${}_r$(T), namely, using the $\sqrt{N/(N-1)}{t}_{N-1}$ approximation for the asymptotic distribution of the t-ratio, is good in almost all cases. Recall that the $\sqrt{N/(N-1)}{t}_{N-1}$ approximation is valid under homoskedasticity. These results are not surprising, given that homoskedastic errors are simulated and that, for the robust estimator, the feasible standard errors based on (32) (or its version under unit root) require both N and T to be large. In fact, when $(N,T)=(50,50)$, the size distortion from RMM${}_r$($NT$) is the smallest compared with other $(N,T)$ combinations with relatively smaller N or T.

Table 2 reports results under cross-sectional heteroskedasticity. Relative performances of these different estimators largely stay the same. Notably, BC and BCJ perform really poorly when $(N,T)=(100,10)$, though they are designed to bias-correct the WG estimator under large N and fixed T. In contrast to the homoskedastic case, HPJ gives a smaller bias and lower RMSE than RMM/RMM${}_r$ when $(N,T)=(100,10)$, but usually performs worse in other cases. In terms of hypothesis testing, when N is relatively large, both RMM(N) and RMM${}_r$(N) work well; when T is relatively large, both RMM(T) and RMM${}_r$(T) provide empirical rejection rates close to the nominal size. Recall that the RMM estimator designed under homoskedasticity is also valid under cross-sectional heteroskedasticity, and so is the related inference procedure. There is pronounced size distortion from HPJ for the unit root case (see Table A2 in Appendix F), but its size performance is reasonably good under the other two parameter configurations when T is not relatively small.

When there is time-series heteroskedasticity, the robust estimator usually dominates all the other estimators in terms of bias and RMSE, as demonstrated in Table 3 (and Table A3 in Appendix F). In terms of hypothesis testing, the large-N RMM${}_r$t-ratio delivers very good size performance when N is relatively large, whereas the $\sqrt{N/(N-1)}{t}_{N-1}$ approximation (RMM${}_r$(T)) provides good results when T is relatively large. The large-N-large-T RMM${}_r$-based inference reports upward size distortions in almost all cases.

Finally, Table 4 provides results under double heteroskedasticity, namely, when both cross-sectional heteroskedasticity and time-series heteroskedasticity are present. RMM${}_r$ is the least biased, except when $(N,T)=(100,10)$, where HPJ is slightly better. On the other hand, RMM, which ignores temporal heteroskedasticity, reports very small bias on many occasions, especially under the second and third parameter configurations (see Table A4 in Appendix F). In terms of RMSE, BCJ and RMM${}_r$ are comparable. In terms of the size performance of the associated t-test, the story is very similar to that reported when there is temporal heteroskedasticity only, namely, under large N, RMM${}_r$(N) is most trustworthy and RMM${}_r$(T) is the one under large T. GMM has the most size distortions in almost all cases, and HPJ gives severe size distortions when there is a unit root.

Summarizing all the simulation results, one can learn the following. (i) When there is no heteroskedasticity, the proposed estimator can be safely used, either the one based on (7) or the robust one based on (25), regardless of N and T. When heteroskedasticity in the time dimension is present, then the robust version is the best. (ii) The presence of a unit root or not has no substantial impact on its performance. (iii) In terms of inference, when N is relatively large or is of comparable size relative to T, the large N-based inference from RMM${}_r$ has very good size performance, regardless of heteroskedasticity. When T is relatively large, the large T-based inference from RMM gives reliable inference under homoskedasticity and cross-sectional heteroskedasticity. (iv) When T is large, the $\sqrt{N/(N-1)}{t}_{N-1}$ approximation for the t-ratio from RMM${}_r$ with the feasible robust variance usually has good size performance, regardless of heteroskedasticity.

6. Conclusions and Directions of Future Research

This paper proposes an estimation strategy that does not rely on instrumental variables. One can view the estimation strategy as using the endogenous lagged dependent variables as their own instruments and then constructing recentered moment conditions by explicitly exploiting the correlation between the endogenous variables and error term in the model. The asymptotic properties of the new estimator are thoroughly investigated under various conditions that relate to the sizes of cross-sectional units and time periods, heterogeneity in the error variance, and the issue of whether a unit root is present. In general, the asymptotic distribution does not require both N and T to be large. Under large T, it resembles the familiar OLS result in traditional regression analysis and its asymptotic variance achieves the efficiency bound under homoskedasticity. Similar to time series autoregressions, the convergence rate of the estimator of the autoregressive parameters is different when there is a unit root under large T, but the standard t-test procedure carries through in hypothesis testing. Monte Carlo simulations demonstrate that it possesses good finite-sample properties in various situations. All the theoretical results and Monte Carlo evidence in this paper suggest two directions for future research.

6.1. Cross-Sectional Correlation

Note that cross-sectional correlation is not explicitly discussed in this paper. If there is a weak cross-sectional correlation (in the sense that the covariance matrix of ${(u_{1t,}\cdots ,u_{Nt})}^{\prime }$ has a bounded norm as $N\to \infty$), then the estimation strategy proposed in this paper still holds, see discussions in Appendix E. However, both N and T need to be large. This form of cross-sectional correlation may cover situations where correlation arises from spatial contiguity as in the spatial econometrics literature. Nevertheless, when there are dominating units (Pesaran and Yang 2021) whose errors are always correlated with those from other units, the assumption of weak correlation is violated. Another situation of violation is when cross-sectional correlation is due to common factors. Under homoskedasticity, for the case of DP(1), De Vos and Everaert (2021) augment the model by the cross-sectional averages of the right-hand side regressors, which can be interpreted as proxies for the unknown factors. Then, they derive the asymptotic bias of the common correlated effects pooled (CCEP) estimator of the main parameter vector under large N and fixed T and design an estimator by matching the CCEP estimator with its asymptotic bias.20 One could have directly focused on the endogeneity of the defactored lagged dependent variables, namely, $({\mathit{I}}_T-\mathit{P}){\mathit{w}}_i$, where $\mathit{P}$ is the projection matrix on the proxies, consisting of cross-sectional averages (including ${\mathbf{1}}_T$). The exact expectation of ${\sum }_{i=1}^N{\mathit{w}}_i^{\prime }({\mathit{I}}_T-\mathit{P}){\mathit{u}}_i$ is not easy to derive, but one can follow De Vos and Everaert (2021) to approximate it and construct the resulting estimator. It is left for future research to extend the robust estimator to cases when there is a strong cross-sectional correlation in addition to heteroskedasticity in higher-order DP with the possibly of a unit root.

6.2. Inference under Heteroskedasticity for Long Panels

It has been observed from Section 5 that all the other estimators in the experiments perform much worse in terms of bias under cross-sectional and temporal heteroskedasticity than the robust estimator when T is not very small. On the other hand, non-negligible size distortions using the feasible variance (32) are also documented and one may need to resort to the approximation of Hansen (2007) for inference purposes, though one has yet to show rigorously that the approximation is still valid in the presence of heteroskedasticity. The re-sampling approach (Kapetanios 2008) may not be applicable under small N and large T. In addition, if one re-samples the data across i (when N is large and T is small) or over t (when T is large and N is small), there is the issue of heteroskedasticity or temporal correlation that one needs to take into account. Designing a practical and reliable inference procedure for the robust estimator in long panels is a second avenue for future research.