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Pooled Bewley Estimator of LongRun Relationships in Dynamic
Heterogenous Panels
Alexander Chudik, M. Hashem Pesaran and Ron P. Smith
Globalization Institute Working Paper 409
June 2021 (Revised November 2023)
Research Department
https://doi.org/10.24149/gwp409r2
Working papers from the Federal Reserve Bank of Dallas are preliminary drafts circulated for professional comment.
The views in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank
of Dallas or the Federal Reserve System. Any errors or omissions are the responsibility of the authors.
Pooled Bewley Estimator of Long-Run Relationships in Dynamic
Heterogenous Panels *
Alexander Chudik†, M. Hashem Pesaran‡ and Ron P. Smith§
May 27, 2021
Revised: 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 crosssectional 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
*
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, alexander.chudik@gmail.com.
‡M. Hashem Pesaran, University of Southern California, USA and Trinity College, Cambridge, UK, mhp1@cam.ac.uk.
§Ron P. Smith, Birkbeck, University of London, United Kingdom, r.smith@bbk.ac.uk.
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 …elds in economics.
Existing single-equation panel estimators in the literature are panel Fully Modi…ed OLS (FMOLS)
by Pedroni (1996, 2001a, 2001b), panel Dynamic OLS (PDOLS) by Mark and Sul (2003), and the
likelihood based Pooled Mean Group (PMG) estimator by Pesaran, Shin, and Smith (1999). Multiequation (system) approach by Breitung (2005), and the related system PMG approach by Chudik,
Pesaran, and Smith (2023) 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
(Bewley, 1979). See also Wickens and Breusch (1988) for a discussion of the Bewley transform.
Our setting is the same as that of Pesaran, Shin, and Smith (1999). Under this setting, any shortrun 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
in…nity jointly such that n =
T
, for 0 <
< 2, where we use the notation
(:) to denote the
1
same order of magnitude asymptotically, namely if ffs g1
s=1 and fgs gs=1 are both positive sequences
of real numbers, then fs =
such that inf s
S0
(fs =gs )
(gs ) if there exists S0
C0 ; and sups
S0
(fs =gs )
1 and positive …nite constants C0 and C1 ,
C1 . 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
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 di¤erent 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.
1
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 su¤er 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) su¤er
from bias in …nite 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 Maddala and Wu (1999).
We found the sieve wild bootstrap procedure to be remarkably e¤ective 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 (Dhaene and Jochmans, 2015) or
sieve wild bootstrap approaches. In contrast to split-panel approaches in panels without stochastic
trends, as, for instance, considered by of Dhaene and Jochmans (2015) or Chudik, Pesaran, and
Yang (2018), in this paper we need to combine the full sample and half-panel subsamples using
di¤erent weighting due to the fact that the rate of convergence of the estimators of long run
p
p
coe¢ cients is faster, at the rate of T n, as compared to the standard rate of nT . We …nd 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 Pesaran, Shin, and Smith
(1999). This application shows that quite a di¤erent conclusion would be reached when using
PB estimator, which does not reject the zero long-run coe¢ cient on in‡ation, in line with the
long-run neutrality of monetary policy, whereas the PMG estimator results in a highly statistically
signi…cant negative long-run coe¢ cient. Estimates of the long-run coe¢ cient on real income are less
diverse across estimators, but the inference on whether a unit long-run coe¢ cient on real income
(as suggested by balanced growth path models in the literature) can be rejected or not depends on
2
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
Pesaran, Shin, and Smith (1999). 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 Pesaran, Shin, and Smith (1999), and consider the following
illustrative model
yit = ci
i (yi;t 1
xi;t
1)
+ uy;it ,
xit = ux;it ,
(1)
(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
zit = ( yit ; xit )0 entering both equations (1)-(2), and the approach is also applicable to
multiple xit ’s with a single long run relationship. We consider the following assumptions:
Assumption 1 (Coe¢ cients) supi j1
ij
< 1.
Assumption 2 (shocks) ux;it
2
xi
, and uy;it is given by
IID 0;
uy;it =
for all i and t, where vit
i ux;it
+ vit ,
(3)
2
vi
, and ux;it is independently distributed of vi0 t0 for all i,i0 ; t,
P
< K and supi;t E jux;it j8 < K, and limn!1 n 1 ni=1 2xi = 2x >
IID 0;
and t0 . In addition, supi;t E jvit j16
P
0 and lim n!1 n 1 ni=1 2xi 2vi = 6
2
i
= ! 2v > 0 exist.
Assumption 3 (Initial values and deterministic terms) The initial values, zi;0 = (yi;0 ; xi;0 )0 ,
3
follow the process
zi0 =
for all i and t, and ci =
k
ik
i i;1
i
i;2
i
+ Ci (L) u0 ,
for all i, where u0 = (uy;i;0 ; ux;i;0 )0 ,
i
=
i;1 ;
i;2
0
,
< K, and Ci (L) is de…ned in Section A.1 in Appendix A.
Remark 1 Assumption 1 requires that
i
6= 0 for all i. In contrast PMG allows
(but not all) units. Although the estimator works for
i
< 2; empirically,
i
i
= 0 for some
< 1 is likely to be the
relevant case.
Remark 2 Assumption 2 allows for ux;it to be correlated with uy;it . Cross-section dependence of
uit is ruled out. Assumption 3 (together with the remaining assumptions) ensure that
(yit
zit and
xit ) 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 !j 1, but only the
case where n is …xed as T ! 1. Joint asymptotics typically requires more stringent assumptions
on the errors and restrictions on the relative expansion rates of n and T .
Substituting …rst (3) for uy;it in (1), and then substituting ux;it =
xit , we obtain the following
ARDL representation for yit
yit = ci
i (yi;t 1
xi;t
1)
+
xit + vit .
i
(4)
The pooled Bewley estimator takes advantage of the Bewley transform (Bewley, 1979). Subtracting
(1
i ) yit
from both sides of (4) and re-arranging, we have
i yit
or (noting that
i
= ci
(1
i)
yit +
i
xit +
xit + vit ;
i
> 0 for all i and multiplying the equation above by
yit =
1
i
ci + xit +
4
0
i
zit +
1
i
vit ;
1
i
)
(5)
zit = ( yit ; xit )0 , and
where
1
=
i
i
i
;
0
i
i
. Further, stacking (5) for t = 1; 2; :::; T , we
have
1
yi =
i
ci
T
+ xi +
T
is T
i
1
+
i
vi ,
(6)
Zi = ( z0i1 ; z0i2 ; :::; z0iT )0 , vi = (vi;1 ; vi;2 ; :::; vi;T )0 ,
where yi = (yi1 ; yi2 ; :::; yiT )0 , xi = (xi1 ; xi2 ; :::; xiT )0 ,
and
Zi
1 vector of ones. De…ne projection matrix M
= IT
T
1
T
0 .
T
This pro-
jection matrix subtracts the period average. Let y
~i = (~
yi1 ; y~i2 ; :::; y~iT )0 = M yi , and similarly
x
~i = (~
xi1 ; x
~i2 ; :::; x
~iT )0 = M xi ,
~i = M
Z
Zi , and v
~i = M vi . Multiplying (6) by M , we have
y
~i = x
~i +
~i
Z
i
1
+
i
v
~i .
Now consider the matrix of instruments
~ i = (~
H
yi;
where yi;
1
= (yi;1 ; yi;1 ; :::; yi;T
(xi;1 ; xi;1 ; :::; xi;T
1)
0
~i ; x
~i; 1 )
1; x
0
1)
= M Hi , Hi = (yi;
1 ; xi ; xi; 1 ) ,
is the data vector on the …rst lag of yit , similarly xi;
. The PB estimator of
n
X
^=
(7)
1
=
is given by
x
~0i Mi x
~i
i=1
!
1
n
X
x
~0i Mi y
~i
i=1
!
,
(8)
~ 0i Pi ,
Z
(9)
where
~i
Pi Z
Mi = Pi
~ 0i Pi Z
~i
Z
1
and
~i H
~ i0H
~i
Pi = H
1
~ 0i ,
H
(10)
~ i.
is the projection matrix associated with H
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 ! 1.
Assumption 4 Let M = IT
~i = M
Z
T
1
T
0 ;
T
y
~ i = M yi , x
~ i = M xi ,
Zi = M ( z0i1 ; z0i2 ; :::; z0iT )0 , where
5
zit = ( yit ; xit )0 . Then there exists T0 2 N
such that the following conditions are satis…ed:
2
min (BiT )
(i) supi2N, T >T0 E
(ii) supi2N, T >T0 E
h
2
min
~ 0H
~ AT
AT H
i
i
i
< K, where Hi = x
~i ; x
~i ; ~i;
0
B T
B
AT = B
B 0
@
0
~i;
1
= ~i;0 ; ~i;1 ; :::; ~i;T
1
~ 0 Pi Z
~ i =T , Pi is given by (10).
Z
i
< K, where BiT =
0
, ~i;t
1
= y~i;t
1
0
T
0
C
C
C,
C
A
0
T
x
~i;t
1
1
0
1=2
,
1
1=2
1:
Remark 4 Under Assumptions 1-3 (and without Assumption 4), we have plimT !1 Bi;T = Bi ,
where Bi is nonsingular (see Lemma A.7 in Appendix A). Similarly, it can be shown that Assump~ 0H
~ AT to exist and to be nonsingular. However, these
tions 1-3 are su¢ cient for plimT !1 AT H
i
i
results are not su¢ cient for the moments of Bi
1
and (AT Hi 0 Hi AT )
1
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 y
~i = x
~i +
p
~i
Z
T n ^
i
+
1
i
~ i = 0, we have
v
~i in (8), and using Mi Z
n
=
~0i Mi x
~i
1Xx
n
T2
i=1
!
1
n
1 Xx
~0i Mi v
~i
p
T i
n
i=1
!
.
(11)
Consider the …rst term on the right side of (11) …rst. Since Mi is an orthogonal projection matrix,
~i =T 2 . The second moments of x
~0i x
~i =T 2 are bounded, and, in addition,
x
~0i Mi x
~i =T 2 is bounded by x
~0i x
P
x
~0i Mi x
~i =T 2 is cross-sectionally independent. It follows that n1 ni=1 x
~0i Mi x
~i =T 2 converges to a
constant, which we denote by ! 2x , as n; T ! 1. Lemma A.4 in Appendix A establishes the
P
expression for ! 2x = 2x =6, where 2x = limn!1 n 1 ni=1 2xi , but the speci…c expression for ! 2x is
not relevant for the inference approach that we adopt below. Consider the second term of (11)
6
next,
n
n
1 Xx
~0i Mi v
~i
p
n
iT
~0i Mi v
~i
1 X x
p
n
iT
=
i=1
i=1
n
X
1
+p
n
E
x
~0i Mi v
~i
iT
E
i=1
x
~0i Mi v
~i
iT
(12)
,
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
n
1 X
p
E
n
x
~0i Mi v
~i
T
i
i=1
as n and T ! 1. This condition holds so long as n =
! 0,
T
(13)
for some 0 <
< 2. See Lemma A.10
for a proof. The asymptotic distribution of the …rst term in (12) is in turn established by Lemma
A.11, see (A.74). The following theorem now follows for the asymptotic distribution of ^ .
Theorem 1 Let (yit ; xit ) be generated by model (1)-(2), suppose Assumptions 1-4 hold, and n; T !
1 such that n =
T
< 2. Consider the PB estimator ^ given by (8). Then,
, for some 0 <
p
T n ^
where ! 2x =
2 =6,
x
2
x
= limn!1 n
1
!d N (0; ) ,
Pn
i=1
2
xi
= ! x 4 ! 2v ,
and ! 2v = lim n!1 n
1
(14)
Pn
i=1
2 2 =
xi vi
6
2
i
.
Remark 5 Like the PMG estimator in Pesaran, Shin, and Smith (1999), the PB estimator will
also work when variables are integrated of order 0 (the I(0) case), which is not pursued in this
p
paper. In the I(0) case, the PB estimator converges at the standard rate of nT .
To conduct inference, let
!
^ 2x = n
1
n
X
x0 Mi xi
i
i=1
and !
^ 2v = n
1
Pn
i=1
x0i Mi v
^i
T
2
, where v
^i = Mi yi
we propose the following estimator of
T2
,
(15)
^ xi , and Mi is de…ned by (9). Accordingly,
:
^ =!
^ x 4!
^ 2v .
7
(16)
2.2
Bias mitigation and bootstrapping critical values for robust inference
When n is su¢ ciently large relative to T , speci…cally when
p
n=T ! K > 0, then
p
nT ^
is
no longer asymptotically distributed with zero mean. The asymptotic bias is due to the nonzero
mean of T
1x
~0i Mi v
~i ,
and it can be of some relevance for …nite 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 su¤er from serious size distortions in …nite
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 ^ is available, denoted as ^b, then the bias-corrected PB estimator
is given by
~=^
^b.
(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. Given ^ , estimate the remaining unknown coe¢ cients in (1)-(2) by least squares, and compute
residuals denoted by u
^y;it ; u
^x;it .
(r)
(r)
(r)
(r)
2. For each r = 1; 2; :::; R, generate new draws for u
^y;it = at u
^y;it , and u
^x;it = at u
^x;it , where
(r)
at
is randomly drawn from Rademacher distribution (Liu, 1988),
(r)
at
=
8
>
<
1, with probability 1/2
>
: 1,
.
with probability 1/2
Given the estimated parameters of (1)-(2) from Step 1, and the initial values fyi1 ; xi1 for i = 1; 2; :::; ng
8
(r)
(r)
generate the simulated series yit ; xit for t = 2; 3; :::; T , and i = 1; 2; :::; n, and the bootstrap
(r)
estimates ^ for r = 1; 2; :::; R.
h
Using simulated data with R = 10; 000, we compute an estimate of the bias ^bR = R
We then compute the
percent critical values using the 1
(r)
(r)
(r)
(r)
t(r) = ~ =se ~
, ~ =^
(r)
simulated data, se ~
=T
percent quantile of
^b is the bias-corrected estimate of
1 n 1=2 ^ (r)
t(r)
1
PR
^ (r)
r=1
R
, where
r=1
using the r-th draw of the
is the corresponding standard error estimate, and ^ (r)
is computed in the same way as ^ in (16) but using the simulated data.
2.2.2
Jackknife bias reduction
The split-panel jackknife bias correction method is given by
~
jk
^ +^
a
b
2
= ~ jk ( ) = ^
^
!
;
(18)
where ^ is the full sample PB estimator, ^ a and ^ b are the …rst and the second half sub-sample PB
estimators, and
is of order O T
is a suitably chosen weighting parameter. In a stationary setting, where the bias
1
,
is chosen to be one, so that
K
T
K
T
K
T =2
= 0; for any arbitrary choice
of K. See, for example, Dhaene and Jochmans (2015) and Chudik, Pesaran, and Yang (2018).
In general, when the bias is of order O (T
K
T
K
(T =2)
K
T
= 0, which yields
need to correct ^ for its O T
2
) for some
= 1= (2
bias, namely
> 0, then
can be chosen to solve
1). Under our setup with I(1) variables, we
= 2, which yields
= 1=3.
jk
Inference using ~ can be conducted based on (16) but with !
^ 2v replaced by
i 12
0h
0M
0 M
n
(1
+
)
x
2
x
v
~i
X
i
ab;i
i
ab;i
1
@
A ,
!
~ 2v = !
^ 2v =
n
T
(19)
i=1
where v
~i = Mi yi
~ jk xi ,
0
B
x0ab;i = @
x0a;i
x0b;i
x0a;i
x0b;i
1
0
1
C
B Ma;i C
A , Mab;i = @
A,
Mb;i
and Ma;i (Mb;i ) are de…ned in the same way as xi , and Mi but using only the …rst
9
i
^ .
(second) half of the sample.
We compute bootstrapped critical values to conduct more accurate and robust small sample
inference. Speci…cally, the percent critical value is computed as the 1
percent quantile of
n
oR
(r)
(r)
(r)
(r)
(r)
tjk
, where tjk = ~ jk =se ~ jk , ~ jk is the jackknife estimate of using the r-th draw of
r=1
(r)
the simulated data generated using the algorithm described in Subsection 2.2.1, se ~ jk
(r)
corresponding standard error estimate, namely se ~ jk
1 n 1=2 ^ (r) ,
jk
= T
is the
4
^ (r) = !
^ x;(r)
!
~ 2v;(r) , in
jk
which !
~ v;(r) and !
^ 2x;(r) are computed using the simulated data, based on expressions (19) and (15),
respectively.
3
3.1
Monte Carlo Evidence
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
generate
i
IIDN (
2 ; I2 ),
and ci =
2
x;i
i
i;2 .
We
IIDU [0:2; 0:3]. We consider two DGPs based on the cross-sectional dependence of
i
errors. In the cross-sectionally independent DGP, we generate uy;it =
2 ;
y;i
i i;1
y;i ey;it ,
ux;it =
x;i ex;it ,
IIDU [0:8; 1:2],
0
1
B ey;it C
@
A
ex;it
IIDN (02 ;
e) ,
0
B 1
@
e
i
i
1
1
C
A , and
i
IIDU [0:3; 0:7] .
In the DGP with cross-sectionally dependent errors, we generate ey;it to contain a factor structure
including strong, semi-strong and weak factors:
ey;it = {i
"y;it +
m
X
i` f`t
`=1
where "y;it
IIDN (0; 1), f`;t
m = 5 factors and
1
"x;it
`
with
`
IIDU 0;
,
max;`
, for ` = 1; 2; :::; m. We choose
= 1; 0:9; 0:8; 0:7; 0:6, for ` = 1; 2; :::; 5, respectively.
P
1=2
2
Scaling constant {i is set to ensure E e2y;it = 1, namely {i = 1 + m
. We generate
`=1 i`
q
2"
ex;it to ensure unit variance and cov (ey;it ; ex;it ) = i . Speci…cally, ex;it = i ey;it + 1
i x;it ,
max;`
= 2n
IIDN (0; 1),
!
`
IIDN (0; 1). Both designs features heteroskedastic (over i) and correlated (over y & x
10
equations) errors, namely E u2y;it =
2 ,
y;i
E u2x;it =
2 ,
x;i
and corr (uy;it ; ux;it ) =
i.
We consider
n; T = 20; 30; 40; 50 and compute RM C = 2000 Monte Carlo replications.
3.2
Bias, RMSE and inference
We report bias, root mean square error (RMSE), size (H0 :
(H1 :
= 1, 5% nominal level) and power
= 0:9, 5% nominal level) …ndings for the PB estimator ^ given by (8), with variance
estimated using (16). Moreover, we also report …ndings 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 Pesaran, Shin, and Smith (1999), panel dynamic OLS (PDOLS) estimator by
Mark and Sul (2003), and the group-mean fully modi…ed OLS (FMOLS) estimator by Pedroni
(1996, 2001b). 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 Rb = 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 di¤erent 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 su¤er from varying degrees of size distortions. The inference based on the
PB estimator is the most accurate. Speci…cally, the size distortions for the PB estimator are lowest
11
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 …ndings for the estimation of long-run coe¢ cient
in experiments with
cross-sectionally independent errors.
Estimators without bias correction and inference conducted using standard critical values.
Bias (
nnT
20
30
100)
40
RMSE (
50
20
30
100)
40
Size (5% level)
50
20
30
40
Power (5% level)
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
54.05 93.25 99.65 100.00
9.95
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
-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
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
50
FMOLS
Notes: DGP is given by
with
= 1 and
i
yit = ci
i
(yi;t
1
xi;t
1)
+ uy;it and
xit = ux;it , for i = 1; 2; :::; n; T = 1; 2; :::; T ,
IIDU [0:2; 0:3]. Errors uy;it , ux;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
Pesaran, Shin, and Smith (1999). PDOLS is panel dynamic OLS estimator by Mark and Sul (2003). FMOLS is the
group-mean fully modi…ed OLS estimator by Pedroni (1996, 2001b). The size and power …ndings are computed
using 5% nominal level and the reported power is the rejection frequency for testing the hypothesis
12
= 0:9.
Table 2: MC …ndings for the estimation of long-run coe¢ cient
in experiments with
cross-sectionally independent errors.
Bias corrected estimators and inference conducted using bootstrapped critical values.
nnT
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
Bias ( 100)
RMSE ( 100)
20
30
40
50
20 30 40 50
Jackknife bias-corrected estimators
PB
-1.60 -0.52 -0.25 -0.14
6.27 4.18 3.10 2.52
-1.34 -0.55 -0.25 -0.18
5.24 3.45 2.58 2.06
-1.45 -0.60 -0.24 -0.15
4.59 3.03 2.29 1.81
-1.53 -0.62 -0.26 -0.15
4.23 2.72 2.07 1.61
PMG
-0.55 -0.20 -0.05 0.01
9.26 5.59 3.78 2.86
-0.10 -0.30 0.02 -0.04
7.35 4.48 3.09 2.28
-0.27 -0.13 -0.01 -0.01
6.73 3.83 2.72 2.06
-0.42 -0.21 -0.06 0.00
6.13 3.43 2.47 1.83
PDOLS
-4.22 -2.56 -1.97 -1.59
8.00 5.05 3.77 3.04
-3.91 -2.55 -1.92 -1.56
6.89 4.42 3.30 2.64
-4.09 -2.69 -1.98 -1.60
6.35 4.17 3.08 2.47
-4.11 -2.70 -2.02 -1.58
6.01 3.91 2.93 2.30
FMOLS
-8.70 -5.07 -3.70 -2.76
11.19 7.19 5.52 4.33
-8.17 -5.02 -3.60 -2.73
10.02 6.60 4.94 3.86
-8.49 -5.22 -3.78 -2.78
9.84 6.40 4.79 3.68
-8.50 -5.28 -3.78 -2.74
9.61 6.26 4.62 3.50
Bootstrap bias-corrected estimators
PB
-1.28 -0.33 -0.16 -0.11
5.87 3.93 2.93 2.39
-0.98 -0.39 -0.15 -0.13
4.90 3.24 2.43 1.93
-1.07 -0.43 -0.13 -0.10
4.24 2.83 2.15 1.70
-1.09 -0.44 -0.15 -0.10
3.89 2.56 1.96 1.52
PMG
-1.28 -0.44 -0.21 -0.11
7.88 4.83 3.41 2.57
-0.88 -0.55 -0.12 -0.13
6.40 3.99 2.79 2.08
-0.96 -0.44 -0.14 -0.11
5.73 3.42 2.47 1.85
-1.02 -0.51 -0.19 -0.10
5.20 3.06 2.24 1.66
PDOLS
-2.19 -0.90 -0.59 -0.42
6.75 4.31 3.12 2.55
-1.89 -0.96 -0.59 -0.43
5.70 3.62 2.63 2.08
-2.00 -1.04 -0.59 -0.41
4.98 3.24 2.34 1.84
-2.00 -1.05 -0.64 -0.40
4.61 2.93 2.14 1.65
FMOLS
-4.59 -1.97 -1.30 -0.80
8.84 5.62 4.28 3.40
-4.19 -2.04 -1.28 -0.84
7.54 4.86 3.61 2.83
-4.36 -2.14 -1.34 -0.81
6.89 4.36 3.22 2.51
-4.41 -2.22 -1.35 -0.78
6.56 4.10 2.98 2.29
Size (5% level)
20
30
40 50
5.95
6.70
5.85
6.20
Power (5% level)
20
30
40
50
5.15
5.05
5.05
4.60
5.40
5.10
4.50
5.65
5.30
5.00
4.90
4.25
25.10
37.45
42.80
51.10
56.75
73.70
84.10
90.80
82.95 94.30
93.10 99.05
98.15 99.85
99.55 100.00
14.90 11.05
14.00 10.95
15.40 9.30
16.45 9.65
9.10
8.50
8.90
9.40
7.20
7.35
8.05
7.55
33.45
45.60
48.90
57.55
57.15
72.05
82.95
88.55
83.05 95.60
93.10 99.30
97.10 99.75
98.85 100.00
27.50
37.30
41.10
45.75
55.25
69.75
75.20
83.75
77.90
91.00
94.95
97.60
1.40 8.10
1.65 10.20
0.75 8.85
0.95 10.55
25.75
35.20
38.70
45.75
8.40
8.70
7.70
7.90
5.80
5.75
5.80
5.90
5.40
5.95
4.90
4.55
4.75
4.70
4.05
3.55
8.70
10.15
10.05
11.30
10.85
9.95
9.20
10.65
5.95
5.00
5.00
5.00
4.55
4.65
3.45
3.75
4.00
3.00
3.25
2.85
1.05
0.50
0.15
0.10
7.25
7.75
7.00
8.95
6.20
6.35
6.55
6.25
6.05
6.35
5.95
6.45
5.75
5.80
5.10
5.55
36.20
52.20
59.75
69.40
67.60
84.00
91.35
95.50
88.30 96.95
96.90 99.55
99.45 99.95
99.90 100.00
14.10 10.45
13.10 10.75
14.95 9.05
15.15 9.85
7.90
8.25
7.80
9.05
6.80
6.85
6.90
6.95
35.25
47.45
53.55
60.90
63.40
77.75
87.60
91.65
86.70 97.30
95.85 99.65
98.40 99.85
99.30 100.00
10.35
10.75
10.05
10.70
7.25
7.70
7.05
8.65
7.25
7.15
7.35
6.80
28.70
39.90
46.40
52.85
64.45
77.10
85.75
91.55
86.65 95.85
95.35 99.30
98.65 99.95
99.50 100.00
10.90 9.20
10.90 9.15
12.05 9.80
14.35 10.50
7.40
7.70
7.75
8.90
20.20
24.85
26.10
30.45
41.10
51.60
60.40
69.45
63.20
78.60
86.25
92.05
17.10
18.50
20.10
23.80
7.80
8.90
8.65
9.40
81.70
93.15
96.80
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.
13
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 …ndings 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 e¤ective 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 …nite 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 Pesaran, Shin, and
Smith (1999), hereafter PSS. The long-run consumption function is assumed to be given by
cit = di +
d
1 yit
+
14
2 it
+ #it ,
d is the
for country i = 1; 2; :::; n, where cit is the logarithm of real consumption per capita, yit
logarithm of real per capita disposable income,
it
is the rate of in‡ation, and #it is an I (0) process.
We take the dataset from PSS, which consists of n = 24 countries and a slightly unbalanced time
period covering 1960-1993. PSS estimate
1
and
2
using an ARDL(1,1,1) speci…cation, which can
be written as error-correcting panel regressions
cit =
i
ci;t
1
d
1 yi;t 1
di
2 i;t 1
+
i1
d
yit
+
i2
for i = 1; 2; :::; n, where all coe¢ cients, except the long-run coe¢ cients
it
1
+ vit ,
and
2
(20)
are country-
speci…c.
Table 3 presents alternative estimates of the long-run coe¢ cients. The upper panel presents
…ndings for estimators without bias correction and standard con…dence intervals. The middle
and lower panels present jackknife and bootstrap bias-corrected estimates with con…dence intervals
based on bootstrapped critical values. Results di¤er widely across di¤erent approaches to estimation
and inference. Depending on which bias correction approach is conducted, the PB estimates of the
long-run coe¢ cient on real income (
on the in‡ation variable (
2)
1)
is estimated to be 0.921 or 0.926, and the long-run coe¢ cient
is estimated to be -0.120 or -0.125. The null hypothesis that the
d is unity cannot be rejected at the 5 percent nominal level, nor is the hypothesis
coe¢ cient on yit
that the long run coe¢ cient on in‡ation is zero. From an economic perspective, unit long-run
real income elasticity and no long-run e¤ects of in‡ation 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 di¤erent conclusion would be reached according to
PMG estimates - namely both the unit coe¢ cients on the real income variable and zero coe¢ cient
on in‡ation 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 in‡ation long run
coe¢ cients. FMOLS estimates of
1
are larger than the other estimates, but the unit coe¢ cient
on the income variable still cannot be rejected. The FMOLS estimates of
2
The choice of estimation method clearly matters in this empirical illustration.
15
are also quite large.
Table 3: Estimated consumption function coe¢ cients for OECD countries
1:
PB
PMG
PDOLS
FMOLS
PB
PMG
PDOLS
FMOLS
PB
PMG
PDOLS
FMOLS
Income 95% Conf. Int.
2 : In‡ation
Estimator without bias correction
.912
[.845,.980]
-.134
.904
[.889,.919]
-.466
.923
[.798,1.047]
-.187
.951
[.942,.959]
-.336
Jackknife bias-corrected estimators
.926
[.835,1.017]
-.120
.915
[.880,.949]
-.403
.940
[.737,1.143]
-.184
.983
[.912,1.053]
-.397
Bootstrap bias-corrected estimators
.921
[.830,1.012]
-.125
.905
[.875,.936]
-.477
.932
[.746,1.118]
-.183
.985
[.941,1.028]
-.438
95% Conf. Int.
[-.260,-.008]
[-.566,-.365]
[-.407,.033]
[-.408,-.265]
[-.345,.105]
[-616.,-.190]
[-.530,.161]
[-1.370,.576]
[-.314,.065]
[-.657,-.297]
[-.499,.133]
[-1.047,.171]
Notes: This table revisits empirical application in Table 1 of Pesaran, Shin, and Smith (1999), reporting estimates of long-run
income elasticity ( 1 ) and in‡ation e¤ect ( 2 ) coe¢ cients and their 95% con…dence intervals in the ARDL(1,1,1) consumption
functions (20) for OECD countries using the dataset from Pesaran, Shin, and Smith (1999). PB stands for pooled Bewley
estimator developed in this paper. PMG is the Pooled Mean Group estimator proposed by Pesaran, Shin, and Smith (1999).
PDOLS is panel dynamic OLS estimator by Mark and Sul (2003). FMOLS is the group-mean fully modi…ed OLS estimator
by Pedroni (1996, 2001b). 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 Chudik, Pesaran,
and Smith (2023), 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 bene…t 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
…xed and T ! 1; we derive the joint (n; T ) asymptotic distribution of the PB estimator, when
both n and T diverge to in…nity jointly such that n =
T
, for 0 <
< 2: This covers a broader
range of empirical applications where both n and T are large. The small sample and asymptotic
16
results suggest that the PB estimator is a useful addition to estimators for long run e¤ects in single
equation dynamic heterogeneous panels, where the direction of long-run causality is known.
17
References
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18
Appendices
A
Mathematical derivations
This appendix is organized in four sections. Section A.1 introduces some notations and de…nitions. 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 de…nitions
0
Let zit = (yit ; xit ) , and de…ne Ci (L) =
P1
`=0
Ci` L` and Ci (L) =
= I2 ,
Ci0
=
Ci`
(
i
1
=
` 1
,
i
I2 )
i
i
`=0
Ci` L` , where
` = 1; 2; ::::,
!
i
0
P1
1
`
i
Ci (1) = Ci0 + Ci1 + ::::: = lim
`!1
,
=
(A.1)
0
0
1
!
,
and
Ci0
Ci`
= Ci0
= Ci;`
1
0
Ci (1) =
1
!
0
`
i)
(1
+ Ci` =
,
0
!
`
i)
(1
0
, for ` = 1; 2; :::.
Model (1)-(2) can be equivalently written as
i
(L) zit = ci + uit ,
0
for i = 1; 2; :::; n and t = 1; 2; :::; T , where ci = (ci ; 0) ,
i
(L) = I2
i L,
(A.2)
and I2 is a 2 2 identity matrix. The lag polynomial i (L) can be re-written in the following (error
correcting) form
L) I2 ,
(A.3)
i (L) =
i L + (1
where
i
=
(I2
i)
i
=
0
i
0
!
.
(A.4)
The VAR model (A.5) can be also rewritten in the following form
i
where ci =
i
i
0
= (ci ; 0) , namely ci =
(L) (zit
i i;1
i)
i
19
i;2 .
= uit ,
(A.5)
Using Granger representation theorem, the process zit under the assumptions 1-3 has representation
yit
=
yi + sit +
1
X
`
i)
(1
(uy;i;t
ux;i;t
`
`) ,
(A.6)
`=0
xit
=
xi
+ sit ,
(A.7)
where
t
X
sit =
ux;it ,
(A.8)
`=1
is the stochastic trend.
A.2
Lemmas: Statements and proofs
0
Lemma A.1 Suppose Assumptions 2 and 3 hold, and consider x
~i = (~
xi;1 ; x
~i;2 ; :::; x
~i;T ) , where x
~it = xit xi ,
PT
Pt
1
xit = s=1 ux;it , and xi = T
t=1 xit . Then
n
1
n
X
x
~0 x
~i
i
T2
i=1
where
2
x
= limn!1 n
Pn
1
i=1
2
x
!p ! 2x =
6
, as n; T ! 1,
(A.9)
2
xi .
Proof. Recall that M = IT T 1 T 0T , where IT is T T identity matrix and T is T 1 vector of
~0i x
~i as
ones. Since x
~i = M xi , and M is symmetric and idempotent (M0 M = M = M0 ) we can write x
0
0
2
0
0
0
0
0
~i xi . Denote Si;T = x
~i xi =T . We have
x
~i x
~i = xi M M xi = xi M xi = x
n
1
n
X
x
~0 x
~i
i
T2
i=1
1
=n
n
X
i=1
=
=
=
n
X
E (Si;T ) + n
1
i=1
Consider E (Si;T ) …rst. Noting that x
~it =
Pt
s=1 ux;it , Si;T can be written as
Si;T
1
Si;T = n
Pt
s=1
[Si;T
E (Si;T )] .
xi , xi = T
xi
t
X
s=1
t
X
T
1
PT
s=1
(T
s + 1) ust , and xit =
3
ux;is 5 ,
3
t
X
s+1
ux;is
ux;is 5 .
T
s=1
s=1
Taking expectations, we obtain
E (Si;T ) =
Using
Pt
s=1
T
s+1
T
=
Pt
s=1
(1
E (Si;T ) =
2
xi
T2
s=T + 1=T ) = t
2
xi
T2
T
X
t=1
t
t+
T
X
t=1
"
t
t
X
T
s=1
#
s+1
.
T
(t + 1) t= (2T ) + t=T , we have
(t + 1) t
2T
20
(A.10)
i=1
ux;it
T
1 X
x
~it xit ,
T 2 t=1
2
!2
T
t
1 X4 X
ux;is
T 2 t=1
s=1
2
!2
T
t
1 X4 X
ux;is
T 2 t=1
s=1
n
X
t
T
=
2
xi
T2
T
X
(t + 1) t
t=1
2T
t
.
T
Finally, noting that
PT
t=1
(t + 1) t = (T + 2) (T + 1) T =3, and
2
xi {T
E (Si;T ) =
PT
t=1
t = (T + 1) T =2, we obtain
< K < 1,
(A.11)
for all T > 0, where
(T + 2) (T + 1) T
6T 3
{T =
(T + 1) T
.
2T 3
(A.12)
In addition, {T ! 1=6, as T ! 1, and
n
n
1X
1X
E (Si;T ) = {T
n i=1
n i=1
2
xi
2
x
!
6
,
as n; T ! 1. This establishes the limit of the …rst term on the right side of (A.10). Consider the second
term next. Since E [Si;T E (Si;T )] = 0, and Si;T is independent over i, we have
E
(
n
1
n
X
[Si;T
)2
E (Si;T )]
i=1
=
n
1 X
2
E Si;T
n2 i=1
n
1 X
2
[E (Si;T )] .
n2 i=1
But it follows from (A.11) that there exist …nite positive constant K1 < 1 (which does not depend on n; T )
2
such that [E (Si;T )] < K1 . In addition, due to existence of uniformly bounded fourth moments of ux;it ,
Pn
2
2
< K2 < 1. Hence, E n 1 i=1 [Si;T E (Si;T )] = O n 1 , which
it also can be shown that E Si;T
P
n
implies n 1 i=1 [Si;T E (Si;T )] !p 0, as n; T ! 1. This completes the proof.
Lemma A.2 Suppose Assumptions 1-2 hold. Then there exists …nite positive constant K that does not
depend on i and/or T such that
!%
T
1X
E
ux;it x
~it
< K,
(A.13)
T t=1
and
E
for % = 4, where x
~it = xit
P1
` 1
[vi;t ` + (
i
i)
`=1 (1
i
T
1X
T t=1
yit x
~it
!%
< K,
Pt
xi , xit =
s=1 ux;it , xi = T
) ux;i;t ` ].
1
(A.14)
PT
t=1
xit , and
yit =
PT
Pt
Proof. Consider
~it =T and % = 2 …rst, and note that x
~it =
t=1 uit x
s=1 ux;is
P
T
T 1 s=1 (T s + 1) ux;is . We have
T
1X
ux;it x
~it
T t=1
!2
=
T
T
1 XX
ux;it ux;it0 x
~it x
~it0 ,
T 2 t=1 0
t =1
=
1
T2
T X
T
X
t=1
ux;it ux;it0
t0 =1
= Ai;T;1 + Ai;T;2
t
X
ux;is
s=1
Ai;T;3
21
Ai;T;4 ,
! 0 t0
X
xi @
ux;is
s=1
i ux;it
+ vit
xi , where xi =
1
xi A ,
where
Ai;T;1
=
T
T
1 XX
ux;it ux;it0
T 2 t=1 0
t =1
Ai;T;2
=
t
X
s=1
T
T
1 XX
ux;it ux;it0 x2i ,
T 2 t=1 0
1
! 0 t0
X
ux;is @
ux;is A ,
s=1
t =1
Ai;T;3
=
T
T
t
X
1 XX
0 xi
u
u
ux;is ,
x;it
x;it
T 2 t=1 0
s=1
t =1
0
Ai;T;4
=
T
T
t
X
1 XX
0 xi
u
u
ux;is .
x;it
x;it
T 2 t=1 0
s=1
t =1
Taking expectations and noting that ux;it is independent of ux;it0 for any t 6= t0 , we have
T X
t 1
X
1
E (Ai;T;1 ) = 2
T
2
ix
+
t=1 t0 =1
T
X
E u4x;it +
T X
t 1
X
t=1 t0 =1
t=1
2
ix
!
.
Under Assumption 2, there exists a …nite constant K that does not depend on i and/or t, such that 2ix < K
and E u4x;it < K. Hence jE (Ai;T;1 )j < K. Similarly, we can bound the remaining elements, jE (Ai;T;j )j <
2
PT
~it < K, where the upper bound K does not depend
K, for j = 2; 3; 4. It now follows that E T1 t=1 ux;it x
on i or T . This establishes (A.13) hold for % = 2. Su¢ cient conditions for (A.13) to hold when % = 4 are:
E A2i;T;j < K for j = 1; 2; 3; 4. These conditions follow from uniformly bounded eights moments of ux;it .
This completes the proof of (A.13). Result (A.14) can be established in the same way by using the …rst
di¤erence of representation (A.6).
Lemma A.3 Suppose Assumptions 1-4 hold, and consider siT given by
~i
siT = x
~0i Z
where Pi is given by (10), and x
~i and
n
~ 0i Pi Z
~i
Z
1
~ 0i x
Z
~i ,
(A.15)
~ i are de…ned below (6). Then,
Z
1
n
X
siT
i=1
T2
!p 0, as n; T ! 1.
(A.16)
Proof. Consider si;T =T , which can be written as
siT
T
where
aiT =
a0iT BiT1 aiT ,
~0 x
Z
i ~i
=
T
and
BiT =
Z0i x
~i
,
T
~ 0 Pi Z
~i
Z
i
.
T
22
(A.17)
(A.18)
(A.19)
Using these notations, we have
n
Using a0iT BiT1 aiT
1
min
(BiT ) a0iT aiT , and Cauchy-Schwarz inequality, we obtain
n
E
n
1 X
E a0iT BiT1 aiT .
nT i=1
1 X siT
n i=1 T 2
E
n
1 X
nT i=1
1 X siT
n i=1 T 2
r h
iq
2
E (a0iT aiT )
E
2
min
(BiT ) .
Lemma A.2 implies the fourth
h moments
i of the individual elements of ai;T are uniformly bounded in i and
2
2
T , which is su¢ cient for E (a0iT aiT ) < K. In addition, E min
(BiT ) < K by Assumption 4. Hence,
there exists K < 1, which does not depend on (n; T ) such that
1
E n
n
X
siT
i=1
<
T2
K
,
T
(A.20)
, as n; T ! 1;
(A.21)
and result (A.16) follows.
Lemma A.4 Suppose Assumptions 1-4 hold. Then
n
1
n
X
x
~0 Mi x
~i
i
T2
i=1
where
2
x
= limn!1 n
1
Pn
i=1
2
xi ,
2
x
!p ! 2x =
6
Mi is de…ned in (9) and x
~i is de…ned below (6).
Proof. Noting that x
~i is one of the column vectors of Hi , we have Pi x
~i = x
~i , and x
~0i Mi x
~i can be written
as
x
~0i Mi x
~i = x
~0i x
~i si;T ,
(A.22)
where si;T is given by (A.15). Su¢ cient conditions for result (A.21) are:
n
1
n
X
x
~0 x
~i
i
i=1
and
n
1
T2
!p ! 2x =
n
X
si;T
i=1
T2
2
x
6
, as n; T ! 1;
(A.23)
!p 0, as n; T ! 1.
(A.24)
Condition (A.23) is established by Lemma A.1, and condition (A.24) is established by Lemma A.3.
Lemma A.5 Let Assumptions 1-3 hold. Then
n
where ! 2v = lim n!1 n
1
Pn
1 Xx
~0i v
~i
p
!d N 0; ! 2v , as n; T ! 1,
n i=1 T i
2 2
xi vi =
i=1
Proof. Recall M = IT T
Since M0 M = M0 , we have
1
T
0
T,
6
2
i
(A.25)
, and x
~i and v
~i are de…ned below (6).
where IT is T
T identity matrix and
x
~0i v
~i = x0i M0 M vi = x0i M0 vi = x
~0i vi .
23
T
is T
1 vector of ones.
x
~0 v
Let Ci = xi i vi and Qi;T = Ci 1 Ti ii . We have E (Qi;T ) = 0, and (under independence of vit over t and
independence of vit and ux;it0 for any t; t0 )
E
2
=
where E vit
(A.12). Hence,
2
vi .
"
x
~0i vi
T i
2
#
T
X
1
=
T2
2
i t=1
1
T2
In addition, (A.11) established that
E
"
2
x
~0i vi
T i
#
=
2
,
E x
~2it E vit
PT
t=1
2 2
vi xi
2 {T
i
E x
~2it =
2
xi {T ,
where {T is given by
= Ci2 {T .
It follows that
E Q2i;T = {T ,
where {T ! 1=6 < 1. Finite fourth moments of ux;it and vit imply Q4i;T is uniformly bounded in T , and
therefore Q2i;T is uniformly integrable in T . We can apply Theorem 3 of Phillips and Moon (1999) to obtain
n
n
1 X
1 Xx
~0i vi
p
Ci Qi;T = p
!d N 0; ! 2v , as n; T ! 1,
n i=1
n i=1 T i
where ! 2v = limn!1 Ci2 {T = lim n!1 n
1
Pn
i=1
2 2
xi vi =
2
i
6
.
1
Lemma A.6 Suppose Assumptions 1-4 hold, and consider qiT =
i
p
~ 0 Pi v
Z
~i = T . Then,
i
4
and
E kqiT k2 < K,
(A.26)
K
jE (qiT )j < p .
T
(A.27)
Proof. Denote the individual elements of 2 1 vector qiT as qiT;j , j = 1; 2. Su¢ cient conditions for (A.26)
to hold are
4
E (qiT;j ) < K, for j = 1; 2.
(A.28)
We establish (A.28) for j = 1 …rst. We have
y
~i0 Pi v
~
p i,
i T
qiT;1 =
where
yi can be written as
y
~i =
where ~i;
~
i i; 1
+
i
x
~ i + vi ,
1
~i H
~
~ 0 and H
~i
~ 0H
~i; 1 x
~i; 1 . Note that Pi = H
H
1 = y
i
i i
~0i; 1 Pi = ~0i; 1 , since x
~i and ~i; 1 can be both obtained as
x
~0i and
~ i . Hence
vectors of H
qiT;1 =
~0i; 1 v
~i
p
+
T
x
~0i v
~
v
~0 P v
~
p i + i pi i
i T
i T
where we simpli…ed notations by introducing & a;iT =
24
~0i;
(A.29)
= (~
yi;
~i ; x
~i; 1 ).
1; x
p
x
~0i Pi =
a linear combinations of the column
& a;iT + & b;iT + & c;iT ,
~i =
1v
Hence
T , & b;iT =
1
i
(A.30)
p
x
~0i v
~i = T and & c;iT =
p
~i = T to denote the individual terms in the expression (A.30) for qiT;1 . Su¢ cient conditions for
v
~i0 Pi v
4
E qiT;1
< K are E & 4s;iT < K for s 2 fa; b; cg.
For s = a, we have
1
i
~0i; 1 v
~i
p
=
T
& a;iT =
where
i; 1
=T
1
PT
t=1
T
1 X
p
T i=1
i;t 1 ,
i; 1
(vit
T
1 X
p
T i=1
vi ) =
i;t 1 vit
+
p
T
i; 1 vi ,
and
1
X
=
it
i;t 1
`=0
1
X
=
(1
`
i)
(1
i) (
(uy;i;t
`
ux;i;t
`
) ux;i;t
i
`) ,
`+
`=0
1
X
(1
i)
`
vit .
`=0
Noting that supi j1
i j < 1 under Assumption 1, and fourth moments of ux;i;t and eights moments of vit
are bounded, we obtain
2
!4 3
T
X
1
5 K,
E4 p
i;t 1 vit
T i=1
and
T2 E
which are su¢ cient conditions for E & 4a;iT
For s = b, we have
& b;iT =
x
~0i v
~
p i =
i T
i
1
p
T
T
X
4
4
i; 1 vi
K,
K.
(ux;it
ux;i ) (vit
vi ) =
i
t=1
1
p
T
T
X
ux;it vit
p
t=1
T
ux;i vi .
i
Using Assumption 2, we obtain the following upper bound
E
4
& 4b;iT
i
T
1X
4
E u4x;it E vit
+
T t=1
4
i
T E u4x;i E vi4
4
where i 4 < K, E u4x;it < K, E vit
< K, E u4x;i < K=T 2 , and E vi4 < K=T 2 .
For s = c, we have
1
~ 0v
~i H
~ 0H
~
H
v
~i0 H
i ~i
i i
v
~i0 Pi v
~i
p
p
& c;iT =
=
.
i T
i T
~ = x
Consider H
~i ; x
~i ; ~i;
i
1
and note that
~ i = (~
H
yi;
where
~i ; x
~i; 1 )
1; x
0
B
B =@ 1
1
25
0
1
~i,
=B H
1
1
C
0 A,
0
K,
(A.31)
1
~i H
~ 0H
~
is nonsingular (for any ). Hence Pi = H
i i
& c;iT
0
0
0
0
0
T
1
~ 0 , and we can write & c;iT as
H
i
.
1
0
0
1=2
T
~ 0v
H
i ~i
C
A.
1=2
1
1
~ i AT AT H
~ i 0H
~ i AT
p v
~i0 H
i T
& c;iT =
1
1
T
B
AT = @
Using the inequality x0 A
1
~ 0H
~
~ H
v
~i0 H
i
i
i
p
=
i T
Consider the scaling matrix
We have
~ 0H
~
~0 = H
~ H
H
i
i
i
i
(A.32)
~ i 0v
AT H
~i
0.
2
x
min
0
(A) kxk , we have
& c;iT
i
1
p
1
min
T
~ i 0H
~ i AT
AT H
~ i 0v
AT H
~i
2
2
.
Using Cauchy-Schwarz inequality, we obtain
E
But
4
i
& 4c;iT
1
4T 2
i
r h
E
< K under Assumption 1, and E
E
~ 0v
Let AT H
i ~i
h
4
min
4
min
& 4c;iT
~ 0H
~ AT
AT H
i
i
~ 0H
~ AT
AT H
i
i
K
T2
r
i
ir
8
~ 0v
E AT H
i ~i
2
.
< K under Assumption 4. It follows
~ 0v
E AT H
i ~i
8
2
.
hviT and consider the individual elements of hviT , denoted as hviT;j for j = 1; 2; 3,
hviT
0
1 0
hviT;1
B
C B
~ i 0v
= AT H
~i = @ hviT;2 A = @
hviT;3
Under Assumption 2, it can be shown that
PT
1
x
~it v~it
T
Pt=1
T
p1
u
~ v~
TP t=1 it it
T
p1
~it
t=1 i;t 1 v
T
1
C
A.
E h8viT;j < K, for j = 1; 2; 3,
~ 0v
which is su¢ cient for E AT H
i ~i
8
2
< K. It follows that
E & 4c;iT <
K
.
T2
(A.33)
This completes the proof of (A.26) for j = 1. Consider next (A.26) for j = 2, and note qiT;2 is the same as
& b;iT , namely
x
~0i Pi v
~
x
~0 v
~
p i = pi i = & b;iT .
qiT;2 =
T
i
i T
But E & 4b;iT < K, see (A.31). This completes the proof of (A.26).
Next we establish (A.27). As before we consider the individual elements of 2
26
1 vector qiT , denoted as
qiT;s for s = 1; 2, separately. For s = 1 we have (using the individual terms in expression (A.30))
jE (qiT;1 )j = E
~0i; 1 v
~i
p
+
T
x
~0i v
~
p i+
i T
v
~i0 Pi v
~
p i
i T
!
jE (& a;iT )j + jE (& b;iT )j + jE (& c;iT )j .
(A.34)
For the …rst term in (A.34), we obtain
jE (& a;iT )j =
E
"
~0i; 1 v
~i
p
=
T
T
1 X
p
E
T i=1
But E
i;t 1 vit
= 0 and E
i; 1 vi
T
1 X
p
T i=1
i;t 1 vit
+
p
i;t 1
TE
i; 1
i; 1 vi
(vit
#
vi )
,
.
< K=T under Assumptions 1-2. Hence,
K
p .
T
jE (& a;iT )j
For the second term in (A.34), we obtain
jE (& b;iT )j =
E
i
1
p
T
T
X
ux;it vit
p
T
ux;i vi
i
t=1
!
,
T
p
1 X
jE (ux;it vit )j + K T jE (ux;i vi )j .
Kp
T t=1
But E (ux;it vit ) = 0 and E (ux;i vi ) = 0 under Assumption 2. Hence
jE (& b;iT )j = 0.
Finally, for the last term we note that
jE (& c;iT )j
E j& c;iT j
r
E & 2c;iT ,
and using result (A.33), we obtain
K
jE (& c;iT )j < p .
T
p
It now follows that jE (qiT;1 )j < K= T , as desired.
Consider jE (qiT;s )j for s = 2 next. We have
jE (qiT;2 )j = jE (& b;iT )j = 0.
This completes the proof of result (A.27).
Lemma A.7 Let Assumptions 1-4 hold, and consider BiT de…ned by (A.19). Then we have
T ' kBiT
Bi k !p 0 as T ! 1, for any ' < 1=2,
27
(A.35)
where
2
it
2
iE
Bi = plim BiT =
it
=
P1
`=0
(1
i)
`
(uy;i;t
ux;i;t
`
!
2
i xi
2
xi
2
i xi
T !1
and
2 2
i xi
+
,
(A.36)
` ).
Proof. We have
BiT =
~ 0 Pi Z
~i
Z
1
i
=
T
T
y
~i0 Pi y
~i
x
~0i Pi y
~i
T
1X 2
u
T t=1 x;it
biT;12 =
Under Assumption 2, ux;it
2
xi
IID 0;
biT;11
biT;21
=
x
~it = ux;it
!
biT;12
biT;22
!
.
ux;i , we have
u2x;i .
with …nite fourth order moments, and therefore
T
1X 2
u
T t=1 x;it
T'
x
~0i , and
x
~0i Pi =
Consider the element biT;22 …rst. Since
!
y
~i0 Pi x
~i
x
~0i Pi x
~i
2
xi
!
p
! 0, for any ' < 1=2.
p
In addition, E u2x;i < K=T , which implies T ' u2x;i ! 0, for any ' < 1=2. It follows
T ' biT;22
p
2
xi
! 0, for any ' < 1=2.
(A.37)
Consider the element biT;11 next. We will use similar arguments as in the proof of Lemma A.6. In particular,
y
~i can be written as in (A.29), and, since Pi ~i; 1 = ~i; 1 and Pi x
~i = x
~i , we have
y
~i0 Pi y
~i
=
T
bi;T;11 =
+
aa;iT
bb;iT
+
cc;iT
+2
bb;iT
=
ab;iT
+2
ac;iT
+2
bc;iT ,
(A.38)
where
aa;iT
=
cc;iT
=
2
i
~0i;
~
1 i; 1
T
v
~i0 Pi v
~i
,
T
,
x
~0i x
~i
,
T
2
i
and the cross-product terms are
ab;iT
=
~0i;
i i
We consider these individual terms
it
=
=
1
X
`=0
1
X
1
x
~i
T
;
ac;iT
=
ij
i
~i
1v
T
,
bc;iT
=
i
x
~0i v
~i
.
T
next. Note that
(1
i)
`
(1
i)
`
(uy;i;t
vi;t
`
ux;i;t
`
+(
`=0
where supi j1
~0i;
i
)
`) ,
1
X
(1
`
i)
ux;i;t
`,
`=0
< 1 under Assumption 1, and innovations vit and uxit have …nite fourth order moments
28
h
under Assumption 2. Hence, T ' T
T'
Noting that
bb;iT
=
2
i bi;T;12 ,
h
1
PT
t=1
2
i;t 1
aa;iT
i
2
i; 1
2
iE
2
i;t 1
E
i
!p 0, E
2
i; 1
< K=T , and we obtain
!p 0, for any ' < 1=2.
(A.39)
and using result (A.37), we have
T'
2 2
i xi
bb;iT
!p 0, for any ' < 1=2.
(A.40)
p
Consider cc;iT and note that cc;iT = pTi & c;iT , where & c;iT = i 1 v
~i0 Pi v
~i = T was introduced in (A.30) in
proof of Lemma A.6. But E & 2c;iT < K
T by (A.33), and it follows
T'
cc;iT
!p 0, for any ' < 1=2.
(A.41)
Using similar arguments, we obtain for the cross-product terms,
T'
ab;iT
!p 0, T '
ac;iT
!p 0, and T '
bc;iT
!p 0, for any ' < 1=2, as T ! 1.
(A.42)
Using (A.39)-(A.42) in (A.38), we obtain
T ' bi;T;11
2
iE
2
it
p
2 2
i xi
! 0, for any ' < 1=2.
(A.43)
Using the same arguments for the last term bi;T;12 = bi;T;21 , we obtain
T ' bi;T;12
2
i xi
p
! 0, for any ' < 1=2.
This completes the proof of (A.35).
Lemma A.8 Let Assumptions 1-4 hold, and consider BiT de…ned by (A.19) and Bi = plimT !1 BiT de…ned
by (A.36). Then we have
T ' BiT1
Bi
1
!p 0; as T ! 1, for any ' < 1=2.
(A.44)
Proof. This proof closely follows proof of Lemma A.8 in Chudik and Pesaran (2013). Let p = Bi 1 ,
q = BiT1 Bi 1 , and r = kBiT Bi k. We suppressed subscripts i; T to simplify the notations, but it is
understood that the terms p; q; r depend on (i; T ). Using the triangle inequality and the submultiplicative
property of matrix norm k:k, we have
q
=
BiT1 (Bi
BiT1
BiT1
BiT ) Bi
1
,
1
1
rp,
rp,
Bi
+ Bi
(p + q) rp.
Subtracting rpq from both sides and multiplying by T ' , we have, for any ' < 1=2,
(1
rp) (T ' q)
29
p2 (T ' r) .
(A.45)
p
Note that T ' r ! 0 by Lemma A.7, and jpj < K since Bi is invertible and min (Bi ) is bounded away from
zero (this follows from observing that both 2xi and E 2it as well as 2i in (A.36) are bounded away from
zero). Hence,
p
(1 rp) ! 1,
(A.46)
and
p
p2 (T ' r) ! 0:
(A.47)
p
(A.45)-(A.47) imply T ' q ! 0. This establishes result (A.44).
Lemma A.9 Let Assumptions 1-4 hold, and consider
iT
~i
x
~0 Z
= i
T
where Pi is given by (10), and x
~i and
Proof. Term
iT
T
de…ned by
~ 0 Pi Z
~i
Z
i
T
!
1
~ 0 Pi v
Z
~
i
p i.
i T
(A.48)
~ i are de…ned below (6). Then
Z
n
1 X
p
nT i=1
as n; T ! 1 such that n =
iT
!p 0,
(A.49)
= a0iT BiT1 qiT ,
(A.50)
for some 0 <
iT
< 2.
can be written as
iT
where aiT is given by (A.18), BiT is given by (A.19), and
~ 0 Pi v
Z
~
i
p i.
i T
qiT =
We have
n
1 X
p
nT i=1
i;T
n
1 X 0
=p
aiT BiT1
nT i=1
Bi
1
(A.51)
n
1 X 0
qiT + p
aiT Bi 1 qiT .
nT i=1
(A.52)
Consider the two terms on the right side of (A.52) in turn. Lemma A.2 established fourth moments of aiT are
bounded, which is su¢ cient for kaiT k = Op (1). Result (A.26) of Lemma A.6 established second moments
of individual elements of qiT are bounded, which is su¢ cient for kqiT k = Op (1). In addition, Lemma A.8
established
T ' BiT1 Bi 1 !p 0 as T ! 1, for any ' < 1=2.
Set ' = (
1) =2 < 1=2 . Then we obtain
n
1 X 0
p
aiT BiT1
nT i=1
Bi
1
qiT
=
p
n
n 1 1X 0
p
aiT T ' BiT1 Bi 1 qiT ,
' n
T
T
i=1
!
p
n
n
1X
1
1
'
kaiT k T BiT
Bi
kqiT k !p 0, (A.53)
T =2 n i=1
p
as n; T ! 1 such that < 2, where we used T T ' = T 1=2+' = T
Consider next the second term on the right side of (A.52). Let
30
p
, and T n=2 < K since n =
T .
1
0
iT = E aiT Bi qiT , and consider the
=2
1=2
variance of (nT )
Pn
i=1
a0iT Bi 1 qiT . By independence of a0iT Bi 1 qiT across i,
n
1 X 0
p
aiT Bi 1 qiT
nT i=1
V ar
!
n
1 X
V ar a0iT Bi 1 qiT ,
nT i=1
=
n
1 X
E a0iT Bi 1 qiT
nT i=1
2
.
(A.54)
Denoting individual elements of Bi 1 as bi;sj , individual elements of aiT as aiT;j , and individual elements of
qiT as qiT;s , for s; j = 1; 2, we have
a0iT Bi 1 qiT
=
2 X
2
X
bi;sj aiT;s qiT;j ,
s=1 j=1
=
bi;11 aiT;1 qiT;1 + bi;21 aiT;2 qiT;1 + bi;12 aiT;1 qiT;2 + bi;22 aiT;2 qiT;2 ,
where
aiT;1
(A.55)
T
T
1X
1X
x
~it y~it =
(xit
=
T t=1
T t=1
xi ) yit ,
(A.56)
T
T
1X
1X
x
~it x
~it =
(xit
T t=1
T t=1
xi ) ux;it ,
(A.57)
aiT;2 =
y
~i0 Pi v
~
p i,
i T
qiT;1 =
and
(A.58)
T
~x;it v~it
1 Xu
qiT;2 = p
.
T t=1
i
(A.59)
Note that Bi is invertible and inf i min (Bi ) is bounded away from zero (this follows from observing that
both 2xi and E 2it as well as 2i in (A.36) are bounded away from zero). It follows supi Bi 1 < K, and
2
therefore bi;sj
< K. Using this result and Cauchy-Schwarz inequality for the individual summands on
the right side of (A.55), we obtain
E
2
a0iT Bi 1 qiT
K
2 X
2
X
s=1 j=1
r
E
a4iT;s
r
4
E qiT;j
< K,
(A.60)
4
where E a4iT;s < K by Lemma A.2, and E qiT;j
< K by result (A.26) of Lemma A.6. Using (A.60) in
(A.54), it follows that
!
n
1 X 0
K
V ar p
aiT Bi 1 qiT < ,
T
nT i=1
and therefore
n
1 X 0
p
aiT Bi 1 qiT
nT i=1
We establish an upper bound for j
j
iT j
iT
!q:m: 0 as n; T ! 1.
next. We have (using (A.55) and noting that bi;sj < K)
iT j < K
2 X
2
X
s=1 j=1
31
jE (aiT;s qiT;j )j .
(A.61)
It follows that if we can show that
K
jE (aiT;s qiT;j )j < p ,
T
(A.62)
holds for all s; j = 1; 2, then
j
K
<p ,
T
iT j
(A.63)
hold. We establish (A.62) for s = j = 2, …rst, which is the most convenient case to consider. We have
T
1X
(xit
T t=1
E (aiT;2 qiT;2 ) = E
T
1 X ux;it vit
p
T t=1
i
xi ) ux;it
!
= 0,
(A.64)
since vit is independently distributed of ux;it0 for any t; t0 . Consider next s = 1, j = 2. We have
E (aiT;1 qiT;2 ) = E
T
1X
(xit
T t=1
T
1 X ux;it vit
p
T t=1
i
xi ) yit
!
,
(A.65)
where (…rst-di¤erencing (A.6) and substituting (3))
yit
=
i ux;it
+ vit
1
X
i
` 1
i)
(1
[vi;t
`
+(
) ux;i;t
i
`] ,
`=1
=
u;it
+
v;it ,
in which
u;it
=
i ux;it
i
(A.66)
1
X
` 1
i)
(1
(
) ux;i;t
i
`,
(A.67)
`=1
and
v;it
= vit
i
1
X
` 1
i)
(1
vi;t
`.
(A.68)
`=1
Hence, E (aiT;1 qiT;2 ) can be written as
E (aiT;1 qiT;2 )
= E
T
1X
(xit
T t=1
+E
xi )
T
1X
(xit
T t=1
u;it
xi )
T
1 X ux;it vit
p
T t=1
i
v;it
!
T
1 X ux;it vit
p
T t=1
i
!
.
The …rst term is equal to 0, since vit is independently distributed of ux;it0 for any t; t0 . Consider the second
term. Noting that E [(xit xi ) ux;is ] < K and i 1 < K for any i; t; s, we obtain
E
T
1X
(xit
T t=1
xi )
v;it
T
1 X ux;it vit
p
T t=1
i
!
But
E
v;is vit =
1
=
8
>
<
>
:
T X
T
X
1
i
E [(xit
T
T
K XX
E
T 3=2 t=1 s=1
v;is vit
T 3=2
0, for s < t,
< K, for s = t,
K s t , for s > t,
2
vi
32
xi ) ux;is ] E
t=1 s=1
.
v;is vit
,
where
supi j1
ij
PT
< 1 by Assumption 1. Hence
E
T
1X
(xit
T t=1
xi )
s=1
E
v;is vit
T
1 X ux;it vit
p
T t=1
i
v;it
!
< K for any t = 1; 2; :::T , and
K
p ,
T
(A.69)
as desired. This establish (A.62) hold for s = 1, j = 2.
Consider next (A.62) for s 2 f1; 2g and j = 1. Using expression (A.30), we can write aiT;s qiT;1 , for
s = 1; 2, as
aiT;s qiT;1 = aiT;s & a;iT + aiT;s & b;iT + aiT;s & c;iT ,
(A.70)
p
p
p
0
where as in the proof of Lemma A.6 & a;iT = ~i; 1 v
~i = T , & b;iT = i 1 x
~0i v
~i = T and & c;iT = i 1 v
~i0 Pi v
~i = T .
Using similar arguments as in establishing (A.69), we obtain
K
jE (aiT;s & a;iT )j < p , for s = 1; 2.
T
1
Noting next that & b;iT =
i
qi;T;2 , it directly follows from results (A.64) and (A.69) that
K
jE (aiT;s & b;iT )j < p , for s = 1; 2.
T
Consider the last term, ai;T;s & c;iT , for s = 1; 2. Using Cauchy-Schwarz inequality we have
jE (aiT;s & c;iT )j
r
E
r
a2iT;s
E & 2c;iT , for s = 1; 2.
But E a2iT;s < K, for s = 1; 2 by Lemma A.2, and E & 2c;iT < K=T is implied by (A.33). Hence
jE (aiT;s & c;iT )j
K
p , for s = 1; 2.
T
This completes the proof of (A.62) for all s; j = 1; 2, and therefore (A.63) holds. Using (A.63), we
n
1 X
p
nT i=1
as n; T ! 1 such that
as n; T ! 1 such that
p
p
iT
n
1 X
p
j
nT i=1
p
n
1 X K
n
p
p
j
<
=
K
! 0,
iT
T
nT i=1 T
(A.71)
n=T ! 0. Results (A.61) and (A.71) imply
n
1 X 0
p
aiT Bi 1 qiT !p 0,
nT i=1
(A.72)
n=T ! 0. Finally, using (A.53) and (A.72) in (A.52), we obtain (A.61), as desired.
Lemma A.10 Let Assumptions 1-4 hold. Then
n
1 X
p
E
n i=1
as n; T ! 1 such that n =
de…ned below (6).
T
for some 0 <
x
~0i Mi v
~i
T
i
! 0;
< 2, where Mi is de…ned in (9), and x
~i and v
~i are
33
Proof. We have
n
n
n
1 Xx
~0i Mi v
~i
1 Xx
~0i v
~i
1 X
p
=p
+p
n i=1
n i=1 i T
nT i=1
iT
iT ,
(A.73)
where iT is de…ned by (A.48). For the …rst term, E (~
x0i v
~i ) = 0. For the second term, Lemma A.9 established
Pn
1
p
!
0,
as
n;
T
!
1
such
that
n
=
T , for some 0 < < 2. Hence it follows that
p
i=1 iT
nT
0
P
x
~
M
v
~
n
i i
i
p1
! 0, as n,T ! 1 such that n =
T , for some 0 < < 2.
i=1 E
n
iT
Lemma A.11 Suppose conditions of Theorem 1 hold. Then
n
where ! 2v = lim n!1 n
1
Pn
~i
1 X x
~0i Mi v
p
n i=1
iT
i=1
2 2
xi vi =
6
2
i
~i
x
~0i Mi v
iT
E
!d N 0; ! 2v ,
(A.74)
, Mi is de…ned by (9), and x
~i and v
~i are de…ned below (6).
Proof. It is convenient to use (A.73) in (A.74) to obtain
n
1 X x
~0i Mi v
~i
p
n i=1
iT
E
n
n
1 Xx
~0i v
~i
1 X
=p
+p
[
n i=1 i T
nT i=1
x
~0i Mi v
~i
iT
E(
iT
iT )] ,
Pn
1
where E (~
x0i v
~i ) = 0. It follows from Lemmas A.9 and A.10 that pnT
E ( iT )] !p 0, as n; T ! 1
i=1 [ iT
such that n =
T , for
some
0
<
<
2.
Hence,
under
the
conditions
of
Theorem
1, the asymptotic
i
Pn h x~0i Mi v~i
Pn x~0i v~i
x
~0i Mi v
~i
1
1
distribution of pn i=1
E
is given by the …rst term, pn i=1 i T , alone. Lemma A.5
iT
iT
establishes the asymptotic normality of this term.
n
~0i Mi v
~i
1 Xx
p
!d N 0; ! 2v ,
T
n i=1
i
(A.75)
as n; T ! 1. This completes the proof.
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 …ndings 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 coe¢ cient , as originally proposed by Pesaran, Shin, and Smith
(1999), is computed by solving the following equations iteratively:
^
P MG
=
n ^2
X
i 0
2 xi Hx;i xi
^
i=1 i
^ = ^0 Hx;i ^
i
i
i
!
1
1 n
X
i=1
^2
i
^ 2i
x0i Hx;i
yi
^0i Hx;i yi , i = 1; 2; :::; n,
34
^ yi;
i
1
,
(B.1)
(B.2)
and
^ 2i = T
1
yi
^^
i i
0
Hx;i
yi
^ ^ , i = 1; 2; :::; n,
i i
(B.3)
0
0
where ^i = yi; 1 xi ^ P M G , xi = (xi;1 ; xi;2 ; :::; xi;T ) , yi = yi yi; 1 , yi = (yi;1 ; yi;2 ; :::; yi;T ) , yi; 1 =
1
0
0
(yi;0 ; yi;1 ; :::; yi;T 1 ) , Hx;i = IT
xi ( x0i xi )
x0i , xi = xi xi; 1 , and xi; 1 = (xi;0 ; xi;1 ; :::; xi;T 1 ) .
To solve (B.1)-(B.3) iteratively, we set ^ P M G;(0) to the pooled Engle-Granger estimator, and given the
initial estimate ^ P M G;(0) , we compute ^i;(0) = yi; 1 xi ^ P M G;(0) , ^ i;(0) and ^ 2i;(0) , for i = 1; 2; :::; n using
(B.2)-(B.3). Next we compute ^ P M G;(1) using (B.1) and given values ^ i;(0) and ^ 2i;(0) . Then we iterate for a given value of ^ P M G;(`) we compute ^i;(`) , ^ i;(`) and ^ 2i;(`) ; and for given values of ^ i;(`) and ^ 2i;(`) we
compute ^ P M G;(`+1) . If convergence is not achieved, we increase ` by one and repeat. We de…ne convergence
^
< 10 4 .
by ^
P M G;(`+1)
P M G;(`)
Inference is conducted using equation (17) of Pesaran, Shin, and Smith (1999). In particular,
p
T n ^P MG
where
n
P MG
=
1X
n i=1
i;0
2 rxi ;xi
i;0
!
N (0;
0
P M G) ,
1
, and rxi ;xi = plimT !1 T
2 0
xi Hx;i xi
.
Standard error of ^ P M G , denoted as se ^ P M G , is estimated as
se
b ^P MG = T
where
^ P MG =
B.2
n
1 X ^ i;0
r^x ;x
n i=1 ^ 2i;0 i i
!
1
n
1=2 ^
P M G,
1
and r^xi ;xi = T
2 0
xi Hx;i xi .
(B.4)
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 ^ e for
e = P M G; P DOLS, and F M OLS, respectively. Bootstrap bias corrected version of these estimators is
given by
~ =^
^be ,
(B.5)
e
e
for e = P M G; P DOLS, and F M OLS, where ^be an estimate of the bias obtained by the following sieve
wild bootstrap algorithm, which resembles the algorithm in Subsection 2.2.1. For e= P M G; P DOLS, and
F M OLS :
1. Compute ^ e . Given ^ e , estimate the remaining unknown coe¢ cients of (1)-(2) by least squares, and
compute residuals u
^ey;it ; u
^ex;it .
e;(r)
(r)
e;(r)
(r)
(r)
2. For each r = 1; 2; :::; R, generate new draws for u
^y;it = at u
^ey;it , and u
^x;it = at u
^ex;it , where at
randomly drawn from Rademacher distribution (Liu, 1988) namely
(r)
at
=
(
1,
1,
with probability 1/2
.
with probability 1/2
35
are
Given the estimated parameters of (1)-(2) from Step 1, and initial values yi1 ; xi1 generate simulated
(r)
e;(r)
e;(r)
data yit ; xit for t = 2; 3; :::; T and i = 1; 2; :::; n. Using the generated data compute ^ e .
i
h
PR
(r)
^ .
3. Compute ^be = R 1 r=1 ^ e
e
The possibility of iterating the algorithm above by using the bias-corrected estimate ~ e in Step 1 is not
considered in this paper.
~
= ~ e k^e se
b ^e = ~e
We conduct inference by using the 1
con…dence interval C1
e
oR
n
(r)
(r)
(r)
(r)
, in which te = ~ e =se
=
T 1 n 1=2 k^e ^ e , where k^e is the 1
percent quantile of te
b ^e
r=1
~ (r) = ^ (r) ^be is the bias-corrected PMG estimate of
T n
in the r-th draw of the
e
e
(r)
bootstrap data in the algorithm above, and ^ e is estimated standard error using the bootstrap data.
1
B.2.1
1=2 ~ (r) ^ (r)
e = e ,
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,
~
jk;e
^
= ~ jk;e ( ) = ^ e
e;a
+ ^ e;b
2
^
e
!
,
for e = P M G; P DOLS, and F M OLS, where ^ e is the full sample estimator, ^ e;a and ^ e;b are the …rst and
the second half sub-sample estimators, and = 1=3 is the same weighting parameter as in Section 2.2.
~
~
^
We conduct inference by using the 1
con…dence interval C1
b ^ e = ~ jk;e
jk;e = jk;e kjk;e se
oR
n
(r)
(r)
(r)
(r)
, in which tjk;e = ~ jk;e =se
b ^e
percent quantile of tjk;e
=
k^jk;e T 1 n 1=2 ^ e , where k^jk is the 1
r=1
(r)
(r)
(r)
T 1 n 1=2 ~ jk;e = ^ e , ~ jk;e is the jackknife bias-corrected estimate of using the r-th draw of the bootstrap
(r)
data generated using the same algorithm as in Subsection B.2, and ^ e is estimated standard error using
the bootstrap data.
B.3
Monte Carlo results for experiments with cross-sectionally dependent errors
36
Table B1: MC …ndings for the estimation of long-run coe¢ cient in experiments with cross-sectionally
dependent errors.
Estimators without bias correction and inference conducted using standard critical values.
nnT
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
Bias (
20
30
PB
-3.83 -1.93
-3.57 -1.98
-3.94 -2.04
-3.88 -1.99
PMG
-1.96 -0.90
-1.60 -1.05
-1.79 -1.02
-1.76 -0.89
PDOLS
-6.20 -4.10
-5.76 -4.15
-6.29 -4.24
-6.12 -4.19
FMOLS
-10.89 -7.25
-10.32 -7.26
-10.94 -7.60
-10.72 -7.42
100)
40
= 1 and
i
RMSE ( 100)
20 30 40 50
Size (5% level)
20
30
40
50
Power (5% level)
20
30
40
50
-1.01
-1.17
-1.10
-1.10
-0.66
-0.69
-0.74
-0.76
7.58
6.67
6.58
6.35
4.92
4.43
4.20
4.01
3.57
3.32
3.08
2.89
2.78
2.46
2.37
2.23
28.00
28.95
33.60
38.30
21.95
23.55
27.25
29.55
17.55
22.10
23.75
25.30
16.80
18.80
22.90
23.30
37.30
45.10
46.60
50.70
64.50
75.25
82.00
85.60
86.90
92.70
96.70
97.75
-0.40
-0.56
-0.50
-0.46
-0.26
-0.31
-0.32
-0.32
9.05
7.76
7.07
6.70
5.49
4.83
4.51
4.21
3.86
3.54
3.27
3.00
3.00
2.58
2.44
2.28
45.75
46.35
49.40
54.30
34.00
37.25
40.80
42.95
26.30
32.75
35.00
34.20
23.90
27.35
29.90
31.55
62.45
70.50
73.05
76.40
78.50
85.25
90.25
93.90
92.45 98.15
95.60 99.65
97.65 99.85
98.90 100.00
-3.08
-3.08
-3.12
-3.09
-2.40
-2.37
-2.44
-2.47
9.50
8.65
8.67
8.46
6.31
5.89
5.82
5.62
4.67
4.39
4.27
4.16
3.67
3.41
3.35
3.27
31.40
34.85
42.95
47.90
27.30
33.45
41.85
45.45
25.70
32.15
38.60
45.15
26.30
33.35
38.80
45.05
22.35
27.95
26.95
31.50
39.95
48.60
52.35
59.55
65.20
77.05
82.15
88.25
85.80
93.55
97.05
98.65
-5.42
-5.29
-5.45
-5.38
-4.20
-4.06
-4.29
-4.30
13.27
12.28
12.57
12.26
9.09
8.70
8.76
8.52
6.79
6.45
6.41
6.28
5.37
5.04
5.10
5.00
85.10
88.60
91.90
93.50
76.30
82.70
88.20
89.15
68.35
76.00
82.85
85.40
63.05
71.35
77.50
82.35
54.40
57.00
60.10
63.55
54.40
60.70
61.75
68.15
73.30
82.20
84.95
89.30
88.35
95.25
96.65
98.80
Notes: DGP is given by
with
50
yit = ci
i
(yi;t
1
xi;t
1)
+ uy;it and
96.85
99.00
99.65
99.95
xit = ux;it , for i = 1; 2; :::; n; T = 1; 2; :::; T ,
IIDU [0:2; 0:3]. Errors uy;it , ux;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 Pesaran, Shin, and Smith (1999). PDOLS is panel dynamic OLS estimator by Mark and Sul (2003). FMOLS is
the group-mean fully modi…ed OLS estimator by Pedroni (1996, 2001b). The size and power …ndings are computed
using 5% nominal level and the reported power is the rejection frequency for testing the hypothesis
37
= 0:9.
Table B2: MC …ndings for the estimation of long-run coe¢ cient in experiments with cross-sectionally
dependent errors.
Bias corrected estimators and inference conducted using bootstrapped critical values.
nnT
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
20
30
40
50
Bias ( 100)
RMSE ( 100)
20
30
40
50
20 30 40 50
Jackknife bias-corrected estimators
PB
-1.57 -0.54 -0.11 -0.01
7.83 5.24 3.92 3.04
-1.29 -0.67 -0.34 -0.05
6.76 4.63 3.59 2.72
-1.67 -0.62 -0.20 -0.06
6.48 4.35 3.36 2.58
-1.54 -0.58 -0.21 -0.09
6.16 4.12 3.15 2.43
PMG
-0.54 -0.15 0.02 0.06
10.66 6.26 4.45 3.44
-0.25 -0.39 -0.19 0.03
9.10 5.50 4.02 2.94
-0.57 -0.27 -0.08 0.03
8.25 5.13 3.78 2.78
-0.41 -0.10 -0.06 0.03
7.81 4.81 3.44 2.63
PDOLS
-4.61 -2.84 -2.06 -1.54
9.89 6.23 4.51 3.51
-4.09 -2.96 -2.15 -1.55
8.86 5.72 4.21 3.20
-4.66 -2.94 -2.13 -1.57
8.62 5.53 4.00 3.06
-4.41 -2.91 -2.11 -1.63
8.30 5.29 3.84 2.95
FMOLS
-8.65 -5.12 -3.61 -2.69
12.18 7.99 5.79 4.56
-8.06 -5.27 -3.57 -2.55
11.00 7.43 5.42 4.14
-8.65 -5.53 -3.70 -2.74
11.08 7.35 5.25 4.08
-8.44 -5.36 -3.63 -2.78
10.72 7.07 5.07 3.94
Bootstrap bias-corrected estimators
PB
-1.26 -0.43 -0.05 -0.01
7.34 4.80 3.56 2.78
-1.04 -0.51 -0.23 -0.05
6.29 4.21 3.24 2.42
-1.31 -0.51 -0.12 -0.08
5.98 3.93 3.01 2.32
-1.26 -0.45 -0.12 -0.09
5.78 3.72 2.78 2.17
PMG
-1.23 -0.44 -0.09 -0.03
9.24 5.54 3.91 3.03
-0.92 -0.60 -0.27 -0.10
7.89 4.84 3.57 2.59
-1.10 -0.56 -0.19 -0.11
7.14 4.51 3.30 2.45
-1.06 -0.45 -0.16 -0.10
6.77 4.23 3.02 2.29
PDOLS
-2.34 -1.13 -0.66 -0.37
8.73 5.46 3.92 3.05
-2.03 -1.29 -0.74 -0.41
7.73 4.85 3.53 2.68
-2.46 -1.28 -0.71 -0.41
7.40 4.68 3.31 2.54
-2.32 -1.22 -0.68 -0.44
7.28 4.41 3.15 2.39
FMOLS
-4.39 -2.01 -1.24 -0.77
10.37 6.73 4.85 3.83
-4.00 -2.25 -1.25 -0.71
9.02 6.02 4.42 3.43
-4.46 -2.44 -1.27 -0.84
8.82 5.67 4.11 3.21
-4.30 -2.27 -1.22 -0.86
8.48 5.41 3.92 3.02
Size (5% level)
20
30
40 50
Power (5% level)
20
30
40
50
7.65
6.30
6.75
6.80
6.55
6.60
6.15
6.20
5.55
6.40
6.20
5.85
4.65
5.25
5.40
4.65
20.55
25.80
25.55
27.65
41.35
51.35
54.30
59.40
65.35
73.90
80.25
84.95
83.50
91.45
94.45
96.70
16.05
14.80
14.10
14.85
10.80
10.40
10.50
11.05
8.55
9.05
9.15
8.65
7.90
6.90
7.80
7.65
29.10
36.30
35.10
39.55
45.05
53.85
59.50
64.65
69.15
76.40
81.25
85.90
84.75
93.40
95.55
97.15
8.75
8.70
8.80
9.85
7.50
7.30
6.55
6.50
6.45
6.55
6.45
6.25
4.95
5.40
4.55
5.10
10.05
12.20
10.00
11.95
21.55
24.05
23.90
24.40
39.00
46.10
47.70
52.10
63.80
73.10
77.80
80.80
11.25
9.65
11.50
12.45
5.65
6.85
6.50
6.70
4.20
5.20
4.55
4.55
3.85
3.35
3.85
3.40
2.00
2.10
1.45
1.75
10.10
10.20
10.15
10.20
7.95
7.35
7.50
7.85
6.45
7.20
6.75
7.05
5.85
5.40
5.80
6.00
29.35
37.20
39.00
42.90
52.15
64.30
69.80
75.25
76.10
85.00
89.80
93.50
90.15
95.90
98.15
99.20
16.00 10.50
14.70 9.95
14.80 10.00
14.90 10.45
7.60
8.45
8.90
8.25
6.70
5.85
7.00
6.00
31.35
38.20
39.35
44.90
51.80
60.85
67.05
71.40
76.10
83.80
88.90
91.75
90.80
97.10
98.10
99.15
12.05 10.05
12.15 8.60
12.65 10.20
15.00 9.60
7.75
8.45
9.05
8.40
6.90
7.65
7.75
7.40
22.40
26.95
26.15
30.15
45.25
51.50
55.55
59.85
71.30
79.30
84.05
88.40
87.75
94.15
96.90
98.55
10.90 7.80
12.00 9.60
12.10 9.60
13.90 10.10
6.85
7.70
9.05
7.80
17.80
20.95
20.10
22.85
31.75
37.25
38.50
44.85
50.55
61.40
67.65
72.20
70.75
82.55
85.95
90.40
18.45
17.25
19.35
21.20
4.55 9.50 25.50
3.95 12.65 31.60
3.35 12.25 32.05
4.40 13.70 34.30
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.
38