# Forward Orthogonal Deviations GMM and the Absence of Large Sample Bias

Robert F. Phillips1112115 G Street, NW, Suite 340, Washington DC, 20052;
phone: 202-994-8619; fax: 202-994-6147; email<EMAIL_ADDRESS>
Department of Economics
George Washington University

(December 2022)

##  1 Introduction

When estimating a regression with panel data, researchers are often concerned
about unobserved individual-specific effects, typically referred to as “fixed
effects”. The concern is that they may be correlated with the explanatory
variables. The fixed effects can be eliminated by subtracting from each
variable its within-group average. If the explanatory variables are strictly
exogenous, then applying ordinary least squares after removing the within-
group averages produces an estimator—the fixed effects estimator—whose
distribution has no asymptotic bias as the number of cross-sectional units
($n$) increases and the number of time periods ($T$) is fixed, or both $n$ and
$T$ increase, or $T$ increases with $n$ fixed.222In cases where $T$ increases,
dependence across time among variables must be restricted [Hansen, 2007].
Consequently, under suitable conditions [see, e.g., Hansen, 2007], regardless
of whether $n$ is large, or $T$ is large, or both are large, inference based
on the fixed effects estimator is reliable.

But this conclusion is predicated on the regressors being strictly exogenous,
which is not true when the model is dynamic. For a dynamic model, although the
fixed effects estimator’s distribution is correctly centered when $\lim
n/T=0$, it has a bias term if $\lim n/T>0$ [Alvarez and Arellano, 2003].
Consequently, if a dynamic panel data model is estimated with the fixed
effects estimator, confidence intervals and test statistics based on the
estimator can be misleading when $n$ is large compared to $T$.

As a result, researchers proposed alternative estimators for dynamic panel
data regressions specifically for the case where $T$ is fixed and
$n\rightarrow\infty$. For example, maximum-likelihood and quasi maximum-
likehood estimators of dynamic panel data regressions have been proposed for
the large-$n$, small-$T$ case; see, for example, Alvarez and Arellano [2004],
Anderson and Hsiao [1981], Binder et al. [2005], Hsiao et al. [2002],
Kruiniger [2013], Moral-Benito [2013], and Phillips (2010, 2018), among
others. Moreover, some well-known generalized method of moments (GMM)
estimators of dynamic panel data models are suitable when $n$ is large
compared to $T$; see, for example, Arellano and Bond [1991], Arellano and
Bover [1995], Blundell and Bond [1998], and Holtz-Eakin et al. [1988]. But a
drawback of these estimators is that their large sample distributions may not
be correctly centered when $T$ is not small compared to $n$ [Alvarez and
Arellano, 2003].

In this paper I provide conditions that, if satisfied, indicate a well-known
class of GMM estimators have no asymptotic bias, even when $T$ is not small
compared to $n$. Like the fixed effects estimator, the data transformation I
examine removes fixed effects by subtracting a within-group average from each
observation. But unlike the fixed effects estimator, the GMM estimators
studied here use only future values for the within-group averages.
Specifically, I examine GMM based on the forward orthogonal deviations (FOD)
transformation (FOD-GMM).

In the panel data literature the FOD transformation dates back to at least
Arellano and Bover [1995]. Arellano and Bover [1995] showed that, under
suitable conditions, FOD-GMM and GMM based on the first difference (FD)
transformation (FD-GMM) are identical. Alvarez and Arellano [2003] exploited
this equivalence to establish the asymptotic properties of a FD/FOD-GMM
estimator of the autoregressive parameter in a first-order autoregressive
(AR(1)) panel data model. In particular, they showed that, if all available
instrumental variables are used, then the FOD-GMM estimator is asymptotically
biased if $\lim T/n=c$, with $0<c<\infty$.

But this conclusion depends on the use of all available instrumental
variables. Bun and Kiviet [2006] and Hsiao and Zhou [2017] provided results
indicating that, if a fixed number of instruments are used each period, FOD-
GMM estimators do not have large sample bias, regardless of the relative rate
of increase in $T$ and $n$. To obtain this conclusion Bun and Kiviet [2006]
and Hsiao and Zhou [2017] studied specific models and instrument choices.

The first contribution of this paper is that it generalizes Bun and Kiviet’s
and Hsiao and Zhou’s work. I provide results indicating FOD-GMM estimators do
not have large sample bias—regardless of the relative rate of increase in $T$
and $n$—for a much larger class of regression problems and instrument choices
than that examined by Bun and Kiviet [2006] and Hsiao and Zhou [2017].
Moreover, a new result provided in this paper indicates that the absence of
large sample bias of FOD-GMM estimators depends, not on using a fixed number
of instrumental variables per period, but rather on how fast the maximum
number of instrumental variables used in a period increases with $T$. Another
contribution provided in this paper is an asymptotic distribution result for
FOD-GMM estimators.

Section 3 provides Monte Carlo evidence that corroborates the analytical
results in Section 2. The Monte Carlo experiments illustrate that the bias of
an FOD-GMM estimator, which uses current and recent lags of predetermined
variables as instrumental variables, decreases as $n$, or $T$, or both
increase. As predicted by the analytical results, the bias tends to be least
when both $n$ and $T$ are large. As a result, confidence intervals are
reliable when $n$ and $T$ are large, and given the close connection between
confidence intervals and t-statistics, we can conclude that t-tests will also
be reliable when $n$ and $T$ are large.

For the sake of comparison, I also provide Monte Carlo results for an FD-GMM
estimator that exploits the same instrumental variables used by the FOD-GMM
estimator. Although the FD- and FOD-GMM estimators are the same for the
estimation problem Alvarez and Arellano [2003] studied, they are not the same
when only current and recent lags of predetermined variables are used as
instruments [see, e.g., Phillips, 2019]. Consequently, we should not expect
their sampling performance to be similar. Indeed, the bias of the FD-GMM
estimator was usually greater than that of the FOD-GMM estimator, and
consequently confidence intervals based on the FD-GMM estimator were, as
expected, much less reliable. Moreover, the FD-GMM estimator was generally
less efficient, though no theoretical explanation for this result is apparent.

##  2 Large sample results

###  2.1 The model and estimator

The regression model studied in this paper is

$y_{i,t}=\boldsymbol{x}_{i,t}^{\prime}\boldsymbol{\beta}+\eta_{i}+v_{i,t}\qquad(t=1,\ldots,T,\;i=1,\ldots,n),$

where $\boldsymbol{x}_{i,t}^{\prime}:=(x_{i,t,1},\ldots,x_{i,t,K})$ and
$\boldsymbol{\beta}^{\prime}:=(\beta_{1},\ldots,\beta_{K})$ are vectors of
regressors and parameters. Some or all of the $x_{i,t,k}$s may be lagged
values of $y_{i,t}$. The term $\eta_{i}$ is an unobserved individual-specific
or fixed effect, and $v_{i,t}$ is an error term.

This paper examines estimation of $\boldsymbol{\beta}$ with FOD-GMM. The FOD
transformation removes from each variable its within-group average over future
periods. For example, the FOD transformed explanatory variables for the $t$th
period are given by
$\ddot{\boldsymbol{X}}_{t}:=c_{t}\left(\boldsymbol{X}_{t}-\overline{\boldsymbol{X}}_{t}\right)$,
where
$\boldsymbol{X}_{t}^{\prime}:=\left(\boldsymbol{x}_{1,t},\ldots,\boldsymbol{x}_{n,t}\right)$,
$\overline{\boldsymbol{X}}_{t}:=\left(T-t\right)^{-1}\sum_{s=1}^{T-t}\boldsymbol{X}_{t+s}$,
and $c_{t}^{2}:=(T-t)/(T-t+1)$. Moreover, the transformed values of the
dependent variable in the $t$th period are
$\ddot{\boldsymbol{y}}_{t}:=c_{t}\left(\boldsymbol{y}_{t}-\overline{\boldsymbol{y}}_{t}\right)$,
where $\boldsymbol{y}_{t}^{\prime}:=\left(y_{1,t},\ldots,y_{n,t}\right)$ and
$\overline{\boldsymbol{y}}_{t}:=\left(T-t\right)^{-1}\sum_{s=1}^{T-t}\boldsymbol{y}_{t+s}$.
The constant $c_{t}$ ensures the transformed errors (the $\ddot{v}_{i,t}$s)
are conditionally homoskedastic and uncorrelated if the original errors (the
$v_{i,t}$s) are conditionally homoskedastic and uncorrelated [Arellano, 2003,
p. 17].

In addition to FOD transformed dependent and explanatory variables, an FOD-GMM
estimator relies on instrumental variables. Let $\boldsymbol{z}_{i,t}$ denote
a $q_{t}\times 1$ vector of instrumental variables for the $i$th individual in
the $t$th period, and let
$\boldsymbol{Z}_{t}^{\prime}:=\left(\boldsymbol{z}_{1,t},\ldots,\boldsymbol{z}_{n,t}\right)$.
The $t$th period projection matrix based on these instruments is
$\boldsymbol{P}_{t}:=\boldsymbol{Z}_{t}\left(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t}\right)^{-1}\boldsymbol{Z}_{t}^{\prime}$.
Using these projections and the transformed regression variables, the FOD-GMM
estimator of $\boldsymbol{\beta}$ is

$\widehat{\boldsymbol{\beta}}:=\left(\sum_{t=1}^{T-1}\ddot{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{P}_{t}\ddot{\boldsymbol{X}}_{t}\right)^{-1}\,\sum_{t=1}^{T-1}\ddot{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{P}_{t}\ddot{\boldsymbol{y}}_{t}$

[Arellano, 2003, p. 154].

###  2.2 When there is no asymptotic bias

The reliability of large sample tests and confidence intervals based on an
FOD-GMM estimator depends on whether or not
$\sqrt{nT}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})$ is asymptotically
unbiased. To see when the distribution of
$\sqrt{nT}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})$ is correctly
centered at a vector of zeros, note that

$\sqrt{nT}\left(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)-\boldsymbol{A}_{n,T}^{-1}E\left(\boldsymbol{b}_{n,T}\right)=\boldsymbol{A}_{n,T}^{-1}\left[\boldsymbol{b}_{n,T}-E\left(\boldsymbol{b}_{n,T}\right)\right],$

where
$\boldsymbol{b}_{n,T}:=(nT)^{-1/2}\sum_{t=1}^{T-1}\ddot{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{P}_{t}\ddot{\boldsymbol{v}}_{t}$,
$\boldsymbol{A}_{n,T}:=(nT)^{-1}\sum_{t=1}^{T-1}\ddot{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{P}_{t}\ddot{\boldsymbol{X}}_{t}$,
$\ddot{\boldsymbol{v}}_{t}:=c_{t}\left(\boldsymbol{v}_{t}-\overline{\boldsymbol{v}}_{t}\right)$,
$\boldsymbol{v}_{t}^{\prime}:=\left(v_{1,t},\ldots,v_{n,t}\right)$, and
$\overline{\boldsymbol{v}}_{t}:=\left(T-t\right)^{-1}\sum_{s=1}^{T-t}\boldsymbol{v}_{t+s}$.
If, as the sample size increases, the mean of $\boldsymbol{b}_{n,T}$ does not
converge to a vector of zeros, then we cannot expect
$\sqrt{nT}\left(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)$ to be
asymptotically unbiased. This point is illustrated with the following example.

The AR(1) panel data model provides a comparatively simple but instructive
case. The AR(1) panel data model is

$y_{i,t}=\beta y_{i,t-1}+\eta_{i}+v_{i,t},$

with $\left|\beta\right|<1$. For this case,
$\sqrt{nT}(\widehat{\beta}-\beta)-E(b_{n,T})/a_{n,T}=[b_{n,T}-E(b_{n,T})]/a_{n,T}$,
where
$a_{n,T}=(nT)^{-1}\sum_{t=1}^{T-1}\ddot{\boldsymbol{y}}_{t-1}^{\prime}\boldsymbol{P}_{t}\ddot{\boldsymbol{y}}_{t-1}$
and
$b_{n,T}=(nT)^{-1/2}\sum_{t=1}^{T-1}\ddot{\boldsymbol{y}}_{t-1}^{\prime}\boldsymbol{P}_{t}\ddot{\boldsymbol{v}}_{t}$,
with
$\ddot{\boldsymbol{y}}_{t-1}:=c_{t}(\boldsymbol{y}_{t-1}-\overline{\boldsymbol{y}}_{t-1})$,
$\boldsymbol{y}_{t-1}^{\prime}:=(y_{1,t-1},\ldots,y_{n,t-1})$, and
$\overline{\boldsymbol{y}}_{t-1}:=\left(T-t\right)^{-1}\sum_{s=1}^{T-t}\boldsymbol{y}_{t-1+s}$.
Alvarez and Arellano [2003] provide conditions that imply

$a_{n,T}\overset{p}{\rightarrow}\frac{\sigma^{2}}{1-\beta^{2}}\qquad\text{and}\qquad
E(b_{n,T})-\sqrt{\frac{T}{n}}\left(\frac{\sigma^{2}}{\beta-1}\right)\rightarrow
0\qquad(n,T\rightarrow\infty),$

where $\sigma^{2}:=\text{var}(v_{i,t})$ (see Alvarez and Arellano, 2003, p.
1128, Lemma 2). Using these results, they showed that, if $\lim(T/n)=c$, with
$0\leq c<\infty$, then

$\sqrt{nT}\left(\widehat{\beta}-\beta\right)-\left[-\sqrt{T/n}\left(1+\beta\right)\right]\overset{d}{\rightarrow}N(0,\,1-\beta^{2}).$

(see Alvarez and Arellano, p. 1129, Theorem 2). Therefore, for the AR(1) panel
data model, $\sqrt{nT}(\widehat{\beta}-\beta)$ has asymptotic bias of
$-\sqrt{c}(1+\beta)$, which is zero only when $c=0$. In other words,
$\sqrt{nT}(\widehat{\beta}-\beta)$ is asymptotically unbiased only if
$E(b_{n,T})\rightarrow 0$, which only occurs when $T/n\rightarrow 0$.

This example is instructive for two reasons. First, it illustrates there will
be asymptotic bias if we do not have $E(b_{n,T})\rightarrow 0$. Hence,
$E(b_{n,T})\rightarrow 0$ is a critical requirement for the absence of
asymptotic bias. The second reason the example is instructive is less obvious
and is the point of this paper. The failure to get $E(b_{n,T})\rightarrow 0$
when $c>0$ is due the number of instrumental variables that are used. Alvarez
and Arellano [2003] assumed all available moment restrictions are exploited.
For the estimation problem they considered, using all available instrumental
variables implies that the maximum number of instrumental variables used per
period increases at the rate $T$ increases. Theorem 1 shows that this matters,
because, according to the theorem, if the maximum number of instrumental
variables used each period increases at a rate no faster than $T^{\delta}$,
with $\delta<1/2$, then $E(\boldsymbol{b}_{n,T})\rightarrow\boldsymbol{0}$
regardless of how $T$ and $n$ increase.

In order to state sufficient conditions for this conclusion some additional
definitions will be used. Specifically, let $\boldsymbol{w}_{i,t}$ be a column
vector consisting of all of the distinct entries in
$\left\\{\left(\boldsymbol{x}_{i,s}^{\prime},\boldsymbol{z}_{i,s}^{\prime}\right);\;s=1,\ldots,t\right\\}$.
That is, $\boldsymbol{w}_{i,t}$ contains all the distinct values of the
explanatory variables and instrumental variables for the $i$th individual from
the first period up to the $t$th period. Next, set
$\boldsymbol{u}_{i}^{\prime}:=(\boldsymbol{w}_{i,T}^{\prime},v_{i,1},\ldots,v_{i,T},\eta_{i})$.
Finally, let
$\gamma_{k,t,s}:=\text{cov}\left(v_{1,t},x_{1,t+s,k}|\boldsymbol{w}_{1,t}\right)$,
and let $b_{n,T,k}$ denote the $k$th entry in $\boldsymbol{b}_{n,T}$
($k=1,\ldots,K$).

Theorem 1 can now be stated.

######  Theorem 1.

Assume $T\geq 3$ and let $q_{T}^{\ast}:=\max_{1\leq t\leq T-1}q_{t}$. Further
assume

  * A1:

the $\boldsymbol{u}_{i}$s are independent and identically distributed
$($i.i.d.$)$ across $i$;

  * A2:

$\text{rank}\left(\boldsymbol{Z}_{t}\right)=q_{t}$ with probability 1;

  * A3:

$E(v_{1,t}|\boldsymbol{w}_{1,t})=0$; and

  * A4:

$\left|\sum_{s=1}^{S}\gamma_{k,t,s}\right|\leq M$ with probability 1, for some
finite $M$ and all $t\geq 1$, $S\geq 1$, and $k=1,\ldots,K$.

Then, if $q_{T}^{\ast}=O(T^{\delta})$, it follows that

$E\left(b_{n,T,k}\right)=O\left(n^{-1/2}T^{\delta-1/2}\ln
T\right)\qquad(k=1,\ldots,K).$

Proofs are provided in Appendix A.

Theorem 1 shows how $E(\boldsymbol{b}_{n,T})$ depends on the maximum number of
instrumental variables used per period. Specifically, if the maximum number of
instrumental variables used per period increases with $T$ at a rate no greater
than $T^{\delta}$, with $\delta<1/2$, the conclusion of Theorem 1 implies
$E(\boldsymbol{b}_{n,T})\rightarrow\boldsymbol{0}$ for all sequences with $n$
increasing, or $T$ increasing, or both increasing.

Theorem 1 relies on several simplifying assumptions. The first three
assumptions are straightforward. The fourth characterizes the linear
association between the error in period $t$ and the $k$th regressor $s$
periods in the future relative to that period, conditional on the available
information at time $t$. Because the conditional covariance between $v_{1,t}$
and $x_{1,t+s,k}$ is not restricted to be zero, Assumption A4 allows for
predetermined regressors. Moreover, if the error term and future values of the
explanatory variables are suitably weakly dependent, the bound on the sum of
covariances in A4 will be satisfied. For example, Assumption A4 is satisfied
by stationary $K$th-order autoregressive panel data models. This fact is
verified in Appendix B of the Supplemental Material [Phillips, 2022]. That
appendix also provides another dynamic panel data model that satisfies
Assumption A4; a model that includes the model considered by Bun and Kiviet
[2006] as a special case.

Bun and Kiviet [2006] and Hsiao and Zhou [2017] provide results that are
related to Theorem 1. But neither Bun and Kiviet [2006] nor Hsiao and Zhou
[2017] examined the possibility that asymptotic bias can vanish as the number
of instrumental variables used per period increases, provided the maximum
number of per period instrumental variables increases slowly enough. Moreover,
the models and instrument choices these earlier papers studied satisfy the
assumptions in Theorem 1 [see Appendix B of the Supplemental Material,
Phillips, 2022], and, therefore, the estimation problems examined in Bun and
Kiviet [2006] and Hsiao and Zhou [2017] are special cases covered by Theorem
1.

The conclusion of Theorem 1 implies
$E(\boldsymbol{b}_{n,T})\rightarrow\boldsymbol{0}$, for all sequences for
which $n$, or $T$, or both increase, if the maximum number of instruments used
per period does not increase with $T$ too rapidly. For some estimation
problems this restriction on the number of instrumental variables rules out
using all available instrumental variables. However, in practice researchers
often use no more than a fixed number of instrumental variables per period. In
which case, if Assumptions A1 through A4 are satisfied, then
$E(\boldsymbol{b}_{n,T})\rightarrow\boldsymbol{0}$ for all sequences for which
$n$, or $T$, or both increase.

The next theorem generalizes results in Hsiao and Zhou [2017]. Hsiao and Zhou
[2017] used sequential-limit analysis to obtain asymptotic distributions for
FOD-GMM estimators of the AR(1) panel data model when one instrumental
variable is used per period. Specifically, they considered limits as
$n\rightarrow\infty$, and then $T\rightarrow\infty$.333Taking limits
sequentially simplifies the analysis. But the simplification is only useful if
it provides reliable approximations for large samples. For some sequences of
$n$ and $T$ increasing, this will not be the case when the presence of
asymptotic bias depends on what happens to $T/n$ as $n$ and $T$ jointly
increase. However, the conclusion of Theorem 1 indicates that, under the
conditions described in this paper, bias disappears, as $n$ and $T$ increase,
regardless of what happens to $T/n$. Theorem 2 generalizes this approach to a
much broader class of regressions and instrument choices.

Theorem 2 relies on additional assumptions. In order to state the first of
these assumptions, let
$\boldsymbol{w}_{1,t}^{\ast\prime}=(\boldsymbol{w}_{1,t}^{\prime},v_{1,0},\ldots,v_{1,t-1})$
$(t\geq 1)$. Assumption A3∗ strengthens Assumption A3:

  * A3∗: $E(v_{1,t}|\boldsymbol{w}_{1,t}^{\ast})=0$ ($t\geq 1$).

Also assume

  * A5: $E\left(v_{1,t}^{2}|\boldsymbol{w}_{1,t}^{\ast}\right)=\sigma^{2}$ ($t\geq 1$);

and

  * A6: the entries in $\boldsymbol{x}_{1,t}$ and $\boldsymbol{z}_{1,t}$ have finite second moments.

Assumption A6 implies the entries in the matrices
$\boldsymbol{R}_{t}:=E(\boldsymbol{z}_{1,t}\ddot{\boldsymbol{x}}_{1,t}^{\prime})$
and $\boldsymbol{Q}_{t}:=E(\boldsymbol{z}_{1,t}\boldsymbol{z}_{1,t}^{\prime})$
are finite. Moreover, Assumption A2 implies $\boldsymbol{Q}_{t}$ is positive
definite. Hence, if Assumptions A2 and A6 are satisfied, we can define
$\boldsymbol{\Omega}_{t}:=\boldsymbol{R}_{t}^{\prime}\boldsymbol{Q}_{t}^{-1}\boldsymbol{R}_{t}$.
Also, let
$\overline{\boldsymbol{\Omega}}_{T}:=T^{-1}\sum_{t=1}^{T-1}\boldsymbol{\Omega}_{t}$.
Finally, let $\omega_{t,j,k}$ denote the ($j,k$)th entry of
$\boldsymbol{\Omega}_{t}$, and set $\nu_{t}:=\max_{j,k}|\omega_{t,j,k}|$. Then
the last set of assumptions for Theorem 2 are

  * A7: rank$(\boldsymbol{R}_{t})=K$ $(t=1,\ldots,T-1)$;

  * A8: $\overline{\boldsymbol{\Omega}}:=\lim_{T\rightarrow\infty}\overline{\boldsymbol{\Omega}}_{T}$ exists, with $\overline{\boldsymbol{\Omega}}$ a positive definite matrix;
and

  * A9: $\lim_{T\rightarrow\infty}T^{-3/2}\sum_{t=1}^{T-1}\nu_{t}^{3/2}=0$.

Theorem 2 can now be stated.

######  Theorem 2.

Assume $q_{T}^{\ast}=O(T^{\delta})$, with $0\leq\delta<1/2$, and assume A1,
A2, A3∗, and A4 through A9 are satisfied. Then

$\sqrt{nT}\left(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)\overset{d}{\rightarrow}N\left(\boldsymbol{0},\sigma^{2}\overline{\boldsymbol{\Omega}}^{-1}\right)\qquad(n,T\rightarrow\infty)_{\text{seq}}.$

The notation “$(n,T\rightarrow\infty)_{\text{seq}}$” means
$n\rightarrow\infty$, then $T\rightarrow\infty$.

##  3 Monte Carlo simulations

This section provides results from Monte Carlo simulations. Using generated
samples, FOD-GMM estimator biases, standard deviations, and confidence
intervals were estimated for different sampling designs. For the sake of
comparison, the bias, standard deviation, and confidence intervals for a FD-
GMM estimator were also calculated.

###  3.1 Samples

Monte Carlo samples were generated using sampling schemes similar to those
described in Bun and Kiviet [2006]. The dependent variable in the regression
model was always generated according to

$y_{i,t}=\beta_{1}y_{i,t-1}+\beta_{2}x_{i,t}+\eta_{i}+v_{i,t}\qquad(t=-49,\ldots,-1,0,1,\ldots,T,\
i=1,\ldots,n),$

with $\beta_{1}\in\left\\{0.25,0.75\right\\}$ and $\beta_{2}=1-\beta_{1}$. The
start-up value $y_{i,-50}$ was set to zero. The error components $v_{i,t}$ and
$\eta_{i}$ were generated independently of one another as i.i.d. standard
normal variates. Moreover, two schemes were used to generate the explanatory
variable $x_{i,t}$. The two schemes—Scheme 1 and Scheme 2—were similar to
those used by Bun and Kiviet [2006].

For Scheme 1, the time series for $x_{i,t}$ was constructed according to

$\begin{split}x_{i,-50}&=\kappa_{1}\eta_{i}+\varepsilon_{i,-50},\\\
x_{i,-49}&=\kappa_{1}\eta_{i}+\frac{1}{1-\rho
L}\left(\varepsilon_{i,-49}+\phi_{1}v_{i,-50}\right)\\\
x_{i,t}&=\kappa_{1}\eta_{i}+\frac{1}{1-\rho
L}\varepsilon_{i,t}+\phi_{1}v_{i,t-1}\qquad(t=-48,\ldots,-1,0,+1,\ldots,T),\end{split}$

with $\rho\in\left\\{0.50,0.95\right\\}$,
$\kappa_{1}\in\left\\{-1,0,+1\right\\}$, and
$\phi_{1}\in\left\\{-1,0,+1\right\\}$. Moreover, the $\varepsilon_{i,t}$s were
generated as
$\varepsilon_{i,t}\overset{i.i.d.}{\sim}U(-\sqrt{12}/2,\,\sqrt{12}/2)$
independently of the $v_{i,t}$s and $\eta_{i}$s.

As for Scheme 2, $x_{i,t}$ was generated as

$\begin{split}x_{i,-50}&=\kappa_{2}\eta_{i}+\varepsilon_{i,-50},\\\
x_{i,t}&=\rho
x_{i,t-1}+\phi_{2}y_{i,t-1}+\kappa_{2}\eta_{i}+\varepsilon_{i,t}\qquad(t=-49,\ldots,-1,0,+1,\ldots,T).\end{split}$

Following Bun and Kiviet [2006], I set
$\phi_{2}=\phi_{1}\beta_{2}(1-\rho)/(1+\beta_{2}\phi_{1})$ and
$\kappa_{2}=\kappa_{1}(1-\rho-\phi_{2})-\phi_{2}/\beta_{2}$.

For both Scheme 1 and Scheme 2, there were 36 experimental Designs, where a
Design is a combination of parameter values. Table 1 lists Designs 1 through
18. For these designs I set $\beta_{1}=0.25$. Designs 19 through 36 were
identical to Designs 1 through 18 except that, for Designs 19 through 36,
$\beta_{1}$ was set to $0.75$.

Table 1: Designs 1 through 18 ($\beta_{1}=0.25)$. | Designs
---|---
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
$\rho$ | $0.50$ | $0.50$ | $0.50$ | $0.50$ | $0.50$ | $0.50$ | $0.50$ | $0.50$ | $0.50$
$\phi_{1}$ | $-1.00$ | $-1.00$ | $-1.00$ | $0.00$ | $0.00$ | $0.00$ | $1.00$ | $1.00$ | $1.00$
$\kappa_{1}$ | $-1.00$ | $0.00$ | $1.00$ | $-1.00$ | $0.00$ | $1.00$ | $-1.00$ | $0.00$ | $1.00$
| Designs
| 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18
$\rho$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$ | $0.95$
$\phi_{1}$ | $-1.00$ | $-1.00$ | $-1.00$ | $0.00$ | $0.00$ | $0.00$ | $1.00$ | $1.00$ | $1.00$
$\kappa_{1}$ | $-1.00$ | $0.00$ | $1.00$ | $-1.00$ | $0.00$ | $1.00$ | $-1.00$ | $0.00$ | $1.00$

The effect of how each series was initialized was eliminated by dropping the
first 50 values of each generated time series. Therefore, estimation was based
on the values $(x_{i,0},y_{i,0}),(x_{i,1},y_{i,1}),\ldots,(x_{i,T},y_{i,T})$
($i=1,\ldots,n$).

Finally, four combinations of $(T,n)$ were considered: small-$T$, small-$n$
$(25,25)$; small-$T$, large-$n$ $(25,100)$; large-$T$, small-$n$ $(100,25)$;
and large-$T$, large-$n$ $(100,100)$. These sample sizes were selected to see
how the bias of the FOD-GMM estimator changed with sample size. The large-$T$,
small-$n$ sample size is not usually considered in the panel data literature,
but the theory in Section 2 predicts that the bias of the FOD-GMM estimator
should be less for $(T,n)=(100,25)$ than for $(T,n)=(25,25)$. Therefore,
experiments were conducted to investigate this prediction.

###  3.2 Estimators

For each Design and sample size, 5,000 samples were generated.444The data were
generated using GAUSS. Moreover, all calculations were performed with GAUSS.
After a sample was generated, an FOD-GMM estimate was calculated using the
instrumental variables
$\boldsymbol{z}_{i,1}^{\prime}=(y_{i,0},x_{i,0},x_{i,1})$ and
$\boldsymbol{z}_{i,t}^{\prime}=(y_{i,t-2},y_{i,t-1},x_{i,t-2},x_{i,t-1},x_{i,t})$
($t=2,\ldots,T-1$, $i=1,\ldots,n$). Then bias and standard deviation estimates
were calculated. I also calculated standard errors by taking the square roots
of the diagonal entries of
$\widehat{\sigma}^{2}\left(\sum_{t=1}^{T-1}\ddot{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{P}_{t}\ddot{\boldsymbol{X}}_{t}\right)^{-1}$,
where
$\widehat{\sigma}^{2}:=\sum_{t=1}^{T-1}(\ddot{\boldsymbol{y}}_{t}-\ddot{\boldsymbol{X}}_{t}\widehat{\boldsymbol{\beta}})^{\prime}(\ddot{\boldsymbol{y}}_{t}-\ddot{\boldsymbol{X}}_{t}\widehat{\boldsymbol{\beta}})/[n(T-1)]$.
Using the standard errors and parameter estimates, 90- and 95-percent
confidence intervals were calculated for $\beta_{1}$ and $\beta_{2}$.

For the sake of comparison, estimates of a FD-GMM estimator’s bias, standard
deviation, and confidence intervals were also calculated for $\beta_{1}$ and
$\beta_{2}$. For the FD-GMM estimator, the transformed data are
$\tilde{\boldsymbol{y}}^{\prime}:=\left(\tilde{\boldsymbol{y}}_{1}^{\prime},\ldots,\tilde{\boldsymbol{y}}_{n}^{\prime}\right)$,
where
$\tilde{\boldsymbol{y}}_{i}^{\prime}:=(y_{i,2}-y_{i,1},\ldots,y_{i,T}-y_{i,T-1})$;
and
$\tilde{\boldsymbol{X}}^{\prime}:=\left(\tilde{\boldsymbol{X}}_{1}^{\prime}\,\ldots\,\tilde{\boldsymbol{X}}_{n}^{\prime}\right)$,
where the $t$th row of matrix $\tilde{\boldsymbol{X}}_{i}$ is
$\tilde{\boldsymbol{x}}_{i,t+1}^{\prime}:=\boldsymbol{x}_{i,t+1}^{\prime}-\boldsymbol{x}_{i,t}^{\prime}$
($t=1,\ldots,T-1$). Moreover, I used the same instrumental variables for the
FD-GMM estimator that I used for the FOD-GMM estimator. Specifically, for the
instrument matrix, I set
$\tilde{\boldsymbol{Z}}^{\prime}:=\left(\tilde{\boldsymbol{Z}}_{1}^{\prime}\,\ldots,\,\tilde{\boldsymbol{Z}}_{n}^{\prime}\right)$,
where $\tilde{\boldsymbol{Z}}_{i}$ is a block-diagonal matrix, with
$\boldsymbol{z}_{i,t}^{\prime}$ in the $t$th-diagonal block
($t=1,\ldots,T-1$). Then, the one-step FD-GMM estimator is

$\widetilde{\boldsymbol{\beta}}:=\left[\tilde{\boldsymbol{X}}^{\prime}\tilde{\boldsymbol{Z}}\left(\sum_{i=1}^{n}\tilde{\boldsymbol{Z}}_{i}^{\prime}\boldsymbol{G}\tilde{\boldsymbol{Z}}_{i}\right)^{-1}\tilde{\boldsymbol{Z}}^{\prime}\tilde{\boldsymbol{X}}\right]^{-1}\tilde{\boldsymbol{X}}^{\prime}\tilde{\boldsymbol{Z}}\left(\sum_{i=1}^{n}\tilde{\boldsymbol{Z}}_{i}^{\prime}\boldsymbol{G}\tilde{\boldsymbol{Z}}_{i}\right)^{-1}\tilde{\boldsymbol{Z}}^{\prime}\tilde{\boldsymbol{y}},$

where $\boldsymbol{G}$ is a $(T-1)\times(T-1)$ matrix with twos running down
the main diagonal, minus ones just above and below the main diagonal, and
zeros everywhere else [Arellano and Bond, 1991]. Standard errors for the
entries in $\widetilde{\boldsymbol{\beta}}$ were calculated using the square
roots of the diagonal entries in
$\widetilde{\sigma}^{2}\left[\tilde{\boldsymbol{X}}^{\prime}\tilde{\boldsymbol{Z}}\left(\sum_{i=1}^{n}\tilde{\boldsymbol{Z}}_{i}^{\prime}\boldsymbol{G}\tilde{\boldsymbol{Z}}_{i}\right)^{-1}\tilde{\boldsymbol{Z}}^{\prime}\tilde{\boldsymbol{X}}\right]^{-1}$,
where
$\widetilde{\sigma}^{2}:=\sum_{i=1}^{n}(\tilde{\boldsymbol{y}}_{i}-\tilde{\boldsymbol{X}}_{i}\widetilde{\boldsymbol{\beta}})^{\prime}(\tilde{\boldsymbol{y}}_{i}-\tilde{\boldsymbol{X}}_{i}\widetilde{\boldsymbol{\beta}})/[2n(T-1)]$.

###  3.3 Simulation results

Two schemes, each with 36 designs and four cases of $(T,n)$, led to a total of
288 Monte Carlo experiments. This produced a large number of bias, standard
deviation, and confidence interval estimates. But the main conclusions drawn
from the experiments can be illustrated with only some of the simulated data.

For example, the bias estimate results were qualitatively similar for Schemes
1 and 2. Therefore, only Scheme 1 bias estimates are provided here.555Scheme 2
bias estimates are provided in Appendix C of the Supplemental Material
[Phillips, 2022]. Tables 2 and 3 provide Scheme 1 bias estimates for
$\sqrt{nT}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})$ and
$\sqrt{nT}(\widetilde{\boldsymbol{\beta}}-\boldsymbol{\beta})$, where
$\widehat{\boldsymbol{\beta}}$ and $\widetilde{\boldsymbol{\beta}}$ denote the
FOD- and FD-GMM estimators. Table 2 provides bias estimates for Designs 1
through 18 ($\beta_{1}=0.25$), and Table 3 provides estimates for Designs 19
through 36 ($\beta_{1}=0.75$).

Table 2: Scheme 1: Bias estimates for Designs 1 through 18 ($\beta_{1}=0.25$).

Designs: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---
($T$, $n$) | Bias of $\sqrt{nT}(\text{FOD estimator of }\beta_{1}-\beta_{1})$
($25$, $25$): | $-0.01$ | $0.00$ | $0.02$ | $-0.23$ | $-0.26$ | $-0.26$ | $-0.25$ | $-0.25$ | $-0.28$
($25$, $100$): | $0.02$ | $0.00$ | $0.02$ | $-0.11$ | $-0.12$ | $-0.16$ | $-0.12$ | $-0.14$ | $-0.18$
($100$, $25$): | $0.01$ | $-0.01$ | $0.01$ | $-0.10$ | $-0.09$ | $-0.11$ | $-0.12$ | $-0.11$ | $-0.13$
($100$, $100$): | $-0.02$ | $0.01$ | $0.01$ | $-0.04$ | $-0.05$ | $-0.06$ | $-0.04$ | $-0.05$ | $-0.08$
| Bias of $\sqrt{nT}(\text{FOD estimator of }\beta_{2}-\beta_{2})$
($25$, $25$): | $0.13$ | $0.11$ | $0.14$ | $0.10$ | $0.10$ | $0.01$ | $0.03$ | $0.01$ | $-0.02$
($25$, $100$): | $0.08$ | $0.08$ | $0.08$ | $0.06$ | $0.03$ | $-0.02$ | $0.01$ | $0.01$ | $-0.02$
($100$, $25$): | $0.05$ | $0.05$ | $0.05$ | $0.07$ | $0.05$ | $0.02$ | $0.02$ | $0.01$ | $-0.01$
($100$, $100$): | $0.03$ | $0.05$ | $0.04$ | $0.03$ | $-0.01$ | $0.04$ | $0.01$ | $0.00$ | $0.02$
| Bias of $\sqrt{nT}(\text{FD estimator of }\beta_{1}-\beta_{1})$
($25$, $25$): | $0.11$ | $0.10$ | $0.12$ | $-0.91$ | $-0.93$ | $-0.94$ | $-0.95$ | $-0.99$ | $-1.00$
($25$, $100$): | $0.10$ | $0.07$ | $0.09$ | $-0.46$ | $-0.48$ | $-0.51$ | $-0.50$ | $-0.54$ | $-0.55$
($100$, $25$): | $0.27$ | $0.25$ | $0.25$ | $-1.54$ | $-1.53$ | $-1.51$ | $-1.66$ | $-1.68$ | $-1.67$
($100$, $100$): | $0.13$ | $0.17$ | $0.16$ | $-0.80$ | $-0.83$ | $-0.81$ | $-0.86$ | $-0.88$ | $-0.90$
| Bias of $\sqrt{nT}(\text{FD estimator of }\beta_{2}-\beta_{2})$
($25$, $25$): | $0.64$ | $0.55$ | $0.57$ | $0.15$ | $0.06$ | $-0.13$ | $-0.02$ | $-0.08$ | $-0.18$
($25$, $100$): | $0.36$ | $0.33$ | $0.32$ | $0.09$ | $0.00$ | $-0.10$ | $-0.02$ | $-0.05$ | $-0.11$
($100$, $25$): | $1.07$ | $1.05$ | $1.02$ | $0.16$ | $0.13$ | $0.05$ | $-0.03$ | $-0.05$ | $-0.10$
($100$, $100$): | $0.57$ | $0.59$ | $0.56$ | $0.06$ | $0.00$ | $0.04$ | $-0.02$ | $-0.05$ | $-0.04$

Table 2 is continued on the next page.

Table 2 continued.

Designs: | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18
---|---|---|---|---|---|---|---|---|---
($T$, $n$) | Bias of $\sqrt{nT}(\text{FOD estimator of }\beta_{1}-\beta_{1})$
($25$, $25$): | $-0.01$ | $-0.03$ | $-0.02$ | $-0.18$ | $-0.22$ | $-0.22$ | $-0.22$ | $-0.25$ | $-0.24$
($25$, $100$): | $0.00$ | $-0.00$ | $0.01$ | $-0.11$ | $-0.10$ | $-0.12$ | $-0.13$ | $-0.16$ | $-0.16$
($100$, $25$): | $-0.02$ | $-0.02$ | $-0.03$ | $-0.10$ | $-0.11$ | $-0.09$ | $-0.10$ | $-0.11$ | $-0.14$
($100$, $100$): | $-0.01$ | $-0.01$ | $-0.03$ | $-0.05$ | $-0.04$ | $-0.05$ | $-0.04$ | $-0.06$ | $-0.06$
| Bias of $\sqrt{nT}(\text{FOD estimator of }\beta_{2}-\beta_{2})$
($25$, $25$): | $0.11$ | $0.12$ | $0.12$ | $0.14$ | $0.17$ | $0.13$ | $0.03$ | $0.06$ | $0.04$
($25$, $100$): | $0.08$ | $0.06$ | $0.08$ | $0.05$ | $0.06$ | $0.09$ | $0.03$ | $0.03$ | $0.01$
($100$, $25$): | $0.04$ | $0.03$ | $0.04$ | $0.10$ | $0.09$ | $0.08$ | $0.06$ | $0.07$ | $0.11$
($100$, $100$): | $0.02$ | $0.02$ | $0.03$ | $0.04$ | $0.03$ | $0.05$ | $0.02$ | $0.04$ | $0.03$
| Bias of $\sqrt{nT}(\text{FD estimator of }\beta_{1}-\beta_{1})$
($25$, $25$): | $0.09$ | $0.08$ | $0.06$ | $-0.72$ | $-0.76$ | $-0.76$ | $-0.79$ | $-0.82$ | $-0.82$
($25$, $100$): | $0.08$ | $0.09$ | $0.07$ | $-0.39$ | $-0.37$ | $-0.39$ | $-0.43$ | $-0.46$ | $-0.45$
($100$, $25$): | $0.08$ | $0.04$ | $0.03$ | $-1.18$ | $-1.20$ | $-1.17$ | $-1.29$ | $-1.30$ | $-1.33$
($100$, $100$): | $0.08$ | $0.06$ | $0.05$ | $-0.63$ | $-0.63$ | $-0.63$ | $-0.67$ | $-0.71$ | $-0.69$
| Bias of $\sqrt{nT}(\text{FD estimator of }\beta_{2}-\beta_{2})$
($25$, $25$): | $0.62$ | $0.61$ | $0.56$ | $0.40$ | $0.31$ | $0.16$ | $0.01$ | $-0.01$ | $-0.09$
($25$, $100$): | $0.38$ | $0.36$ | $0.33$ | $0.15$ | $0.12$ | $0.09$ | $0.01$ | $-0.01$ | $-0.06$
($100$, $25$): | $0.84$ | $0.79$ | $0.80$ | $0.61$ | $0.57$ | $0.53$ | $0.22$ | $0.20$ | $0.23$
($100$, $100$): | $0.48$ | $0.46$ | $0.47$ | $0.25$ | $0.23$ | $0.24$ | $0.08$ | $0.08$ | $0.08$

Note: Each bias estimate is based on 5,000 samples.

Table 3: Scheme 1: Bias estimates for Designs 19 through 36
($\beta_{1}=0.75$).

Designs: | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27
---|---|---|---|---|---|---|---|---|---
($T$, $n$) | Bias of $\sqrt{nT}(\text{FOD estimator of }\beta_{1}-\beta_{1})$
($25$, $25$): | $-1.11$ | $-1.41$ | $-1.50$ | $-0.90$ | $-1.11$ | $-1.23$ | $-0.75$ | $-0.94$ | $-1.06$
($25$, $100$): | $-0.78$ | $-1.41$ | $-1.83$ | $-0.63$ | $-0.87$ | $-1.13$ | $-0.45$ | $-0.63$ | $-0.86$
($100$, $25$): | $-0.43$ | $-0.54$ | $-0.60$ | $-0.38$ | $-0.48$ | $-0.52$ | $-0.30$ | $-0.37$ | $-0.45$
($100$, $100$): | $-0.29$ | $-0.46$ | $-0.62$ | $-0.21$ | $-0.34$ | $-0.39$ | $-0.16$ | $-0.24$ | $-0.34$
| Bias of $\sqrt{nT}(\text{FOD estimator of }\beta_{2}-\beta_{2})$
($25$, $25$): | $-0.19$ | $-0.42$ | $-0.56$ | $0.19$ | $0.01$ | $-0.20$ | $0.16$ | $0.17$ | $0.09$
($25$, $100$): | $-0.16$ | $-0.49$ | $-0.81$ | $0.16$ | $0.01$ | $-0.29$ | $0.11$ | $0.13$ | $0.06$
($100$, $25$): | $-0.05$ | $-0.10$ | $-0.19$ | $0.15$ | $0.05$ | $-0.04$ | $0.12$ | $0.09$ | $0.07$
($100$, $100$): | $-0.02$ | $-0.12$ | $-0.26$ | $0.09$ | $0.02$ | $-0.05$ | $0.07$ | $0.09$ | $0.06$
| Bias of $\sqrt{nT}(\text{FD estimator of }\beta_{1}-\beta_{1})$
($25$, $25$): | $-3.80$ | $-4.68$ | $-5.30$ | $-3.01$ | $-3.34$ | $-3.64$ | $-2.73$ | $-2.98$ | $-3.16$
($25$, $100$): | $-2.30$ | $-3.16$ | $-4.17$ | $-1.76$ | $-2.00$ | $-2.37$ | $-1.59$ | $-1.75$ | $-1.91$
($100$, $25$): | $-4.31$ | $-4.39$ | $-4.55$ | $-3.65$ | $-3.73$ | $-3.75$ | $-3.08$ | $-3.13$ | $-3.13$
($100$, $100$): | $-2.50$ | $-2.61$ | $-2.78$ | $-2.05$ | $-2.11$ | $-2.16$ | $-1.72$ | $-1.74$ | $-1.78$
| Bias of $\sqrt{nT}(\text{FD estimator of }\beta_{2}-\beta_{2})$
($25$, $25$): | $-1.03$ | $-1.63$ | $-2.12$ | $0.06$ | $-0.45$ | $-1.01$ | $0.28$ | $0.09$ | $-0.16$
($25$, $100$): | $-0.70$ | $-1.19$ | $-1.82$ | $0.01$ | $-0.30$ | $-0.76$ | $0.16$ | $0.05$ | $-0.11$
($100$, $25$): | $-1.05$ | $-1.14$ | $-1.30$ | $0.07$ | $-0.14$ | $-0.27$ | $0.49$ | $0.39$ | $0.32$
($100$, $100$): | $-0.66$ | $-0.76$ | $-0.90$ | $-0.03$ | $-0.13$ | $-0.20$ | $0.25$ | $0.23$ | $0.17$

Table 3 is continued on the next page.

Table 3 continued.

Designs: | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36
---|---|---|---|---|---|---|---|---|---
($T$, $n$) | Bias of $\sqrt{nT}(\text{FOD estimator of }\beta_{1}-\beta_{1})$
($25$, $25$): | $-0.47$ | $-0.48$ | $-0.50$ | $-0.69$ | $-0.71$ | $-0.66$ | $-0.66$ | $-0.71$ | $-0.70$
($25$, $100$): | $-0.32$ | $-0.33$ | $-0.33$ | $-0.45$ | $-0.48$ | $-0.47$ | $-0.45$ | $-0.47$ | $-0.48$
($100$, $25$): | $-0.16$ | $-0.18$ | $-0.17$ | $-0.25$ | $-0.27$ | $-0.25$ | $-0.25$ | $-0.29$ | $-0.26$
($100$, $100$): | $-0.10$ | $-0.10$ | $-0.11$ | $-0.15$ | $-0.15$ | $-0.15$ | $-0.16$ | $-0.15$ | $-0.17$
| Bias of $\sqrt{nT}(\text{FOD estimator of }\beta_{2}-\beta_{2})$
($25$, $25$): | $0.13$ | $0.10$ | $0.09$ | $0.29$ | $0.26$ | $0.20$ | $0.16$ | $0.15$ | $0.12$
($25$, $100$): | $0.03$ | $0.03$ | $0.01$ | $0.16$ | $0.15$ | $0.06$ | $0.09$ | $0.06$ | $0.02$
($100$, $25$): | $0.11$ | $0.09$ | $0.08$ | $0.18$ | $0.19$ | $0.16$ | $0.16$ | $0.18$ | $0.15$
($100$, $100$): | $0.06$ | $0.04$ | $0.05$ | $0.12$ | $0.10$ | $0.09$ | $0.09$ | $0.09$ | $0.08$
| Bias of $\sqrt{nT}(\text{FD estimator of }\beta_{1}-\beta_{1})$
($25$, $25$): | $-1.65$ | $-1.73$ | $-1.71$ | $-2.02$ | $-2.01$ | $-1.92$ | $-1.92$ | $-1.94$ | $-1.95$
($25$, $100$): | $-1.00$ | $-1.12$ | $-1.14$ | $-1.09$ | $-1.14$ | $-1.19$ | $-1.08$ | $-1.08$ | $-1.08$
($100$, $25$): | $-1.62$ | $-1.63$ | $-1.62$ | $-1.98$ | $-2.00$ | $-1.97$ | $-1.96$ | $-1.99$ | $-1.95$
($100$, $100$): | $-0.96$ | $-0.94$ | $-0.99$ | $-1.10$ | $-1.11$ | $-1.11$ | $-1.08$ | $-1.08$ | $-1.07$
| Bias of $\sqrt{nT}(\text{FD estimator of }\beta_{2}-\beta_{2})$
($25$, $25$): | $-0.07$ | $-0.21$ | $-0.26$ | $0.22$ | $-0.05$ | $-0.26$ | $0.17$ | $0.04$ | $-0.14$
($25$, $100$): | $-0.17$ | $-0.28$ | $-0.33$ | $0.08$ | $-0.07$ | $-0.34$ | $0.08$ | $-0.02$ | $-0.13$
($100$, $25$): | $0.34$ | $0.30$ | $0.26$ | $0.76$ | $0.71$ | $0.60$ | $0.64$ | $0.61$ | $0.54$
($100$, $100$): | $0.10$ | $0.08$ | $0.05$ | $0.35$ | $0.28$ | $0.24$ | $0.32$ | $0.27$ | $0.27$

Note: Each bias estimate is based on 5,000 samples.

According to the results in Section 2, the FOD-GMM estimator’s bias is
predicted to disappear as both $T$ and $n$ increase. This claim is confirmed
by the data in Tables 2 and 3. For some designs the absolute bias estimates
for $\sqrt{nT}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})$ are small for
all sample sizes. On the other hand, for other designs the bias estimates are
substantial when $(T,n)=(25,25)$ but much smaller when $(T,n)=(100,100)$.
Moreover, $T$ and $n$ need not both be large in order to get a reduction in
absolute bias. The analytical results in Section 2 predicts that the bias
becomes less as either $T$ or $n$ increase, and the simulation data confirm
this prediction. The absolute bias of the FOD-GMM estimator is usually smaller
for both the $(T,n)=(25,100)$ and the $(T,n)=(100,25)$ sample sizes than for
the $(T,n)=(25,25)$ sample size. Bias is usually smallest, however, when both
$T$ and $n$ are large.

Table 4 shows the effect of bias on the reliability of confidence intervals.
The table provides the coverage of estimated 95-percent confidence intervals
for Scheme 1.666For Scheme 2, qualitatively similar results are provided in
Appendix C of the Supplemental Material [Phillips, 2022]. Both 95-percent and
90-percent confidence intervals were calculated, but, for the sake of brevity,
only the coverage of the estimated 95-percent confidence intervals is
provided. I report results for only 95-percent confidence intervals because
they are representative. In particular, when the coverage of a 95-percent
confidence interval approximated 95 percent, the coverage of the 90-percent
confidence interval likewise approximated 90 percent. On the other hand, if
the coverage of a 95-percent confidence interval did not approximate 95
percent well, the 90-percent confidence interval did not approximate 90
percent. Additionally, again for the sake of brevity, I report the coverage of
95-percent confidence intervals only for the $(T,n)=(25,25)$ and
$(T,n)=(100,100)$ sample sizes, for these are arguably the extreme cases in
terms of the FOD-GMM estimator’s bias. Therefore, these sample sizes should
provide the extremes in terms of poor and good approximate coverage of the FOD
confidence intervals. Specifically, the coverage should be best for the
large-$T$, large-$n$ cases and poorest for the small-$T$, small-$n$ cases.

Table 4: Scheme 1: Coverage of FD and FOD 95-percent confidence intervals.

| Designs: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
$(T,n)$ |  |  |  |  |  |  |  |  |  |
|  | Percentage of FOD intervals covering $\beta_{1}$
$(25,25)$: |  | 95.1 | 94.9 | 95.3 | 93.8 | 93.8 | 94.6 | 94.4 | 94.5 | 93.9
$(100,100)$: |  | 95.1 | 95.4 | 95.2 | 95.1 | 95.3 | 94.6 | 94.7 | 94.7 | 95.2
|  | Percentage of FOD intervals covering $\beta_{2}$
$(25,25)$: |  | 94.5 | 94.7 | 94.5 | 94.2 | 94.8 | 95.2 | 94.9 | 95.1 | 94.6
$(100,100)$: |  | 94.5 | 94.8 | 95.3 | 95.0 | 95.3 | 95.4 | 94.6 | 94.9 | 95.0
|  | Percentage of FD intervals covering $\beta_{1}$
$(25,25)$: |  | 94.6 | 94.3 | 94.1 | 84.0 | 83.7 | 84.2 | 81.9 | 80.5 | 79.6
$(100,100)$: |  | 94.6 | 95.3 | 94.9 | 87.4 | 87.6 | 86.0 | 84.4 | 83.5 | 83.6
|  | Percentage of FD intervals covering $\beta_{2}$
$(25,25)$: |  | 90.4 | 91.6 | 91.2 | 93.9 | 94.4 | 93.9 | 94.3 | 94.7 | 93.4
$(100,100)$: |  | 92.0 | 91.9 | 92.4 | 94.3 | 94.7 | 95.4 | 94.7 | 94.6 | 94.7
| Designs: | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18
$(T,n)$ |  |  |  |  |  |  |  |  |  |
|  | Percentage of FOD intervals covering $\beta_{1}$
$(25,25)$: |  | 95.0 | 95.1 | 95.0 | 93.8 | 94.2 | 94.6 | 94.2 | 94.4 | 94.3
$(100,100)$: |  | 94.8 | 95.5 | 95.1 | 95.1 | 95.3 | 95.5 | 94.8 | 94.7 | 95.0
|  | Percentage of FOD intervals covering $\beta_{2}$
$(25,25)$: |  | 94.1 | 95.0 | 94.8 | 94.4 | 95.1 | 94.7 | 94.7 | 95.2 | 95.4
$(100,100)$: |  | 95.0 | 95.1 | 95.3 | 94.7 | 94.6 | 95.1 | 94.9 | 94.9 | 94.7
|  | Percentage of FD intervals covering $\beta_{1}$
$(25,25)$: |  | 94.4 | 94.4 | 94.8 | 87.3 | 86.7 | 87.2 | 85.5 | 85.1 | 84.3
$(100,100)$: |  | 94.6 | 95.5 | 95.0 | 88.8 | 89.7 | 89.7 | 88.1 | 87.1 | 88.5
|  | Percentage of FD intervals covering $\beta_{2}$
$(25,25)$: |  | 91.0 | 90.2 | 91.2 | 92.9 | 94.1 | 94.2 | 94.1 | 94.3 | 94.2
$(100,100)$: |  | 92.2 | 92.5 | 92.1 | 94.3 | 94.3 | 94.9 | 94.8 | 94.9 | 94.8

Table 4 is continued on the next page.

Table 4 continued.

| Designs: | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27
---|---|---|---|---|---|---|---|---|---|---
$(T,n)$ |  |  |  |  |  |  |  |  |  |
|  | Percentage of FOD intervals covering $\beta_{1}$
$(25,25)$: |  | 88.1 | 88.1 | 87.9 | 87.5 | 85.5 | 85.2 | 88.6 | 85.7 | 85.2
$(100,100)$: |  | 94.7 | 94.3 | 93.2 | 94.6 | 94.1 | 94.2 | 95.1 | 93.7 | 94.3
|  | Percentage of FOD intervals covering $\beta_{2}$
$(25,25)$: |  | 93.7 | 92.9 | 92.3 | 94.5 | 94.5 | 93.8 | 95.0 | 94.8 | 95.2
$(100,100)$: |  | 95.0 | 94.4 | 94.3 | 94.8 | 95.4 | 94.3 | 95.1 | 94.2 | 95.2
|  | Percentage of FD intervals covering $\beta_{1}$
$(25,25)$: |  | 50.8 | 43.7 | 38.7 | 41.8 | 38.1 | 33.9 | 44.5 | 38.4 | 35.3
$(100,100)$: |  | 72.3 | 71.1 | 69.6 | 64.8 | 63.7 | 63.2 | 67.4 | 66.5 | 65.6
|  | Percentage of FD intervals covering $\beta_{2}$
$(25,25)$: |  | 81.3 | 70.3 | 63.0 | 93.6 | 90.7 | 82.1 | 93.2 | 94.4 | 92.3
$(100,100)$: |  | 90.9 | 89.2 | 88.0 | 94.6 | 94.5 | 94.2 | 94.2 | 93.2 | 94.4
| Designs: | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36
$(T,n)$ |  |  |  |  |  |  |  |  |  |
|  | Percentage of FOD intervals covering $\beta_{1}$
$(25,25)$: |  | 91.5 | 92.0 | 91.8 | 88.6 | 88.2 | 89.3 | 89.1 | 87.9 | 88.6
$(100,100)$: |  | 94.3 | 95.1 | 94.2 | 94.9 | 94.5 | 94.6 | 94.9 | 94.8 | 94.7
|  | Percentage of FOD intervals covering $\beta_{2}$
$(25,25)$: |  | 95.1 | 94.4 | 94.5 | 93.4 | 93.6 | 93.6 | 94.4 | 94.2 | 94.4
$(100,100)$: |  | 94.8 | 94.6 | 94.9 | 94.4 | 94.6 | 94.4 | 94.3 | 94.6 | 95.0
|  | Percentage of FD intervals covering $\beta_{1}$
$(25,25)$: |  | 75.2 | 74.7 | 75.8 | 58.5 | 59.1 | 60.9 | 59.3 | 56.2 | 56.1
$(100,100)$: |  | 88.4 | 88.6 | 87.4 | 81.8 | 81.5 | 80.4 | 78.6 | 78.6 | 78.6
|  | Percentage of FD intervals covering $\beta_{2}$
$(25,25)$: |  | 90.7 | 91.8 | 91.2 | 90.0 | 89.8 | 89.3 | 92.0 | 92.0 | 91.6
$(100,100)$: |  | 94.9 | 94.7 | 94.8 | 93.7 | 94.1 | 94.4 | 92.6 | 93.1 | 93.2

Note: Each coverage estimate is based on 5,000 samples.

This prediction is confirmed by the data in Table 4. For Designs 1 through 18
the estimated coverage of the FOD 95-percent confidence intervals approximates
95 percent in most cases. This is true for both the $(T,n)=(25,25)$ and
$(T,n)=(100,100)$ sample sizes. However, when $T$ and $n$ are both small, the
approximation deteriorates for Designs 19 through 36, especially for
$\beta_{1}$, which is $0.75$ for those designs. On the other hand, even when
$\beta_{1}=0.75$, the coverage of the FOD 95-percent confidence intervals is
usually close to 95 percent when $T$ and $n$ are both large.

Finally, the standard deviations of the FD-GMM and FOD-GMM estimators were
also estimated. Rather than report all of the standard deviation estimates,
the main conclusion drawn from calculating them can be summarized briefly: not
only did the FOD transformation generally produce less biased estimates, it
led to more efficient estimates. As already noted, there were a total of 288
experiments, and for each experiment the standard deviations of the FOD-GMM
estimators of $\beta_{1}$ and $\beta_{2}$ were estimated, for a total of 576
FOD standard deviation estimates. Similarly, 576 standard deviation estimates
were obtained for the FD-GMM estimators of $\beta_{1}$ and $\beta_{2}$. When
the standard deviation estimates of the FD- and FOD-GMM estimators were
compared, only nine FD standard deviation estimates, out of a total of 576,
were smaller than the corresponding FOD standard deviation estimates. In other
words, FOD-GMM was rarely less efficient than FD-GMM. Moreover, for those
cases for which the FOD-GMM estimator was less efficient, the maximum
reduction in standard deviation from using the FD-GMM estimator rather than
the FOD-GMM estimator was only about seven percent. On the other hand, for
over 98 percent of the experiments the FOD-GMM estimator was more efficient,
and for those experiments the maximum reduction in standard deviation from
using the FOD-GMM estimator rather than the FD-GMM estimator was about 47
percent. The median reduction was approximately 19 percent.

However, this dominance of the FOD-GMM estimator in terms of efficiency may be
specific to the experiments considered here. Unlike the reductions in bias
from using the FOD transformation, I know of no theoretical explanation for
why FOD-GMM should be superior to FD-GMM in terms of efficiency. For the
experiments considered in this section, neither the FD-GMM estimator nor the
FOD-GMM estimator exploited all available moment restrictions. But the two
estimators exploited the same number of moment restrictions, and both are
efficient for the moment restrictions they exploited. However, the
restrictions they exploited differ.

##  4 Conclusion

When using panel data to estimate a dynamic regression model, researchers are
often concerned about unobserved time-invariant effects. These effects can be
removed with a suitable transformation. Many transformations are available,
but due to historical priority, differencing the data has become a widely
adopted approach. An alternative is the FOD transformation. The FOD
transformation is computationally fast, even when both $n$ and $T$ are large
[Phillips, 2020], which is an advantage for simulation studies and
bootstrapping. Moreover, the analytical and Monte Carlo results provided in
this paper indicate that FOD-GMM estimators have little large sample bias,
regardless of whether $n$ is large, or $T$ is large, or both are large. This
conclusion is useful because the reliability of large sample confidence
intervals and test statistics based on an estimator depend on the estimator
being asymptotically unbiased. Key restrictions for FOD-GMM estimators to have
little bias are that the regressors be predetermined, the instrumental
variables should not be weak, and the number of instruments used in any given
period cannot increase with $T$ too fast.

## Appendix A: Proofs

The proofs of Theorems 1 and 2 rely on a few preliminary results. Lemma A.1 is
used in the proof of Theorem 1. Theorem 2, on the other hand, draws on Lemma
A.2.

######  Lemma A.1.

Assume $\text{rank}\left(\boldsymbol{Z}_{t}\right)=q_{t}$ with probability 1
and define
$p_{n,t}:=n\,\boldsymbol{z}_{1,t}^{\prime}\left(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t}\right)^{-1}\boldsymbol{z}_{1,t}$.
If $\boldsymbol{z}_{i,t}$ is identically distributed across $i$, then
$E(p_{n,t})=q_{t}$.

Proof: By a well-known result for the trace of a projection matrix, we have
$\text{tr}\left(\boldsymbol{P}_{t}\right)=\text{rank}\left(\boldsymbol{Z}_{t}\right)$.
And $\text{rank}\left(\boldsymbol{Z}_{t}\right)=q_{t}$ with probability 1 by
assumption. But
$\text{tr}\left(\boldsymbol{P}_{t}\right)=\sum_{i=1}^{n}\boldsymbol{z}_{i,t}^{\prime}\left(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t}\right)^{-1}\boldsymbol{z}_{i,t}$.
Hence,
$\sum_{i=1}^{n}E\left[\boldsymbol{z}_{i,t}^{\prime}\left(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t}\right)^{-1}\boldsymbol{z}_{i,t}\right]=q_{t}$.
Therefore, if $\boldsymbol{z}_{i,t}$ is identically distributed across $i$,
then
$nE\left[\boldsymbol{z}_{1,t}^{\prime}\left(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t}\right)^{-1}\boldsymbol{z}_{1,t}\right]=q_{t}$.

######  Lemma A.2.

Assume $E(v_{1,t}|\boldsymbol{w}_{1,t}^{\ast})=0$ and
$E(v_{1,t}^{2}|\boldsymbol{w}_{1,t}^{\ast})=\sigma^{2}$ $(t\geq 1)$. Also,
assume
$E(\boldsymbol{z}_{1,t}\boldsymbol{z}_{1,t}^{\prime})=\boldsymbol{Q}_{t}$
$(t\geq 1)$. Then, for $t\geq 1$, $r\geq 0$, and $1\leq t+r\leq T-1$, we have
$E(\ddot{v}_{1,t}\ddot{v}_{1,t+r}\boldsymbol{z}_{1,t}\boldsymbol{z}_{1,t+r}^{\prime})=\sigma^{2}\boldsymbol{Q}_{t}$
or $\boldsymbol{0}$ according as $r=0$ or $r>0$.

Proof: Note that
$E(\ddot{v}_{1,t}\ddot{v}_{1,t+r}\boldsymbol{z}_{1,t}\boldsymbol{z}_{1,t+r}^{\prime})=E[E(\ddot{v}_{1,t}\ddot{v}_{1,t+r}|\boldsymbol{w}_{1,t+r}^{\ast})\boldsymbol{z}_{1,t}\boldsymbol{z}_{1,t+r}^{\prime}]$.
Moreover,

$\begin{split}E(\ddot{v}_{1,t}\ddot{v}_{1,t+r}|\boldsymbol{w}_{1,t+r}^{\ast})&=c_{t}c_{t+r}E\left(v_{1,t}v_{1,t+r}|\boldsymbol{w}_{1,t+r}^{\ast}\right)\\\
&-\frac{c_{t}c_{t+r}}{T-t-r}\sum_{s=1}^{T-t-r}E\left(v_{1,t}v_{1,t+r+s}|\boldsymbol{w}_{1,t+r}^{\ast}\right)\\\
&-c_{t}c_{t+r}\frac{1}{T-t}\sum_{s=1}^{T-t}E\left(v_{1,t+s}v_{1,t+r}|\boldsymbol{w}_{1,t+r}^{\ast}\right)\\\
&+\frac{c_{t}c_{t+r}}{(T-t)(T-t-r)}\sum_{s=1}^{T-t}\,\sum_{j=1}^{T-t-r}E\left(v_{1,t+s}v_{1,t+r+j}|\boldsymbol{w}_{1,t+r}^{\ast}\right).\end{split}$
(A.1)

In order to evaluate the right-hand side of Eq. (A.1), note that the
assumptions of the lemma imply that the $\ddot{v}_{1,t}$s are conditionally
homoskedastic and uncorrelated across $t$. Specifically, for $\ell=k=j$, we
have $E(v_{1,\ell}v_{1,k}|\boldsymbol{w}_{1,j}^{\ast})=\sigma^{2}$. For
$\ell=k>j$, we get
$E[E(v_{1,\ell}v_{1,k}|\boldsymbol{w}_{1,k}^{\ast})|\boldsymbol{w}_{1,j}^{\ast}]=E(\sigma^{2}|\boldsymbol{w}_{1,j}^{\ast})=\sigma^{2}$.
If $\ell<k=j$, then
$E(v_{1,\ell}v_{1,k}|\boldsymbol{w}_{1,j}^{\ast})=v_{1,\ell}E(v_{1,k}|\boldsymbol{w}_{1,k}^{\ast})=0$.
Finally, if $\ell<k$ and $k>j$, then
$E(v_{1,\ell}v_{1,k}|\boldsymbol{w}_{1,j}^{\ast})=E[v_{1,\ell}E(v_{1,k}|\boldsymbol{w}_{1,k}^{\ast})|\boldsymbol{w}_{1,j}^{\ast}]=E(v_{1,\ell}0|\boldsymbol{w}_{1,j}^{\ast})=0$.
Hence,

$E(v_{1,\ell}v_{1,k}|\boldsymbol{w}_{1,j}^{\ast})=\sigma^{2}\text{ or }0\text{
according as }\ell=k\text{ or }\ell<k\quad(k\geq j).$ (A.2)

Suppose $r=0$. Then from (A.2), we see that the first term on the right-hand
side of Eq. (A.1) is $c_{t}^{2}\sigma^{2}$, the second and third terms are
zero, and the last term is $c_{t}^{2}\sigma^{2}/(T-t)$. Hence,
$E(\ddot{v}_{1,t}^{2}|\boldsymbol{w}_{1,t}^{\ast})=\sigma^{2}$, and thus
$E(\ddot{v}_{1,t}^{2}\boldsymbol{z}_{1,t}\boldsymbol{z}_{1,t}^{\prime}|\boldsymbol{w}_{1,t}^{\ast})=\sigma^{2}\boldsymbol{Q}_{t}$.

Suppose $r>0$. Then the first two terms on the right-hand side of (A.1) are
zero. The third term equals $-c_{t}c_{t+r}\sigma^{2}/(T-t)$, and the last term
is $c_{t}c_{t+r}\sigma^{2}/(T-t)$. Hence,
$E(\ddot{v}_{1,t}\ddot{v}_{1,t+r}|\boldsymbol{w}_{1,t+r}^{\ast})=0$, and thus
$E(\ddot{v}_{1,t}\ddot{v}_{1,t+r}\boldsymbol{z}_{1,t}\boldsymbol{z}_{1,t+r}^{\prime}|\boldsymbol{w}_{1,t+r}^{\ast})=\boldsymbol{0}$.

### A.1 Theorem 1 proof

First note that

$\sqrt{nT}\boldsymbol{b}_{n,T}=\sum_{t=1}^{T-1}c_{t}^{2}\boldsymbol{X}_{t}^{\prime}\boldsymbol{P}_{t}\left(\boldsymbol{v}_{t}-\overline{\boldsymbol{v}}_{t}\right)-\sum_{t=1}^{T-1}c_{t}^{2}\overline{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{P}_{t}\left(\boldsymbol{v}_{t}-\overline{\boldsymbol{v}}_{t}\right).$
(A.3)

Moreover, Assumption A1 implies $(\boldsymbol{w}_{i,t}^{\prime},v_{i,t+s})$ is
independent of $\boldsymbol{w}_{j,t}$, for $i\neq j$. Thus, if we set
$\boldsymbol{w}_{t}^{\prime}:=(\boldsymbol{w}_{1,t}^{\prime},\ldots,\boldsymbol{w}_{n,t}^{\prime})$,
then
$E\left(v_{i,t+s}|\boldsymbol{w}_{t}\right)=E\left(v_{i,t+s}|\boldsymbol{w}_{i,t}\right)$.
And, A1, A3, and the law of iterated expectations imply
$E\left(v_{i,t+s}|\boldsymbol{w}_{i,t}\right)=E\left[E\left(v_{i,t+s}|\boldsymbol{w}_{i,t+s}\right)|\boldsymbol{w}_{i,t}\right]=E\left(0|\boldsymbol{w}_{i,t}\right)=0$
($s\geq 0$). Hence,
$E\left(\boldsymbol{v}_{t}-\overline{\boldsymbol{v}}_{t}|\boldsymbol{w}_{t}\right)=\boldsymbol{0}$,
and thus
$E\left[\boldsymbol{X}_{t}^{\prime}\boldsymbol{P}_{t}\left(\boldsymbol{v}_{t}-\overline{\boldsymbol{v}}_{t}\right)\right]=E\left[\boldsymbol{X}_{t}^{\prime}\boldsymbol{P}_{t}E\left(\boldsymbol{v}_{t}-\overline{\boldsymbol{v}}_{t}|\boldsymbol{w}_{t}\right)\right]=\boldsymbol{0}$.
From this observation and Eq. (A.3), it follows that

$\sqrt{nT}E(\boldsymbol{b}_{n,T})=-\sum_{t=1}^{T-1}\frac{1}{T-t+1}\left(\boldsymbol{s}_{1,T-t}-\boldsymbol{s}_{2,T-t}\right),$
(A.4)

where
$\boldsymbol{s}_{1,T-t}:=\sum_{s=1}^{T-t}E\left(\boldsymbol{X}_{t+s}^{\prime}\boldsymbol{P}_{t}\boldsymbol{v}_{t}\right)$
and
$\boldsymbol{s}_{2,T-t}:=\sum_{s=1}^{T-t}E\left(\boldsymbol{X}_{t+s}^{\prime}\boldsymbol{P}_{t}\overline{\boldsymbol{v}}_{t}\right)$.

In order to evaluate $\boldsymbol{s}_{1,T-t}$, note that the $k$th entry of
$E\left(\boldsymbol{X}_{t+s}^{\prime}\boldsymbol{P}_{t}\boldsymbol{v}_{t}\right)$
is

$\sum_{i=1}^{n}E\left(x_{i,t+s,k}\boldsymbol{z}_{i,t}^{\prime}(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t})^{-1}\boldsymbol{Z}_{t}^{\prime}\boldsymbol{v}_{t}\right)=\sum_{i=1}^{n}E\left[\boldsymbol{z}_{i,t}^{\prime}(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t})^{-1}\sum_{j=1}^{n}\boldsymbol{z}_{j,t}E\left(v_{j,t}x_{i,t+s,k}|\boldsymbol{w}_{t}\right)\right]$
(A.5)

And, by the law of iterated expectations,
$E\left(v_{j,t}x_{i,t+s,k}|\boldsymbol{w}_{t}\right)=E\left[x_{i,t+s,k}E\left(v_{j,t}|\boldsymbol{w}_{t},\boldsymbol{x}_{i,t+s}\right)|\boldsymbol{w}_{t}\right]$.
Given $(\boldsymbol{w}_{j,t}^{\prime},v_{j,t})$ is independent of
$\boldsymbol{w}_{i,T}$ for $j\neq i$, it follows that
$E\left(v_{j,t}|\boldsymbol{w}_{t},\boldsymbol{x}_{i,t+s}\right)=E\left(v_{j,t}|\boldsymbol{w}_{j,t}\right)$
for $j\neq i$. Moreover, $E\left(v_{j,t}|\boldsymbol{w}_{j,t}\right)=0$ by A1
and A3. Hence, $E\left(v_{j,t}x_{i,t+s,k}|\boldsymbol{w}_{t}\right)=0$ for
$j\neq i$. On the other hand,
$E\left(v_{j,t}x_{i,t+s,k}|\boldsymbol{w}_{t}\right)=E\left(v_{i,t}x_{i,t+s,k}|\boldsymbol{w}_{t}\right)$
for $j=i$. Hence,
$\sum_{j=1}^{n}\boldsymbol{z}_{j,t}E\left(v_{j,t}x_{i,t+s,k}|\boldsymbol{w}_{t}\right)=\boldsymbol{z}_{i,t}E\left(v_{i,t}x_{i,t+s,k}|\boldsymbol{w}_{t}\right)$,
which implies the right-hand side of (A.5) is

$\sum_{i=1}^{n}E\left[\boldsymbol{z}_{i,t}^{\prime}(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t})^{-1}\boldsymbol{z}_{i,t}E\left(v_{i,t}x_{i,t+s,k}|\boldsymbol{w}_{t}\right)\right]=E\left[p_{n,t}E\left(v_{1,t}x_{1,t+s,k}|\boldsymbol{w}_{t}\right)\right],$
(A.6)

where the equality in Eq. (A.6) follows from the fact that the
$\boldsymbol{u}_{i}$s are identically distributed and
$p_{n,t}=n\,\boldsymbol{z}_{1,t}^{\prime}\left(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t}\right)^{-1}\boldsymbol{z}_{1,t}$.
Moreover, A1 implies
$E\left(v_{1,t}x_{1,t+s,k}|\boldsymbol{w}_{t}\right)=E\left(v_{1,t}x_{1,t+s,k}|\boldsymbol{w}_{1,t}\right)$.
Furthermore, from A3 and the definition of conditional covariance, we have
$\gamma_{k,t,s}=E\left(v_{1,t}x_{1,t+s,k}|\boldsymbol{w}_{1,t}\right)$. The
preceding observations imply the $k$th entry of
$E\left(\boldsymbol{X}_{t+s}^{\prime}\boldsymbol{P}_{t}\boldsymbol{v}_{t}\right)$
is $E\left(p_{n,t}\gamma_{k,t,s}\right)$. This conclusion and the definition
of $\boldsymbol{s}_{1,T-t}$ implies the $k$th entry in
$\boldsymbol{s}_{1,T-t}$ is

$s_{1,T-t,k}=E\left(p_{n,t}\sum_{s=1}^{T-t}\gamma_{k,t,s}\right)\qquad(1\leq
t\leq T-1).$

Moreover,

$\left|s_{1,T-t,k}\right|\leq
E\left(p_{n,t}\left|\sum_{s=1}^{T-t}\gamma_{k,t,s}\right|\right)\leq
E\left(p_{n,t}\right)M=q_{t}M,$ (A.7)

where the second inequality in (A.7) follows from A4 and the equality on the
far right-hand side follows from Lemma A.1.

Next, the entries in $\boldsymbol{s}_{2,T-t}$ are evaluated. To that end, note
that by arguments similar to those used to establish Eq. (A.6), the $k$th
entry in
$E\left(\boldsymbol{X}_{t+s}^{\prime}\boldsymbol{P}_{t}\overline{\boldsymbol{v}}_{t}\right)$
is $E\left(p_{n,t}\overline{v}_{1,t}x_{1,t+s,k}\right)$. Moreover,
$E\left(p_{n,t}v_{1,t+r}x_{1,t+s,k}\right)=E\left[p_{n,t}x_{1,t+s,k}E\left(v_{1,t+r}|\boldsymbol{w}_{t+s}\right)\right]=0$
if $r\geq s$. Hence, for $s=1$, we get
$\left(T-t\right)E\left(p_{n,t}\overline{v}_{1,t}x_{1,t+s,k}\right)=E(p_{n,t}x_{1,t+1,k}\sum_{r=1}^{T-t}{v}_{1,t+r})=0$.
And, when $t=T-1$, it must be that $s=1$. Thus, for $t=T-1$ and $s=1$, we get
$\left(T-t\right)E\left(p_{n,t}\overline{v}_{1,t}x_{1,t+s,k}\right)=E\left(p_{n,T-1}v_{1,T}x_{1,T,k}\right)=0$.
On the other hand, for $1\leq t\leq T-2$ and $s\geq 2$, we have

$\begin{split}(T-t)E\left(p_{n,t}\overline{v}_{1,t}x_{1,t+s,k}\right)&=E\left(p_{n,t}x_{1,t+s,k}\sum_{r=1}^{T-t}v_{1,t+r}\right)\\\
&=E\left(p_{n,t}x_{1,t+s,k}\sum_{r=1}^{s-1}v_{1,t+r}\right)\\\
&=E\left[p_{n,t}\sum_{r=1}^{s-1}E\left(v_{1,t+r}x_{1,t+r+(s-r),k}|\boldsymbol{w}_{t+r}\right)\right]\\\
&=E\left[p_{n,t}\sum_{r=1}^{s-1}\gamma_{k,t+r,s-r}\right].\end{split}$ (A.8)

Moreover, for $T\geq 3$,

$\begin{split}\sum_{s=2}^{T-t}\sum_{r=1}^{s-1}\gamma_{k,t+r,s-r}&=\underset{(r=1)}{\gamma_{k,t+1,1}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad(s=2)\\\
&+\underset{(r=1)}{\gamma_{k,t+1,2}}+\underset{(r=2)}{\gamma_{k,t+2,1}}\qquad\qquad\qquad\qquad\qquad\qquad\quad(s=3)\\\
&+\cdots\\\
&+\underset{(r=1)}{\gamma_{k,t+1,T-t-1}}+\underset{(r=2)}{\gamma_{k,t+2,T-t-2}}+\cdots+\underset{(r=T-t-1)}{\gamma_{k,T-1,1}}\qquad(s=T-t)\\\
&=\sum_{r=1}^{T-t-1}\sum_{j=1}^{T-t-r}\gamma_{k,t+r,j}.\end{split}$ (A.9)

It follows from Eq.s (A.8) and (A.9),
$E\left(p_{n,T-1}v_{1,T}x_{1,T,k}\right)=0$, and the definition of
$\boldsymbol{s}_{2,T-t}$ that, for $T\geq 3$, the $k$th entry in
$\boldsymbol{s}_{2,T-t}$ is

$\begin{split}s_{2,T-t,k}&=\frac{1}{T-t}E\left(p_{n,t}\sum_{r=1}^{T-t-1}\sum_{j=1}^{T-t-r}\gamma_{k,t+r,j}\right)\qquad(1\leq
t\leq T-2)\\\
&=0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(t=T-1).\end{split}$

Hence, $\left|s_{2,T-t,k}\right|=0$ for $t=T-1$, and, for $1\leq t\leq T-2$,
we have

$\begin{split}\left|s_{2,T-t,k}\right|&\leq\frac{1}{T-t}E\left(p_{n,t}\sum_{r=1}^{T-t-1}\left|\sum_{j=1}^{T-t-r}\gamma_{k,t+r,j}\right|\right)\\\
&\leq E\left(p_{n,t}\right)\frac{1}{T-t}\sum_{r=1}^{T-t-1}M\\\ &\leq
q_{t}M.\end{split}$ (A.10)

Expression (A.4), an obvious inequality, and the inequalities in (A.7) and
(A.10) imply

$\left|E\left(b_{n,T,k}\right)\right|\leq\frac{1}{\sqrt{nT}}\sum_{t=1}^{T-1}\frac{1}{T-t+1}\left(\left|s_{1,T-t,k}\right|+\left|s_{2,T-t,k}\right|\right)\leq\frac{2}{\sqrt{nT}}M\sum_{t=1}^{T-1}\frac{q_{t}}{T-t+1}.$

Moreover, given $q_{t}\leq q_{T}^{\ast}$, for all $t$, we have
$\sum_{t=1}^{T-1}q_{t}/(T-t+1)\leq q_{T}^{\ast}\sum_{t=1}^{T-1}(T-t+1)^{-1}$.
And, $\sum_{t=1}^{T-1}(T-t+1)^{-1}=\sum_{t=1}^{T}t^{-1}-1$. Furthermore,
$\sum_{t=1}^{T}t^{-1}-1\leq\ln T$ for $T\geq 1$ [see,e.g., Havil, 2003, p.
47]. The foregoing implies that, for $T\geq 3$,

$\left|E\left(b_{n,T,k}\right)\right|\leq 2M\frac{q_{T}^{\ast}}{\sqrt{nT}}\ln
T\qquad(k=1,\ldots,K).$ (A.11)

Now suppose $q_{T}^{\ast}=O(T^{\delta})$. This assumption and (A.11) imply

$E\left(b_{n,T,k}\right)=O\left(n^{-1/2}T^{\delta-1/2}\ln
T\right)\qquad(k=1,\ldots,K).$

### A.2 Theorem 2 proof:

Note that

$\sqrt{nT}\left(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right)-\boldsymbol{A}_{n,T}^{-1}E\left(\boldsymbol{b}_{n,T}\right)=\boldsymbol{A}_{n,T}^{-1}\left[\boldsymbol{b}_{n,T}-E\left(\boldsymbol{b}_{n,T}\right)\right].$

If $q_{T}^{\ast}=O(T^{\delta})$, with $0\leq\delta<1/2$, it follows from
Theorem 1 that
$\boldsymbol{b}_{n,T}-E(\boldsymbol{b}_{n,T})=\boldsymbol{b}_{n,T}+o(1)$ for
all sequences of $n$ and $T$—for example, for $n\rightarrow\infty$ with $T$
fixed or $T\rightarrow\infty$. Consequently, we can ignore
$E(\boldsymbol{b}_{n,T})$ when considering the asymptotic behavior of
$\boldsymbol{b}_{n,T}$ and that of
$\sqrt{nT}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})$.

Note that Assumption A3∗ implies
$E(\boldsymbol{z}_{1,t}\ddot{v}_{1,t})=\boldsymbol{0}$. Moreover, from Lemma
A.2, it follows that
$\text{Var}(\boldsymbol{z}_{1,t}\ddot{v}_{1,t})=\sigma^{2}\boldsymbol{Q}_{t}$.
Therefore, by the central limit theorem for i.i.d. random vectors, we have
$\boldsymbol{Z}_{t}^{\prime}\ddot{\boldsymbol{v}}_{t}/\sqrt{n}\overset{d}{\rightarrow}N(\boldsymbol{0},\sigma^{2}\boldsymbol{Q}_{t})$,
as $n\rightarrow\infty$. Hence,
$\boldsymbol{Z}_{t}^{\prime}\ddot{\boldsymbol{v}}_{t}/\sqrt{n}=O_{p}(1)$ for
each $t$. Moreover, A1, A2, A6, and the law of large numbers implies
$\ddot{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{Z}_{t}\left(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t}\right)^{-1}\overset{p}{\rightarrow}\boldsymbol{R}_{t}^{\prime}\boldsymbol{Q}_{t}^{-1}$,
as $n\rightarrow\infty$, for each $t$. It follows from the preceding that

$\begin{split}\boldsymbol{b}_{n,T}-\frac{1}{\sqrt{nT}}\sum_{t=1}^{T-1}\boldsymbol{R}_{t}^{\prime}\boldsymbol{Q}_{t}^{-1}\boldsymbol{Z}_{t}^{\prime}\ddot{\boldsymbol{v}}_{t}&=\frac{1}{\sqrt{T}}\sum_{t=1}^{T-1}\left(\ddot{\boldsymbol{X}}_{t}^{\prime}\boldsymbol{Z}_{t}\left(\boldsymbol{Z}_{t}^{\prime}\boldsymbol{Z}_{t}\right)^{-1}-\boldsymbol{R}_{t}^{\prime}\boldsymbol{Q}_{t}^{-1}\right)\frac{1}{\sqrt{n}}\boldsymbol{Z}_{t}^{\prime}\ddot{\boldsymbol{v}}_{t}\\\
&=o_{p}(1)\qquad(n\rightarrow\infty).\end{split}$

Therefore, the asymptotic distribution of $\boldsymbol{b}_{n,T}$, as
$n\rightarrow\infty$, is the same as the asymptotic distribution of
$\sum_{t=1}^{T-1}\boldsymbol{R}_{t}^{\prime}\boldsymbol{Q}_{t}\boldsymbol{Z}_{t}^{\prime}\ddot{\boldsymbol{v}}_{t}/\sqrt{nT}$,
as $n\rightarrow\infty$.

To evaluate the latter, note that if we set
$\boldsymbol{\epsilon}_{i,t}:=\boldsymbol{R}_{t}^{\prime}\boldsymbol{Q}_{t}^{-1}\boldsymbol{z}_{i,t}\ddot{v}_{i,t}$,
then
$\boldsymbol{R}_{t}^{\prime}\boldsymbol{Q}_{t}^{-1}\boldsymbol{Z}_{t}^{\prime}\ddot{\boldsymbol{v}}_{t}=\sum_{i=1}^{n}\boldsymbol{\epsilon}_{i,t}$.
Next, define the $K(T-1)\times 1$ vectors
$\boldsymbol{\epsilon}_{i}:=(\boldsymbol{\epsilon}_{i1}^{\prime},\ldots,\boldsymbol{\epsilon}_{i,T-1}^{\prime})^{\prime}$
($i=1,\ldots,n$). Assumption A3∗ implies
$E(\boldsymbol{\epsilon}_{1})=\boldsymbol{0}$. Moreover, the conclusion of
Lemma A.2 implies the entries in $\boldsymbol{\epsilon}_{1,s}$ are
uncorrelated with the entries in $\boldsymbol{\epsilon}_{1,t}$ for $s\neq t$.
Hence, the variance-covariance matrix of $\boldsymbol{\epsilon}_{1}$ is block
diagonal. In particular,
$\text{Var}(\boldsymbol{\epsilon}_{1})=\sigma^{2}\boldsymbol{\Omega}$, where
$\boldsymbol{\Omega}:=\text{diag}(\boldsymbol{\Omega}_{1},\ldots,\boldsymbol{\Omega}_{T-1})$.
Moreover, Assumptions A2 and A7 imply $\boldsymbol{\Omega}$ is positive
definite. And, A1 implies the $\boldsymbol{\epsilon}_{i}$s are i.i.d across
$i$. It follows from the central limit theorem for i.i.d. random vectors that,
as $n\rightarrow\infty$, we have
$\sum_{i=1}^{n}\boldsymbol{\epsilon}_{i}/\sqrt{n}\overset{d}{\rightarrow}\boldsymbol{\epsilon}^{\ast}$,
with $\boldsymbol{\epsilon}^{\ast}\sim
N(\boldsymbol{0},\sigma^{2}\boldsymbol{\Omega})$.

Partition $\boldsymbol{\epsilon}^{\ast}$ so that
$\boldsymbol{\epsilon}^{\ast}=(\boldsymbol{\epsilon}_{1}^{\ast\prime},\ldots,\boldsymbol{\epsilon}_{T-1}^{\ast\prime})^{\prime}$,
where
$\boldsymbol{\epsilon}_{1}^{\ast},\ldots,\boldsymbol{\epsilon}_{T-1}^{\ast}$
are $K\times 1$ vectors, and recall that $\boldsymbol{\Omega}$ is block
diagonal; that is, the entries of $\boldsymbol{\epsilon}_{s}^{\ast}$ are
uncorrelated with the entries in $\boldsymbol{\epsilon}_{t}^{\ast}$ for $s\neq
t$. Moreover, as shown in the last paragraph, the entries in
$\boldsymbol{\epsilon}^{\ast}$ are multivariate normal. Hence,
$\boldsymbol{\epsilon}_{s}^{\ast}$ is independent of
$\boldsymbol{\epsilon}_{t}^{\ast}$ for $s\neq t$.

Let $\boldsymbol{a}:=(a_{1},\ldots,a_{K})^{\prime}$ be a vector of constants
with $\boldsymbol{a}\neq\boldsymbol{0}$, and define
$\mathcal{Y}_{t}=\boldsymbol{a}^{\prime}\boldsymbol{\epsilon}_{t}^{\ast}$
$(t=1,\ldots,T-1)$. The variable $\mathcal{Y}_{t}$ is independent across $t$,
with mean $E(\mathcal{Y}_{t})=0$ and variance
$\sigma_{\mathcal{Y},t}^{2}=\sigma^{2}\boldsymbol{a}^{\prime}\boldsymbol{\Omega}_{t}\boldsymbol{a}$.
Let $s_{T-1}^{2}=\sum_{t=1}^{T-1}\sigma_{\mathcal{Y},t}^{2}$ and
$\mathcal{Z}_{T-1}=\sum_{t=1}^{T-1}\mathcal{Y}_{t}/s_{T-1}$. Then, by
Lyapunov’s central limit theorem,
$\mathcal{Z}_{T-1}\overset{d}{\rightarrow}N(0,1)$, as $T\rightarrow\infty$, if
$s_{T-1}^{-3}\sum_{t=1}^{T-1}E\left|\mathcal{Y}_{t}\right|^{3}\rightarrow 0$,
as $T\rightarrow\infty$ [see, e.g., Sen and Singer, 1993, Theorem 3.3.2]. But
$\mathcal{Y}_{t}$ is normal, and thus
$E\left|\mathcal{Y}_{t}\right|^{3}=\sigma_{\mathcal{Y},t}^{3}2^{3/2}\Gamma(2)/\sqrt{\pi}<\infty$
[Winkelbauer, 2014]. Thus,
$s_{T-1}^{-3}\sum_{t=1}^{T-1}E\left|\mathcal{Y}_{t}\right|^{3}\rightarrow 0$,
as $T\rightarrow\infty$, if

$\frac{1}{s_{T-1}^{3}}\sum_{t=1}^{T-1}\sigma_{\mathcal{Y},t}^{3}\rightarrow
0,\qquad T\rightarrow\infty.$ (A.12)

In order to verify (A.12) note that

$\frac{1}{s_{T-1}^{3}}\sum_{t=1}^{T-1}\sigma_{\mathcal{Y},t}^{3}=\frac{1}{\left(\boldsymbol{a}^{\prime}\overline{\boldsymbol{\Omega}}_{T}\boldsymbol{a}\right)^{3/2}}\frac{1}{T^{3/2}}\sum_{t=1}^{T-1}\left(\boldsymbol{a}^{\prime}\boldsymbol{\Omega}_{t}\boldsymbol{a}\right)^{3/2}.$
(A.13)

Moreover,

$\left(\boldsymbol{a}^{\prime}\boldsymbol{\Omega}_{t}\boldsymbol{a}\right)^{3/2}\leq
K\sum_{j=1}^{K}\sum_{k=1}^{K}\left|a_{j}a_{k}\omega_{t,j,k}\right|^{3/2}\leq\nu_{t}^{3/2}K\sum_{j=1}^{K}\sum_{k=1}^{K}\left|a_{j}a_{k}\right|^{3/2},$
(A.14)

where the first inequality in (A.14) follows from the cr-inequality [see,
e.g., Sen and Singer, 1993, p. 21] and the second from the definition of
$\nu_{t}$. It follows from (A.13), (A.14), Assumption A8, which implies
$\lim_{T\rightarrow\infty}\boldsymbol{a}^{\prime}\overline{\boldsymbol{\Omega}}_{T}\boldsymbol{a}=\boldsymbol{a}^{\prime}\overline{\boldsymbol{\Omega}}\boldsymbol{a}>0$,
and Assumption A9 that

$\frac{1}{s_{T-1}^{3}}\sum_{t=1}^{T-1}\sigma_{\mathcal{Y},t}^{3}\leq\frac{1}{\left(\boldsymbol{a}^{\prime}\overline{\boldsymbol{\Omega}}_{T}\boldsymbol{a}\right)^{3/2}}K\sum_{j=1}^{K}\sum_{k=1}^{K}\left|a_{j}a_{k}\right|^{3/2}\frac{1}{T^{3/2}}\sum_{t=1}^{T-1}\nu_{t}^{3/2}\rightarrow
0,\qquad T\rightarrow\infty.$

In other words, (A.12) is satisfied, and therefore
$\mathcal{Z}_{T-1}\overset{d}{\rightarrow}N(0,1)$, as $T\rightarrow\infty$.
Upon applying the Cram$\acute{\text{e}}$r-Wold Theorem [Sen and Singer, 1993,
Theorem 3.2.4], it follows that
$\sum_{t=1}^{T-1}\boldsymbol{\epsilon}_{t}^{\ast}/\sqrt{T}\overset{d}{\rightarrow}N(\boldsymbol{0},\sigma^{2}\overline{\boldsymbol{\Omega}})$,
as $T\rightarrow\infty$.

The preceding verifies
$\boldsymbol{b}_{n,T}\overset{d}{\rightarrow}N(\boldsymbol{0},\sigma^{2}\overline{\boldsymbol{\Omega}})$
$(n,T\rightarrow\infty)_{\text{seq}}$. Moreover,
$\boldsymbol{A}_{n,T}\overset{p}{\rightarrow}\overline{\boldsymbol{\Omega}}$
$(n,T\rightarrow\infty)_{\text{seq}}$. Hence,
$\sqrt{nT}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta})=\boldsymbol{A}_{n,T}^{-1}\boldsymbol{b}_{n,T}\overset{d}{\rightarrow}N(\boldsymbol{0},\sigma^{2}\overline{\boldsymbol{\Omega}}^{-1})$
$(n,T\rightarrow\infty)_{\text{seq}}$.

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