3.1 The Anderson–Hsiao Estimator

The first estimator we consider relies on the approach taken by Anderson and Hsiao (Reference Anderson and Hsiao1981, Reference Anderson and Hsiao1982), which is based on the first difference equation. In this section, we provide the asymptotic properties of the AH estimator.

The first step in the AH estimator is to eliminate the individual effects using first differences. Let us rewrite the approximated model (2) by

$$ \begin{align*} y_{it}=\mu _{i}+x_{it}(p)^{\prime }\alpha (p)+u_{it,p} , \end{align*} $$

where $x_{it}(p)=(y_{i,t-1},\ldots ,y_{i,t-p})^{\prime }$ and $ \alpha (p)=( \alpha _{1},\ldots ,\alpha _{p})^{\prime } $ . Differencing to remove the individual specific effect $\mu _{i}$ yields

(3) $$ \begin{align} \Delta y_{it}=\alpha (p)^{\prime }\Delta x_{it}(p)+\Delta u_{it,p}, \end{align} $$

where $\Delta $ is the difference operator so that $\Delta y_{it}=y_{it}-y_{i,t-1}$ , $\Delta x_{it} (p) = x_{it} (p) - x_{i,t-1} (p),$ and $\Delta u_{it} = u_{it} - u_{i,t-1}$ .

The AH estimator is based on the following moment conditions. For $\Delta \epsilon _{it} = \epsilon _{it} - \epsilon _{i,t-1}$ ,

$$ \begin{align*} E(y_{i,t-s}\Delta \epsilon _{it})=0\text{ for }2\leq s\leq p+1 \text{ and } t= p+2, \dots, T. \end{align*} $$

For each t, there are p moment conditions. Because the dimension of $\alpha (p)$ is p, there are as many moment conditions as the parameters for each t. Given we cannot recover $\Delta \epsilon _{it}$ because of $b_{it}$ , we use $\Delta u_{it}$ , although this yields moment conditions that are only approximately valid. Solving the sample analog of the set of moment conditions, we obtain the AH estimator:

$$ \begin{align*} \hat{\alpha}_{AH}(p)=\left( \sum_{i=1}^{N} \sum_{t=p+2}^T Z_{it}(p)\Delta x_{it}^{\prime }(p)\right) ^{-1}\sum_{i=1}^{N} \sum_{t=p+2}^T Z_{it}(p)\Delta y_{it}, \end{align*} $$

where $Z_{it}(p)=(y_{i,t-2}, \dots , y_{i,t-p-1})^{\prime }$ .

We make the following additional assumption to analyze the asymptotic properties of the estimator. Let $\gamma _{j}=E(w_{it} w_{i,t-j})$ , where $w_{it}=y_{it}-\eta _{i}=\sum _{k=0}^{\infty }\psi _{k}\epsilon _{i,t-k}$ . Define

$$ \begin{align*} \Gamma _{\Delta }=E(Z_{it}(p)\Delta x_{it}(p)^{\prime })=\left( \begin{array}{cccc} \gamma _{1}-\gamma _{0} & \gamma _{0}-\gamma _{1} & \cdots & \gamma _{p-2}-\gamma _{p-1} \\ \gamma _{2}-\gamma _{1} & \gamma _{1}-\gamma _{0} & & \vdots \\ \vdots & & \ddots & \vdots \\ \gamma _{p}-\gamma _{p-1} & \cdots & \cdots & \gamma_{1}-\gamma _{0} \end{array} \right). \end{align*} $$

The invertibility requires that the process is not too persistent. Section 9 of the Supplementary Material demonstrates the invertibility when $w_{it}$ is an i.i.d. process. Typically, $\lambda _p$ is $O(\sqrt {p})$ and $\lambda _p>1$ , in general with $p>1$ . In Section 9 of the Supplementary Material, we derive the exact value of $\lambda _p$ in a simple case. Note that this assumption allows $\lambda _p$ to explode as $p\to \infty $ .

We first prove the consistency of $\hat {\alpha }_{AH}(p)$ .

Theorem 1. Suppose that Assumptions 1 and 2 are satisfied. If $N\rightarrow \infty $ , $T -p\rightarrow \infty , $ and $p\to \infty $ with $\lambda _p \sqrt {p} \sum _{k=p+1}^{\infty } |\alpha _k | \to 0$ and $\lambda _p^2 p^2 / (N (T-p)) \to 0$ , we have

$$ \begin{align*} ||\hat{\alpha}_{AH}(p)-\alpha (p)||\rightarrow _{p}0. \end{align*} $$

Next, we show the asymptotic normality of a linear combination of the estimated AR parameters. Let $\ell _{p}$ be an arbitrary sequence of $p\times 1$ vectors such that $0<M_{1}\leq ||\ell _{p}||^{2}=\ell _{p}^{\prime }\ell _{p}\leq M_{2}<\infty $ , for some constants $M_1, M_2>0$ .

Theorem 2. Suppose that Assumptions 1 and 2 are satisfied. If $N\rightarrow \infty $ , $T -p \rightarrow \infty , $ and $p\to \infty $ with $(T-p) \lambda _p^2 p / N^2 =O(1)$ , $\lambda _p^4 p^4 / (N(T-p)) \to 0$ , $\lambda _p \sqrt {N(T-p)p} \sum _{k=p+1}^{\infty } |\alpha _k | \to 0 $ , and $\lambda _p^2 p^{3/2} \sum _{k=p+1}^{\infty } |\alpha _k | \to 0$ , we have

$$ \begin{align*} \sqrt{N(T-p)}\left[ \ell_{p}^{\prime }\hat{\alpha}_{AH}(p)-\ell_{p}^{\prime }\alpha (p)\right] /v_{p,AH}\to_d N(0,1), \end{align*} $$

where $v_{p,AH}^{2}=\ell _{p}^{\prime }\Gamma _{\Delta }^{-1} B_p\Gamma _{\Delta }^{-1} \ell _p$ and $B_p=(T-p)^{-1} \sum _{t=p+2}^{T} \sum _{t'=\max (t-1,p+2)}^{\min (t+1,T)} E(\Delta \epsilon _{it}\Delta \epsilon _{it'} Z_{it}(p)Z_{it'}(p)^{\prime }) $ .

The theorems require some conditions regarding the rates at which N and T tend to infinity. These conditions, in particular $(T-p) \lambda _p^2 p / N^2 =O(1)$ , imply that N should be sufficiently large compared with T. The requirement of large N is expected given that IV estimation for dynamic panel data models is originally developed to perform well when N is large but T is small. Note that $v_{p,AH}$ is of order $\lambda _p$ and the convergence rate of the AH estimator is slower than $\sqrt {NT}$ . Roughly speaking, this slow convergence rate is caused by correlations across instruments which result in a weak set of moment conditions. The above theorem is established under $T-p \to \infty $ . An inspection of the proof reveals that we need a different approach to handle the case with $p=T$ . It is thus beyond the scope of the article.

We mainly examine conditions on N and T and let p be appropriately chosen when we discuss the asymptotic results, including those presented later. The empirical setting gives N and T, and they are not the choice of researchers. In contrast, p can be chosen. Thus, conditions on $ N $ and $ T $ are more relevant to examining whether a particular method is applicable in a given setting.

3.2 The GMM Estimator

As the second IV estimator, we consider the GMM estimator based on Holtz-Eakin et al. (Reference Holtz-Eakin, Newey and Rosen1988) and Arellano and Bond (Reference Arellano and Bond1991). For the GMM estimator, we apply the forward filter to the variables to eliminate the individual effects.Footnote ⁷ Let

$$ \begin{align*} y_{it}^{\ast }=\sqrt{\frac{T-t}{T-t+1}}\left( y_{it}-\frac{1}{T-t} (y_{i,t+1}+\dots +y_{iT})\right) , \end{align*} $$

and $x_{it}^{\ast }(p)$ and $u_{it,p}^{\ast }$ be similarly defined.Footnote ⁸ In this article, a variable with $ \ast $ superscript is a variable transformed by the forward filter even when it is not explicitly mentioned. By rewriting the model (2) in terms of the transformed variables, we have

(4) $$ \begin{align} y_{it}^* = x_{it}^* (p)^{\prime }\alpha (p) + u_{it,p}^* \end{align} $$

so that the individual effect $\mu _{i}$ is eliminated.

The GMM estimator exploits the following moment conditions:

$$ \begin{align*} E(y_{i,t-s}\epsilon _{it}^{\ast })=0\text{ for }s\geq 1. \end{align*} $$

We note that there are $T-p-1$ equations (one equation for each period) to be estimated and the equation for $y_{it}^{\ast }$ has $t-1$ instruments. Therefore, there are $\sum _{t=p+1}^{T-1}(t-1)=(T-2)(T-1)/2-(p-1)p/2$ moment conditions in total, and the number of moment conditions can be very large, even when T is not very large.

We now define the GMM estimator. Let $Z_{it}=(y_{i,t-1},\dots ,y_{i1})^{\prime }$ be the set of IV for the equation with $y_{it}^{\ast }$ as the dependent variable. The GMM estimator, $\hat {\alpha }_{G}(p)$ , is

$$ \begin{align*} \hat{\alpha}_{G}(p)=\left( \sum_{t=p+1}^{T-1}x_{t}^{\ast }(p)^{\prime }M_{t}x_{t}^{\ast }(p)\right) ^{-1}\sum_{t=p+1}^{T-1}x_{t}^{\ast }(p)^{\prime }M_{t}y_{t}^{\ast }, \end{align*} $$

where $M_{t}=Z_{t}(Z_{t}^{\prime }Z_{t})^{-1}Z_{t}^{\prime }$ .

To investigate the asymptotic properties of the GMM estimator, we employ the following two additional assumptions.

Assumption 3 imposes a tighter moment condition on individual effects. This assumption is used to control the variability of $ \sum _{t=p+1}^{T-1}x_{t}^{\ast }(p)^{\prime }M_{t}x_{t}^{\ast }(p)$ . Assumption 4 is satisfied, for example, when the true process is a finite order ARMA model. This assumption is employed to control the many-moment bias. The following theorem shows the consistency of $\hat {\alpha }_{G}(p)$ .

Theorem 3. Suppose that Assumptions 1, 3, and 4 are satisfied. If $N \to \infty $ , $T\to \infty ,$ and $p\to \infty $ with $T/N \to 0$ , $p^2 /T \to 0$ , and $ p^{1/2}\sum _{k=p+1}^{\infty }|\alpha _{k}|\rightarrow 0$ as $p\rightarrow \infty $ , we have

$$ \begin{align*} || \hat{\alpha}_G(p)-\alpha (p) || \to_p 0. \end{align*} $$

Next, we show the asymptotic normality of a linear combination of the estimated AR parameters. Let $v_{p}^{2}=\sigma ^{2}\ell _{p}^{\prime }\Gamma _{p}^{-1}\ell _{p}$ , where

$$ \begin{align*} \Gamma_p = \begin{pmatrix} \gamma_0 & \gamma_1 & \dots & \gamma_{p-1} \\ \gamma_1 & \gamma_0 & \dots & \gamma_{p-2} \\ \dots & \dots & \dots & \dots \\ \gamma_{p-1} & \gamma_{p-2} & \dots & \gamma_0 \end{pmatrix} \end{align*} $$

is the variance–covariance matrix of the vector $(w_{it},\dots ,w_{i,t-p+1})^{\prime }$ . Note that Assumption 1(iii) guarantees that the maximum eigenvalue of $\Gamma _p^{-1}$ is bounded, which implies that $v_p^2$ is bounded away from zero. The following theorem provides the asymptotic normality of $\ell _{p}^{\prime }\hat {\alpha }_{G}(p)$ .

Theorem 4. Suppose that Assumptions 1, 3, and 4 are satisfied. If $N \to \infty $ , $T\to \infty ,$ and $p \to \infty $ with $\sqrt {NT} \sum _{k=p+1}^{\infty }|\alpha _k| \to 0$ , $p^2T /N \to 0, $ and $p^3 \log T /T \to 0 $ , we have

$$ \begin{align*} \sqrt{N(T-p)}\left[\ell_p^{\prime }\hat{\alpha}_G(p)-\ell_p^{\prime }\alpha (p) \right] /v_{p} \to_d N(0,1). \end{align*} $$

To assess the effect of an infinite lag order, we compare Theorem 4 with Theorem 2 of Alvarez and Arellano (Reference Alvarez and Arellano2003). The assumptions underlying our Theorem 4 are similar to those in their Theorem 2, which requires $N,T \to \infty $ , $(\log T)^2 /N \to 0$ , and $T/N \to c$ . However, due to the large lag order in our setting, we may not be able to accommodate as large a value of T as Alvarez and Arellano (Reference Alvarez and Arellano2003). Both results achieve the same rate of convergence $\sqrt {NT}$ for the asymptotic distribution, which matches our Theorem 2’s rate of $\sqrt {N(T-p)}$ since p is of smaller order than T. The key difference lies in the additional conditions our theorem imposes on the lag order p: $\sqrt {NT} \sum _{k=p+1}^{\infty }|\alpha _k| \to 0$ , $p^2T /N \to 0$ , and $p^3 \log T /T \to 0$ . These conditions ensure that p remains sufficiently smaller than T, ruling out choices such as $p=T$ .

It should be noted that our asymptotic normality results for the GMM estimator are derived under the assumption that N grows at a rate faster than T. In general, the GMM estimator suffers from bias caused by the presence of many moment conditions. For instance, the GMM estimator in Alvarez and Arellano (Reference Alvarez and Arellano2003) exhibits an asymptotic bias of $(1+\alpha )/N$ . However, our assumption on N and T allows us to ignore the many-moment bias. If this assumption does not hold, the asymptotic distribution would not be centered around zero. As an alternative, we may be able to relax the condition on the relative magnitude of N and T and derive the asymptotic distribution that explicitly evaluates the many moments bias term as in Alvarez and Arellano (Reference Alvarez and Arellano2003). However, deriving the exact form of many-moment bias in the current setting is very difficult and should be considered as a separate analysis. Instead of deriving the exact formula and correcting the bias of the GMM estimator, in the next section, we consider the DFIV estimator that is free from many-moment bias.

3.3 The Double Filter Instrumental Variables Estimator

As the third IV estimator, we consider the DFIV estimator. The idea behind the DFIV estimator is similar to that behind the GMM estimator except for the choice of instruments. The instruments are constructed by subtracting the average of past realizations from the regressors. Let $z_{it}(p)=(y_{i,t-1},\dots ,y_{i,t-p})'$ and

$$ \begin{align*} h_{it}(p)=\sqrt{\frac{T-t}{T-t+1}}\left( z_{it}(p)-\frac{1}{t-p-1} (z_{i,t-1}(p)+\dots +z_{i,p+1}(p))\right). \end{align*} $$

The choice of $h_{it}(p)$ as instruments for estimating equation (4) can be motivated by the following observation. In finite order AR models, the optimal instruments for $x_{it}^{\ast }(p)$ can be approximated by $( y_{i,t-1} - \eta _i , \dots , y_{i,t-p} - \eta _i)'$ when T is large (see, Hayakawa, Reference Hayakawa2009, and also Arellano, Reference Arellano2016). The instrument $h_{it}(p)$ may be regarded as an approximation to this optimal instrument, using the average of past realizations of $y_{it}$ in place of $\eta _{i}$ . We use only the past realizations so that the moment conditions become valid even when T is small. This estimator is called the double filter (DF) IV estimator because both the regressor and the instrument are filtered (but using different filters).

The DFIV estimator is given by

$$ \begin{align*} \hat{\alpha}_{DF}(p)=\left( \sum_{t=p+2}^{T-1}h_{t}(p)^{\prime }x_{t}^{\ast }(p)\right) ^{-1}\sum_{t=p+2}^{T-1}h_{t}(p)^{\prime }y_{t}^{\ast }. \end{align*} $$

Hayakawa (Reference Hayakawa2009) shows that for finite order AR models, this estimator is consistent regardless of the relative magnitude of N and T and is efficient when $T\rightarrow \infty $ under Gaussianity. We apply the DFIV estimator to estimate the infinite order AR models.

The following theorems provide the consistency of $\hat {\alpha }_{DF}(p)$ and the asymptotic normality of $\ell _{p}^{\prime }\hat {\alpha }_{DF}(p)$ .

Theorem 5. Suppose that Assumption 1 is satisfied. If $N \to \infty $ , $T\to \infty ,$ and $p\to \infty $ with $ p^2 /T \to 0$ and $p^{1/2}\sum _{k=p+1}^{\infty }|\alpha _{k}|\rightarrow 0$ as $p\rightarrow \infty $ , we have

$$ \begin{align*} || \hat{\alpha}_{DF}(p)-\alpha (p) || \to_p 0. \end{align*} $$

Theorem 6. Suppose that Assumption 1 is satisfied. If $N \to \infty $ , $T\to \infty ,$ and $p \to \infty $ with $ \sqrt {NT} \sum _{k=p+1}^{\infty }|\alpha _k| \to 0$ and $p^3/T \to 0 $ , we have

$$ \begin{align*} \sqrt{N(T-p)}\left[\ell_p^{\prime }\hat{\alpha}_{DF}(p)-\ell_p^{\prime }\alpha (p) \right] /v_{p} \to_d N(0,1). \end{align*} $$

Theorem 6 shows that the convergence rate and asymptotic variance of the DFIV estimator are identical to those of the GMM estimator. The most notable result is that, in contrast to the GMM estimator, the asymptotic normality of the DFIV estimator does not require any condition on the relative rates of growth for N and T. This implies that the same asymptotic variance can be obtained under more general settings by the DFIV estimator. Therefore, the DFIV estimator may behave better than the GMM estimator when T is relatively large. The theorem requires that p be chosen to satisfy $p^3/T\to 0$ .Footnote ⁹ Thus, setting $p=T$ is not allowed, and p should be sufficiently smaller than T.

3.4 Extension With Exogenous Regressors

In this section, we consider a more general model that includes additional exogenous regressors, which is crucial for the scope of empirical applications. We estimate the model using our preferred DFIV estimator.

We assume that panel data $\{ \{y_{it}\}_{t=1}^{T}\}_{i=1}^{N}$ are generated from an AR process of possibly infinite order with individual specific effects and exogenous regressors:

(5) $$ \begin{align} y_{it}=\mu_{i}+\sum_{k=1}^{\infty }\alpha _{k}y_{i,t-k}+ g_{it}'\beta + \epsilon _{it}. \end{align} $$

We make the following set of assumptions.

Assumption 5(i) extends the i.i.d assumption of Assumption 1(i) to include $g_{it}$ . We note that while $\epsilon _{it}$ is i.i.d. both across individuals and over time, $g_{it}$ is allowed to be serially correlated. Assumption 5(ii) states that $g_{it}$ is strictly exogenous. The time-series properties of $g_{it}$ are mentioned in Assumption 5(iii). It states that the exogenous regressors are strictly stationary and of short memory. This assumption guarantees that $y_{it}$ is strictly stationary and of short memory. To see this, we introduce an infinite order representation of $y_{it}$ :

$$ \begin{align*} y_{it}=\frac{\mu_i + E(g_{it})'\beta }{1-\sum_{k=1}^{\infty }\alpha _{k} }+ \sum_{k=0}^{\infty }\psi _{k}( (g_{i,t-k} - E(g_{i,t-k}))'\beta +\epsilon_{i,t-k}). \end{align*} $$

The innovation term $(g_{i,t-k} - E(g_{i,t-k}))'\beta +\epsilon _{i,t-k}$ is strictly stationary and of short memory under this assumption together with the i.i.d. assumption of $\epsilon _{it}$ . Because $\psi _k$ is absolutely summable, $y_{it}$ is strictly stationary and of short memory. Assumption 5(iv) strengthens (iii) by imposing the uniform boundedness of the long-run variances of $g_{it}$ .

To estimate (5), we consider the following approximated model:

(6) $$ \begin{align} y_{it}=\mu _{i}+\sum_{k=1}^{p}\alpha _{k}y_{i,t-k}+ g_{it}'\beta + u_{it,p}, \end{align} $$

where $u_{it,p}=b_{it,p}+\epsilon _{it}$ with the “truncation error” $b_{it,p}=\sum _{k=p+1}^{\infty }\alpha _{k}y_{i,t-k}$ .

This approximated model is estimated by the DFIV estimator. By rewriting the model (6) in terms of the transformed variables, we have

$$ \begin{align*} y_{it}^* = x_{it}^* (p)^{\prime }\alpha (p) + g_{it}^{\ast \prime} \beta + u_{it,p}^* \end{align*} $$

so that the individual effect $\mu _{i}$ is eliminated. The DFIV estimator is given by

$$ \begin{align*} \begin{pmatrix} \hat{\alpha}_{DF}(p) \\ \hat{\beta}_{DF} \end{pmatrix} =\left( \sum_{t=p+2}^{T-1} \begin{pmatrix} h_{t}(p)' \\ g_{t}^{\ast \prime} \end{pmatrix} \begin{pmatrix} x_{t}^{\ast }(p) & g_{t}^{\ast} \end{pmatrix} \right) ^{-1}\sum_{t=p+2}^{T-1} \begin{pmatrix} h_{t}(p)' \\ g_{t}^{\ast \prime} \end{pmatrix} y_{t}^{\ast }, \end{align*} $$

where $g_{t}^* = (g_{1t}^* ,\dots , g_{Nt}^*)'$ .

The following theorems provide the consistency of the DFIV estimator and the asymptotic normality of its linear combination.

Theorem 7. Suppose that Assumptions 1 and 5 are satisfied. If $N \to \infty $ , $T\to \infty ,$ and $p\to \infty $ with $p^2 /T \to 0$ , we have

$$ \begin{align*} \left\Vert \begin{pmatrix} \hat{\alpha}_{DF}(p) \\ \hat{\beta}_{DF} \end{pmatrix}- \begin{pmatrix} \alpha (p) \\ \beta \end{pmatrix} \right\Vert \to_p 0. \end{align*} $$

Theorem 8. Suppose that Assumptions 1 and 5 are satisfied. If $N \to \infty $ , $T\to \infty ,$ and $p \to \infty $ with $\sqrt {NT} \sum _{k=p+1}^{\infty }|\alpha _k| \to 0$ and $p^3/T \to 0 $ , we have

$$ \begin{align*} \sqrt{N(T-p)}\left[\ell_p^{\prime }\begin{pmatrix} \hat{\alpha}_{DF}(p) \\ \hat{\beta}_{DF} \end{pmatrix} -\ell_p^{\prime }\begin{pmatrix} \alpha (p) \\ \beta \end{pmatrix} \right] /v_{p}^x \to_d N(0,1), \end{align*} $$

where $v_{p}^x$ is defined in Appendix E.

This theorem states that the DFIV estimator is asymptotically normal, and its convergence rate is $\sqrt {NT}$ even for models with exogenous regressors.

3.5 Standard Errors for the Estimators

In this section, we discuss how to construct standard errors for the estimators. We note that the estimators are asymptotically normal so that standard errors can be constructed in a usual manner once the asymptotic variance is estimated. For the DFIV estimator, a cluster-robust standard error is also developed.

We first consider the case of the AH estimator. A consistent estimator of the asymptotic variance of $\hat {\alpha }_{AH}(p)$ can be constructed by replacing $\Gamma _{\Delta }$ and $B_p$ in $v_{p,AH}^{2}=\ell _p'\Gamma _{\Delta }^{-1}B_p \Gamma _{\Delta }^{-1}\ell _p$ with their estimators. A consistent estimator for $\Gamma _{\Delta }$ is given by $\hat {\Gamma }_{\Delta } =\sum _{i=1}^{N}\sum _{t=p+2}^{T} Z_{it}(p) \Delta x_{it}(p)^{\prime }/(N(T-p))$ . We propose estimating $B_p$ by

$$ \begin{align*} \hat{B}_p=&\left( \frac{1}{2N(T-p)} \sum_{i=1}^N \sum_{t=p+2}^T (\Delta \hat{u}_{it})^2\right) \\ & \times \frac{1}{N(T-p)} \sum_{i=1}^{N} \sum_{t=p+2}^{T} \sum_{t'=\max (t-1,p+2)}^{\min(t+1,T)} (2- 3\mathbf{1}_{(|t-t'|=1)} )Z_{it}(p)Z_{it'}(p)^{\prime }, \end{align*} $$

where $\Delta \hat {u}_{it}=\Delta y_{it}-\Delta x_{it}(p)^{\prime }\hat {\alpha }_{AH}(p)$ and $\mathbf {1}_{(\cdot )}$ is an indicator function.

Next, we consider the cases of the GMM and DFIV estimators. According to Theorems 4 and 6, $\hat \alpha _G (p)$ and $\hat \alpha _{DF} (p)$ share a common asymptotic variance given by $v_p^2 = \sigma ^{2}\ell _{p}^{\prime }\Gamma _{p}^{-1}\ell _{p}$ . Natural estimators for $v_p^2$ are constructed by

$$ \begin{align*} \hat{v}_{p,G}^{2} = \left( \frac{1}{N(T-p)}\sum_{i=1}^{N} \sum_{t=p+1}^{T-1}(y_{it}^{\ast }-x_{it}^{\ast }(p)^{\prime }\hat{\alpha}_{G}(p))^{2}\right) \ell _{p}^{\prime }(\hat{\Gamma}_{p}^{G})^{-1}\ell _{p}, \end{align*} $$

where

$$ \begin{align*} \hat{\Gamma}_{p}^{G}=\frac{1}{N(T-p)}\sum_{t=p+1}^{T-1}x_{t}^{\ast }(p)^{\prime }M_{t}x_{t}^{\ast }(p) \end{align*} $$

for the GMM estimator, and

$$ \begin{align*} \hat{v}_{p,DF}^{2} = & \left( \frac{1}{N(T-p)}\sum_{i=1}^{N} \sum_{t=p+2}^{T-1}(y_{it}^{\ast }-x_{it}^{\ast }(p)^{\prime }\hat{\alpha}_{DF}(p))^{2}\right)\ell _{p}^{\prime }(\hat{\Gamma}_{p}^{DF})^{-1} \\ &\times \frac{1}{N(T-p)} \sum_{t=p+2}^{T-1} h_t(p)' h_t(p) (\hat{\Gamma}_{p}^{DF '})^{-1}\ell _{p}, \end{align*} $$

where

$$ \begin{align*} \hat{\Gamma}_{p}^{DF}=\frac{1}{ N(T-p)}\sum_{t=p+2}^{T-1}h_{t}(p)^{\prime }x_{t}^{\ast }(p) \end{align*} $$

for the DFIV estimator.

For the DFIV estimator, we also develop cluster-robust asymptotic variance estimators (Arellano, Reference Arellano1987) to calculate standard errors. Although the theoretical framework assumes homoskedasticity, it is important to account for heteroskedasticity in real data analyses. Additionally, while the innovation term in the panel AR( $\infty $ ) process is intended to be serially uncorrelated, truncation error may introduce some serial dependence in finite samples. Therefore, employing robust asymptotic variance estimators is a sensible approach for practical applications. Here, we consider cluster-robust standard errors for the DFIV estimator only because it is our preferred estimator.

The cluster-robust variance estimator for the DFIV estimator is

$$ \begin{align*} \tilde{v}_{p,DF}^{2} = \ell _{p}' (\hat{\Gamma}_{p}^{DF})^{-1} \left( \frac{1}{N(T-p)} \sum_{i=1}^N \sum_{t=p+2}^{T-1} \sum_{t' = p+2}^{T-1} \hat u_{it, D}^* \hat u_{it', D}^* h_{it}(p) h_{it'}(p) \right) (\hat{\Gamma}_{p}^{DF \prime})^{-1} \ell _{p}, \end{align*} $$

where

$$ \begin{align*} \hat u_{it', D}^* = y_{it}^{\ast }-x_{it}^{\ast }(p)^{\prime }\hat{\alpha}_{DF}(p) \end{align*} $$

and

$$ \begin{align*} \hat{\Gamma}_{p}^{DF}=\frac{1}{N(T-p)}\sum_{t=p+2}^{T-1}h_{t}(p)^{\prime }x_{t}^{\ast }(p). \end{align*} $$

The next theorem shows that $\hat {v}_{p,DF}^{2}$ is a consistent variance estimator.

3.6 Comparison across the IV Estimators

We now compare the theoretical properties of the three IV estimators considered in the previous sections. All the three estimators (AH, GMM, and DFIV) are consistent and asymptotically normal. However, the asymptotic variances and the bias properties differ as do the required conditions on the relative magnitude of N and T. We prefer the DFIV estimator because its asymptotic bias and variance are small, and it does not require strong assumptions on the ratio of N and T.

We first compare the asymptotic variances of the three estimators. The GMM and DFIV estimators have a common asymptotic variance. However, the AH estimator has a different asymptotic variance and is thus inefficient. Moreover, while $v_{p} = O(1)$ , we may have $v_{AH,p} = O(\lambda _p)$ which may diverge. Thus, even the rate of convergence for the AH estimator may be slower than the other two estimators.

Next, we consider the bias properties. While we impose conditions such that bias does not appear in the asymptotic distributions, it is expected that the AH and the DFIV estimators are less biased than the GMM estimator in finite samples. To see this, we utilize a convenient decomposition formula introduced in Lee et al. (Reference Lee, Okui and Shintani2018).Footnote ¹⁰ As $u_{it,p}=b_{it,p}+\epsilon _{it}$ , the differenced error is given as $\Delta u_{it,p}=\Delta b_{it,p}+\Delta \epsilon _{it}$ . Thus, the total bias of the AH estimator can be decomposed as

$$ \begin{align*} E\left( \hat{\alpha}_{AH}(p) - \alpha_{AH}(p) \right) &= E\left( (\hat{\Gamma}_{\Delta})^{-1}\frac{1}{N(T-p)}\sum_{i=1}^{N} \sum_{t=p+2}^{T}Z_{it}(p)\Delta u_{it,p} \right) \\ &= \underbrace{E\left( (\hat{\Gamma}_{\Delta})^{-1}\frac{1}{N(T-p)} \sum_{i=1}^{N}\sum_{t=p+2}^{T}Z_{it}(p)\Delta b_{it,p} \right)}_{\text{ truncation bias}}\\&\quad + \underbrace{E\left( (\hat{\Gamma}_{\Delta})^{-1}\frac{1}{N(T-p)}\sum_{i=1}^{N}\sum_{t=p+2}^{T}Z_{it}(p)\Delta \epsilon_{it} \right) }_{\text{fundamental bias}}. \end{align*} $$

The first term is the bias that arises because we estimate the AR model with a truncated lag length, not the true infinite order AR model. We refer to this term as “truncation bias.” We call the second term “fundamental bias” because this part of the bias is present even if we estimate the true finite order AR model with the correct lag length. While the truncation bias may not be negligible in finite samples, it vanishes in our asymptotic analysis because of our assumption that $|\alpha _{k}|\rightarrow 0$ sufficiently fast. We expect that the AH estimator has small fundamental bias because $E( Z_{it}(p)\Delta \epsilon _{it})=0$ . For the GMM estimator, as the transformed error $u^{\ast }_{it,p}$ is the sum of $ b^{\ast }_{it,p}=\sqrt {(T-t)/(T-t+1)}( b_{it,p}-\sum _{\tau =t+1}^{T}b_{i\tau ,p}/(T-t))$ and $\epsilon ^{\ast }_{it}=\sqrt {(T-t)/(T-t+1)}( \epsilon _{it}-\sum _{\tau =t+1}^{T}\epsilon _{i\tau }/(T-t))$ , the total bias can be decomposed as

$$ \begin{align*} E\left( \hat{\alpha}_{G}(p) - \alpha (p) \right) &= E\left( (\hat{\Gamma}_p^G)^{-1}\frac{1}{N(T-p)}\sum_{t=p+1}^{T}x^{\ast} _{t}(p)^{\prime }M_t u^{\ast}_{t,p} \right) \\ &= \underbrace{E\left( (\hat{\Gamma}_p^G)^{-1}\frac{1}{N(T-p)} \sum_{t=p+1}^{T}x^{\ast}_{t}(p)^{\prime }M_t b^{\ast}_{t,p} \right)}_{\text{ truncation bias}}\\ &\quad + \underbrace{E\left( (\hat{\Gamma}_p^G)^{-1}\frac{1}{N(T-p) }\sum_{t=p+1}^{T}x^{\ast}_{t}(p)^{\prime }M_t\epsilon^{\ast}_{t} \right) }_{\text{fundamental bias}}. \end{align*} $$

The GMM estimator is biased because $E(x^{\ast }_{t}(p)^{\prime }M_t\epsilon ^{\ast }_{t}) \neq 0$ . A large T leads to many moment conditions so that $M_t$ has a large trace, which results in a large $E(x^{\ast }_{t}(p)^{\prime }M_t\epsilon ^{\ast }_{t}) $ . That is, the GMM estimator suffers from “many-moment bias,” which is included in fundamental bias. This is the reason we employ $T/N \to 0$ to derive the asymptotic distribution of the GMM estimator. The bias for the DFIV estimator can be decomposed similarly. The fundamental bias for the DFIV estimator vanishes asymptotically without imposing any condition on the relative rate for N and T. This is because $E(h_t (p)' \epsilon _t^* )=0 $ .

All estimators require different assumptions on the relative magnitude of N and T.Footnote ¹¹ Any violation of the conditions on the relative magnitude of N and T would change the asymptotic properties. The AH estimator requires that N grows sufficiently faster than T as discussed in Section 3.1. The GMM estimator also requires that N grows faster than T (see Section 3.2). Note that while both the AH and GMM estimators require N to be relatively large, the underlying reasons differ. The GMM estimator suffers from the many moments bias when N is smaller than T. For the AH estimator, the underlying mechanism is more tenuous. Roughly speaking, the condition is used to control the variability of $Z_{it}(p)$ . In contrast, the DFIV estimator does not restrict the relative magnitude of N and T (see Section 3.3).

Overall, the DFIV estimator is expected to exhibit the smallest fundamental bias and is more efficient than the AH estimator. The AH estimator is also expected to exhibit the smallest fundamental bias, but it is inefficient. Moreover, the DFIV estimator is asymptotically normal without imposing any conditions on the relative rates of N and T. The GMM estimator is as efficient as the DFIV estimator but suffers from “many-moment bias,” necessitating that $T/N \rightarrow 0$ to disregard this bias.

These theoretical results lead us to recommend the DFIV estimator. We also investigate the finite-sample properties of these estimators through simulations whose results are presented in Section 4.

3.7 Comparison between IV Estimation and FE Estimation

In this article, we have considered the estimators based on the IV approach. Another popular estimation approach in the literature on dynamic panel data models is FE estimation. In this section, we compare the IV estimation with the FE estimation considered by Lee et al. (Reference Lee, Okui and Shintani2018). We argue that the IV estimation would be more suitable than the FE estimation when N is relatively large, which is the case in many empirical applications. We focus on comparison with the GMM and DFIV estimators because the AH estimator is less efficient.

Using the same framework as the current article, Lee et al. (Reference Lee, Okui and Shintani2018) considered the FE estimator and its bias-corrected version. The FE estimator can also be written as OLS in orthogonal deviations (see Arellano and Bover, Reference Arellano and Bover1995; Alvarez and Arellano, Reference Alvarez and Arellano2003). Applying OLS to (4) yields the FE estimator:

$$ \begin{align*} \hat{\alpha}_{F}(p)=\left( \sum_{t=p+1}^{T} x^{\ast}_{t}(p)^{\prime }x^{\ast}_{t}(p)\right) ^{-1}\sum_{t=p+1}^{T}x^{\ast}_{t}(p)^{\prime }y^{\ast}_{t}. \end{align*} $$

We note that $\hat {\alpha }_{F}(p)= \alpha (p) + \left ( \sum _{t=p+1}^{T} x^{\ast }_{t}(p)^{\prime }x^{\ast }_{t}(p)\right ) ^{-1}\sum _{t=p+1}^{T}x^{\ast }_{t}(p)^{\prime } u^{\ast }_t$ . For any value of T, $E(y_{i,t-s}^{\ast }u_{it}^{\ast })\neq 0$ because $y_{i,t-s}^{\ast }$ contains $y_{i,t-s}, \dots , y_{i,T}$ and $u_{it}^{\ast }$ contains $u_{i,t+1},\dots ,u_{i,T}$ . As a consequence, $E (x^{\ast }_{t}(p)^{\prime } u^{\ast }_t) \neq 0$ , and $\hat {\alpha }_{F}(p)$ is inconsistent for fixed T as N goes to infinity. Conversely, the bias disappears as $T \to \infty $ . The FE estimator is shown to be consistent and asymptotically normal (Theorems 1 and 2 in Lee et al., Reference Lee, Okui and Shintani2018) under the double asymptotic, however it is asymptotically biased. Lee et al. (Reference Lee, Okui and Shintani2018) proposed the bias correction by first estimating the bias by $\hat {B} = (\hat {\sigma }^{2}/(1-\sum _{k=1}^{p}\tilde { \alpha }_{k}))\iota _{p}/(T-p)$ , where $\hat {\sigma }^{2}$ is a consistent estimator $\sigma ^2$ , $\iota _{p}$ is the $p\times 1$ vector of ones, and $\tilde {\alpha }_{k}$ ’s are FE estimators for $\alpha _{k}$ ’s. Then, the bias-corrected FE (BCFE) estimator is constructed as $\hat {\alpha }_{BF}(p)=\hat {\alpha }_{F}(p)+(\hat {\Gamma }_{p}^{F})^{-1}\hat {B}$ , where $\hat {\Gamma }_{p}^{F} = \sum _{t=p+1}^{T} x^{\ast }_{t}(p)^{\prime }x^{\ast }_{t}(p)/ (N(T-p))$ . The BCFE estimator $\hat {\alpha }_{BF}(p)$ is shown to be consistent: $|| \hat {\alpha }_{BF}(p)-\alpha (p) || \to _p 0$ , and asymptotically normal:

$$ \begin{align*} \sqrt{N(T-p)}\left[\ell_p^{\prime }(\hat{\alpha}_{BF}(p)-\alpha (p)) \right] /v_{p}\to_d N(0,1). \end{align*} $$

This result also shows that the BCFE estimator can effectively eliminate the asymptotic bias. The asymptotic variance $v_p$ is identical to that of the GMM and DFIV estimators (see Lee et al., Reference Lee, Okui and Shintani2018 for details).

The key difference between the FE and IV estimators is the moment conditions on which they are based. As discussed, the FE estimator is based on the moment condition $E (x_t^*(p)' \epsilon _t^* ) =0$ , which is only valid under $T\to \infty $ . This causes a bias which corresponds to the well-known incidental parameter bias of the FE estimator in dynamic panel data models with finite AR lags (see, e.g., Nickell, Reference Nickell1981; Kiviet, Reference Kiviet1995; Hahn and Kuersteiner, Reference Hahn and Kuersteiner2002; Lee, Reference Lee2012). The BCFE estimator reduces this bias, but still requires a condition that T is sufficiently large. In contrast, the moment conditions for the IV estimators given by

$$\begin{align*}E(y_{i,t-s} \epsilon_{it}^{\ast})=0\;\;\;\text{for } s \ge 1 \end{align*}$$

hold even when T is fixed. This means that the IV estimators based on such moment conditions will be consistent for a fixed T as N goes to infinity (Arellano and Bond, Reference Arellano and Bond1991; Arellano and Bover, Reference Arellano and Bover1995).

The theory thus suggests that when N is large, the IV approach provides more sensible estimates. Indeed, Lee et al. (Reference Lee, Okui and Shintani2018) show that the BCFE estimator requires $p^2 N/T^3 \to 0$ . Among the IV approach, the DFIV estimator is more preferable. The GMM estimator requires that N grows at a rate faster than T; otherwise, the bias will not vanish asymptotically. In contrast, the DFIV estimator is free of any asymptotic bias without imposing any condition on the relative rate of growth for N and T.

We investigate the finite-sample properties of these estimators through simulations in Section 4. Overall, the BCFE estimator has small bias and high precision but poor coverage probability, and it requires a sufficiently large T. In contrast, the IV estimators exhibit small bias and good coverage probability, but their finite sample distributions are more dispersed, and the GMM estimator requires a sufficiently large N.

Thus, we can conclude that if our concern is coverage probability or bias, then the DFIV estimator is recommended. On the other hand, if we care primarily about precision, then we recommend the BCFE estimator if T is not small compared to N and the GMM estimator when N is large and T is small.

3.8 Lag Selection

For the proposed estimators to be used in empirical applications, we need to choose the lag order of the approximated model, p. We consider the general-to-specific rule, which uses the t-statistic for the coefficient of the highest lag order in the model to test for the significance of the coefficients in each step.Footnote ¹²

Specifically, let $e_p$ be the $p\times 1$ vector whose p-th element is $1$ and other elements are zero. Let

$$ \begin{align*} t_p( \hat{\alpha} (p)) = \sqrt{NT} e_p^{\prime }\hat{\alpha} (p) / \hat{v}_p, \end{align*} $$

where $\hat {\alpha } (p)$ and $\hat {v}_p$ are estimators of $\alpha (p)$ and $ v_p$ with $\ell _p =e_p$ , respectively. The statistic $t_p (\hat {\alpha } (p)) $ is the t-test statistic for the null hypothesis $\alpha _p =0$ based on estimator $\hat {\alpha } (p)$ .

We a priori set the maximum value of possible p, denoted $p_{\max }$ . Then, the general-to-specific procedure chooses $\hat {p}$ as the maximum value of p such that $| t_p (\hat {\alpha } (p))|> z_{0.5\alpha } $ , where $z_{0.5\alpha }$ is the upper $0.5\alpha $ quantile of the standard normal distribution, for $p= 1, 2, \dots , p_{\max } $ .

We make the following additional assumptions.

These assumptions allow us to use the high-dimensional central limit theorem and the Fuk–Nagaev inequality. Assumption 6 states that the tail of the distribution of the innovation term $\epsilon _{it}$ vanishes exponentially. This assumption guarantees that any finite moment of $\epsilon _{it}$ exists and is stronger than the part of Assumption 1(i) related to $\epsilon _{it}$ . It also implies that $w_{it}$ has a distribution whose tail decays exponentially fast. Assumption 6(ii) imposes the mixing condition on $w_{it}$ . Given that $w_{it}$ is a linear process with i.i.d. innovations $\epsilon _{it}$ , this condition can be implied by some conditions on the distribution of $\epsilon _{it}$ and the properties of $\alpha _{k}, k = 1, \dots $ . However, such conditions tend to be complicated (see, e.g., Withers, Reference Withers1981) and we decide to state the assumptions in terms of $w_{it}$ directly.

The following theorem shows that the lag order selected by the general-to-specific procedure achieves an appropriate rate. The proof is in the Supplementary Material. We focus on the case in which $\hat \alpha (p) = \hat \alpha _{DF} (p)$ . Let $\hat p$ be the lag order chosen by the general-to-specific procedure for the DFIV estimator.

We set $p_{\max }$ such that the order of $p_{\max }$ satisfies the conditions for the asymptotic normality of each estimator. In the simulation, we set $p_{\max }=[12(T/100)^{1/4}] (=O(T^{1/4})),$ where $[x]$ is the integer part of x. In the case of least squares estimation, the order of p chosen by the sequential t tests is known to be the same as that of $p_{\max }$ (see Ng and Perron, Reference Ng and Perron1995; Lee et al., Reference Lee, Okui and Shintani2018).

4.1 Design 1: Basic Comparisons

We generate samples from the ARMA(1,1) model of the following form:

$$\begin{align*}y_{it}=\eta _{i}+\phi y_{i,t-1}+\epsilon _{it}+\theta \epsilon _{i,t-1}, \end{align*}$$

where $\phi =\{0.5,0.99\}$ , $\theta =0.4,$ and $\eta _{i}\sim N(0,1)$ is independent across i, $\epsilon _{it}\sim N(0,1)$ is independent across i and t.Footnote ¹³ The individual effect $\eta _{i}$ and idiosyncratic error $\epsilon _{it}$ are also independently drawn. For each process, $y_{i0}$ ’s are generated from the (conditional) stationary distribution:

$$\begin{align*}y_{i0}|\eta _{i}\sim N\left( \frac{\eta _{i}}{1-\phi },\frac{1+\theta ^{2}+2\phi \theta }{1-\phi ^{2}}\right). \end{align*}$$

The pairs of N and T we consider are taken from the set $\{25,50,100\}$ . All the Monte Carlo simulation results are based on 10,000 replications.

We estimate the first AR coefficient $\alpha _{1}$ and the sum of the AR coefficients (SAR) $ \sum _{k=1}^{\infty }\alpha _{k}$ using four alternative estimators: (i) the BCFE estimator $\hat {\alpha }_{BF}(p)$ ; (ii) the AH estimator $\hat {\alpha }_{AH}(p)$ ; (iii) the GMM estimator $\hat {\alpha }_{G}(p)$ ; and (iv) the DFIV estimator $\hat {\alpha }_{DF}(p)$ . When $\phi =0.5$ (DGP1), true $\alpha _{1}$ and SAR are 0.9 and 0.643, respectively. When $\phi =0.99$ (DGP2), the impulse response function is hump-shaped with true $\alpha _{1}$ being 1.390 and the process becomes highly persistent with the true SAR being near unity at 0.993. As illustrated in Figure 1, the autocorrelation function under DGP2 displays a markedly slow rate of decay, indicating extremely high persistence in the series.

Figure 1

Autocorrelation function of DGP1 and DGP2.

For the choice of lag length p in the approximated AR models, we consider both the fixed and automatically selected rules. For the fixed rule, we follow a conventional rule of thumb from the time-series literature and use $ p=[12(T/100)^{1/4}],$ where $[x]$ is the integer part of x. This fixed rule provides $p=8$ , $10,$ and $12$ for $T=25$ , $50,$ and $100$ , respectively. The automatic lag selection rule corresponds to the general-to-specific procedure described in Section 3.8 with the maximum lag set at $p_{\max }=[12(T/100)^{1/4}]$ and the significance level set at $\alpha =0.1$ . This implies that the automatic procedure always selects lags shorter than or equal to those using the fixed rule. At the same time, it should be noted that both the fixed and automatically selected rules satisfy the required conditions in the theoretical analysis.

Table 1 shows the median bias, interquartile range (iqr), and coverage probability of an asymptotic 95% confidence interval when $N=100$ and $T=\{25,50,100\}$ .Footnote ¹⁴ The interquartile range (iqr) represents the difference between the 75th and 25th percentiles and is robust to outliers. The results clearly illustrate the bias properties of the estimators and are consistent with the theoretical predictions. For DGP1, the AH and the DFIV estimators have smaller bias than the GMM estimator. The bias of the GMM estimator is likely from the many-moment bias and higher-order bias not covered in our theoretical analysis. The bias of the DFIV estimator is the smallest among almost all the estimators for DGP1. The bias-correction procedure suggested in Lee et al. (Reference Lee, Okui and Shintani2018) reduces the bias of the FE estimator. The bias of the BCFE estimator is smaller than that of the GMM estimator for DGP1, and is comparable to that of the DFIV estimator under automatic lag selection for DGP2. All estimators have larger bias under DGP2, but it quickly decreases as p increases. Overall, the choice of lag selection method has little effect on the relative size of the median bias among estimators, whereas the selected lags from the automatic procedure clearly depend on the DGPs and estimators.

Table 1

Finite-sample performance of the estimators when N=100

Note: Median of finite-sample bias (median bias), interquartile range (iqr), and coverage probability of 95% confidence interval (cp) of bias-corrected fixed effects (BCFE), Anderson–Hsiao (AH), GMM, and double filter IV (DFIV) estimators. Lag length is selected either by the sequential rule (automatic lag) with the maximum lag set at $[12(T/100)^{1/4}]$ or by the fixed rule (fixed lag) of $[12(T/100)^{1/4}]$ . 10,000 replications.

The small bias property of the IV estimators is obtained at the cost of dispersion so that the interquartile ranges of the IV estimators are often very large. For both DGPs, the dispersion of BCFE is smaller than that of the AH, GMM, and the DFIV estimators. The dispersion measures of the GMM and the two other IV estimators, the AH and the DFIV, are comparable for DGP1 (with that of the AH being slightly larger), but the dispersion measures of the AH and the DFIV estimators are much larger than that of the GMM estimator for DGP2.

For the standard errors required in constructing the asymptotic confidence intervals of the GMM estimator $\hat {\alpha }_{G}(p)$ and the DFIV estimator $\hat {\alpha }_{DF}(p)$ , we utilize the variance estimators $\hat {v}_{p,G}^{2}$ and $\hat v_{p,DF}^2$ provided in Section 3.5. For the BCFE estimator $\hat \alpha _{BF}(p)$ , we use similarly defined variance estimator

$$ \begin{align*} \hat{v}_{p,BF}^{2} =\left( \frac{1}{N(T-p)}\sum_{i=1}^{N}\sum_{t=p+1}^{T}( \tilde{y}_{it}-\tilde{x}_{it}(p)^{\prime }\hat{\alpha}_{BF}(p))^{2}\right) \ell _{p}^{\prime }(\hat{\Gamma}_{p}^{F})^{-1}\ell _{p}. \end{align*} $$

In terms of the coverage probability, the AH and DFIV estimators perform best for both DGPs. The performance of the BCFE estimator improves as T increases. However, its coverage probability of the confidence intervals for DGP2 is near zero. Finally, for either DGP1 or DGP2, the confidence intervals for the GMM estimator do not perform well because of the many-moment bias and higher-order bias.

4.2 Design 2: Decompositions

To evaluate the source of the finite-sample bias, we conduct an additional simulation exercise. As discussed in Section 3.6, the bias of each IV estimator is decomposed into the truncation bias and the fundamental bias. In the simulation, we can directly evaluate the relative contribution of each component because information about the true process is available. More specifically, the biases of the AH estimator, the GMM estimator, and the DFIV estimation in the simulation can be decomposed as

$$ \begin{align*} \frac{1}{R}\sum_{r=1}^{R}\left( \hat{\alpha}_{AH}^{(r)}(p)-\alpha (p)\right) &=\underbrace{\frac{1}{R}\sum_{r=1}^{R}\left( (\hat{\Gamma}_{\Delta}^{(r)}) ^{-1} \frac{1}{N(T-p)} \sum_{i=1}^{N} \sum_{t=p+2}^T Z_{it}^{(r)}(p)\Delta b_{it,p}^{(r)} \right) }_{\text{truncation bias}} \\ & \quad+\underbrace{\frac{ 1}{R}\sum_{r=1}^{R} \left( ( \hat{\Gamma}_{\Delta}^{(r)} ) ^{-1}\frac{1}{N(T-p)} \sum_{i=1}^{N} \sum_{t=p+2}^T Z_{it}^{(r)}(p)\Delta \epsilon_{it}^{(r)} \right), }_{\text{fundamental bias}} \\ \frac{1}{R}\sum_{r=1}^{R}\left( \hat{\alpha}_{G}^{(r)}(p)-\alpha (p)\right) &=\underbrace{\frac{1}{R}\sum_{r=1}^{R}\left( (\hat{\Gamma}_{p}^{G(r)})^{-1}\frac{1}{N(T-p)}\sum_{t=p+1}^{T-1}x_{t}^{\ast (r) }(p)^{\prime }M_{t}^{(r)}b_{t,p}^{\ast (r)}\right) }_{\text{truncation bias}} \\ & \quad+\underbrace{\frac{1}{R}\sum_{r=1}^{R}\left( (\hat{\Gamma}_{p}^{G(r)})^{-1}\frac{1}{N(T-p)} \sum_{t=p+1}^{T-1}x_{t}^{\ast (r)}(p)^{\prime }M_{t}^{(r)}\epsilon _{t}^{\ast (r)}\right) }_{\text{fundamental bias}}, \\ \frac{1}{R}\sum_{r=1}^{R}\left( \hat{\alpha}_{DF}^{(r)}(p)-\alpha (p)\right) &=\underbrace{\frac{1}{R}\sum_{r=1}^{R}\left( (\hat{\Gamma}_{p}^{DF(r)})^{-1}\frac{1}{N(T-p)}\sum_{t=p+2}^{T-1}h_{t}^{(r)}(p)^{\prime }b_{t,p}^{\ast (r) }\right) }_{\text{truncation bias}} \\ & \quad+\underbrace{\frac{ 1}{R}\sum_{r=1}^{R}\left( (\hat{\Gamma}_{p}^{DF(r)})^{-1}\frac{1}{N(T-p)} \sum_{t=p+2}^{T-1}h_{t}^{(r)}(p)^{\prime }\epsilon _{t}^{\ast (r)}\right) }_{\text{fundamental bias}}, \end{align*} $$

respectively, where the superscript r signifies the r-th simulated observation in R replications.

Table 2 provides such a decomposition of the finite-sample bias of the AH, GMM, and DFIV estimators when the data are generated from DGP1 and DGP2 with $N=\{25,50,100\}$ and $T=25$ . Since we expect the decreasing contribution of the truncation bias as lag length increases, we report the bias decomposition when the model is estimated using $p=\{2,4,8\}$ .Footnote ¹⁵ From the table, we observe that these two types of bias appear very different across the three estimators. For given N and p, the relative contribution of truncation bias in the total bias is the smallest for the GMM estimator and largest for the AH estimator. However, the GMM estimator exhibits a large fundamental bias when N is small. In contrast, the bias of the AH and the DFIV estimators comes solely from truncation bias. An important observation is that for the GMM estimator, there is a trade-off in the value of p such that as p increases, the truncation bias quickly vanishes but the fundamental bias increases. Unlike the GMM estimator, the fundamental biases of the AH and the DFIV estimators remain negligible, even when $p=8$ .

Table 2

Decomposition of the finite-sample bias of the estimators when $T=25$

Note: Means of the components of the finite-sample bias of the Anderson–Hsiao (AH), GMM, and double filter IV (DFIV) estimators. The total finite-sample bias (total) is decomposed into truncation bias (trunc) and fundamental bias (fund). 10,000 replications.

Overall, the simulation results can be summarized as follows. If we care primarily about bias and dispersion, then we recommend the BCFE estimator. Alternatively, if our concern is bias and coverage probability, then the DFIV estimator is recommended.

4.3 Design 3: BCFE vs. DFIV Estimators

We now consider a design that enables a more in-depth comparison between the BCFE and DFIV estimators. Lee et al. (Reference Lee, Okui and Shintani2018) show that the BCFE estimator performs well when T is large. As discussed above, the DFIV estimator performs well when N is large relative to T. The theoretical results show that the DFIV estimator also remains effective as T increases. Thus, it is interesting to examine how the DFIV estimator compares to the BCFE estimator when T is large.

We use the same DGP setting as before, with $T = \{25, 50, 100\}$ and $N=25$ . We consider both homoskedastic and cluster-robust standard errors. For cluster-robust standard errors for the BCFE estimator, we use

$$ \begin{align*} \tilde{v}_{p,BF}^{2} \;=\; \ell_{p}'\,(\hat{\Gamma}_{p}^{F})^{-1} \left( \frac{1}{N(T-p)} \sum_{i=1}^{N}\sum_{t=p+1}^{T}\sum_{t'=p+1}^{T} \hat u_{it}^{\,*}\,\hat u_{it'}^{\,*}\; \tilde x_{it}(p)\,\tilde x_{it'}(p)^{\prime} \right) (\hat{\Gamma}_{p}^{F\prime})^{-1}\,\ell_{p}, \end{align*} $$

where

$$ \begin{align*} \hat u_{it}^{\,*} \;=\; \tilde y_{it} - \tilde x_{it}(p)^{\prime}\hat{\alpha}_{BF}(p), \qquad \hat{\Gamma}_{p}^{F} \;=\; \frac{1}{N(T-p)}\sum_{t=p+1}^{T}\tilde x_{t}(p)^{\prime}\tilde x_{t}(p). \end{align*} $$

We estimate the models using the BCFE estimator and the DFIV estimator. The lag order is chosen by a general-to-specific rule starting from $p_{ ext{max}} = [12\,(T/100)^{1/4}]$ , based on the t-statistic computed with either homoskedastic or cluster-robust standard errors. Table 3 presents median bias, IQR, 95% confidence interval coverage, and mean lag order.

Table 3

Finite-sample performance of the BCFE and DFIV estimators when N=25

Note: Median of finite-sample bias (median bias), interquartile range (iqr), and coverage probability (cp) of the 95% confidence interval. Lag length is selected by the sequential general-to-specific rule with the maximum lag set at $[12(T/100)^{1/4}]$ . 10,000 replications.

The bias of the DFIV estimator is smaller than that of the BCFE estimator under DGP1 in all cases. However, as T increases, the bias of the BCFE estimator decreases and becomes comparable to that of the DFIV estimator when $T=100$ . In contrast, under DGP2, the BCFE estimator has smaller bias than the DFIV estimator for $T=25$ and $T=50$ . As T increases, the bias of the DFIV estimator decreases more rapidly and becomes smaller than that of the BCFE when $T=100$ .

For both DGPs, the BCFE estimator has less dispersion than the DFIV estimator in all cases. DFIV’s dispersion is especially large under DGP2, while BCFE consistently shows low dispersion.

The BCFE estimator outperforms the DFIV estimator in terms of coverage probability under DGP1. However, under DGP2, coverage by the BCFE estimator is poor. Although coverage for the AR coefficient improves as T increases, the performance for the SAR component remains poor, even at larger values of T.

4.4 Design 4: GARCH Errors

Lastly, we consider a design with heteroskedastic errors. We have developed cluster-robust standard errors for the DFIV estimator, though our theoretical analyses assume homoskedasticity. Although asymptotic analyses with heteroskedastic errors are beyond the scope of this article, we use simulations to assess the performance of cluster-robust standard errors.

We retain the same ARMA(1,1) structure with individual effects but allow the shocks to follow a conditionally heteroskedastic GARCH(1,1) process:

$$\begin{align*}y_{it}=\eta _{i}+\phi y_{i,t-1}+u_{it}+\theta u_{i,t-1}, \end{align*}$$

where $u_{it}=\sqrt {h_{it}}\,\varepsilon _{it}$ with $\varepsilon _{it}\sim \text {i.i.d. }N(0,1)$ , and $h_{it}=\omega +\alpha _{garch}\,u_{i,t-1}^2+\beta _{garch}\,h_{i,t-1}$ with $\omega $ chosen so that $E(h_{it})=1$ (thus $E(u_{it}^2)=1$ ). We again consider DGP1 with $(\phi ,\theta )=(0.5,0.4)$ and DGP2 with $(\phi ,\theta )=(0.99,0.4)$ . We consider two GARCH specifications: a moderate case with $(\alpha _{garch},\beta _{garch})=(0.1,0.8)$ and a near-IGARCH case with $(\alpha _{garch},\beta _{garch})=(0.35,0.64)$ .

To achieve stationarity within the data, we employ burn-in periods and discard initial observations. For the burn-in period, we set the first-period variance to its long-run average: $h_{i1}=E(h_{it})=\omega /(1-\alpha _{garch}-\beta _{garch})$ . We then discard 2,000 pre-sample observations in the moderate case and 3,000 in the near-IGARCH case.

For each DGP and GARCH specification, we set $T = \{25, 50, 100 \}$ with $N=100$ . We estimate AR(p) models using the DFIV estimator. Results are shown for both homoskedastic and cluster-robust standard errors. The lag order is fixed by the rule $p=[ 12\,(T/100)^{1/4}]$ . Table 4 reports median bias, IQR, and coverage probability of the nominal 95% confidence interval.

Table 4

Finite-sample performance of the DFIV estimators under GARCH errors when N=100

Note: Median of finite-sample bias (median bias), interquartile range (iqr), and coverage probability of 95% confidence interval based on two standard errors (cp) of double filter IV (DFIV) estimator. Lag length is selected by the fixed rule (fixed lag) of $[12(T/100)^{1/4}]$ . 10,000 replications.

For DGP1, the coverage probability using cluster-robust standard errors is approximately 95% under the moderate GARCH case. However, it drops substantially under the near-IGARCH case. In contrast, the coverage probability with homoskedastic standard errors is already low for the moderate GARCH case. The deterioration further intensifies for the near-IGARCH case.

For DGP2, the coverage probability with cluster-robust standard errors remains high, while it is low with homoskedastic standard errors. Under the near-IGARCH, both estimators show improved coverage compared to DGP1.