# Pooled Bewley Estimator of Long Run Relationships in Dynamic Heterogenous
Panels††thanks: The views expressed in this paper are those of the authors and
do not necessarily reflect those of the Federal Reserve Bank of Dallas or the
Federal Reserve System. Monte Carlo simulations in this paper were computed
using computational resources provided by the Big-Tex High Performance
Computing Group at the Federal Reserve Bank of Dallas. We are grateful for
comments by two referees and the editor.

Alexander Chudik
Federal Reserve Bank of Dallas Email<EMAIL_ADDRESS>M. Hashem
Pesaran
University of Southern California, USA and Trinity College, Cambridge, UK
Email<EMAIL_ADDRESS>Ron P. Smith
Birkbeck, University of London, United Kingdom Email<EMAIL_ADDRESS>
(October 30, 2023)

###### Abstract

Using a transformation of the autoregressive distributed lag model due to
Bewley, a novel pooled Bewley (PB) estimator of long-run coefficients for
dynamic panels with heterogeneous short-run dynamics is proposed. The PB
estimator is directly comparable to the widely used Pooled Mean Group (PMG)
estimator, and is shown to be consistent and asymptotically normal. Monte
Carlo simulations show good small sample performance of PB compared to the
existing estimators in the literature, namely PMG, panel dynamic OLS (PDOLS),
and panel fully-modified OLS (FMOLS). Application of two bias-correction
methods and a bootstrapping of critical values to conduct inference robust to
cross-sectional dependence of errors are also considered. The utility of the
PB estimator is illustrated in an empirical application to the aggregate
consumption function.

Keywords: Heterogeneous dynamic panels; I(1) regressors; pooled mean group
estimator (PMG), Autoregressive-Distributed Lag model (ARDL), Bewley
transform, PDOLS, FMOLS, bias correction, robust inference, cross-sectional
dependence.

JEL Classification: C12, C13, C23, C33

##  1 Introduction

Estimation of cointegrating relationships in panels with heterogeneous short-
run dynamics is important for empirical research in open economy
macroeconomics as well as in other fields in economics. Existing single-
equation panel estimators in the literature are panel Fully Modified OLS
(FMOLS) by Pedroni1996 (Pedroni1996, Pedroni2001, Pedroni2001ReStat), panel
Dynamic OLS (PDOLS) by MarkSul2003, and the likelihood based Pooled Mean Group
(PMG) estimator by PesaranShinSmith1999. Multi-equation (system) approach by
Breitung2005, and the related system PMG approach by ChudikPesaranSmith2022
are another contributions in the literature on estimating cointegrating
vectors in a panel context. In this paper, we propose a pooled Bewley (PB)
estimator of long-run relationships, relying on the Bewley transform of an
autoregressive distributed lag (ARDL) model (Bewley1979, Bewley1979). See also
WickensBreusch1988 (WickensBreusch1988) for a discussion of the Bewley
transform.

Our setting is the same as that of PesaranShinSmith1999. Under this setting,
any short-run feedbacks between the outcome variable ($y$) and regressors
($x$) are allowed, but the direction of the long run causality is assumed to
go from $x$ to $y$. Hence, as with the PMG, the PB estimator allows for
heterogeneity in short-run feedbacks, but restricts the direction of long run
causality. The PB estimator is computed analytically, and does not rely on
numerical maximization of the likelihood function that underlies the PMG
estimation. We derive the asymptotic distribution of the PB estimator when the
cross-section dimension ($n$) and the time dimension ($T$) diverge to infinity
jointly such that $n=\Theta\left(T^{\theta}\right)$, for $0<\theta<2$, where
we use the notation $\Theta\left(.\right)$ to denote the same order of
magnitude asymptotically, namely if $\left\\{f_{s}\right\\}_{s=1}^{\infty}$
and $\left\\{g_{s}\right\\}_{s=1}^{\infty}$ are both positive sequences of
real numbers, then $f_{s}=\ominus\left(g_{s}\right)$ if there exists
$S_{0}\geq 1$ and positive finite constants $C_{0}$ and $C_{1}$, such that
$\inf_{s\geq S_{0}}\left(f_{s}/g_{s}\right)\geq C_{0},$ and $\sup_{s\geq
S_{0}}\left(f_{s}/g_{s}\right)\leq C_{1}$. Our asymptotic analysis is an
advance over the theoretical results currently available for PMG, PDOLS and
FMOLS estimators where it is assumed that $n$ is small relative to $T$ (which
corresponds to the case where $\theta$ is close to zero).

How well individual estimators work in samples of interest in practice where
$n$ and/or $T$ are often less than 50 is a different matter, which we shed
light on using Monte Carlo experiments. Monte Carlo evidence shows PB
estimator can be superior to PMG, PDOLS and FMOLS, in terms of its overall
precision as measured by the Root Mean Square Error (RMSE), and in terms of
accuracy of inference as measured by size distortions. These experiments
reveal PB is a useful addition to the literature.

Monte Carlo evidence also shows that the time dimension is very important for
the performance of these estimators, and that all of the four estimators under
consideration suffer from the same two drawbacks: small sample bias and size
distortion. Although the size distortions are found to be less serious for the
PB estimator in our experiments, all four estimators exhibit notable over-
rejections in sample sizes relevant in practice. In addition, all four
estimators (perhaps unsurprisingly) suffer from bias in finite samples, albeit
a rather small one. Both drawbacks diminish as $T$ is increased relative to
$n$.

To conduct reliable inference regardless of the cross-sectional dependence of
errors, we make use of the sieve wild bootstrap procedure. To accommodate
cross-sectional dependence, we resample the cross-section vectors of
residuals, an idea that was originally proposed by MaddalaWu1999. We found the
sieve wild bootstrap procedure to be remarkably effective for all four
estimators, regardless of cross-sectional dependence of errors, and we
therefore recommend using it in empirical research.

Regarding the small sample bias, we consider the application of two bias-
correction methods taken from the literature, relying either on split-panel
jackknife (DhaeneJochmansy2015, DhaeneJochmansy2015) or sieve wild bootstrap
approaches. In contrast to split-panel approaches in panels without stochastic
trends, as, for instance, considered by of DhaeneJochmansy2015 or
ChudikPesaranYang2018, in this paper we need to combine the full sample and
half-panel subsamples using different weighting due to the fact that the rate
of convergence of the estimators of long run coefficients is faster, at the
rate of $T\sqrt{n}$, as compared to the standard rate of $\sqrt{nT}$. We find
that both of these approaches can be helpful in reducing the bias (for all
four estimators). However, given that the bias is small to begin with, the
value of bias correction methods is limited.

The relevance of choosing a particular estimation approach is illustrated in
the context of a consumption function application for OECD economies taken
from PesaranShinSmith1999. This application shows that quite a different
conclusion would be reached when using PB estimator, which does not reject the
zero long-run coefficient on inflation, in line with the long-run neutrality
of monetary policy, whereas the PMG estimator results in a highly
statistically significant negative long-run coefficient. Estimates of the
long-run coefficient on real income are less diverse across estimators, but
the inference on whether a unit long-run coefficient on real income (as
suggested by balanced growth path models in the literature) can be rejected or
not depends on the choice of a particular estimator.

The remainder of this paper is organized as follows. Section 2 presents the
model and assumptions, introduces the PB estimator, and provides asymptotic
results. Application of bias correction methods and bootstrapping critical
values are also discussed in Section 2. Section 3 presents Monte Carlo
evidence. Section 4 revisits the aggregate consumption function empirical
application in PesaranShinSmith1999. Section 5 concludes. Mathematical
derivations and proofs are provided in Appendix A. Details on the
implementation of individual estimators and bootstrapping, and additional
Monte Carlo results are provided in Appendix B.

##  2 Pooled Bewley estimator of long-run relationships

We adopt the same setting as in PesaranShinSmith1999, and consider the
following illustrative model

$\displaystyle\Delta y_{it}$ $\displaystyle=$ $\displaystyle
c_{i}-\alpha_{i}\left(y_{i,t-1}-\beta x_{i,t-1}\right)+u_{y,it}\text{,}$ (1)
$\displaystyle\Delta x_{it}$ $\displaystyle=$ $\displaystyle u_{x,it}\text{,}$
(2)

for $i=1,2,...,n$, and $t=1,2,...,T$. For expositional clarity and notational
simplicity, we focus on a single regressor and one lag, but it is understood
that our analysis is applicable to multiple lags of
$\Delta\mathbf{z}_{it}=\left(\Delta y_{it},\Delta x_{it}\right)^{\prime}$
entering both equations (1)-(2), and the approach is also applicable to
multiple $x_{it}$’s with a single long run relationship. We consider the
following assumptions:

###### Assumption 1

(Coefficients)  $\sup_{i}\left|1-\alpha_{i}\right|<1$.

###### Assumption 2

(shocks) $u_{x,it}\sim IID\left(0,\sigma_{xi}^{2}\right)$, and $u_{y,it}$ is
given by

$u_{y,it}=\delta_{i}u_{x,it}+v_{it}\text{,}$ (3)

for all $i$ and $t$, where $v_{it}\sim IID\left(0,\sigma_{vi}^{2}\right)$, and
$u_{x,it}$ is independently distributed of $v_{i^{\prime}t^{\prime}}$ for all
$i$,$i^{\prime},t$, and $t^{\prime}$. In addition,
$\sup_{i,t}E\left|v_{it}\right|^{16}<K$ and
$\sup_{i,t}E\left|u_{x,it}\right|^{8}<K$, and
$\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}=\sigma_{x}^{2}>0$
and
$\lim{}_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}\sigma_{vi}^{2}/\left(6\alpha_{i}^{2}\right)=\omega_{v}^{2}>0$
exist.

###### Assumption 3

(Initial values and deterministic terms) The initial values,
$\mathbf{z}_{i,0}=\left(y_{i,0},x_{i,0}\right)^{\prime}$, follow the process

$\mathbf{z}_{i0}=\mathbf{\mu}_{i}+\mathbf{C}_{i}^{\ast}\left(L\right)\mathbf{u}_{0}\text{,}$

for all $i$ and $t$, and $c_{i}=$
$\alpha_{i}\mu_{i,1}-\alpha_{i}\beta\mu_{i,2}$ for all $i$, where
$\mathbf{u}_{0}=\left(u_{y,i,0},u_{x,i,0}\right)^{\prime}$,
$\mathbf{\mu}_{i}=\left(\mu_{i,1},\mu_{i,2}\right)^{\prime}$,
$\left\|\mathbf{\mu}_{i}\right\|<K$, and $\mathbf{C}_{i}^{\ast}\left(L\right)$
is defined in Section A.1 in Appendix A.

###### Remark 1

Assumption 1 requires that $\alpha_{i}\neq 0$ for all $i$. In contrast PMG
allows $\alpha_{i}=0$ for some (but not all) units. Although the estimator
works for $\alpha_{i}<2,$ empirically, $\alpha_{i}<1$ is likely to be the
relevant case.

###### Remark 2

Assumption 2 allows for $u_{x,it}$ to be correlated with $u_{y,it}$. Cross-
section dependence of $\mathbf{u}_{it}$ is ruled out. Assumption 3 (together
with the remaining assumptions) ensure that $\Delta\mathbf{z}_{it}$ and
$\left(y_{it}-\beta x_{it}\right)$ are covariance stationary.

###### Remark 3

In comparing our assumptions with the rest of the literature, it should be
noted that the rest of the literature does not consider the case of joint
convergence $n,T\rightarrow_{j}\infty$, but only the case where $n$ is fixed
as $T\rightarrow\infty$. Joint asymptotics typically requires more stringent
assumptions on the errors and restrictions on the relative expansion rates of
$n$ and $T$.

Substituting first (3) for $u_{y,it}$ in (1), and then substituting
$u_{x,it}=\Delta x_{it}$, we obtain the following ARDL representation for
$y_{it}$

$\Delta y_{it}=c_{i}-\alpha_{i}\left(y_{i,t-1}-\beta
x_{i,t-1}\right)+\delta_{i}\Delta x_{it}+v_{it}\text{.}$ (4)

The pooled Bewley estimator takes advantage of the Bewley transform
(Bewley1979, Bewley1979). Subtracting $\left(1-\alpha_{i}\right)y_{it}$ from
both sides of (4) and re-arranging, we have

$\alpha_{i}y_{it}=c_{i}-\left(1-\alpha_{i}\right)\Delta y_{it}+\alpha_{i}\beta
x_{it}+\delta_{i}\Delta x_{it}+v_{it},$

or (noting that $\alpha_{i}>0$ for all $i$ and multiplying the equation above
by $\alpha_{i}^{-1}$)

$y_{it}=\alpha_{i}^{-1}c_{i}+\beta
x_{it}+\mathbf{\psi}_{i}^{\prime}\Delta\mathbf{z}_{it}+\alpha_{i}^{-1}v_{it},$
(5)

where $\Delta\mathbf{z}_{it}=\left(\Delta y_{it},\Delta
x_{it}\right)^{\prime}$, and
$\mathbf{\psi}_{i}=\left(-\frac{1-\alpha_{i}}{\alpha_{i}},\frac{\delta_{i}}{\alpha_{i}}\right)^{\prime}$.
Further, stacking (5) for $t=1,2,...,T$, we have

$\mathbf{y}_{i}=\alpha_{i}^{-1}c_{i}\mathbf{\tau}_{T}+\mathbf{x}_{i}\beta+\Delta\mathbf{Z}_{i}\mathbf{\psi}_{i}+\alpha_{i}^{-1}\mathbf{v}_{i}\text{,}$
(6)

where $\mathbf{y}_{i}=\left(y_{i1},y_{i2},...,y_{iT}\right)^{\prime}$,
$\mathbf{x}_{i}=\left(x_{i1},x_{i2},...,x_{iT}\right)^{\prime}$,
$\Delta\mathbf{Z}_{i}=\left(\Delta\mathbf{z}_{i1}^{\prime},\Delta\mathbf{z}_{i2}^{\prime},...,\Delta\mathbf{z}_{iT}^{\prime}\right)^{\prime}$,
$\mathbf{v}_{i}=\left(v_{i,1},v_{i,2},...,v_{i,T}\right)^{\prime}$, and
$\mathbf{\tau}_{T}$ is $T\times 1$ vector of ones. Define projection matrix
$\mathbf{M}_{\tau}=\mathbf{I}_{T}-T^{-1}\mathbf{\tau}_{T}\mathbf{\tau}_{T}^{\prime}$.
This projection matrix subtracts the period average. Let
$\mathbf{\tilde{y}}_{i}=\left(\tilde{y}_{i1},\tilde{y}_{i2},...,\tilde{y}_{iT}\right)^{\prime}=\mathbf{M}_{\tau}\mathbf{y}_{i}$,
and similarly
$\mathbf{\tilde{x}}_{i}=\left(\tilde{x}_{i1},\tilde{x}_{i2},...,\tilde{x}_{iT}\right)^{\prime}=\mathbf{M}_{\tau}\mathbf{x}_{i}$,
$\Delta\mathbf{\tilde{Z}}_{i}=\mathbf{M}_{\tau}\Delta\mathbf{Z}_{i}$, and
$\mathbf{\tilde{v}}_{i}=\mathbf{M}_{\tau}\mathbf{v}_{i}$. Multiplying (6) by
$\mathbf{M}_{\tau}$, we have

$\mathbf{\tilde{y}}_{i}=\mathbf{\tilde{x}}_{i}\beta+\Delta\mathbf{\tilde{Z}}_{i}\mathbf{\psi}_{i}+\alpha_{i}^{-1}\mathbf{\tilde{v}}_{i}\text{.}$

Now consider the matrix of instruments

$\mathbf{\tilde{H}}_{i}=\left(\mathbf{\tilde{y}}_{i,-1},\mathbf{\tilde{x}}_{i},\mathbf{\tilde{x}}_{i,-1}\right)=\mathbf{M}_{\tau}\mathbf{H}_{i}\text{,
}\mathbf{H}_{i}=\left(\mathbf{y}_{i,-1},\mathbf{x}_{i},\mathbf{x}_{i,-1}\right)\text{,}$
(7)

where $\mathbf{y}_{i,-1}=\left(y_{i,1},y_{i,1},...,y_{i,T-1}\right)^{\prime}$
is the data vector on the first lag of $y_{it}$, similarly
$\mathbf{x}_{i,-1}=\left(x_{i,1},x_{i,1},...,x_{i,T-1}\right)^{\prime}$. The
PB estimator of $\beta$ is given by

$\hat{\beta}=\left(\sum_{i=1}^{n}\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{x}}_{i}\right)^{-1}\left(\sum_{i=1}^{n}\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{y}}_{i}\right)\text{,}$
(8)

where

$\mathbf{M}_{i}=\mathbf{P}_{i}-\mathbf{P}_{i}\Delta\mathbf{\tilde{Z}}_{i}\left(\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{Z}}_{i}\right)^{-1}\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\text{,}$
(9)

and

$\mathbf{P}_{i}=\mathbf{\tilde{H}}_{i}\left(\mathbf{\tilde{H}}_{i}{}^{\prime}\mathbf{\tilde{H}}_{i}\right)^{-1}\mathbf{\tilde{H}}_{i}^{\prime}\text{,}$
(10)

is the projection matrix associated with $\mathbf{\tilde{H}}_{i}$.

In addition to Assumptions 1-3, we also require the following high-level
conditions to hold in the derivations of the asymptotic distribution of the PB
estimator under the joint asymptotics $n,T\rightarrow\infty$.

###### Assumption 4

Let
$\mathbf{M}_{\tau}=\mathbf{I}_{T}-T^{-1}\mathbf{\tau}_{T}\mathbf{\tau}_{T}^{\prime},$
$\mathbf{\tilde{y}}_{i}=\mathbf{M}_{\tau}\mathbf{y}_{i}$,
$\mathbf{\tilde{x}}_{i}=\mathbf{M}_{\tau}\mathbf{x}_{i}$,
$\Delta\mathbf{\tilde{Z}}_{i}=\mathbf{M}_{\tau}\Delta\mathbf{Z}_{i}=\mathbf{M}_{\tau}\left(\Delta\mathbf{z}_{i1}^{\prime},\Delta\mathbf{z}_{i2}^{\prime},...,\Delta\mathbf{z}_{iT}^{\prime}\right)^{\prime}$,
where $\Delta\mathbf{z}_{it}=\left(\Delta y_{it},\Delta
x_{it}\right)^{\prime}$. Then there exists $T_{0}\in\mathbb{N}$ such that the
following conditions are satisfied:

  1. ($i$)

$\sup_{i\in\mathbb{N}\text{,
}T>T_{0}}E\left[\lambda_{\min}^{-2}\left(\mathbf{B}_{iT}\right)\right]<K$,
where
$\mathbf{B}_{iT}=\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{Z}}_{i}/T$,
$\mathbf{P}_{i}$ is given by (10).

  2. ($ii$)

$\sup_{i\in\mathbb{N}\text{,
}T>T_{0}}E\left[\lambda_{\min}^{-2}\left(\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{H}}_{i}^{\ast}\mathbf{A}_{T}\right)\right]<K$,
where
$\mathbf{H}_{i}^{\ast}=\left(\mathbf{\tilde{x}}_{i},\Delta\mathbf{\tilde{x}}_{i},\mathbf{\tilde{\xi}}_{i,-1}\right)$,

$\mathbf{A}_{T}=\left(\begin{array}[]{ccc}T^{-1}&0&0\\\ 0&T^{-1/2}&0\\\
0&0&T^{-1/2}\end{array}\right)\text{,}$

$\mathbf{\tilde{\xi}}_{i,-1}=\left(\tilde{\xi}_{i,0},\tilde{\xi}_{i,1},...,\tilde{\xi}_{i,T-1}\right)^{\prime}$,
$\tilde{\xi}_{i,t-1}=\tilde{y}_{i,t-1}-\beta\tilde{x}_{i,t-1}.$

###### Remark 4

Under Assumptions 1-3 (and without Assumption 4), we have
$\mathop{\mathrm{p}lim}_{T\rightarrow\infty}\mathbf{B}_{i,T}=\mathbf{B}_{i}$,
where $\mathbf{B}_{i}$ is nonsingular (see Lemma A.7 in Appendix A).
Similarly, it can be shown that Assumptions 1-3 are sufficient for
$\mathop{\mathrm{p}lim}_{T\rightarrow\infty}\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{H}}_{i}^{\ast}\mathbf{A}_{T}$
to exist and to be nonsingular. However, these results are not sufficient for
the moments of $\left\|\mathbf{B}_{i}^{-1}\right\|$ and
$\left\|\left(\mathbf{A}_{T}\mathbf{H}_{i}^{\ast\prime}\mathbf{H}_{i}^{\ast}\mathbf{A}_{T}\right)^{-1}\right\|$
to exist, which we require for the derivations of the asymptotic distribution
of the PB estimator. This is ensured by Assumption 4.

###  2.1 Asymptotic results

Substituting
$\mathbf{\tilde{y}}_{i}=\mathbf{\tilde{x}}_{i}\beta+\Delta\mathbf{\tilde{Z}}_{i}\mathbf{\psi}_{i}+\alpha_{i}^{-1}\mathbf{\tilde{v}}_{i}$
in (8), and using $\mathbf{M}_{i}\Delta\mathbf{\tilde{Z}}_{i}=\mathbf{0}$, we
have

$T\sqrt{n}\left(\hat{\beta}-\beta\right)=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{x}}_{i}}{T^{2}}\right)^{-1}\left(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{T\alpha_{i}}\right)\text{.}$
(11)

Consider the first term on the right side of (11) first. Since
$\mathbf{M}_{i}$ is an orthogonal projection matrix,
$\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{x}}_{i}/T^{2}$ is
bounded by $\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}/T^{2}$. The
second moments of
$\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}/T^{2}$ are bounded,
and, in addition,
$\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{x}}_{i}/T^{2}$ is
cross-sectionally independent. It follows that
$\frac{1}{n}\sum_{i=1}^{n}\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{x}}_{i}/T^{2}$
converges to a constant, which we denote by $\omega_{x}^{2}$, as
$n,T\rightarrow\infty$. Lemma A.4 in Appendix A establishes the expression for
$\omega_{x}^{2}=\sigma_{x}^{2}/6$, where
$\sigma_{x}^{2}=\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}$,
but the specific expression for $\omega_{x}^{2}$ is not relevant for the
inference approach that we adopt below. Consider the second term of (11) next,

$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}-E\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\right)\right]$
$\displaystyle+\frac{1}{\sqrt{n}}\sum_{i=1}^{n}E\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\right)\text{,}$

The term in the square brackets has zero mean and is independently distributed
over $i$. For the asymptotic distribution to be correctly centered we need

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}E\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\right)\rightarrow
0\text{,}$ (13)

as $n$ and $T\rightarrow\infty$. This condition holds so long as
$n=\Theta\left(T^{\theta}\right)$ for some $0<\theta<2$. See Lemma A.10 for a
proof. The asymptotic distribution of the first term in (2.1) is in turn
established by Lemma A.11, see (A.76). The following theorem now follows for
the asymptotic distribution of $\hat{\beta}$.

###### Theorem 1

Let $\left(y_{it},x_{it}\right)$ be generated by model (1)-(2), suppose
Assumptions 1-4 hold, and $n,T\rightarrow\infty$ such that
$n=\Theta\left(T^{\theta}\right)$, for some $0<\theta<2$. Consider the PB
estimator $\hat{\beta}$ given by (8). Then,

$T\sqrt{n}\left(\hat{\beta}-\beta\right)\rightarrow_{d}N\left(0,\Omega\right)\text{,
}\Omega=\omega_{x}^{-4}\omega_{v}^{2}\text{,}$ (14)

where $\omega_{x}^{2}=\sigma_{x}^{2}/6$,
$\sigma_{x}^{2}=\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}$
and
$\omega_{v}^{2}=\lim{}_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}\sigma_{vi}^{2}/\left(6\alpha_{i}^{2}\right)$.

###### Remark 5

Like the PMG estimator in PesaranShinSmith1999, the PB estimator will also
work when variables are integrated of order 0 (the I(0) case), which is not
pursued in this paper. In the I(0) case, the PB estimator converges at the
standard rate of $\sqrt{nT}$.

To conduct inference, let

$\hat{\omega}_{x}^{2}=n^{-1}\sum_{i=1}^{n}\frac{\mathbf{x}_{i}^{\prime}\mathbf{M}_{i}\mathbf{x}_{i}}{T^{2}}\text{,}$
(15)

and
$\hat{\omega}_{v}^{2}=n^{-1}\sum_{i=1}^{n}\left(\frac{\mathbf{x}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\hat{v}}_{i}^{\ast}}{T}\right)^{2}$,
where
$\mathbf{\hat{v}}_{i}^{\ast}=\mathbf{M}_{i}\left(\mathbf{y}_{i}-\hat{\beta}\mathbf{x}_{i}\right)$,
and $\mathbf{M}_{i}$ is defined by (9). Accordingly, we propose the following
estimator of $\Omega$:

$\hat{\Omega}=\hat{\omega}_{x}^{-4}\hat{\omega}_{v}^{2}\text{.}$ (16)

###  2.2 Bias mitigation and bootstrapping critical values for robust
inference

When $n$ is sufficiently large relative to $T$, specifically when
$\sqrt{n}/T\rightarrow K>0$, then $\sqrt{n}T\left(\hat{\beta}-\beta\right)$ is
no longer asymptotically distributed with zero mean. The asymptotic bias is
due to the nonzero mean of
$T^{-1}\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}$,
and it can be of some relevance for finite sample performance, as the Monte
Carlo evidence in Section 3 illustrates. Monte Carlo evidence also reveals
that the inference based on PB and other existing estimators in the literature
can suffer from serious size distortions in finite samples. To deal with these
problems, we consider bootstrapping critical values using sieve wild bootstrap
for more accurate and more robust inference that allows for cross-sectional
dependence of errors. In addition, we also consider two bias-correction
techniques - a bootstrap one as well as the split-panel jackknife method. The
same bias-correction methods are also applied to the three other estimators,
namely PMG, PDOLS, and FMOLS, considered in the paper. In what follows we
focus on the PB estimator. A description of bias-corrections applied to the
other three estimators are given in Section B.2 of Appendix B.

####  2.2.1 Bootstrap bias reduction

Once an estimate of the bias of $\hat{\beta}$ is available, denoted as
$\hat{b}$, then the bias-corrected PB estimator is given by

$\tilde{\beta}=\hat{\beta}-\hat{b}\text{.}$ (17)

One possibility of estimating the bias in the literature is by bootstrap. We
adopt the following sieve wild bootstrap algorithm for generating simulated
data.

  1. 1.

Given $\hat{\beta}$, estimate the remaining unknown coefficients in (1)-(2) by
least squares, and compute residuals denoted by
$\hat{u}_{y,it},\hat{u}_{x,it}$.

  2. 2.

For each $r=1,2,...,R$, generate new draws for
$\hat{u}_{y,it}^{\left(r\right)}=a_{t}^{\left(r\right)}\hat{u}_{y,it}$, and
$\hat{u}_{x,it}^{\left(r\right)}=a_{t}^{\left(r\right)}\hat{u}_{x,it}$, where
$a_{t}^{\left(r\right)}$ is randomly drawn from Rademacher distribution
(Liu1988, Liu1988),

$a_{t}^{\left(r\right)}=\left\\{\begin{array}[]{cc}-1\text{,}&\text{with
probability 1/2}\\\ 1\text{,}&\text{with probability
1/2}\end{array}\right.\text{.}$

Given the estimated parameters of (1)-(2) from Step 1, and the initial values
$\left\\{y_{i1},x_{i1}\text{ for }i=1,2,...,n\right\\}$ generate the simulated
series $y_{it}^{\left(r\right)},x_{it}^{\left(r\right)}$ for $t=2,3,...,T$ ,
and $i=1,2,...,n$, and the bootstrap estimates $\hat{\beta}^{(r)}$ for
$r=1,2,...,R$.

Using simulated data with $R=10,000$, we compute an estimate of the bias
$\hat{b}_{R}=\left[R^{-1}\sum_{r=1}^{R}\hat{\beta}^{\left(r\right)}-\hat{\beta}\right]$.
We then compute the $\alpha$ percent critical values using the $1-\alpha$
percent quantile of
$\left\\{\left|t^{\left(r\right)}\right|\right\\}_{r=1}^{R}$, where
$t^{\left(r\right)}=\tilde{\beta}^{\left(r\right)}/se\left(\tilde{\beta}^{\left(r\right)}\right)$,
$\tilde{\beta}^{\left(r\right)}=\hat{\beta}^{\left(r\right)}-\hat{b}$ is the
bias-corrected estimate of $\beta$ using the $r$-$th$ draw of the simulated
data,
$se\left(\tilde{\beta}^{\left(r\right)}\right)=T^{-1}n^{-1/2}\hat{\Omega}^{\left(r\right)}$
is the corresponding standard error estimate, and
$\hat{\Omega}^{\left(r\right)}$ is computed in the same way as $\hat{\Omega}$
in (16) but using the simulated data.

####  2.2.2 Jackknife bias reduction

The split-panel jackknife bias correction method is given by

$\tilde{\beta}_{jk}=\tilde{\beta}_{jk}\left(\kappa\right)=\hat{\beta}-\kappa\left(\frac{\hat{\beta}_{a}+\hat{\beta}_{b}}{2}-\hat{\beta}\right),$
(18)

where $\hat{\beta}$ is the full sample PB estimator, $\hat{\beta}_{a}$ and
$\hat{\beta}_{b}$ are the first and the second half sub-sample PB estimators,
and $\kappa$ is a suitably chosen weighting parameter. In a stationary
setting, where the bias is of order $O\left(T^{-1}\right)$, $\kappa$ is chosen
to be one, so that
$\frac{K}{T}-\kappa\cdot\left(\frac{K}{T/2}-\frac{K}{T}\right)=0,$ for any
arbitrary choice of $K$. See, for example, DhaeneJochmansy2015 and
ChudikPesaranYang2018.

In general, when the bias is of order $O\left(T^{-\epsilon}\right)$ for some
$\epsilon>0$, then $\kappa$ can be chosen to solve
$\frac{K}{T^{\epsilon}}-\kappa\cdot\left(\frac{K}{\left(T/2\right)^{\epsilon}}-\frac{K}{T^{\epsilon}}\right)=0$,
which yields $\kappa=1/\left(2^{\epsilon}-1\right)$. Under our setup with I(1)
variables, we need to correct $\hat{\beta}$ for its $O\left(T^{-2}\right)$
bias, namely $\epsilon=2$, which yields $\kappa=1/3$.

Inference using $\tilde{\beta}^{jk}$ can be conducted based on (16) but with
$\hat{\omega}_{v}^{2}$ replaced by

$\tilde{\omega}_{v}^{2}=\hat{\omega}_{v}^{2}=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{\left[\left(1+\kappa\right)\mathbf{x}_{i}^{\prime}\mathbf{M}_{i}-2\kappa\mathbf{x}_{ab,i}^{\prime}\mathbf{M}_{ab,i}\right]\mathbf{\tilde{v}}_{i}^{\ast}}{T}\right)^{2}\text{,}$
(19)

where
$\mathbf{\tilde{v}}_{i}^{\ast}=\mathbf{M}_{i}\left(\mathbf{y}_{i}-\tilde{\beta}^{jk}\mathbf{x}_{i}\right)$,

$\mathbf{x}_{ab,i}^{\prime}=\left(\begin{array}[]{c}\mathbf{x}_{a,i}^{\prime}\\\
\mathbf{x}_{b,i}^{\prime}\end{array}\right)\text{,
}\mathbf{M}_{ab,i}=\left(\begin{array}[]{c}\mathbf{M}_{a,i}\\\
\mathbf{M}_{b,i}\end{array}\right)\text{,}$

$\mathbf{x}_{a,i}^{\prime}$ $\left(\mathbf{x}_{b,i}^{\prime}\right)$ and
$\mathbf{M}_{a,i}$ ($\mathbf{M}_{b,i}$) are defined in the same way as
$\mathbf{x}_{i}$, and $\mathbf{M}_{i}$ but using only the first (second) half
of the sample.

We compute bootstrapped critical values to conduct more accurate and robust
small sample inference. Specifically, the $\alpha$ percent critical value is
computed as the $1-\alpha$ percent quantile of
$\left\\{\left|t_{jk}^{\left(r\right)}\right|\right\\}_{r=1}^{R}$, where
$t_{jk}^{\left(r\right)}=\tilde{\beta}_{jk}^{\left(r\right)}/se\left(\tilde{\beta}_{jk}^{\left(r\right)}\right)$,
$\tilde{\beta}_{jk}^{\left(r\right)}$ is the jackknife estimate of $\beta$
using the $r$-$th$ draw of the simulated data generated using the algorithm
described in Subsection 2.2.1,
$se\left(\tilde{\beta}_{jk}^{\left(r\right)}\right)$ is the corresponding
standard error estimate, namely
$se\left(\tilde{\beta}_{jk}^{\left(r\right)}\right)=T^{-1}n^{-1/2}\hat{\Omega}_{jk}^{\left(r\right)}$,
$\hat{\Omega}_{jk}^{\left(r\right)}=\hat{\omega}_{x,\left(r\right)}^{-4}\tilde{\omega}_{v,\left(r\right)}^{2}$,
in which $\tilde{\omega}_{v,\left(r\right)}$ and
$\hat{\omega}_{x,\left(r\right)}^{2}$ are computed using the simulated data,
based on expressions (19) and (15), respectively.

##  3 Monte Carlo Evidence

###  3.1 Design

The Data Generating Process (DGP) is given by (1)-(2), for $i=1,2,...,n,$
$T=1,2,...,T$, with starting values satisfying Assumption 3 with
$\mathbf{\mu}_{i}\sim IIDN\left(\mathbf{\tau}_{2},\mathbf{I}_{2}\right)$, and
$c_{i}=$ $\alpha_{i}\mu_{i,1}-\alpha_{i}\beta\mu_{i,2}$. We generate
$\alpha_{i}\sim IIDU\left[0.2,0.3\right]$. We consider two DGPs based on the
cross-sectional dependence of errors. In the cross-sectionally independent
DGP, we generate $u_{y,it}=\sigma_{y,i}e_{y,it}$,
$u_{x,it}=\sigma_{x,i}e_{x,it}$, $\sigma_{y,i}^{2},\sigma_{x,i}^{2}\sim
IIDU\left[0.8,1.2\right]$,

$\left(\begin{array}[]{c}e_{y,it}\\\ e_{x,it}\end{array}\right)\sim
IIDN\left(\mathbf{0}_{2},\mathbf{\Sigma}_{e}\right)\text{,
}\mathbf{\Sigma}_{e}\sim\left(\begin{array}[]{cc}1&\rho_{i}\\\
\rho_{i}&1\end{array}\right)\text{, and }\rho_{i}\sim
IIDU\left[0.3,0.7\right]\text{.}$

In the DGP with cross-sectionally dependent errors, we generate $e_{y,it}$ to
contain a factor structure including strong, semi-strong and weak factors:

$e_{y,it}=\varkappa_{i}\left(\varepsilon_{y,it}+\sum_{\ell=1}^{m}\gamma_{i\ell}f_{\ell
t}\right)\text{, }$

where $\varepsilon_{y,it}\sim IIDN\left(0,1\right)$, $f_{\ell,t}\sim
IIDN\left(0,1\right)$, $\gamma_{\ell}\sim
IIDU\left[0,\gamma_{\max,\ell}\right]$, for $\ell=1,2,...,m$. We choose $m=5$
factors and $\gamma_{\max,\ell}=2n^{\alpha_{\ell}-1}$ with
$\alpha_{\ell}=1,0.9,0.8,0.7,0.6$, for $\ell=1,2,...,5$, respectively. Scaling
constant $\varkappa_{i}$ is set to ensure $E\left(e_{y,it}^{2}\right)=1$,
namely
$\varkappa_{i}=\left(1+\sum_{\ell=1}^{m}\gamma_{i\ell}^{2}\right)^{-1/2}$. We
generate $e_{x,it}$ to ensure unit variance and
$cov\left(e_{y,it},e_{x,it}\right)=\rho_{i}$. Specifically,
$e_{x,it}=\rho_{i}e_{y,it}+\sqrt{1-\rho_{i}^{2}}\varepsilon_{x,it}$,
$\varepsilon_{x,it}\sim IIDN\left(0,1\right)$. Both designs features
heteroskedastic (over $i$) and correlated (over $y$ & $x$ equations) errors,
namely $E\left(u_{y,it}^{2}\right)=\sigma_{y,i}^{2}$,
$E\left(u_{x,it}^{2}\right)=\sigma_{x,i}^{2}$, and
$corr\left(u_{y,it},u_{x,it}\right)=\rho_{i}$. We consider $n,T=20,30,40,50$
and compute $R_{MC}=2000$ Monte Carlo replications.

###  3.2 Bias, RMSE and inference

We report bias, root mean square error (RMSE), size ($H_{0}:\beta=1$, 5%
nominal level) and power ($H_{1}:\beta=0.9$, 5% nominal level) findings for
the PB estimator $\hat{\beta}$ given by (8), with variance estimated using
(16). Moreover, we also report findings for the two bias corrected versions of
PB estimator as described in Subsection 2.2 with bootstrapped critical values
for inference robust to error cross-sectional dependence. We compare the
performance of the PB estimator with the PMG estimator by
PesaranShinSmith1999, panel dynamic OLS (PDOLS) estimator by MarkSul2003, and
the group-mean fully modified OLS (FMOLS) estimator by Pedroni1996
(Pedroni1996, Pedroni2001ReStat). Similarly to the PB estimator, we also
consider jackknife and bootstrap based bias-corrected versions of the PMG,
PDOLS and FMOLS estimators with cross-sectionally robust bootstrapped critical
values, described in Appendix B. We use $R_{b}=10,000$ bootstrap replications
(within each MC replication) for bootstrap bias correction and for computation
of robust and more accurate bootstrapped critical values.

###  3.3 Findings

Table 1 report the results for the original (without bias-correction)
estimators. PB estimator stands out as the most precise estimator in terms of
having the lowest RMSE values among the four estimators. The second best is
PMG estimator with RMSE values 1 to 21 percent larger compared with the PB
estimator, the third is PDOLS with RMSE values 23 to 66 percent larger
compared with PB, and the FMOLS comes last with RMSE values 95 to 180 percent
larger compared with PB. In terms of the bias alone, the ordering of the
estimators is slightly different with PMG and PB switching their places. For
$T=20$, the bias of PMG estimator is -0.016 to -0.020, the bias of PB
estimator is in the range -0.034 to -0.037, the bias of the PDOLS estimator is
in the range -0.052 to -0.056 and the bias of the FMOLS estimator is in the
range -0.104 to -0.110. For such a small value of $T$, the bias is not very
large, and, as expected, it declines with an increase in $T$.

All four estimators suffer from varying degrees of size distortions. The
inference based on the PB estimator is the most accurate. Specifically, the
size distortions for the PB estimator are lowest among the four estimators -
with reported size in the range between 9.9 and 25.2 percent, exceeding the
chosen nominal value of 5 percent. Size distortions diminish with an increase
in $T$.

Table 1: MC findings for the estimation of long-run coefficient $\beta$ in
experiments with cross-sectionally independent errors.

Estimators without bias correction and inference conducted using standard
critical values.

| Bias ($\mathbf{\times}$ 100) |  |  | RMSE ($\mathbf{\times}$ 100) |  |  | Size (5% level) |  |  | Power (5% level)
---|---|---|---|---|---|---|---|---|---|---
$n\backslash T$ | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50
| PB
20 | -3.69 | -1.75 | -1.07 | -0.73 |  |  | 6.43 | 4.12 | 3.04 | 2.45 |  |  | 18.40 | 13.35 | 11.80 | 11.40 |  |  | 34.00 | 68.30 | 89.95 | 97.70
30 | -3.39 | -1.79 | -1.04 | -0.74 |  |  | 5.55 | 3.54 | 2.58 | 2.03 |  |  | 19.40 | 14.50 | 11.95 | 10.10 |  |  | 43.70 | 81.50 | 96.50 | 99.60
40 | -3.56 | -1.87 | -1.06 | -0.74 |  |  | 5.18 | 3.25 | 2.33 | 1.81 |  |  | 21.25 | 15.55 | 12.30 | 10.45 |  |  | 45.90 | 87.35 | 99.05 | 99.95
50 | -3.58 | -1.90 | -1.09 | -0.74 |  |  | 4.96 | 3.05 | 2.18 | 1.66 |  |  | 25.20 | 15.55 | 13.40 | 9.95 |  |  | 54.05 | 93.25 | 99.65 | 100.00
| PMG
20 | -1.97 | -0.89 | -0.51 | -0.32 |  |  | 7.77 | 4.79 | 3.40 | 2.57 |  |  | 39.45 | 28.15 | 21.40 | 17.85 |  |  | 63.40 | 82.20 | 93.75 | 98.95
30 | -1.56 | -0.97 | -0.41 | -0.33 |  |  | 6.32 | 3.99 | 2.78 | 2.08 |  |  | 41.10 | 28.45 | 22.50 | 16.55 |  |  | 71.20 | 89.60 | 98.10 | 99.85
40 | -1.64 | -0.86 | -0.44 | -0.31 |  |  | 5.71 | 3.44 | 2.47 | 1.85 |  |  | 43.10 | 29.25 | 23.05 | 18.10 |  |  | 77.25 | 95.15 | 99.60 | 100.00
50 | -1.70 | -0.93 | -0.48 | -0.31 |  |  | 5.23 | 3.10 | 2.26 | 1.67 |  |  | 42.25 | 28.60 | 23.75 | 16.85 |  |  | 81.00 | 97.00 | 99.80 | 100.00
| PDOLS
20 | -5.60 | -3.64 | -2.82 | -2.32 |  |  | 7.93 | 5.28 | 4.02 | 3.31 |  |  | 21.75 | 19.10 | 17.30 | 19.30 |  |  | 17.35 | 43.15 | 72.95 | 90.15
30 | -5.25 | -3.60 | -2.75 | -2.26 |  |  | 7.05 | 4.81 | 3.64 | 2.96 |  |  | 24.10 | 23.10 | 23.10 | 24.20 |  |  | 21.40 | 55.90 | 84.85 | 97.30
40 | -5.47 | -3.77 | -2.84 | -2.31 |  |  | 6.78 | 4.69 | 3.53 | 2.86 |  |  | 29.90 | 30.60 | 28.30 | 30.10 |  |  | 21.70 | 62.60 | 91.35 | 99.10
50 | -5.46 | -3.78 | -2.88 | -2.30 |  |  | 6.57 | 4.52 | 3.45 | 2.75 |  |  | 35.10 | 34.30 | 34.40 | 36.45 |  |  | 25.25 | 72.40 | 95.85 | 99.85
| FMOLS
20 | -11.01 | -7.16 | -5.45 | -4.25 |  |  | 12.56 | 8.44 | 6.55 | 5.18 |  |  | 89.25 | 78.25 | 69.90 | 64.15 |  |  | 45.00 | 56.60 | 79.05 | 92.60
30 | -10.44 | -7.06 | -5.30 | -4.17 |  |  | 11.58 | 7.99 | 6.09 | 4.83 |  |  | 93.80 | 86.15 | 77.90 | 71.95 |  |  | 44.90 | 63.20 | 88.75 | 98.20
40 | -10.78 | -7.31 | -5.50 | -4.26 |  |  | 11.59 | 8.00 | 6.08 | 4.77 |  |  | 97.45 | 92.95 | 85.65 | 82.20 |  |  | 44.60 | 67.85 | 92.70 | 99.20
50 | -10.76 | -7.35 | -5.50 | -4.22 |  |  | 11.44 | 7.91 | 5.98 | 4.65 |  |  | 98.70 | 96.25 | 91.35 | 86.85 |  |  | 46.10 | 73.70 | 95.35 | 99.85

Notes: DGP is given by $\Delta y_{it}=c_{i}-\alpha_{i}\left(y_{i,t-1}-\beta
x_{i,t-1}\right)+u_{y,it}$ and $\Delta x_{it}=u_{x,it}$, for $i=1,2,...,n,$
$T=1,2,...,T$, with $\beta=1$ and $\alpha_{i}\sim IIDU\left[0.2,0.3\right]$.
Errors $u_{y,it}$, $u_{x,it}$ are cross-sectionally independent,
heteroskedastic over $i$, and correlated over $y$ & $x$ equations. See Section
3.1 for complete description of the DGP. The pooled Bewley estimator is given
by (8), with variance estimated using (16). PMG is the Pooled Mean Group
estimator proposed by PesaranShinSmith1999. PDOLS is panel dynamic OLS
estimator by MarkSul2003. FMOLS is the group-mean fully modified OLS estimator
by Pedroni1996 (Pedroni1996, Pedroni2001ReStat). The size and power findings
are computed using 5% nominal level and the reported power is the rejection
frequency for testing the hypothesis $\beta=0.9$.

Table 2: MC findings for the estimation of long-run coefficient $\beta$ in
experiments with cross-sectionally independent errors.

Bias corrected estimators and inference conducted using bootstrapped critical
values.

| Bias ($\mathbf{\times}$ 100) |  |  | RMSE ($\mathbf{\times}$ 100) |  |  | Size (5% level) |  |  | Power (5% level)
---|---|---|---|---|---|---|---|---|---|---
$n\backslash T$ | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50
| Jackknife bias-corrected estimators
| PB
20 | -1.60 | -0.52 | -0.25 | -0.14 |  |  | 6.27 | 4.18 | 3.10 | 2.52 |  |  | 5.95 | 5.15 | 5.40 | 5.30 |  |  | 25.10 | 56.75 | 82.95 | 94.30
30 | -1.34 | -0.55 | -0.25 | -0.18 |  |  | 5.24 | 3.45 | 2.58 | 2.06 |  |  | 6.70 | 5.05 | 5.10 | 5.00 |  |  | 37.45 | 73.70 | 93.10 | 99.05
40 | -1.45 | -0.60 | -0.24 | -0.15 |  |  | 4.59 | 3.03 | 2.29 | 1.81 |  |  | 5.85 | 5.05 | 4.50 | 4.90 |  |  | 42.80 | 84.10 | 98.15 | 99.85
50 | -1.53 | -0.62 | -0.26 | -0.15 |  |  | 4.23 | 2.72 | 2.07 | 1.61 |  |  | 6.20 | 4.60 | 5.65 | 4.25 |  |  | 51.10 | 90.80 | 99.55 | 100.00
| PMG
20 | -0.55 | -0.20 | -0.05 | 0.01 |  |  | 9.26 | 5.59 | 3.78 | 2.86 |  |  | 14.90 | 11.05 | 9.10 | 7.20 |  |  | 33.45 | 57.15 | 83.05 | 95.60
30 | -0.10 | -0.30 | 0.02 | -0.04 |  |  | 7.35 | 4.48 | 3.09 | 2.28 |  |  | 14.00 | 10.95 | 8.50 | 7.35 |  |  | 45.60 | 72.05 | 93.10 | 99.30
40 | -0.27 | -0.13 | -0.01 | -0.01 |  |  | 6.73 | 3.83 | 2.72 | 2.06 |  |  | 15.40 | 9.30 | 8.90 | 8.05 |  |  | 48.90 | 82.95 | 97.10 | 99.75
50 | -0.42 | -0.21 | -0.06 | 0.00 |  |  | 6.13 | 3.43 | 2.47 | 1.83 |  |  | 16.45 | 9.65 | 9.40 | 7.55 |  |  | 57.55 | 88.55 | 98.85 | 100.00
| PDOLS
20 | -4.22 | -2.56 | -1.97 | -1.59 |  |  | 8.00 | 5.05 | 3.77 | 3.04 |  |  | 8.40 | 5.80 | 5.40 | 4.75 |  |  | 8.70 | 27.50 | 55.25 | 77.90
30 | -3.91 | -2.55 | -1.92 | -1.56 |  |  | 6.89 | 4.42 | 3.30 | 2.64 |  |  | 8.70 | 5.75 | 5.95 | 4.70 |  |  | 10.15 | 37.30 | 69.75 | 91.00
40 | -4.09 | -2.69 | -1.98 | -1.60 |  |  | 6.35 | 4.17 | 3.08 | 2.47 |  |  | 7.70 | 5.80 | 4.90 | 4.05 |  |  | 10.05 | 41.10 | 75.20 | 94.95
50 | -4.11 | -2.70 | -2.02 | -1.58 |  |  | 6.01 | 3.91 | 2.93 | 2.30 |  |  | 7.90 | 5.90 | 4.55 | 3.55 |  |  | 11.30 | 45.75 | 83.75 | 97.60
| FMOLS
20 | -8.70 | -5.07 | -3.70 | -2.76 |  |  | 11.19 | 7.19 | 5.52 | 4.33 |  |  | 10.85 | 5.95 | 4.55 | 4.00 |  |  | 1.05 | 1.40 | 8.10 | 25.75
30 | -8.17 | -5.02 | -3.60 | -2.73 |  |  | 10.02 | 6.60 | 4.94 | 3.86 |  |  | 9.95 | 5.00 | 4.65 | 3.00 |  |  | 0.50 | 1.65 | 10.20 | 35.20
40 | -8.49 | -5.22 | -3.78 | -2.78 |  |  | 9.84 | 6.40 | 4.79 | 3.68 |  |  | 9.20 | 5.00 | 3.45 | 3.25 |  |  | 0.15 | 0.75 | 8.85 | 38.70
50 | -8.50 | -5.28 | -3.78 | -2.74 |  |  | 9.61 | 6.26 | 4.62 | 3.50 |  |  | 10.65 | 5.00 | 3.75 | 2.85 |  |  | 0.10 | 0.95 | 10.55 | 45.75
| Bootstrap bias-corrected estimators
| PB
20 | -1.28 | -0.33 | -0.16 | -0.11 |  |  | 5.87 | 3.93 | 2.93 | 2.39 |  |  | 7.25 | 6.20 | 6.05 | 5.75 |  |  | 36.20 | 67.60 | 88.30 | 96.95
30 | -0.98 | -0.39 | -0.15 | -0.13 |  |  | 4.90 | 3.24 | 2.43 | 1.93 |  |  | 7.75 | 6.35 | 6.35 | 5.80 |  |  | 52.20 | 84.00 | 96.90 | 99.55
40 | -1.07 | -0.43 | -0.13 | -0.10 |  |  | 4.24 | 2.83 | 2.15 | 1.70 |  |  | 7.00 | 6.55 | 5.95 | 5.10 |  |  | 59.75 | 91.35 | 99.45 | 99.95
50 | -1.09 | -0.44 | -0.15 | -0.10 |  |  | 3.89 | 2.56 | 1.96 | 1.52 |  |  | 8.95 | 6.25 | 6.45 | 5.55 |  |  | 69.40 | 95.50 | 99.90 | 100.00
| PMG
20 | -1.28 | -0.44 | -0.21 | -0.11 |  |  | 7.88 | 4.83 | 3.41 | 2.57 |  |  | 14.10 | 10.45 | 7.90 | 6.80 |  |  | 35.25 | 63.40 | 86.70 | 97.30
30 | -0.88 | -0.55 | -0.12 | -0.13 |  |  | 6.40 | 3.99 | 2.79 | 2.08 |  |  | 13.10 | 10.75 | 8.25 | 6.85 |  |  | 47.45 | 77.75 | 95.85 | 99.65
40 | -0.96 | -0.44 | -0.14 | -0.11 |  |  | 5.73 | 3.42 | 2.47 | 1.85 |  |  | 14.95 | 9.05 | 7.80 | 6.90 |  |  | 53.55 | 87.60 | 98.40 | 99.85
50 | -1.02 | -0.51 | -0.19 | -0.10 |  |  | 5.20 | 3.06 | 2.24 | 1.66 |  |  | 15.15 | 9.85 | 9.05 | 6.95 |  |  | 60.90 | 91.65 | 99.30 | 100.00
| PDOLS
20 | -2.19 | -0.90 | -0.59 | -0.42 |  |  | 6.75 | 4.31 | 3.12 | 2.55 |  |  | 10.35 | 7.80 | 7.25 | 7.25 |  |  | 28.70 | 64.45 | 86.65 | 95.85
30 | -1.89 | -0.96 | -0.59 | -0.43 |  |  | 5.70 | 3.62 | 2.63 | 2.08 |  |  | 10.75 | 8.90 | 7.70 | 7.15 |  |  | 39.90 | 77.10 | 95.35 | 99.30
40 | -2.00 | -1.04 | -0.59 | -0.41 |  |  | 4.98 | 3.24 | 2.34 | 1.84 |  |  | 10.05 | 8.65 | 7.05 | 7.35 |  |  | 46.40 | 85.75 | 98.65 | 99.95
50 | -2.00 | -1.05 | -0.64 | -0.40 |  |  | 4.61 | 2.93 | 2.14 | 1.65 |  |  | 10.70 | 9.40 | 8.65 | 6.80 |  |  | 52.85 | 91.55 | 99.50 | 100.00
| FMOLS
20 | -4.59 | -1.97 | -1.30 | -0.80 |  |  | 8.84 | 5.62 | 4.28 | 3.40 |  |  | 17.10 | 10.90 | 9.20 | 7.40 |  |  | 20.20 | 41.10 | 63.20 | 81.70
30 | -4.19 | -2.04 | -1.28 | -0.84 |  |  | 7.54 | 4.86 | 3.61 | 2.83 |  |  | 18.50 | 10.90 | 9.15 | 7.70 |  |  | 24.85 | 51.60 | 78.60 | 93.15
40 | -4.36 | -2.14 | -1.34 | -0.81 |  |  | 6.89 | 4.36 | 3.22 | 2.51 |  |  | 20.10 | 12.05 | 9.80 | 7.75 |  |  | 26.10 | 60.40 | 86.25 | 96.80
50 | -4.41 | -2.22 | -1.35 | -0.78 |  |  | 6.56 | 4.10 | 2.98 | 2.29 |  |  | 23.80 | 14.35 | 10.50 | 8.90 |  |  | 30.45 | 69.45 | 92.05 | 99.20

Notes: See the notes to Table 1. Bias-corrected versions of the PB estimator
are described in Subsection 2.2. Bias-corrected versions of the PMG, PDOLS and
FMOLS estimator are described in Appendix B. Inference is conducted using
bootstrapped critical values.

We consider next the bias-corrected versions of the four estimators with
inference carried out using robust bootstrap critical values. Upper panel of
Table 2 reports findings for estimators corrected for bias using the jackknife
procedure, and the bottom panel reports on bootstrap bias corrected
estimators. Bias correction did not change the overall ranking of estimators –
PB continues to be the most precise (lowest RMSE). Both bias correction
approaches are quite effective in reducing the bias. The bias of PB and PMG
estimators for any of the two bias corrections are very low. In addition to
reducing the bias, in many cases the bias-correction also resulted in reduced
RMSE values. In the case of the PB estimator, using bootstrap bias correction
resulted in improved RMSE performance for all choices of $n,T$ \- by about 2
to 22 percent. Results in Table 2 also show notable improvement to inference
comes from using bootstrapped critical values - with PB having virtually no
size distortions and size distortions of the remaining estimators are
relatively minor.

Last but not least, we consider the DGP with cross-sectionally correlated
errors. The corresponding results, reported in Tables B1 and B2 in Appendix B,
reveal the same ranking of the four estimators, and, importantly, the
bootstrapped critical values continue to deliver correct size, despite the
error cross-sectional dependence.

The Monte Carlo results show that PB estimator can perform better (in terms of
overall precision as measured by RMSE, and in terms of accuracy of inference)
than existing estimators (PMG, PDOLS, and FMOLS) in finite sample sizes of
interest, whether or not bias correction is considered. Of’ course, our
results do not imply that PB estimator will always be better, but that it can
be a useful addition to the existing literature as a complement to PMG, PDOLS,
and FMOLS estimators. Bias corrections and bootstrapping critical values are
helpful for all four estimators, resulting not only in reduced bias, but
sometimes also in better RMSE. In all cases, they result in more accurate
inference in our experiments.

##  4 Empirical Application

This section revisits consumption function empirical application undertaken by
PesaranShinSmith1999, hereafter PSS. The long-run consumption function is
assumed to be given by

$c_{it}=d_{i}+\beta_{1}y_{it}^{d}+\beta_{2}\pi_{it}+\vartheta_{it}\text{,}$

for country $i=1,2,...,n$, where $c_{it}$ is the logarithm of real consumption
per capita, $y_{it}^{d}$ is the logarithm of real per capita disposable
income, $\pi_{it}$ is the rate of inflation, and $\vartheta_{it}$ is an
$I\left(0\right)$ process. We take the dataset from PSS, which consists of
$n=24$ countries and a slightly unbalanced time period covering 1960-1993. PSS
estimate $\beta_{1}$ and $\beta_{2}$ using an ARDL(1,1,1) specification, which
can be written as error-correcting panel regressions

$\Delta
c_{it}=-\alpha_{i}\left(c_{i,t-1}-d_{i}-\beta_{1}y_{i,t-1}^{d}-\beta_{2}\pi_{i,t-1}\right)+\delta_{i1}\Delta
y_{it}^{d}+\delta_{i2}\Delta\pi_{it}+v_{it}\text{,}$ (20)

for $i=1,2,...,n$, where all coefficients, except the long-run coefficients
$\beta_{1}$ and $\beta_{2}$ are country-specific.

Table 3 presents alternative estimates of the long-run coefficients. The upper
panel presents findings for estimators without bias correction and standard
confidence intervals. The middle and lower panels present jackknife and
bootstrap bias-corrected estimates with confidence intervals based on
bootstrapped critical values. Results differ widely across different
approaches to estimation and inference. Depending on which bias correction
approach is conducted, the PB estimates of the long-run coefficient on real
income ($\beta_{1}$) is estimated to be 0.921 or 0.926, and the long-run
coefficient on the inflation variable ($\beta_{2}$) is estimated to be -0.120
or -0.125. The null hypothesis that the coefficient on $y_{it}^{d}$ is unity
cannot be rejected at the 5 percent nominal level, nor is the hypothesis that
the long run coefficient on inflation is zero. From an economic perspective,
unit long-run real income elasticity and no long-run effects of inflation on
consumption seem both plausible - the former hypothesis is in line with
balanced growth path models, and the latter in line with monetary policy
neutrality in the long-run. A different conclusion would be reached according
to PMG estimates - namely both the unit coefficients on the real income
variable and zero coefficient on inflation would be rejected at the 5 percent
nominal level. The results based on the PDOLS are in line with the PB
estimates and do not reject unit real income and zero inflation long run
coefficients. FMOLS estimates of $\beta_{1}$ are larger than the other
estimates, but the unit coefficient on the income variable still cannot be
rejected. The FMOLS estimates of $\beta_{2}$ are also quite large. The choice
of estimation method clearly matters in this empirical illustration.

Table 3: Estimated consumption function coefficients for OECD countries

|  $\beta_{1}$: Income | 95% Conf. Int. |  $\beta_{2}$: Inflation | 95% Conf. Int.
---|---|---|---|---
| Estimator without bias correction
PB | .912 | [.845,.980] | -.134 | [-.260,-.008]
PMG | .904 | [.889,.919] | -.466 | [-.566,-.365]
PDOLS | .923 | [.798,1.047] | -.187 | [-.407,.033]
FMOLS | .951 | [.942,.959] | -.336 | [-.408,-.265]
| Jackknife bias-corrected estimators
PB | .926 | [.835,1.017] | -.120 | [-.345,.105]
PMG | .915 | [.880,.949] | -.403 | [-616.,-.190]
PDOLS | .940 | [.737,1.143] | -.184 | [-.530,.161]
FMOLS | .983 | [.912,1.053] | -.397 | [-1.370,.576]
| Bootstrap bias-corrected estimators
PB | .921 | [.830,1.012] | -.125 | [-.314,.065]
PMG | .905 | [.875,.936] | -.477 | [-.657,-.297]
PDOLS | .932 | [.746,1.118] | -.183 | [-.499,.133]
FMOLS | .985 | [.941,1.028] | -.438 | [-1.047,.171]

Notes: This table revisits empirical application in Table 1 of
PesaranShinSmith1999, reporting estimates of long-run income elasticity
($\beta_{1}$) and inflation effect ($\beta_{2}$) coefficients and their 95%
confidence intervals in the ARDL(1,1,1) consumption functions (20) for OECD
countries using the dataset from PesaranShinSmith1999. PB stands for pooled
Bewley estimator developed in this paper. PMG is the Pooled Mean Group
estimator proposed by PesaranShinSmith1999. PDOLS is panel dynamic OLS
estimator by MarkSul2003. FMOLS is the group-mean fully modified OLS estimator
by Pedroni1996 (Pedroni1996, Pedroni2001ReStat). Description of bias
correction methods is provided in Subsection 2.2 for PB estimator and in
Appendix B for PMG, PDOLS and FMOLS estimators. Inference in the case of
original estimators uncorrected for bias is conducted using the standard
asymptotic critical values, and it is valid only when errors are not cross-
sectionally dependent. Inference in the case of bias-corrected estimators is
conducted using bootstrapped critical values following ChudikPesaranSmith2022,
and it is robust to cross-section dependence of errors.

##  5 Conclusion

This paper proposes the pooled Bewley (PB) estimator of long-run relationships
in heterogeneous dynamic panels. Relative to existing estimators in the
literature – namely PMG, PDOLS and FMOLS – Monte Carlo evidence reveals that
PB can perform well in small samples. While we developed the asymptotic theory
of PB estimator under a similar setting to the PMG estimator, notably we
assumed cross-sectionally independent errors, we have also shown the benefit
of bootstrapping critical values for inference when errors are cross-
sectionally correlated for all four estimators.

While the asymptotic distribution of the other estimators are derived for the
case where $n$ is fixed and $T\rightarrow\infty,$ we derive the joint
$\left(n,T\right)$ asymptotic distribution of the PB estimator, when both $n$
and $T$ diverge to infinity jointly such that
$n=\Theta\left(T^{\theta}\right)$, for $0<\theta<2.$ This covers a broader
range of empirical applications where both $n$ and $T$ are large. The small
sample and asymptotic results suggest that the PB estimator is a useful
addition to estimators for long run effects in single equation dynamic
heterogeneous panels, where the direction of long-run causality is known.

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Appendices

##  Appendix A Mathematical derivations

This appendix is organized in four sections. Section A.1 introduces some
notations and definitions. Section A.2 presents lemmas and proofs needed for
the proof of Theorem 1 presented in the body of the paper.

###  A.1 Notations and definitions

Let $\mathbf{z}_{it}=\left(y_{it},x_{it}\right)^{\prime}$, and define
$\mathbf{C}_{i}\left(L\right)=\sum_{\ell=0}^{\infty}\mathbf{C}_{i\ell}L^{\ell}$
and
$\mathbf{C}_{i}^{\ast}\left(L\right)=\sum_{\ell=0}^{\infty}\mathbf{C}_{i\ell}^{\ast}L^{\ell}$,
where

$\displaystyle\mathbf{C}_{i0}$ $\displaystyle=$
$\displaystyle\mathbf{I}_{2}\text{,}$ $\displaystyle\mathbf{C}_{i\ell}$
$\displaystyle=$
$\displaystyle\left(\mathbf{\Phi}_{i}-\mathbf{I}_{2}\right)\mathbf{\Phi}_{i}^{\ell-1}\text{,
}\ell=1,2,....\text{,}$
$\mathbf{\Phi}_{i}=\left(\begin{array}[]{cc}1-\alpha_{i}&\alpha_{i}\beta\\\
0&1\end{array}\right)\text{,}$ (A.1)
$\mathbf{C}_{i}(1)=\mathbf{C}_{i0}+\mathbf{C}_{i1}+.....=\lim_{\ell\rightarrow\infty}\mathbf{\Phi}_{i}^{\ell}=\left(\begin{array}[]{cc}0&\beta\\\
0&1\end{array}\right)\text{,}$

and

$\displaystyle\mathbf{C}_{i0}^{\ast}$ $\displaystyle=$
$\displaystyle\mathbf{C}_{i0}-\mathbf{C}_{i}(1)=\left(\begin{array}[]{cc}1&-\beta\\\
0&0\end{array}\right)\text{,}$ $\displaystyle\mathbf{C}_{i\ell}^{\ast}$
$\displaystyle=$
$\displaystyle\mathbf{C}_{i,\ell-1}^{\ast}+\mathbf{C}_{i\ell}=\left(\begin{array}[]{cc}\left(1-\alpha_{i}\right)^{\ell}&-\left(1-\alpha_{i}\right)^{\ell}\beta\\\
0&0\end{array}\right)\text{, for }\ell=1,2,...\text{.}$

Model (1)-(2) can be equivalently written as

$\mathbf{\Phi}_{i}\left(L\right)\mathbf{z}_{it}=\mathbf{c}_{i}+\mathbf{u}_{it}\text{,}$

for $i=1,2,...,n$ and $t=1,2,...,T$, where
$\mathbf{c}_{i}=\left(c_{i},0\right)^{\prime}$,

$\mathbf{\Phi}_{i}\left(L\right)=\mathbf{I}_{2}-\mathbf{\Phi}_{i}L\text{,}$
(A.4)

and $\mathbf{I}_{2}$ is a $2\times 2$ identity matrix. The lag polynomial
$\mathbf{\Phi}_{i}\left(L\right)$ can be re-written in the following (error
correcting) form

$\mathbf{\Phi}_{i}\left(L\right)=-\Pi_{i}L+\left(1-L\right)\mathbf{I}_{2}\text{,}$
(A.5)

where

$\Pi_{i}=-\left(\mathbf{I}_{2}-\mathbf{\Phi}_{i}\right)=\left(\begin{array}[]{cc}-\alpha_{i}&\alpha_{i}\beta\\\
0&0\end{array}\right)\text{.}$ (A.6)

The VAR model (A.7) can be also rewritten in the following form

$\mathbf{\Phi}_{i}\left(L\right)\left(\mathbf{z}_{it}-\mathbf{\mu}_{i}\right)=\mathbf{u}_{it}\text{,}$
(A.7)

where $\mathbf{c}_{i}=-\Pi_{i}\mathbf{\mu}_{i}=\left(c_{i},0\right)^{\prime}$,
namely $c_{i}=\alpha_{i}\mu_{i,1}-\alpha_{i}\beta\mu_{i,2}$.

Using Granger representation theorem, the process $\mathbf{z}_{it}$ under the
assumptions 1-3 has representation

$\displaystyle y_{it}$ $\displaystyle=$ $\displaystyle\mu_{yi}+\beta
s_{it}+\sum_{\ell=0}^{\infty}\left(1-\alpha_{i}\right)^{\ell}\left(u_{y,i,t-\ell}-\beta
u_{x,i,t-\ell}\right)\text{,}$ (A.8) $\displaystyle x_{it}$ $\displaystyle=$
$\displaystyle\mu_{xi}+s_{it}\text{,}$ (A.9)

where

$s_{it}=\sum_{\ell=1}^{t}u_{x,it}\text{,}$ (A.10)

is the stochastic trend.

###  A.2 Lemmas: Statements and proofs

###### Lemma A.1

Suppose Assumptions 2 and 3 hold, and consider
$\mathbf{\tilde{x}}_{i}=\left(\tilde{x}_{i,1},\tilde{x}_{i,2},...,\tilde{x}_{i,T}\right)^{\prime}$,
where $\tilde{x}_{it}=x_{it}-\bar{x}_{i}$, $x_{it}=\sum_{s=1}^{t}u_{x,it}$,
and $\bar{x}_{i}=T^{-1}\sum_{t=1}^{T}x_{it}$. Then

$n^{-1}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}}{T^{2}}\rightarrow_{p}\omega_{x}^{2}=\frac{\sigma_{x}^{2}}{6}\text{,
as }n,T\rightarrow\infty\text{,}$ (A.11)

where
$\sigma_{x}^{2}=\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}$.

Proof. Recall that
$\mathbf{M}_{\tau}=\mathbf{I}_{T}-T^{-1}\mathbf{\tau}_{T}\mathbf{\tau}_{T}^{\prime}$,
where $\mathbf{I}_{T}$ is $T\times T$ identity matrix and $\mathbf{\tau}_{T}$
is $T\times 1$ vector of ones. Since
$\mathbf{\tilde{x}}_{i}=\mathbf{M}_{\tau}\mathbf{x}_{i}$, and
$\mathbf{M}_{\tau}$ is symmetric and idempotent
($\mathbf{M}_{\tau}^{\prime}\mathbf{M}_{\tau}=\mathbf{M}_{\tau}=\mathbf{M}_{\tau}^{\prime}$)
we can write $\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}$ as
$\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}=\mathbf{x}_{i}^{\prime}\mathbf{M}_{\tau}^{\prime}\mathbf{M}_{\tau}\mathbf{x}_{i}=\mathbf{x}_{i}^{\prime}\mathbf{M}_{\tau}^{\prime}\mathbf{x}_{i}=\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{x}_{i}$.
Denote $S_{i,T}=\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{x}_{i}/T^{2}$. We have

$n^{-1}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}}{T^{2}}=n^{-1}\sum_{i=1}^{n}S_{i,T}=n^{-1}\sum_{i=1}^{n}E\left(S_{i,T}\right)+n^{-1}\sum_{i=1}^{n}\left[S_{i,T}-E\left(S_{i,T}\right)\right]\text{.}$
(A.12)

Consider $E\left(S_{i,T}\right)$ first. Noting that
$\tilde{x}_{it}=\sum_{s=1}^{t}u_{x,it}-\bar{x}_{i}$,
$\bar{x}_{i}=T^{-1}\sum_{s=1}^{T}\left(T-s+1\right)u_{st}$, and
$x_{it}=\sum_{s=1}^{t}u_{x,it}$, $S_{i,T}$ can be written as

$\displaystyle S_{i,T}$ $\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\tilde{x}_{it}x_{it}\text{,}$
$\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\left[\left(\sum_{s=1}^{t}u_{x,is}\right)^{2}-\bar{x}_{i}\sum_{s=1}^{t}u_{x,is}\right]\text{,}$
$\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\left[\left(\sum_{s=1}^{t}u_{x,is}\right)^{2}-\sum_{s=1}^{t}\frac{T-s+1}{T}u_{x,is}\cdot\sum_{s=1}^{t}u_{x,is}\right]\text{.}$

Taking expectations, we obtain

$E\left(S_{i,T}\right)=\frac{\sigma_{xi}^{2}}{T^{2}}\sum_{t=1}^{T}\left[t-\sum_{s=1}^{t}\frac{T-s+1}{T}\right]\text{.}$

Using
$\sum_{s=1}^{t}\frac{T-s+1}{T}=\sum_{s=1}^{t}\left(1-s/T+1/T\right)=t-\left(t+1\right)t/\left(2T\right)+t/T$,
we have

$E\left(S_{i,T}\right)=\frac{\sigma_{xi}^{2}}{T^{2}}\sum_{t=1}^{T}\left[t-t+\frac{\left(t+1\right)t}{2T}-\frac{t}{T}\right]=\frac{\sigma_{xi}^{2}}{T^{2}}\sum_{t=1}^{T}\frac{\left(t+1\right)t}{2T}-\frac{t}{T}\text{.}$

Finally, noting that
$\sum_{t=1}^{T}\left(t+1\right)t=\left(T+2\right)\left(T+1\right)T/3$, and
$\sum_{t=1}^{T}t=\left(T+1\right)T/2$, we obtain

$E\left(S_{i,T}\right)=\sigma_{xi}^{2}\varkappa_{T}<K<\infty\text{,}$ (A.13)

for all $T>0$, where

$\varkappa_{T}=\left[\frac{\left(T+2\right)\left(T+1\right)T}{6T^{3}}-\frac{\left(T+1\right)T}{2T^{3}}\right]\text{.}$
(A.14)

In addition, $\varkappa_{T}\rightarrow 1/6$, as $T\rightarrow\infty$, and

$\frac{1}{n}\sum_{i=1}^{n}E\left(S_{i,T}\right)=\varkappa_{T}\frac{1}{n}\sum_{i=1}^{n}\sigma_{xi}^{2}\rightarrow\frac{\sigma_{x}^{2}}{6}\text{,}$

as $n,T\rightarrow\infty$. This establishes the limit of the first term on the
right side of (A.12). Consider the second term next. Since
$E\left[S_{i,T}-E\left(S_{i,T}\right)\right]=0$, and $S_{i,T}$ is independent
over $i$, we have

$E\left\\{n^{-1}\sum_{i=1}^{n}\left[S_{i,T}-E\left(S_{i,T}\right)\right]\right\\}^{2}=\frac{1}{n^{2}}\sum_{i=1}^{n}E\left(S_{i,T}^{2}\right)-\frac{1}{n^{2}}\sum_{i=1}^{n}\left[E\left(S_{i,T}\right)\right]^{2}\text{.}$

But it follows from (A.13) that there exist finite positive constant
$K_{1}<\infty$ (which does not depend on $n,T$) such that
$\left[E\left(S_{i,T}\right)\right]^{2}<K_{1}$. In addition, due to existence
of uniformly bounded fourth moments of $u_{x,it}$, it also can be shown that
$E\left(S_{i,T}^{2}\right)<K_{2}<\infty$. Hence,
$E\left\\{n^{-1}\sum_{i=1}^{n}\left[S_{i,T}-E\left(S_{i,T}\right)\right]\right\\}^{2}=O\left(n^{-1}\right)$,
which implies
$n^{-1}\sum_{i=1}^{n}\left[S_{i,T}-E\left(S_{i,T}\right)\right]\rightarrow_{p}0$,
as $n,T\rightarrow\infty$. This completes the proof.

###### Lemma A.2

Suppose Assumptions 1-2 hold. Then there exists finite positive constant $K$
that does not depend on $i$ and/or $T$ such that

$E\left(\frac{1}{T}\sum_{t=1}^{T}u_{x,it}\tilde{x}_{it}\right)^{\varrho}<K\text{,}$
(A.15)

and

$E\left(\frac{1}{T}\sum_{t=1}^{T}\Delta
y_{it}\tilde{x}_{it}\right)^{\varrho}<K\text{,}$ (A.16)

for $\varrho=4$, where $\tilde{x}_{it}=x_{it}-\bar{x}_{i}$,
$x_{it}=\sum_{s=1}^{t}u_{x,it}$, $\bar{x}_{i}=T^{-1}\sum_{t=1}^{T}x_{it}$, and
$\Delta
y_{it}=\delta_{i}u_{x,it}+v_{it}-\alpha_{i}\sum_{\ell=1}^{\infty}\left(1-\alpha_{i}\right)^{\ell-1}\left[v_{i,t-\ell}+\left(\delta_{i}-\beta\right)u_{x,i,t-\ell}\right]$.

Proof. Consider $\sum_{t=1}^{T}u_{it}\tilde{x}_{it}/T$ and $\varrho=2$ first,
and note that $\tilde{x}_{it}=\sum_{s=1}^{t}u_{x,is}-\bar{x}_{i}$, where
$\bar{x}_{i}=T^{-1}\sum_{s=1}^{T}\left(T-s+1\right)u_{x,is}$. We have

$\displaystyle\left(\frac{1}{T}\sum_{t=1}^{T}u_{x,it}\tilde{x}_{it}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{T}u_{x,it}u_{x,it^{\prime}}\tilde{x}_{it}\tilde{x}_{it^{\prime}}\text{,}$
$\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{T}u_{x,it}u_{x,it^{\prime}}\left(\sum_{s=1}^{t}u_{x,is}-\bar{x}_{i}\right)\left(\sum_{s=1}^{t^{\prime}}u_{x,is}-\bar{x}_{i}\right)\text{,}$
$\displaystyle=$ $\displaystyle
A_{i,T,1}+A_{i,T,2}-A_{i,T,3}-A_{i,T,4}\text{,}$

where

$\displaystyle A_{i,T,1}$ $\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{T}u_{x,it}u_{x,it^{\prime}}\left(\sum_{s=1}^{t}u_{x,is}\right)\left(\sum_{s=1}^{t^{\prime}}u_{x,is}\right)\text{,}$
$\displaystyle A_{i,T,2}$ $\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{T}u_{x,it}u_{x,it^{\prime}}\bar{x}_{i}^{2}\text{,}$
$\displaystyle A_{i,T,3}$ $\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{T}u_{x,it}u_{x,it^{\prime}}\bar{x}_{i}\sum_{s=1}^{t}u_{x,is}\text{,}$
$\displaystyle A_{i,T,4}$ $\displaystyle=$
$\displaystyle\frac{1}{T^{2}}\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{T}u_{x,it}u_{x,it^{\prime}}\bar{x}_{i}\sum_{s=1}^{t^{\prime}}u_{x,is}\text{.}$

Taking expectations and noting that $u_{x,it}$ is independent of
$u_{x,it^{\prime}}$ for any $t\neq t^{\prime}$, we have

$E\left(A_{i,T,1}\right)=\frac{1}{T^{2}}\left(\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{t-1}\sigma_{ix}^{2}+\sum_{t=1}^{T}E\left(u_{x,it}^{4}\right)+\sum_{t=1}^{T}\sum_{t^{\prime}=1}^{t-1}\sigma_{ix}^{2}\right)\text{.}$

Under Assumption 2, there exists a finite constant $K$ that does not depend on
$i$ and/or $t$, such that $\sigma_{ix}^{2}<K$ and
$E\left(u_{x,it}^{4}\right)<K$. Hence
$\left|E\left(A_{i,T,1}\right)\right|<K$. Similarly, we can bound the
remaining elements, $\left|E\left(A_{i,T,j}\right)\right|<K$, for $j=2,3,4$.
It now follows that
$E\left(\frac{1}{T}\sum_{t=1}^{T}u_{x,it}\tilde{x}_{it}\right)^{2}<K$, where
the upper bound $K$ does not depend on $i$ or $T$. This establishes (A.15)
hold for $\varrho=2$. Sufficient conditions for (A.15) to hold when
$\varrho=4$ are: $E\left(A_{i,T,j}^{2}\right)<K$ for $j=1,2,3,4$. These
conditions follow from uniformly bounded eights moments of $u_{x,it}$. This
completes the proof of (A.15). Result (A.16) can be established in the same
way by using the first difference of representation (A.8).

###### Lemma A.3

Suppose Assumptions 1-4 hold, and consider $s_{iT}$ given by

$s_{iT}=\mathbf{\tilde{x}}_{i}^{\prime}\Delta\mathbf{\tilde{Z}}_{i}\left(\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{Z}}_{i}\right)^{-1}\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}\text{,}$
(A.17)

where $\mathbf{P}_{i}$ is given by (10), and $\mathbf{\tilde{x}}_{i}$ and
$\Delta\mathbf{\tilde{Z}}_{i}$ are defined below (6). Then,

$n^{-1}\sum_{i=1}^{n}\frac{s_{iT}}{T^{2}}\rightarrow_{p}0\text{, as
}n,T\rightarrow\infty\text{. }$ (A.18)

Proof. Consider $s_{i,T}/T$, which can be written as

$\frac{s_{iT}}{T}\leq\mathbf{a}_{iT}^{\prime}\mathbf{B}_{iT}^{-1}\mathbf{a}_{iT}\text{,}$
(A.19)

where

$\mathbf{a}_{iT}=\frac{\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}}{T}=\frac{\Delta\mathbf{Z}_{i}^{\prime}\mathbf{\tilde{x}}_{i}}{T}\text{,}$
(A.20)

and

$\mathbf{B}_{iT}=\frac{\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{Z}}_{i}}{T}\text{.}$
(A.21)

Using these notations, we have

$E\left|\frac{1}{n}\sum_{i=1}^{n}\frac{s_{iT}}{T^{2}}\right|\leq\frac{1}{nT}\sum_{i=1}^{n}E\left|\mathbf{a}_{iT}^{\prime}\mathbf{B}_{iT}^{-1}\mathbf{a}_{iT}\right|\text{.}$

Using
$\left|\mathbf{a}_{iT}^{\prime}\mathbf{B}_{iT}^{-1}\mathbf{a}_{iT}\right|\leq\lambda_{\min}^{-1}\left(\mathbf{B}_{iT}\right)\mathbf{a}_{iT}^{\prime}\mathbf{a}_{iT}$,
and Cauchy-Schwarz inequality, we obtain

$E\left|\frac{1}{n}\sum_{i=1}^{n}\frac{s_{iT}}{T^{2}}\right|\leq\frac{1}{nT}\sum_{i=1}^{n}\sqrt{E\left[\left(\mathbf{a}_{iT}^{\prime}\mathbf{a}_{iT}\right)^{2}\right]}\sqrt{E\left[\lambda_{\min}^{-2}\left(\mathbf{B}_{iT}\right)\right]}\text{.}$

Lemma A.2 implies the fourth moments of the individual elements of
$\mathbf{a}_{i,T}$ are uniformly bounded in $i$ and $T$, which is sufficient
for
$E\left[\left(\mathbf{a}_{iT}^{\prime}\mathbf{a}_{iT}\right)^{2}\right]<K$. In
addition, $E\left[\lambda_{\min}^{-2}\left(\mathbf{B}_{iT}\right)\right]<K$ by
Assumption 4. Hence, there exists $K<\infty$, which does not depend on
$\left(n,T\right)$ such that

$E\left|n^{-1}\sum_{i=1}^{n}\frac{s_{iT}}{T^{2}}\right|<\frac{K}{T}\text{,}$
(A.22)

and result (A.18) follows.

###### Lemma A.4

Suppose Assumptions 1-4 hold. Then

$n^{-1}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{x}}_{i}}{T^{2}}\rightarrow_{p}\omega_{x}^{2}=\frac{\sigma_{x}^{2}}{6}\text{,
as }n,T\rightarrow\infty,$ (A.23)

where
$\sigma_{x}^{2}=\lim_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}$,
$\mathbf{M}_{i}$ is defined in (9) and $\mathbf{\tilde{x}}_{i}$ is defined
below (6).

Proof. Noting that $\mathbf{\tilde{x}}_{i}$ is one of the column vectors of
$\mathbf{H}_{i}$, we have
$\mathbf{P}_{i}\mathbf{\tilde{x}}_{i}=\mathbf{\tilde{x}}_{i}$, and
$\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{x}}_{i}$ can be
written as

$\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{x}}_{i}=\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}-s_{i,T}\text{,}$
(A.24)

where $s_{i,T}$ is given by (A.17). Sufficient conditions for result (A.23)
are:

$n^{-1}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{x}}_{i}}{T^{2}}\rightarrow_{p}\omega_{x}^{2}=\frac{\sigma_{x}^{2}}{6}\text{,
as }n,T\rightarrow\infty,$ (A.25)

and

$n^{-1}\sum_{i=1}^{n}\frac{s_{i,T}}{T^{2}}\rightarrow_{p}0\text{, as
}n,T\rightarrow\infty\text{. }$ (A.26)

Condition (A.25) is established by Lemma A.1, and condition (A.26) is
established by Lemma A.3.

###### Lemma A.5

Let Assumptions 1-3 hold. Then

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{T\alpha_{i}}\rightarrow_{d}N\left(0,\omega_{v}^{2}\right)\text{,
as }n,T\rightarrow\infty\text{,}$ (A.27)

where
$\omega_{v}^{2}=\lim{}_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}\sigma_{vi}^{2}/\left(6\alpha_{i}^{2}\right)$,
and $\mathbf{\tilde{x}}_{i}$ and $\mathbf{\tilde{v}}_{i}$ are defined below
(6).

Proof. Recall
$\mathbf{M}_{\tau}=\mathbf{I}_{T}-T^{-1}\mathbf{\tau}_{T}\mathbf{\tau}_{T}^{\prime}$,
where $\mathbf{I}_{T}$ is $T\times T$ identity matrix and $\mathbf{\tau}_{T}$
is $T\times 1$ vector of ones. Since
$\mathbf{M}_{\tau}^{\prime}\mathbf{M}_{\tau}=\mathbf{M}_{\tau}^{\prime}$, we
have

$\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}=\mathbf{x}_{i}^{\prime}\mathbf{M}_{\tau}^{\prime}\mathbf{M}_{\tau}\mathbf{v}_{i}=\mathbf{x}_{i}^{\prime}\mathbf{M}_{\tau}^{\prime}\mathbf{v}_{i}=\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{v}_{i}\text{.}$

Let $C_{i}=\frac{\sigma_{xi}\sigma_{vi}}{\alpha_{i}}$ and
$Q_{i,T}=C_{i}^{-1}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{v}_{i}}{T\alpha_{i}}$.
We have $E\left(Q_{i,T}\right)=0$, and (under independence of $v_{it}$ over
$t$ and independence of $v_{it}$ and $u_{x,it^{\prime}}$ for any
$t,t^{\prime}$)

$E\left[\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{v}_{i}}{T\alpha_{i}}\right)^{2}\right]=\frac{1}{T^{2}\alpha_{i}^{2}}\sum_{t=1}^{T}E\left(\tilde{x}_{it}^{2}\right)E\left(v_{it}^{2}\right)\text{,}$

where $E\left(v_{it}^{2}\right)=\sigma_{vi}^{2}$. In addition, (A.13)
established that
$\frac{1}{T^{2}}\sum_{t=1}^{T}E\left(\tilde{x}_{it}^{2}\right)=\sigma_{xi}^{2}\varkappa_{T}$,
where $\varkappa_{T}$ is given by (A.14). Hence,

$E\left[\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{v}_{i}}{T\alpha_{i}}\right)^{2}\right]=\frac{\sigma_{vi}^{2}\sigma_{xi}^{2}}{\alpha_{i}^{2}}\varkappa_{T}=C_{i}^{2}\varkappa_{T}\text{.}$

It follows that

$E\left(Q_{i,T}^{2}\right)=\varkappa_{T}\text{,}$

where $\varkappa_{T}\rightarrow 1/6<\infty$. Finite fourth moments of
$u_{x,it}$ and $v_{it}$ imply $Q_{i,T}^{4}$ is uniformly bounded in $T$, and
therefore $Q_{i,T}^{2}$ is uniformly integrable in $T$. We can apply Theorem 3
of PhillipsMoon1999 to obtain

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}C_{i}Q_{i,T}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{v}_{i}}{T\alpha_{i}}\rightarrow_{d}N\left(0,\omega_{v}^{2}\right)\text{,
as }n,T\rightarrow\infty\text{,}$

where
$\omega_{v}^{2}=\lim_{n\rightarrow\infty}C_{i}^{2}\varkappa_{T}=\lim{}_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}\sigma_{vi}^{2}/\left(6\alpha_{i}^{2}\right)$.

###### Lemma A.6

Suppose Assumptions 1-4 hold, and consider
$\mathbf{q}_{iT}=\alpha_{i}^{-1}\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}/\sqrt{T}$.
Then,

$E\left\|\mathbf{q}_{iT}\right\|_{2}^{4}<K\text{,}$ (A.28)

and

$\left|E\left(\mathbf{q}_{iT}\right)\right|<\frac{K}{\sqrt{T}}\text{.}$ (A.29)

Proof. Denote the individual elements of $2\times 1$ vector $\mathbf{q}_{iT}$
as $q_{iT,j}$, $j=1,2$. Sufficient conditions for (A.28) to hold are

$E\left(q_{iT,j}\right)^{4}<K\text{, for }j=1,2\text{.}$ (A.30)

We establish (A.30) for $j=1$ first. We have

$q_{iT,1}=\frac{\Delta\mathbf{\tilde{y}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}\text{,}$

where $\Delta\mathbf{y}_{i}$ can be written as

$\Delta\mathbf{\tilde{y}}_{i}=-\alpha_{i}\mathbf{\tilde{\xi}}_{i,-1}+\delta_{i}\Delta\mathbf{\tilde{x}}_{i}+\mathbf{v}_{i}\text{,}$
(A.31)

where
$\mathbf{\tilde{\xi}}_{i,-1}=\mathbf{\tilde{y}}_{i,-1}-\mathbf{\tilde{x}}_{i,-1}$.
Note that
$\mathbf{P}_{i}=\mathbf{\tilde{H}}_{i}\left(\mathbf{\tilde{H}}_{i}^{\prime}\mathbf{\tilde{H}}_{i}\right)^{-1}\mathbf{\tilde{H}}_{i}^{\prime}$
and
$\mathbf{\tilde{H}}_{i}=\left(\mathbf{\tilde{y}}_{i,-1},\mathbf{\tilde{x}}_{i},\mathbf{\tilde{x}}_{i,-1}\right)$.
Hence
$\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{P}_{i}=\Delta\mathbf{\tilde{x}}_{i}^{\prime}$
and
$\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{P}_{i}=\mathbf{\tilde{\xi}}_{i,-1}^{\prime}$,
since $\Delta\mathbf{\tilde{x}}_{i}$ and $\mathbf{\tilde{\xi}}_{i,-1}$ can be
both obtained as a linear combinations of the column vectors of
$\mathbf{\tilde{H}}_{i}$. Hence

$q_{iT,1}=-\frac{\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{\tilde{v}}_{i}}{\sqrt{T}}+\frac{\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}+\frac{\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}\equiv\varsigma_{a,iT}+\varsigma_{b,iT}+\varsigma_{c,iT}\text{,}$
(A.32)

where we simplified notations by introducing
$\varsigma_{a,iT}=-\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{\tilde{v}}_{i}/\sqrt{T}$,
$\varsigma_{b,iT}=\alpha_{i}^{-1}\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}/\sqrt{T}$
and
$\varsigma_{c,iT}=\alpha_{i}^{-1}\Delta\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}/\sqrt{T}$
to denote the individual terms in the expression (A.32) for $q_{iT,1}$.
Sufficient conditions for $E\left(q_{iT,1}^{4}\right)<K$ are
$E\left(\varsigma_{s,iT}^{4}\right)<K$ for $s\in\left\\{a,b,c\right\\}$.

For $s=a$, we have

$\varsigma_{a,iT}=-\frac{\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{\tilde{v}}_{i}}{\sqrt{T}}=-\frac{1}{\sqrt{T}}\sum_{i=1}^{T}\left(\xi_{i,t-1}-\bar{\xi}_{i,-1}\right)\left(v_{it}-\bar{v}_{i}\right)=-\frac{1}{\sqrt{T}}\sum_{i=1}^{T}\xi_{i,t-1}v_{it}+\sqrt{T}\bar{\xi}_{i,-1}\bar{v}_{i}\text{,}$

where $\bar{\xi}_{i,-1}=T^{-1}\sum_{t=1}^{T}\xi_{i,t-1}$, and

$\displaystyle\xi_{it}$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{\infty}\left(1-\alpha_{i}\right)^{\ell}\left(u_{y,i,t-\ell}-\beta
u_{x,i,t-\ell}\right)\text{,}$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{\infty}\left(1-\alpha_{i}\right)^{\ell}\left(\delta_{i}-\beta\right)u_{x,i,t-\ell}+\sum_{\ell=0}^{\infty}\left(1-\alpha_{i}\right)^{\ell}v_{it}\text{.}$

Noting that $\sup_{i}\left|1-\alpha_{i}\right|<1$ under Assumption 1, and
fourth moments of $u_{x,i,t}$ and eights moments of $v_{it}$ are bounded, we
obtain

$E\left[\left(\frac{1}{\sqrt{T}}\sum_{i=1}^{T}\xi_{i,t-1}v_{it}\right)^{4}\right]\leq
K\text{,}$

and

$T^{2}\cdot E\left(\bar{\xi}_{i,-1}^{4}\bar{v}_{i}^{4}\right)\leq K\text{,}$

which are sufficient conditions for $E\left(\varsigma_{a,iT}^{4}\right)\leq
K$.

For $s=b$, we have

$\varsigma_{b,iT}=\frac{\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}=\frac{1}{\alpha_{i}\sqrt{T}}\sum_{t=1}^{T}\left(u_{x,it}-\bar{u}_{x,i}\right)\left(v_{it}-\bar{v}_{i}\right)=\frac{1}{\alpha_{i}\sqrt{T}}\sum_{t=1}^{T}u_{x,it}v_{it}-\frac{\sqrt{T}}{\alpha_{i}}\bar{u}_{x,i}\bar{v}_{i}\text{.}$

Using Assumption 2, we obtain the following upper bound

$E\left(\varsigma_{b,iT}^{4}\right)\leq\left|\alpha_{i}^{-4}\right|\frac{1}{T}\sum_{t=1}^{T}E\left(u_{x,it}^{4}\right)E\left(v_{it}^{4}\right)+\left|\alpha_{i}^{-4}\right|TE\left(\bar{u}_{x,i}^{4}\right)E\left(\bar{v}_{i}^{4}\right)\leq
K\text{,}$ (A.33)

where $\left|\alpha_{i}^{-4}\right|<K$, $E\left(u_{x,it}^{4}\right)<K$,
$E\left(v_{it}^{4}\right)<K$, $E\left(\bar{u}_{x,i}^{4}\right)<K/T^{2}$, and
$E\left(\bar{v}_{i}^{4}\right)<K/T^{2}$.

For $s=c$, we have

$\varsigma_{c,iT}=\frac{\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}=\frac{\Delta\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{\tilde{H}}_{i}\left(\mathbf{\tilde{H}}_{i}^{\prime}\mathbf{\tilde{H}}_{i}\right)^{-1}\mathbf{\tilde{H}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}\text{.}$

Consider
$\mathbf{\tilde{H}}_{i}^{\ast}=\left(\mathbf{\tilde{x}}_{i},\Delta\mathbf{\tilde{x}}_{i},\mathbf{\tilde{\xi}}_{i,-1}\right)$
and note that

$\mathbf{\tilde{H}}_{i}=\left(\mathbf{\tilde{y}}_{i,-1},\mathbf{\tilde{x}}_{i},\mathbf{\tilde{x}}_{i,-1}\right)=\mathbf{B}^{\ast}\mathbf{\tilde{H}}_{i}^{\ast}\text{,}$

where

$\mathbf{B}^{\ast}=\left(\begin{array}[]{ccc}-\beta&\beta&1\\\ 1&0&0\\\
-1&1&0\end{array}\right)\text{,}$

is nonsingular (for any $\beta$). Hence
$\mathbf{P}_{i}=\mathbf{\tilde{H}}_{i}\left(\mathbf{\tilde{H}}_{i}^{\prime}\mathbf{\tilde{H}}_{i}\right)^{-1}\mathbf{\tilde{H}}_{i}^{\prime}=\mathbf{\tilde{H}}_{i}^{\ast}\left(\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{H}}_{i}^{\ast}\right)^{-1}\mathbf{\tilde{H}}_{i}^{\ast\prime}$,
and we can write $\varsigma_{c,iT}$ as

$\varsigma_{c,iT}=\frac{\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{\tilde{H}}_{i}^{\ast}\left(\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{H}}_{i}^{\ast}\right)^{-1}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}\text{.}$

Consider the scaling matrix

$\mathbf{A}_{T}=\left(\begin{array}[]{ccc}T^{-1}&0&0\\\ 0&T^{-1/2}&0\\\
0&0&T^{-1/2}\end{array}\right)\text{.}$ (A.34)

We have

$\varsigma_{c,iT}=\frac{1}{\alpha_{i}\sqrt{T}}\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{\tilde{H}}_{i}^{\ast}\mathbf{A}_{T}\left(\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{H}}_{i}^{\ast}\mathbf{A}_{T}\right)^{-1}\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{v}}_{i}\geq
0\text{.}$

Using the inequality
$\mathbf{x}^{\prime}\mathbf{A}^{-1}\mathbf{x}\leq\lambda_{\min}\left(\mathbf{A}\right)\left\|\mathbf{x}\right\|^{2}$,
we have

$0\leq\varsigma_{c,iT}\leq\frac{1}{\alpha_{i}\sqrt{T}}\lambda_{\min}^{-1}\left(\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{H}}_{i}^{\ast}\mathbf{A}_{T}\right)\left\|\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{v}}_{i}\right\|_{2}^{2}\text{.}$

Using Cauchy-Schwarz inequality, we obtain

$E\left(\varsigma_{c,iT}^{4}\right)\leq\frac{1}{\alpha_{i}^{4}T^{2}}\sqrt{E\left[\lambda_{\min}^{-4}\left(\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{H}}_{i}^{\ast}\mathbf{A}_{T}\right)\right]}\sqrt{E\left\|\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{v}}_{i}\right\|_{2}^{8}}\text{.}$

But $\alpha_{i}^{-4}<K$ under Assumption 1, and
$E\left[\lambda_{\min}^{-4}\left(\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{H}}_{i}^{\ast}\mathbf{A}_{T}\right)\right]<K$
under Assumption 4. It follows

$E\left(\varsigma_{c,iT}^{4}\right)\leq\frac{K}{T^{2}}\sqrt{E\left\|\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{v}}_{i}\right\|_{2}^{8}}\text{.}$

Let
$\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{v}}_{i}\equiv\mathbf{h}_{viT}$
and consider the individual elements of $\mathbf{h}_{viT}$, denoted as
$h_{viT,j}$ for $j=1,2,3$,

$\mathbf{h}_{viT}=\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{v}}_{i}=\left(\begin{array}[]{c}h_{viT,1}\\\
h_{viT,2}\\\
h_{viT,3}\end{array}\right)=\left(\begin{array}[]{c}\frac{1}{T}\sum_{t=1}^{T}\tilde{x}_{it}\tilde{v}_{it}\\\
\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\tilde{u}_{it}\tilde{v}_{it}\\\
\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\bar{\xi}_{i,t-1}\tilde{v}_{it}\end{array}\right)\text{.}$

Under Assumption 2, it can be shown that

$E\left(h_{viT,j}^{8}\right)<K\text{, for }j=1,2,3\text{,}$

which is sufficient for
$E\left\|\mathbf{A}_{T}\mathbf{\tilde{H}}_{i}^{\ast\prime}\mathbf{\tilde{v}}_{i}\right\|_{2}^{8}<K$.
It follows that

$E\left(\varsigma_{c,iT}^{4}\right)<\frac{K}{T^{2}}\text{.}$ (A.35)

This completes the proof of (A.28) for $j=1$. Consider next (A.28) for $j=2$,
and note $q_{iT,2}$ is the same as $\varsigma_{b,iT}$, namely

$q_{iT,2}=\frac{\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}=\frac{\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}=\varsigma_{b,iT}\text{.}$

But $E\left(\varsigma_{b,iT}^{4}\right)<K$, see (A.33). This completes the
proof of (A.28).

Next we establish (A.29). As before we consider the individual elements of
$2\times 1$ vector $\mathbf{q}_{iT}$, denoted as $q_{iT,s}$ for $s=1,2$,
separately. For $s=1$ we have (using the individual terms in expression
(A.32))

$\left|E\left(q_{iT,1}\right)\right|=\left|E\left(-\frac{\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{\tilde{v}}_{i}}{\sqrt{T}}+\frac{\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}+\frac{\Delta\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}\right)\right|\leq\left|E\left(\varsigma_{a,iT}\right)\right|+\left|E\left(\varsigma_{b,iT}\right)\right|+\left|E\left(\varsigma_{c,iT}\right)\right|\text{.}$
(A.36)

For the first term in (A.36), we obtain

$\displaystyle\left|E\left(\varsigma_{a,iT}\right)\right|$ $\displaystyle=$
$\displaystyle\left|E\left[-\frac{\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{\tilde{v}}_{i}}{\sqrt{T}}=-\frac{1}{\sqrt{T}}\sum_{i=1}^{T}\left(\xi_{i,t-1}-\bar{\xi}_{i,-1}\right)\left(v_{it}-\bar{v}_{i}\right)\right]\right|\text{,}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{T}}\sum_{i=1}^{T}\left|E\left(\xi_{i,t-1}v_{it}\right)\right|+\sqrt{T}E\left|\bar{\xi}_{i,-1}\bar{v}_{i}\right|\text{.}$

But $E\left(\xi_{i,t-1}v_{it}\right)=0$ and
$E\left|\bar{\xi}_{i,-1}\bar{v}_{i}\right|<K/T$ under Assumptions 1-2. Hence,

$\left|E\left(\varsigma_{a,iT}\right)\right|\leq\frac{K}{\sqrt{T}}\text{.}$

For the second term in (A.36), we obtain

$\displaystyle\left|E\left(\varsigma_{b,iT}\right)\right|$ $\displaystyle=$
$\displaystyle\left|E\left(\frac{1}{\alpha_{i}\sqrt{T}}\sum_{t=1}^{T}u_{x,it}v_{it}-\frac{\sqrt{T}}{\alpha_{i}}\bar{u}_{x,i}\bar{v}_{i}\right)\right|\text{,}$
$\displaystyle\leq$ $\displaystyle
K\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\left|E\left(u_{x,it}v_{it}\right)\right|+K\sqrt{T}\left|E\left(\bar{u}_{x,i}\bar{v}_{i}\right)\right|\text{.}$

But $E\left(u_{x,it}v_{it}\right)=0$ and
$E\left(\bar{u}_{x,i}\bar{v}_{i}\right)=0$ under Assumption 2. Hence

$\left|E\left(\varsigma_{b,iT}\right)\right|=0\text{.}$

Finally, for the last term we note that

$\left|E\left(\varsigma_{c,iT}\right)\right|\leq
E\left|\varsigma_{c,iT}\right|\leq\sqrt{E\left(\varsigma_{c,iT}^{2}\right)}\text{,}$

and using result (A.35), we obtain

$\left|E\left(\varsigma_{c,iT}\right)\right|<\frac{K}{\sqrt{T}}\text{.}$

It now follows that $\left|E\left(q_{iT,1}\right)\right|<K/\sqrt{T}$, as
desired.

Consider $\left|E\left(q_{iT,s}\right)\right|$ for $s=2$ next. We have

$\left|E\left(q_{iT,2}\right)\right|=\left|E\left(\varsigma_{b,iT}\right)\right|=0\text{.}$

This completes the proof of result (A.29).

###### Lemma A.7

Let Assumptions 1-4 hold, and consider $\mathbf{B}_{iT}$ defined by (A.21).
Then we have

$T^{\varphi}\left\|\mathbf{B}_{iT}-\mathbf{B}_{i}\right\|\rightarrow_{p}0\text{
as }T\rightarrow\infty\text{, for any }\varphi<1/2\text{,}$ (A.37)

where

$\mathbf{B}_{i}=\mathop{\mathrm{p}lim}_{T\rightarrow\infty}\mathbf{B}_{iT}=\left(\begin{array}[]{cc}\alpha_{i}^{2}E\left(\xi_{it}^{2}\right)+\delta_{i}^{2}\sigma_{xi}^{2}&\delta_{i}\sigma_{xi}^{2}\\\
\delta_{i}\sigma_{xi}^{2}&\sigma_{xi}^{2}\end{array}\right)\text{,}$ (A.38)

and
$\xi_{it}=\sum_{\ell=0}^{\infty}\left(1-\alpha_{i}\right)^{\ell}\left(u_{y,i,t-\ell}-\beta
u_{x,i,t-\ell}\right)$.

Proof. We have

$\mathbf{B}_{iT}=\frac{\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{Z}}_{i}}{T}=\frac{1}{T}\left(\begin{array}[]{cc}\Delta\mathbf{\tilde{y}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{y}}_{i}&\Delta\mathbf{\tilde{y}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{x}}_{i}\\\
\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{y}}_{i}&\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{x}}_{i}\end{array}\right)=\left(\begin{array}[]{cc}b_{iT,11}&b_{iT,12}\\\
b_{iT,21}&b_{iT,22}\end{array}\right)\text{.}$

Consider the element $b_{iT,22}$ first. Since
$\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{P}_{i}=\Delta\mathbf{\tilde{x}}_{i}^{\prime}$,
and $\Delta\tilde{x}_{it}=u_{x,it}-\bar{u}_{x,i}$, we have

$b_{iT,12}=\left(\frac{1}{T}\sum_{t=1}^{T}u_{x,it}^{2}\right)-\bar{u}_{x,i}^{2}\text{.}$

Under Assumption 2, $u_{x,it}\sim IID\left(0,\sigma_{xi}^{2}\right)$ with
finite fourth order moments, and therefore

$T^{\varphi}\left(\frac{1}{T}\sum_{t=1}^{T}u_{x,it}^{2}-\sigma_{xi}^{2}\right)\overset{p}{\rightarrow}0\text{,
for any }\varphi<1/2\text{. }$

In addition, $E\left(\bar{u}_{x,i}^{2}\right)<K/T$, which implies
$T^{\varphi}\bar{u}_{x,i}^{2}\overset{p}{\rightarrow}0$, for any
$\varphi<1/2$. It follows

$T^{\varphi}\left(b_{iT,22}-\sigma_{xi}^{2}\right)\overset{p}{\rightarrow}0\text{,
for any }\varphi<1/2\text{. }$ (A.39)

Consider the element $b_{iT,11}$ next. We will use similar arguments as in the
proof of Lemma A.6. In particular, $\Delta\mathbf{\tilde{y}}_{i}$ can be
written as in (A.31), and, since
$\mathbf{P}_{i}\mathbf{\tilde{\xi}}_{i,-1}=\mathbf{\tilde{\xi}}_{i,-1}$ and
$\mathbf{P}_{i}\Delta\mathbf{\tilde{x}}_{i}=\Delta\mathbf{\tilde{x}}_{i}$, we
have

$b_{i,T,11}=\frac{\Delta\mathbf{\tilde{y}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{y}}_{i}}{T}=\zeta_{aa,iT}+\zeta_{bb,iT}+\zeta_{cc,iT}+2\zeta_{ab,iT}+2\zeta_{ac,iT}+2\zeta_{bc,iT}\text{,}$
(A.40)

where

$\displaystyle\zeta_{aa,iT}$ $\displaystyle=$
$\displaystyle\alpha_{i}^{2}\frac{\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{\tilde{\xi}}_{i,-1}}{T}\text{,
}\zeta_{bb,iT}=\delta_{i}^{2}\frac{\Delta\mathbf{\tilde{x}}_{i}^{\prime}\Delta\mathbf{\tilde{x}}_{i}}{T}\text{,}$
$\displaystyle\zeta_{cc,iT}$ $\displaystyle=$
$\displaystyle\frac{\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{T}\text{,}$

and the cross-product terms are

$\zeta_{ab,iT}=\alpha_{i}\delta_{i}\frac{\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\Delta\mathbf{\tilde{x}}_{i}}{T},\text{
}\zeta_{ac,iT}=\alpha_{i}\frac{\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{\tilde{v}}_{i}}{T}\text{,
}\zeta_{bc,iT}=\delta_{i}\frac{\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{T}\text{.}$

We consider these individual terms $\zeta$ next. Note that

$\displaystyle\xi_{it}$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{\infty}\left(1-\alpha_{i}\right)^{\ell}\left(u_{y,i,t-\ell}-\beta
u_{x,i,t-\ell}\right)\text{,}$ $\displaystyle=$
$\displaystyle\sum_{\ell=0}^{\infty}\left(1-\alpha_{i}\right)^{\ell}v_{i,t-\ell}+\left(\delta_{i}-\beta\right)\sum_{\ell=0}^{\infty}\left(1-\alpha_{i}\right)^{\ell}u_{x,i,t-\ell}\text{,}$

where $\sup_{i}\left|1-\alpha_{i}\right|<1$ under Assumption 1, and
innovations $v_{it}$ and $u_{xit}$ have finite fourth order moments under
Assumption 2. Hence,
$T^{\varphi}\left[T^{-1}\sum_{t=1}^{T}\xi_{i,t-1}^{2}-E\left(\xi_{i,t-1}^{2}\right)\right]\rightarrow_{p}0$,
$E\left(\bar{\xi}_{i,-1}^{2}\right)<K/T$, and we obtain

$T^{\varphi}\left[\zeta_{aa,iT}-\alpha_{i}^{2}E\left(\bar{\xi}_{i,-1}^{2}\right)\right]\rightarrow_{p}0\text{,
for any }\varphi<1/2\text{.}$ (A.41)

Noting that $\zeta_{bb,iT}=\delta_{i}^{2}b_{i,T,12}$, and using result (A.39),
we have

$T^{\varphi}\left[\zeta_{bb,iT}-\delta_{i}^{2}\sigma_{xi}^{2}\right]\rightarrow_{p}0\text{,
for any }\varphi<1/2\text{.}$ (A.42)

Consider $\zeta_{cc,iT}$ and note that
$\zeta_{cc,iT}=\frac{\alpha_{i}}{\sqrt{T}}\varsigma_{c,iT}$, where
$\varsigma_{c,iT}=\alpha_{i}^{-1}\Delta\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}/\sqrt{T}$
was introduced in (A.32) in proof of Lemma A.6. But
$E\left(\varsigma_{c,iT}^{2}\right)<\frac{K}{T}$ by (A.35), and it follows

$T^{\varphi}\zeta_{cc,iT}\rightarrow_{p}0\text{, for any }\varphi<1/2\text{.}$
(A.43)

Using similar arguments, we obtain for the cross-product terms,

$T^{\varphi}\zeta_{ab,iT}\rightarrow_{p}0\text{,
}T^{\varphi}\zeta_{ac,iT}\rightarrow_{p}0\text{, and
}T^{\varphi}\zeta_{bc,iT}\rightarrow_{p}0\text{, for any }\varphi<1/2\text{,
as }T\rightarrow\infty\text{.}$ (A.44)

Using (A.41)-(A.44) in (A.40), we obtain

$T^{\varphi}\left(b_{i,T,11}-\alpha_{i}^{2}E\left(\xi_{it}^{2}\right)-\delta_{i}^{2}\sigma_{xi}^{2}\right)\overset{p}{\rightarrow}0\text{,
for any }\varphi<1/2\text{. }$ (A.45)

Using the same arguments for the last term $b_{i,T,12}=b_{i,T,21}$, we obtain

$T^{\varphi}\left(b_{i,T,12}-\delta_{i}\sigma_{xi}^{2}\right)\overset{p}{\rightarrow}0\text{,
for any }\varphi<1/2\text{. }$

This completes the proof of (A.37).

###### Lemma A.8

Let Assumptions 1-4 hold, and consider $\mathbf{B}_{iT}$ defined by (A.21) and
$\mathbf{B}_{i}=\mathop{\mathrm{p}lim}_{T\rightarrow\infty}\mathbf{B}_{iT}$
defined by (A.38). Then we have

$T^{\varphi}\left\|\mathbf{B}_{iT}^{-1}-\mathbf{B}_{i}^{-1}\right\|\rightarrow_{p}0,\text{
as }T\rightarrow\infty\text{, for any }\varphi<1/2\text{.}$ (A.46)

Proof. This proof closely follows proof of Lemma A.8 in PesaranChudik2010. Let
$p=\left\|\mathbf{B}_{i}^{-1}\right\|$,
$q=\left\|\mathbf{B}_{iT}^{-1}-\mathbf{B}_{i}^{-1}\right\|$, and
$r=\left\|\mathbf{B}_{iT}-\mathbf{B}_{i}\right\|$. We suppressed subscripts
$i,T$ to simplify the notations, but it is understood that the terms $p,q,r$
depend on $\left(i,T\right)$. Using the triangle inequality and the
submultiplicative property of matrix norm $\left\|.\right\|$, we have

$\displaystyle q$ $\displaystyle=$
$\displaystyle\left\|\mathbf{B}_{iT}^{-1}\left(\mathbf{B}_{i}-\mathbf{B}_{iT}\right)\mathbf{B}_{i}^{-1}\right\|\text{,}$
$\displaystyle\leq$
$\displaystyle\left\|\mathbf{B}_{iT}^{-1}\right\|rp\text{,}$
$\displaystyle\leq$
$\displaystyle\left\|\left(\mathbf{B}_{iT}^{-1}-\mathbf{B}_{i}^{-1}\right)+\mathbf{B}_{i}^{-1}\right\|rp\text{,}$
$\displaystyle\leq$ $\displaystyle\left(p+q\right)rp\text{.}$

Subtracting $rpq$ from both sides and multiplying by $T^{\varphi}$, we have,
for any $\varphi<1/2$,

$\left(1-rp\right)\left(T^{\varphi}q\right)\leq
p^{2}\left(T^{\varphi}r\right)\text{.}$ (A.47)

Note that $T^{\varphi}r\overset{p}{\rightarrow}0$ by Lemma A.7, and
$\left|p\right|<K$ since $\mathbf{B}_{i}$ is invertible and
$\lambda_{\min}\left(\mathbf{B}_{i}\right)$ is bounded away from zero (this
follows from observing that both $\sigma_{xi}^{2}$ and
$E\left(\xi_{it}^{2}\right)$ as well as $\alpha_{i}^{2}$ in (A.38) are bounded
away from zero). Hence,

$\left(1-rp\right)\overset{p}{\rightarrow}1\text{,}$ (A.48)

and

$p^{2}\left(T^{\varphi}r\right)\overset{p}{\rightarrow}0.$ (A.49)

(A.47)-(A.49) imply $T^{\varphi}q\overset{p}{\rightarrow}0$. This establishes
result (A.46).

###### Lemma A.9

Let Assumptions 1-4 hold, and consider $\xi_{iT}$ defined by

$\xi_{iT}=\frac{\mathbf{\tilde{x}}_{i}^{\prime}\Delta\mathbf{\tilde{Z}}_{i}}{T}\left(\frac{\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\Delta\mathbf{\tilde{Z}}_{i}}{T}\right)^{-1}\frac{\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}\text{.}$
(A.50)

where $\mathbf{P}_{i}$ is given by (10), and $\mathbf{\tilde{x}}_{i}$ and
$\Delta\mathbf{\tilde{Z}}_{i}$ are defined below (6). Then

$\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\xi_{iT}\rightarrow_{p}0\text{,}$ (A.51)

as $n,T\rightarrow\infty$ such that $n=\Theta\left(T^{\theta}\right)$ for some
$0<\theta<2$.

Proof. Term $\xi_{iT}$ can be written as

$\xi_{iT}=\mathbf{a}_{iT}^{\prime}\mathbf{B}_{iT}^{-1}\mathbf{q}_{iT}\text{,}$
(A.52)

where $\mathbf{a}_{iT}$ is given by (A.20), $\mathbf{B}_{iT}$ is given by
(A.21), and

$\mathbf{q}_{iT}=\frac{\Delta\mathbf{\tilde{Z}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}\text{.}$
(A.53)

We have

$\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\xi_{i,T}=\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\mathbf{a}_{iT}^{\prime}\left(\mathbf{B}_{iT}^{-1}-\mathbf{B}_{i}^{-1}\right)\mathbf{q}_{iT}+\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}\text{.}$
(A.54)

Consider the two terms on the right side of (A.54) in turn. Lemma A.2
established fourth moments of $\mathbf{a}_{iT}$ are bounded, which is
sufficient for $\left\|\mathbf{a}_{iT}\right\|=O_{p}\left(1\right)$. Result
(A.28) of Lemma A.6 established second moments of individual elements of
$\mathbf{q}_{iT}$ are bounded, which is sufficient for
$\left\|\mathbf{q}_{iT}\right\|=O_{p}\left(1\right)$. In addition, Lemma A.8
established

$T^{\varphi}\left\|\mathbf{B}_{iT}^{-1}-\mathbf{B}_{i}^{-1}\right\|\rightarrow_{p}0\text{
as }T\rightarrow\infty\text{, for any }\varphi<1/2\text{.}$

Set $\varphi=\left(\theta-1\right)/2<1/2$ . Then we obtain

$\displaystyle\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\mathbf{a}_{iT}^{\prime}\left(\mathbf{B}_{iT}^{-1}-\mathbf{B}_{i}^{-1}\right)\mathbf{q}_{iT}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{n}}{\sqrt{T}}\frac{1}{T^{\varphi}}\frac{1}{n}\sum_{i=1}^{n}\mathbf{a}_{iT}^{\prime}\left[T^{\varphi}\left(\mathbf{B}_{iT}^{-1}-\mathbf{B}_{i}^{-1}\right)\right]\mathbf{q}_{iT}\text{,}$
(A.55) $\displaystyle\leq$
$\displaystyle\frac{\sqrt{n}}{T^{\theta/2}}\left(\frac{1}{n}\sum_{i=1}^{n}\left\|\mathbf{a}_{iT}\right\|\left(T^{\varphi}\left\|\mathbf{B}_{iT}^{-1}-\mathbf{B}_{i}^{-1}\right\|\right)\left\|\mathbf{q}_{iT}\right\|\right)\rightarrow_{p}0\text{,}$

as $n,T\rightarrow\infty$ such that $\theta<2$, where we used
$\sqrt{T}T^{\varphi}=T^{1/2+\varphi}=T^{\theta/2}$, and
$\frac{\sqrt{n}}{T^{\theta/2}}<K$ since $n=\Theta\left(T^{\theta}\right)$.

Consider next the second term on the right side of (A.54). Let
$\mu_{iT}^{\ast}=E\left(\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}\right)$,
and consider the variance of
$\left(nT\right)^{-1/2}\sum_{i=1}^{n}\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}$.
By independence of
$\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}$ across $i$,

$\displaystyle
Var\left(\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{nT}\sum_{i=1}^{n}Var\left(\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}\right)\text{,}$
(A.56) $\displaystyle\leq$
$\displaystyle\frac{1}{nT}\sum_{i=1}^{n}E\left(\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}\right)^{2}\text{.}$

Denoting individual elements of $\mathbf{B}_{i}^{-1}$ as $b_{i,sj}^{-}$,
individual elements of $\mathbf{a}_{iT}$ as $a_{iT,j}$, and individual
elements of $\mathbf{q}_{iT}$ as $q_{iT,s}$, for $s,j=1,2$, we have

$\displaystyle\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}$
$\displaystyle=$
$\displaystyle\sum_{s=1}^{2}\sum_{j=1}^{2}b_{i,sj}^{-}a_{iT,s}q_{iT,j}\text{,}$
(A.57) $\displaystyle=$ $\displaystyle
b_{i,11}^{-}a_{iT,1}q_{iT,1}+b_{i,21}^{-}a_{iT,2}q_{iT,1}+b_{i,12}^{-}a_{iT,1}q_{iT,2}+b_{i,22}^{-}a_{iT,2}q_{iT,2}\text{,}$

where

$a_{iT,1}=\frac{1}{T}\sum_{t=1}^{T}\tilde{x}_{it}\Delta\tilde{y}_{it}=\frac{1}{T}\sum_{t=1}^{T}\left(x_{it}-\bar{x}_{i}\right)\Delta
y_{it}\text{,}$ (A.58)
$a_{iT,2}=\frac{1}{T}\sum_{t=1}^{T}\tilde{x}_{it}\Delta\tilde{x}_{it}=\frac{1}{T}\sum_{t=1}^{T}\left(x_{it}-\bar{x}_{i}\right)u_{x,it}\text{,}$
(A.59)
$q_{iT,1}=\frac{\Delta\mathbf{\tilde{y}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}\sqrt{T}}\text{,}$
(A.60)

and

$q_{iT,2}=\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\frac{\tilde{u}_{x,it}\tilde{v}_{it}}{\alpha_{i}}\text{.}$
(A.61)

Note that $\mathbf{B}_{i}$ is invertible and
$\inf_{i}\lambda_{\min}\left(\mathbf{B}_{i}\right)$ is bounded away from zero
(this follows from observing that both $\sigma_{xi}^{2}$ and
$E\left(\xi_{it}^{2}\right)$ as well as $\alpha_{i}^{2}$ in (A.38) are bounded
away from zero). It follows $\sup_{i}\left\|\mathbf{B}_{i}^{-1}\right\|<K$,
and therefore $\left|\left(b_{i,sj}^{-}\right)^{2}\right|<K$. Using this
result and Cauchy-Schwarz inequality for the individual summands on the right
side of (A.57), we obtain

$E\left(\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}\right)^{2}\leq
K\sum_{s=1}^{2}\sum_{j=1}^{2}\sqrt{E\left(a_{iT,s}^{4}\right)}\sqrt{E\left(q_{iT,j}^{4}\right)}<K\text{,}$
(A.62)

where $E\left(a_{iT,s}^{4}\right)<K$ by Lemma A.2, and
$E\left(q_{iT,j}^{4}\right)<K$ by result (A.28) of Lemma A.6. Using (A.62) in
(A.56), it follows that

$Var\left(\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}\right)<\frac{K}{T}\text{,}$

and therefore

$\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\left(\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}-\mu_{iT}^{\ast}\right)\rightarrow_{q.m.}0\text{
as }n,T\rightarrow\infty\text{.}$ (A.63)

We establish an upper bound for $\left|\mu_{iT}^{\ast}\right|$ next. We have
(using (A.57) and noting that $\left|b_{i,sj}^{-}\right|<K$)

$\left|\mu_{iT}^{\ast}\right|<K\cdot\sum_{s=1}^{2}\sum_{j=1}^{2}\left|E\left(a_{iT,s}q_{iT,j}\right)\right|\text{.}$

It follows that if we can show that

$\left|E\left(a_{iT,s}q_{iT,j}\right)\right|<\frac{K}{\sqrt{T}}\text{, }$
(A.64)

holds for all $s,j=1,2$, then

$\left|\mu_{iT}^{\ast}\right|<\frac{K}{\sqrt{T}}\text{,}$ (A.65)

hold. We establish (A.64) for $s=j=2$, first, which is the most convenient
case to consider. We have

$E\left(a_{iT,2}q_{iT,2}\right)=E\left(\frac{1}{T}\sum_{t=1}^{T}\left(x_{it}-\bar{x}_{i}\right)u_{x,it}\cdot\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\frac{u_{x,it}v_{it}}{\alpha_{i}}\right)=0\text{,}$
(A.66)

since $v_{it}\,$is independently distributed of $u_{x,it^{\prime}}$ for any
$t,t^{\prime}$. Consider next $s=1$, $j=2$. We have

$E\left(a_{iT,1}q_{iT,2}\right)=E\left(\frac{1}{T}\sum_{t=1}^{T}\left(x_{it}-\bar{x}_{i}\right)\Delta
y_{it}\cdot\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\frac{u_{x,it}v_{it}}{\alpha_{i}}\right)\text{,}$
(A.67)

where (first-differencing (A.8) and substituting (3))

$\displaystyle\Delta y_{it}$ $\displaystyle=$
$\displaystyle\delta_{i}u_{x,it}+v_{it}-\alpha_{i}\sum_{\ell=1}^{\infty}\left(1-\alpha_{i}\right)^{\ell-1}\left[v_{i,t-\ell}+\left(\delta_{i}-\beta\right)u_{x,i,t-\ell}\right]\text{,}$
(A.68) $\displaystyle=$ $\displaystyle\eta_{u,it}+\eta_{v,it}\text{,}$

in which

$\eta_{u,it}=\delta_{i}u_{x,it}-\alpha_{i}\sum_{\ell=1}^{\infty}\left(1-\alpha_{i}\right)^{\ell-1}\left(\delta_{i}-\beta\right)u_{x,i,t-\ell}\text{,}$
(A.69)

and

$\eta_{v,it}=v_{it}-\alpha_{i}\sum_{\ell=1}^{\infty}\left(1-\alpha_{i}\right)^{\ell-1}v_{i,t-\ell}\text{.}$
(A.70)

Hence, $E\left(a_{iT,1}q_{iT,2}\right)$ can be written as

$\displaystyle E\left(a_{iT,1}q_{iT,2}\right)$ $\displaystyle=$ $\displaystyle
E\left(\frac{1}{T}\sum_{t=1}^{T}\left(x_{it}-\bar{x}_{i}\right)\eta_{u,it}\cdot\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\frac{u_{x,it}v_{it}}{\alpha_{i}}\right)$
$\displaystyle+E\left(\frac{1}{T}\sum_{t=1}^{T}\left(x_{it}-\bar{x}_{i}\right)\eta_{v,it}\cdot\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\frac{u_{x,it}v_{it}}{\alpha_{i}}\right)\text{.}$

The first term is equal to $0$, since $v_{it}\,$is independently distributed
of $u_{x,it^{\prime}}$ for any $t,t^{\prime}$. Consider the second term.
Noting that $E\left[\left(x_{it}-\bar{x}_{i}\right)u_{x,is}\right]<K$ and
$\left|\alpha_{i}^{-1}\right|<K$ for any $i,t,s$, we obtain

$\displaystyle
E\left(\frac{1}{T}\sum_{t=1}^{T}\left(x_{it}-\bar{x}_{i}\right)\eta_{v,it}\cdot\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\frac{u_{x,it}v_{it}}{\alpha_{i}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{T^{3/2}}\sum_{t=1}^{T}\sum_{s=1}^{T}\alpha_{i}^{-1}E\left[\left(x_{it}-\bar{x}_{i}\right)u_{x,is}\right]E\left(\eta_{v,is}v_{it}\right)\text{,}$
$\displaystyle\leq$
$\displaystyle\frac{K}{T^{3/2}}\sum_{t=1}^{T}\sum_{s=1}^{T}E\left(\eta_{v,is}v_{it}\right)\text{.}$

But

$E\left(\eta_{v,is}v_{it}\right)=\left\\{\begin{array}[]{c}0\text{, for
}s<t\text{,}\\\ \sigma_{vi}^{2}<K\text{, for }s=t\text{,}\\\ \leq
K\rho^{s-t}\text{, for }s>t\text{,}\end{array}\right.$

where $\rho\equiv\sup_{i}\left|1-\alpha_{i}\right|<1$ by Assumption 1. Hence
$\left|\sum_{s=1}^{T}E\left(\eta_{v,is}v_{it}\right)\right|<K$ for any
$t=1,2,...T$, and

$\left|E\left(\frac{1}{T}\sum_{t=1}^{T}\left(x_{it}-\bar{x}_{i}\right)\eta_{v,it}\cdot\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\frac{u_{x,it}v_{it}}{\alpha_{i}}\right)\right|\leq\frac{K}{\sqrt{T}}\text{,}$
(A.71)

as desired. This establish (A.64) hold for $s=1$, $j=2$.

Consider next (A.64) for $s\in\left\\{1,2\right\\}$ and $j=1$. Using
expression (A.32), we can write $a_{iT,s}q_{iT,1}$, for $s=1,2$, as

$a_{iT,s}q_{iT,1}=a_{iT,s}\varsigma_{a,iT}+a_{iT,s}\varsigma_{b,iT}+a_{iT,s}\varsigma_{c,iT}\text{,}$
(A.72)

where as in the proof of Lemma A.6
$\varsigma_{a,iT}=-\mathbf{\tilde{\xi}}_{i,-1}^{\prime}\mathbf{\tilde{v}}_{i}/\sqrt{T}$,
$\varsigma_{b,iT}=\alpha_{i}^{-1}\Delta\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}/\sqrt{T}$
and
$\varsigma_{c,iT}=\alpha_{i}^{-1}\Delta\mathbf{\tilde{v}}_{i}^{\prime}\mathbf{P}_{i}\mathbf{\tilde{v}}_{i}/\sqrt{T}$.
Using similar arguments as in establishing (A.71), we obtain

$\left|E\left(a_{iT,s}\varsigma_{a,iT}\right)\right|<\frac{K}{\sqrt{T}}\text{,
for }s=1,2\text{.}$

Noting next that $\varsigma_{b,iT}=\alpha_{i}^{-1}q_{i,T,2}$, it directly
follows from results (A.66) and (A.71) that

$\left|E\left(a_{iT,s}\varsigma_{b,iT}\right)\right|<\frac{K}{\sqrt{T}}\text{,
for }s=1,2\text{.}$

Consider the last term, $a_{i,T,s}\varsigma_{c,iT}$, for $s=1,2$. Using
Cauchy-Schwarz inequality we have

$\left|E\left(a_{iT,s}\varsigma_{c,iT}\right)\right|\leq\sqrt{E\left(a_{iT,s}^{2}\right)}\sqrt{E\left(\varsigma_{c,iT}^{2}\right)}\text{,
for }s=1,2\text{. }$

But $E\left(a_{iT,s}^{2}\right)<K$, for $s=1,2$ by Lemma A.2, and
$E\left(\varsigma_{c,iT}^{2}\right)<K/T$ is implied by (A.35). Hence

$\left|E\left(a_{iT,s}\varsigma_{c,iT}\right)\right|\leq\frac{K}{\sqrt{T}}\text{,
for }s=1,2\text{. }$

This completes the proof of (A.64) for all $s,j=1,2$, and therefore (A.65)
holds. Using (A.65), we

$\left|\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\mu_{iT}^{\ast}\right|\leq\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\left|\mu_{iT}^{\ast}\right|<\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\frac{K}{\sqrt{T}}=K\frac{\sqrt{n}}{T}\rightarrow
0\text{,}$ (A.73)

as $n,T\rightarrow\infty$ such that $\sqrt{n}/T\rightarrow 0$. Results (A.63)
and (A.73) imply

$\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\mathbf{a}_{iT}^{\prime}\mathbf{B}_{i}^{-1}\mathbf{q}_{iT}\rightarrow_{p}0\text{,}$
(A.74)

as $n,T\rightarrow\infty$ such that $\sqrt{n}/T\rightarrow 0$. Finally, using
(A.55) and (A.74) in (A.54), we obtain (A.63), as desired.

###### Lemma A.10

Let Assumptions 1-4 hold. Then

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}E\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\right)\rightarrow
0,$

as $n,T\rightarrow\infty$ such that $n=\Theta\left(T^{\theta}\right)$ for some
$0<\theta<2$, where $\mathbf{M}_{i}$ is defined in (9), and
$\mathbf{\tilde{x}}_{i}$ and $\mathbf{\tilde{v}}_{i}$ are defined below (6).

Proof. We have

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}+\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\xi_{iT}\text{,}$
(A.75)

where $\xi_{iT}$ is defined by (A.50). For the first term,
$E\left(\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}\right)=0$. For
the second term, Lemma A.9 established
$\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\xi_{iT}\rightarrow_{p}0$, as
$n,T\rightarrow\infty$ such that $n=\Theta\left(T^{\theta}\right)$, for some
$0<\theta<2$. Hence it follows that
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}E\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\right)\rightarrow
0$, as $n$,$T\rightarrow\infty$ such that $n=\Theta\left(T^{\theta}\right)$,
for some $0<\theta<2$.

###### Lemma A.11

Suppose conditions of Theorem 1 hold. Then

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}-E\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\right)\right]\rightarrow_{d}N\left(0,\omega_{v}^{2}\right)\text{,}$
(A.76)

where
$\omega_{v}^{2}=\lim{}_{n\rightarrow\infty}n^{-1}\sum_{i=1}^{n}\sigma_{xi}^{2}\sigma_{vi}^{2}/\left(6\alpha_{i}^{2}\right)$,
$\mathbf{M}_{i}$ is defined by (9), and $\mathbf{\tilde{x}}_{i}$ and
$\mathbf{\tilde{v}}_{i}$ are defined below (6).

Proof. It is convenient to use (A.75) in (A.76) to obtain

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}-E\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\right)\right]=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}+\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\left[\xi_{iT}-E\left(\xi_{iT}\right)\right]\text{,}$

where $E\left(\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}\right)=0$.
It follows from Lemmas A.9 and A.10 that
$\frac{1}{\sqrt{nT}}\sum_{i=1}^{n}\left[\xi_{iT}-E\left(\xi_{iT}\right)\right]\rightarrow_{p}0$,
as $n,T\rightarrow\infty$ such that $n=\Theta\left(T^{\theta}\right)$, for
some $0<\theta<2$. Hence, under the conditions of Theorem 1, the asymptotic
distribution of
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}-E\left(\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\right)\right]$
is given by the first term,
$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}$,
alone. Lemma A.5 establishes the asymptotic normality of this term.

$\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\frac{\mathbf{\tilde{x}}_{i}^{\prime}\mathbf{M}_{i}\mathbf{\tilde{v}}_{i}}{\alpha_{i}T}\rightarrow_{d}N\left(0,\omega_{v}^{2}\right)\text{,}$
(A.77)

as $n,T\rightarrow\infty$. This completes the proof.

##  Appendix B Estimation algorithms

This appendix describes implementation of the Pooled Mean Group (PMG)
estimator, which we compute iteratively, in Section B.1. Section B.2 discusses
implementation of bias-correction methods and bootstrapping of critical values
for the PMG, PDOLS and FMOLS estimators. Section B.3 provides tables with
Monte Carlo findings for experiments with cross-sectionally dependent errors.

###  B.1 Computation of PMG estimator

Consider the same illustrative panel ARDL model as in the paper, namely the
model given by equations (1)-(2). PMG estimator of the long-run coefficient
$\beta$, as originally proposed by PesaranShinSmith1999, is computed by
solving the following equations iteratively:

$\hat{\beta}_{PMG}=-\left(\sum_{i=1}^{n}\frac{\hat{\phi}_{i}^{2}}{\hat{\sigma}_{i}^{2}}\mathbf{x}_{i}^{\prime}\mathbf{H}_{x,i}\mathbf{x}_{i}\right)^{-1}\sum_{i=1}^{n}\frac{\hat{\phi}_{i}^{2}}{\hat{\sigma}_{i}^{2}}\mathbf{x}_{i}^{\prime}\mathbf{H}_{x,i}\left(\Delta\mathbf{y}_{i}-\hat{\phi}_{i}\mathbf{y}_{i,-1}\right)\text{,}$
(B.1)
$\hat{\phi}_{i}=\left(\mathbf{\hat{\xi}}_{i}^{\prime}\mathbf{H}_{x,i}\mathbf{\hat{\xi}}_{i}\right)^{-1}\mathbf{\hat{\xi}}_{i}^{\prime}\mathbf{H}_{x,i}\Delta\mathbf{y}_{i}\text{,
}i=1,2,...,n\text{,}$ (B.2)

and

$\hat{\sigma}_{i}^{2}=T^{-1}\left(\Delta\mathbf{y}_{i}-\hat{\phi}_{i}\mathbf{\hat{\xi}}_{i}\right)^{\prime}\mathbf{H}_{x,i}\left(\Delta\mathbf{y}_{i}-\hat{\phi}_{i}\mathbf{\hat{\xi}}_{i}\right)\text{,
}i=1,2,...,n\text{,}$ (B.3)

where
$\mathbf{\hat{\xi}}_{i}=\mathbf{y}_{i,-1}-\mathbf{x}_{i}\hat{\beta}_{PMG}$,
$\mathbf{x}_{i}=\left(x_{i,1},x_{i,2},...,x_{i,T}\right)^{\prime}$,
$\Delta\mathbf{y}_{i}=\mathbf{y}_{i}-\mathbf{y}_{i,-1}$,
$\mathbf{y}_{i}=\left(y_{i,1},y_{i,2},...,y_{i,T}\right)^{\prime}$,
$\mathbf{y}_{i,-1}=\left(y_{i,0},y_{i,1},...,y_{i,T-1}\right)^{\prime}$,
$\mathbf{H}_{x,i}=\mathbf{I}_{T}-\Delta\mathbf{x}_{i}\left(\Delta\mathbf{x}_{i}^{\prime}\Delta\mathbf{x}_{i}\right)^{-1}\Delta\mathbf{x}_{i}^{\prime}$,
$\Delta\mathbf{x}_{i}=\mathbf{x}_{i}-\mathbf{x}_{i,-1}$, and
$\mathbf{x}_{i,-1}\mathbf{=}\left(x_{i,0},x_{i,1},...,x_{i,T-1}\right)^{\prime}$.

To solve (B.1)-(B.3) iteratively, we set $\hat{\beta}_{PMG,\left(0\right)}$ to
the pooled Engle-Granger estimator, and given the initial estimate
$\hat{\beta}_{PMG,\left(0\right)}$, we compute
$\mathbf{\hat{\xi}}_{i,\left(0\right)}=\mathbf{y}_{i,-1}-\mathbf{x}_{i}\hat{\beta}_{PMG,\left(0\right)}$,
$\hat{\phi}_{i,\left(0\right)}$ and $\hat{\sigma}_{i,\left(0\right)}^{2}$, for
$i=1,2,...,n$ using (B.2)-(B.3). Next we compute
$\hat{\beta}_{PMG,\left(1\right)}$ using (B.1) and given values
$\hat{\phi}_{i,\left(0\right)}$ and $\hat{\sigma}_{i,\left(0\right)}^{2}$.
Then we iterate - for a given value of $\hat{\beta}_{PMG,\left(\ell\right)}$
we compute $\mathbf{\hat{\xi}}_{i,\left(\ell\right)}$,
$\hat{\phi}_{i,\left(\ell\right)}$ and
$\hat{\sigma}_{i,\left(\ell\right)}^{2}$; and for given values of
$\hat{\phi}_{i,\left(\ell\right)}$ and
$\hat{\sigma}_{i,\left(\ell\right)}^{2}$ we compute
$\hat{\beta}_{PMG,\left(\ell+1\right)}$. If convergence is not achieved, we
increase $\ell$ by one and repeat. We define convergence by
$\left|\hat{\beta}_{PMG,\left(\ell+1\right)}-\hat{\beta}_{PMG,\left(\ell\right)}\right|<10^{-4}$.

Inference is conducted using equation (17) of PesaranShinSmith1999. In
particular,

$T\sqrt{n}\left(\hat{\beta}_{PMG}-\beta_{0}\right)\sim
N\left(0,\Omega_{PMG}\right)\text{, }$

where

$\Omega_{PMG}=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{\phi_{i,0}}{\sigma_{i,0}^{2}}r_{x_{i},x_{i}}\right)^{-1}\text{,
and
}r_{x_{i},x_{i}}=plim_{T\rightarrow\infty}T^{-2}\mathbf{x}_{i}^{\prime}\mathbf{H}_{x,i}\mathbf{x}_{i}\text{
.}$

Standard error of $\hat{\beta}_{PMG}$, denoted as
$se\left(\hat{\beta}_{PMG}\right)$, is estimated as

$\widehat{se}\left(\hat{\beta}_{PMG}\right)=T^{-1}n^{-1/2}\hat{\Omega}_{PMG}\text{,
}$

where

$\hat{\Omega}_{PMG}=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{\hat{\phi}_{i,0}}{\hat{\sigma}_{i,0}^{2}}\hat{r}_{x_{i},x_{i}}\right)^{-1}\text{
and
}\hat{r}_{x_{i},x_{i}}=T^{-2}\mathbf{x}_{i}^{\prime}\mathbf{H}_{x,i}\mathbf{x}_{i}\text{.}$
(B.4)

###  B.2 Bias-corrected PMG, PDOLS and FMOLS estimators, and bootstrapped
critical values

Similarly to the bootstrap bias-corrected PB estimator, we consider the
following bootstrap bias-corrected PMG, PDOLS and FMOLS estimators. Let the
original (uncorrected) estimators be denoted as $\hat{\beta}_{e}$ for
$e=PMG,PDOLS$, and $FMOLS$, respectively. Bootstrap bias corrected version of
these estimators is given by

$\tilde{\beta}_{e}=\hat{\beta}_{e}-\hat{b}_{e}\text{,}$ (B.5)

for $e=PMG,PDOLS$, and $FMOLS$, where $\hat{b}_{e}$ an estimate of the bias
obtained by the following sieve wild bootstrap algorithm, which resembles the
algorithm in Subsection 2.2.1. For e$=PMG,PDOLS$, and $FMOLS:$

  1. 1.

Compute $\hat{\beta}_{e}$. Given $\hat{\beta}_{e}$, estimate the remaining
unknown coefficients of (1)-(2) by least squares, and compute residuals
$\hat{u}_{y,it}^{e},\hat{u}_{x,it}^{e}$.

  2. 2.

For each $r=1,2,...,R$, generate new draws for
$\hat{u}_{y,it}^{e,\left(r\right)}=a_{t}^{\left(r\right)}\hat{u}_{y,it}^{e}$,
and
$\hat{u}_{x,it}^{e,\left(r\right)}=a_{t}^{\left(r\right)}\hat{u}_{x,it}^{e}$,
where $a_{t}^{\left(r\right)}$ are randomly drawn from Rademacher distribution
(Liu1988, Liu1988) namely

$a_{t}^{\left(r\right)}=\left\\{\begin{array}[]{cc}-1\text{,}&\text{with
probability 1/2}\\\ 1\text{,}&\text{with probability
1/2}\end{array}\right.\text{.}$

Given the estimated parameters of (1)-(2) from Step 1, and initial values
$y_{i1},x_{i1}$ generate simulated data
$y_{it}^{e,\left(r\right)},x_{it}^{e,\left(r\right)}$ for $t=2,3,...,T$ and
$i=1,2,...,n$. Using the generated data compute
$\hat{\beta}_{e}^{\left(r\right)}$.

  3. 3.

Compute
$\hat{b}_{e}=\left[R^{-1}\sum_{r=1}^{R}\hat{\beta}_{e}^{\left(r\right)}-\hat{\beta}_{e}\right]$.

The possibility of iterating the algorithm above by using the bias-corrected
estimate $\tilde{\beta}_{e}$ in Step 1 is not considered in this paper.

We conduct inference by using the $1-\alpha$ confidence interval
$C_{1-\alpha}\left(\tilde{\beta}_{e}\right)=\tilde{\beta}_{e}\pm\hat{k}_{e}\widehat{se}\left(\hat{\beta}_{e}\right)=\tilde{\beta}_{e}\pm
T^{-1}n^{-1/2}\hat{k}_{e}\hat{\Omega}_{e}$, where $\hat{k}_{e}$ is the
$1-\alpha$ percent quantile of
$\left\\{\left|t_{e}^{\left(r\right)}\right|\right\\}_{r=1}^{R}$, in which
$t_{e}^{\left(r\right)}=\tilde{\beta}_{e}^{\left(r\right)}/\widehat{se}\left(\hat{\beta}_{e}^{\left(r\right)}\right)=T^{-1}n^{-1/2}\tilde{\beta}_{e}^{\left(r\right)}/\hat{\Omega}_{e}^{\left(r\right)}$,
$\tilde{\beta}_{e}^{\left(r\right)}=\hat{\beta}_{e}^{\left(r\right)}-\hat{b}_{e}$
is the bias-corrected PMG estimate of $\beta$ in the $r$-$th$ draw of the
bootstrap data in the algorithm above, and $\hat{\Omega}_{e}^{\left(r\right)}$
is estimated standard error using the bootstrap data.

####  B.2.1 Jackknife bias-corrections

We consider similar jackknife bias correction for PMG, PDOLS and FMOLS
estimator as for the PB estimator in Section 2.2. In particular,

$\tilde{\beta}_{jk,e}=\tilde{\beta}_{jk,e}\left(\kappa\right)=\hat{\beta}_{e}-\kappa\left(\frac{\hat{\beta}_{e,a}+\hat{\beta}_{e,b}}{2}-\hat{\beta}_{e}\right)\text{,}$

for $e=PMG,PDOLS$, and $FMOLS$, where $\hat{\beta}_{e}$ is the full sample
estimator, $\hat{\beta}_{e,a}$ and $\hat{\beta}_{e,b}$ are the first and the
second half sub-sample estimators, and $\kappa=1/3$ is the same weighting
parameter as in Section 2.2.

We conduct inference by using the $1-\alpha$ confidence interval
$C_{1-\alpha}\left(\tilde{\beta}_{jk,e}\right)=\tilde{\beta}_{jk,e}\pm\hat{k}_{jk,e}\widehat{se}\left(\hat{\beta}_{e}\right)=\tilde{\beta}_{jk,e}\pm\hat{k}_{jk,e}T^{-1}n^{-1/2}\hat{\Omega}_{e}$,
where $\hat{k}_{jk}$ is the $1-\alpha$ percent quantile of
$\left\\{\left|t_{jk,e}^{\left(r\right)}\right|\right\\}_{r=1}^{R}$, in which
$t_{jk,e}^{\left(r\right)}=\tilde{\beta}_{jk,e}^{\left(r\right)}/\widehat{se}\left(\hat{\beta}_{e}^{\left(r\right)}\right)=T^{-1}n^{-1/2}\tilde{\beta}_{jk,e}^{\left(r\right)}/\hat{\Omega}_{e}^{\left(r\right)}$,
$\tilde{\beta}_{jk,e}^{\left(r\right)}$ is the jackknife bias-corrected
estimate of $\beta$ using the $r$-$th$ draw of the bootstrap data generated
using the same algorithm as in Subsection B.2, and
$\hat{\Omega}_{e}^{\left(r\right)}$ is estimated standard error using the
bootstrap data.

###  B.3 Monte Carlo results for experiments with cross-sectionally dependent
errors

Table B1: MC findings for the estimation of long-run coefficient $\beta$ in
experiments with cross-sectionally dependent errors.

Estimators without bias correction and inference conducted using standard
critical values.

| Bias ($\mathbf{\times}$ 100) |  |  | RMSE ($\mathbf{\times}$ 100) |  |  | Size (5% level) |  |  | Power (5% level)
---|---|---|---|---|---|---|---|---|---|---
$n\backslash T$ | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50
| PB
20 | -3.83 | -1.93 | -1.01 | -0.66 |  |  | 7.58 | 4.92 | 3.57 | 2.78 |  |  | 28.00 | 21.95 | 17.55 | 16.80 |  |  | 37.30 | 64.50 | 86.90 | 96.85
30 | -3.57 | -1.98 | -1.17 | -0.69 |  |  | 6.67 | 4.43 | 3.32 | 2.46 |  |  | 28.95 | 23.55 | 22.10 | 18.80 |  |  | 45.10 | 75.25 | 92.70 | 99.00
40 | -3.94 | -2.04 | -1.10 | -0.74 |  |  | 6.58 | 4.20 | 3.08 | 2.37 |  |  | 33.60 | 27.25 | 23.75 | 22.90 |  |  | 46.60 | 82.00 | 96.70 | 99.65
50 | -3.88 | -1.99 | -1.10 | -0.76 |  |  | 6.35 | 4.01 | 2.89 | 2.23 |  |  | 38.30 | 29.55 | 25.30 | 23.30 |  |  | 50.70 | 85.60 | 97.75 | 99.95
| PMG
20 | -1.96 | -0.90 | -0.40 | -0.26 |  |  | 9.05 | 5.49 | 3.86 | 3.00 |  |  | 45.75 | 34.00 | 26.30 | 23.90 |  |  | 62.45 | 78.50 | 92.45 | 98.15
30 | -1.60 | -1.05 | -0.56 | -0.31 |  |  | 7.76 | 4.83 | 3.54 | 2.58 |  |  | 46.35 | 37.25 | 32.75 | 27.35 |  |  | 70.50 | 85.25 | 95.60 | 99.65
40 | -1.79 | -1.02 | -0.50 | -0.32 |  |  | 7.07 | 4.51 | 3.27 | 2.44 |  |  | 49.40 | 40.80 | 35.00 | 29.90 |  |  | 73.05 | 90.25 | 97.65 | 99.85
50 | -1.76 | -0.89 | -0.46 | -0.32 |  |  | 6.70 | 4.21 | 3.00 | 2.28 |  |  | 54.30 | 42.95 | 34.20 | 31.55 |  |  | 76.40 | 93.90 | 98.90 | 100.00
| PDOLS
20 | -6.20 | -4.10 | -3.08 | -2.40 |  |  | 9.50 | 6.31 | 4.67 | 3.67 |  |  | 31.40 | 27.30 | 25.70 | 26.30 |  |  | 22.35 | 39.95 | 65.20 | 85.80
30 | -5.76 | -4.15 | -3.08 | -2.37 |  |  | 8.65 | 5.89 | 4.39 | 3.41 |  |  | 34.85 | 33.45 | 32.15 | 33.35 |  |  | 27.95 | 48.60 | 77.05 | 93.55
40 | -6.29 | -4.24 | -3.12 | -2.44 |  |  | 8.67 | 5.82 | 4.27 | 3.35 |  |  | 42.95 | 41.85 | 38.60 | 38.80 |  |  | 26.95 | 52.35 | 82.15 | 97.05
50 | -6.12 | -4.19 | -3.09 | -2.47 |  |  | 8.46 | 5.62 | 4.16 | 3.27 |  |  | 47.90 | 45.45 | 45.15 | 45.05 |  |  | 31.50 | 59.55 | 88.25 | 98.65
| FMOLS
20 | -10.89 | -7.25 | -5.42 | -4.20 |  |  | 13.27 | 9.09 | 6.79 | 5.37 |  |  | 85.10 | 76.30 | 68.35 | 63.05 |  |  | 54.40 | 54.40 | 73.30 | 88.35
30 | -10.32 | -7.26 | -5.29 | -4.06 |  |  | 12.28 | 8.70 | 6.45 | 5.04 |  |  | 88.60 | 82.70 | 76.00 | 71.35 |  |  | 57.00 | 60.70 | 82.20 | 95.25
40 | -10.94 | -7.60 | -5.45 | -4.29 |  |  | 12.57 | 8.76 | 6.41 | 5.10 |  |  | 91.90 | 88.20 | 82.85 | 77.50 |  |  | 60.10 | 61.75 | 84.95 | 96.65
50 | -10.72 | -7.42 | -5.38 | -4.30 |  |  | 12.26 | 8.52 | 6.28 | 5.00 |  |  | 93.50 | 89.15 | 85.40 | 82.35 |  |  | 63.55 | 68.15 | 89.30 | 98.80

Notes: DGP is given by $\Delta y_{it}=c_{i}-\alpha_{i}\left(y_{i,t-1}-\beta
x_{i,t-1}\right)+u_{y,it}$ and $\Delta x_{it}=u_{x,it}$, for $i=1,2,...,n,$
$T=1,2,...,T$, with $\beta=1$ and $\alpha_{i}\sim IIDU\left[0.2,0.3\right]$.
Errors $u_{y,it}$, $u_{x,it}$ are cross-sectionally dependent, heteroskedastic
over $i$, and also correlated over $y$ & $x$ equations. See Section 3.1 for
complete description of the DGP. The pooled Bewley estimator is given by (8),
with variance estimated using (16). PMG is the Pooled Mean Group estimator
proposed by PesaranShinSmith1999. PDOLS is panel dynamic OLS estimator by
MarkSul2003. FMOLS is the group-mean fully modified OLS estimator by
Pedroni1996 (Pedroni1996, Pedroni2001ReStat). The size and power findings are
computed using 5% nominal level and the reported power is the rejection
frequency for testing the hypothesis $\beta=0.9$.

Table B2: MC findings for the estimation of long-run coefficient $\beta$ in
experiments with cross-sectionally dependent errors.

Bias corrected estimators and inference conducted using bootstrapped critical
values.

| Bias ($\mathbf{\times}$ 100) |  |  | RMSE ($\mathbf{\times}$ 100) |  |  | Size (5% level) |  |  | Power (5% level)
---|---|---|---|---|---|---|---|---|---|---
$n\backslash T$ | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50 |  |  | 20 | 30 | 40 | 50
| Jackknife bias-corrected estimators
| PB
20 | -1.57 | -0.54 | -0.11 | -0.01 |  |  | 7.83 | 5.24 | 3.92 | 3.04 |  |  | 7.65 | 6.55 | 5.55 | 4.65 |  |  | 20.55 | 41.35 | 65.35 | 83.50
30 | -1.29 | -0.67 | -0.34 | -0.05 |  |  | 6.76 | 4.63 | 3.59 | 2.72 |  |  | 6.30 | 6.60 | 6.40 | 5.25 |  |  | 25.80 | 51.35 | 73.90 | 91.45
40 | -1.67 | -0.62 | -0.20 | -0.06 |  |  | 6.48 | 4.35 | 3.36 | 2.58 |  |  | 6.75 | 6.15 | 6.20 | 5.40 |  |  | 25.55 | 54.30 | 80.25 | 94.45
50 | -1.54 | -0.58 | -0.21 | -0.09 |  |  | 6.16 | 4.12 | 3.15 | 2.43 |  |  | 6.80 | 6.20 | 5.85 | 4.65 |  |  | 27.65 | 59.40 | 84.95 | 96.70
| PMG
20 | -0.54 | -0.15 | 0.02 | 0.06 |  |  | 10.66 | 6.26 | 4.45 | 3.44 |  |  | 16.05 | 10.80 | 8.55 | 7.90 |  |  | 29.10 | 45.05 | 69.15 | 84.75
30 | -0.25 | -0.39 | -0.19 | 0.03 |  |  | 9.10 | 5.50 | 4.02 | 2.94 |  |  | 14.80 | 10.40 | 9.05 | 6.90 |  |  | 36.30 | 53.85 | 76.40 | 93.40
40 | -0.57 | -0.27 | -0.08 | 0.03 |  |  | 8.25 | 5.13 | 3.78 | 2.78 |  |  | 14.10 | 10.50 | 9.15 | 7.80 |  |  | 35.10 | 59.50 | 81.25 | 95.55
50 | -0.41 | -0.10 | -0.06 | 0.03 |  |  | 7.81 | 4.81 | 3.44 | 2.63 |  |  | 14.85 | 11.05 | 8.65 | 7.65 |  |  | 39.55 | 64.65 | 85.90 | 97.15
| PDOLS
20 | -4.61 | -2.84 | -2.06 | -1.54 |  |  | 9.89 | 6.23 | 4.51 | 3.51 |  |  | 8.75 | 7.50 | 6.45 | 4.95 |  |  | 10.05 | 21.55 | 39.00 | 63.80
30 | -4.09 | -2.96 | -2.15 | -1.55 |  |  | 8.86 | 5.72 | 4.21 | 3.20 |  |  | 8.70 | 7.30 | 6.55 | 5.40 |  |  | 12.20 | 24.05 | 46.10 | 73.10
40 | -4.66 | -2.94 | -2.13 | -1.57 |  |  | 8.62 | 5.53 | 4.00 | 3.06 |  |  | 8.80 | 6.55 | 6.45 | 4.55 |  |  | 10.00 | 23.90 | 47.70 | 77.80
50 | -4.41 | -2.91 | -2.11 | -1.63 |  |  | 8.30 | 5.29 | 3.84 | 2.95 |  |  | 9.85 | 6.50 | 6.25 | 5.10 |  |  | 11.95 | 24.40 | 52.10 | 80.80
| FMOLS
20 | -8.65 | -5.12 | -3.61 | -2.69 |  |  | 12.18 | 7.99 | 5.79 | 4.56 |  |  | 11.25 | 5.65 | 4.20 | 3.85 |  |  | 2.00 | 4.55 | 9.50 | 25.50
30 | -8.06 | -5.27 | -3.57 | -2.55 |  |  | 11.00 | 7.43 | 5.42 | 4.14 |  |  | 9.65 | 6.85 | 5.20 | 3.35 |  |  | 2.10 | 3.95 | 12.65 | 31.60
40 | -8.65 | -5.53 | -3.70 | -2.74 |  |  | 11.08 | 7.35 | 5.25 | 4.08 |  |  | 11.50 | 6.50 | 4.55 | 3.85 |  |  | 1.45 | 3.35 | 12.25 | 32.05
50 | -8.44 | -5.36 | -3.63 | -2.78 |  |  | 10.72 | 7.07 | 5.07 | 3.94 |  |  | 12.45 | 6.70 | 4.55 | 3.40 |  |  | 1.75 | 4.40 | 13.70 | 34.30
| Bootstrap bias-corrected estimators
| PB
20 | -1.26 | -0.43 | -0.05 | -0.01 |  |  | 7.34 | 4.80 | 3.56 | 2.78 |  |  | 10.10 | 7.95 | 6.45 | 5.85 |  |  | 29.35 | 52.15 | 76.10 | 90.15
30 | -1.04 | -0.51 | -0.23 | -0.05 |  |  | 6.29 | 4.21 | 3.24 | 2.42 |  |  | 10.20 | 7.35 | 7.20 | 5.40 |  |  | 37.20 | 64.30 | 85.00 | 95.90
40 | -1.31 | -0.51 | -0.12 | -0.08 |  |  | 5.98 | 3.93 | 3.01 | 2.32 |  |  | 10.15 | 7.50 | 6.75 | 5.80 |  |  | 39.00 | 69.80 | 89.80 | 98.15
50 | -1.26 | -0.45 | -0.12 | -0.09 |  |  | 5.78 | 3.72 | 2.78 | 2.17 |  |  | 10.20 | 7.85 | 7.05 | 6.00 |  |  | 42.90 | 75.25 | 93.50 | 99.20
| PMG
20 | -1.23 | -0.44 | -0.09 | -0.03 |  |  | 9.24 | 5.54 | 3.91 | 3.03 |  |  | 16.00 | 10.50 | 7.60 | 6.70 |  |  | 31.35 | 51.80 | 76.10 | 90.80
30 | -0.92 | -0.60 | -0.27 | -0.10 |  |  | 7.89 | 4.84 | 3.57 | 2.59 |  |  | 14.70 | 9.95 | 8.45 | 5.85 |  |  | 38.20 | 60.85 | 83.80 | 97.10
40 | -1.10 | -0.56 | -0.19 | -0.11 |  |  | 7.14 | 4.51 | 3.30 | 2.45 |  |  | 14.80 | 10.00 | 8.90 | 7.00 |  |  | 39.35 | 67.05 | 88.90 | 98.10
50 | -1.06 | -0.45 | -0.16 | -0.10 |  |  | 6.77 | 4.23 | 3.02 | 2.29 |  |  | 14.90 | 10.45 | 8.25 | 6.00 |  |  | 44.90 | 71.40 | 91.75 | 99.15
| PDOLS
20 | -2.34 | -1.13 | -0.66 | -0.37 |  |  | 8.73 | 5.46 | 3.92 | 3.05 |  |  | 12.05 | 10.05 | 7.75 | 6.90 |  |  | 22.40 | 45.25 | 71.30 | 87.75
30 | -2.03 | -1.29 | -0.74 | -0.41 |  |  | 7.73 | 4.85 | 3.53 | 2.68 |  |  | 12.15 | 8.60 | 8.45 | 7.65 |  |  | 26.95 | 51.50 | 79.30 | 94.15
40 | -2.46 | -1.28 | -0.71 | -0.41 |  |  | 7.40 | 4.68 | 3.31 | 2.54 |  |  | 12.65 | 10.20 | 9.05 | 7.75 |  |  | 26.15 | 55.55 | 84.05 | 96.90
50 | -2.32 | -1.22 | -0.68 | -0.44 |  |  | 7.28 | 4.41 | 3.15 | 2.39 |  |  | 15.00 | 9.60 | 8.40 | 7.40 |  |  | 30.15 | 59.85 | 88.40 | 98.55
| FMOLS
20 | -4.39 | -2.01 | -1.24 | -0.77 |  |  | 10.37 | 6.73 | 4.85 | 3.83 |  |  | 18.45 | 10.90 | 7.80 | 6.85 |  |  | 17.80 | 31.75 | 50.55 | 70.75
30 | -4.00 | -2.25 | -1.25 | -0.71 |  |  | 9.02 | 6.02 | 4.42 | 3.43 |  |  | 17.25 | 12.00 | 9.60 | 7.70 |  |  | 20.95 | 37.25 | 61.40 | 82.55
40 | -4.46 | -2.44 | -1.27 | -0.84 |  |  | 8.82 | 5.67 | 4.11 | 3.21 |  |  | 19.35 | 12.10 | 9.60 | 9.05 |  |  | 20.10 | 38.50 | 67.65 | 85.95
50 | -4.30 | -2.27 | -1.22 | -0.86 |  |  | 8.48 | 5.41 | 3.92 | 3.02 |  |  | 21.20 | 13.90 | 10.10 | 7.80 |  |  | 22.85 | 44.85 | 72.20 | 90.40

Notes: See the notes to Table B1. Bias-corrected versions of the PB estimator
are described in Subsection 2.2. Bias-corrected versions of the PMG, PDOLS and
FMOLS estimator are described in Appendix B. Inference is conducted using
bootstrapped critical values.