- Email: [email protected]
On the bias of the OLS estimator in a nonstationary dynamic panel data model
~-'~:z. '"
1/5
STATISTICS& PROBABILrrY LETTERS ELSEVIER
Statistics & Probability Letters 38 (1998) 145-150
On the bias of the OLS estimator in a nonstationary dynamic panel data model I Jean-Yves
Pitarakis *
Department of Eeonomics, University of Readiny, PO Box 218, Whitekniyhts, RG6 6AA, Readin 9, UK Received 1 April 1997; received in revised form l July 1997
Abstract
This paper derives the joint moment generating function of two quadratic forms appearing in the OLS estimator from a dynamic panel data model. The result is then used to investigate the effect of the cross-section dimension on the asymptotic bias. (~) 1998 Elsevier Science B.V. All rights reserved
Keywords: Panel data; Moment generating function; Unit roots
1. Introduction
Since the early 1980s a considerable body of research in the time series econometrics literature has focused on the properties of estimators and test statistics arising from models containing variables having unit roots in their autoregressive representation. Most of the theoretical work in this area has concentrated mainly on aggregate economic time series. More recently, the growing availability of highly disaggregate data has also directed research interests towards panel data models combining both individual and time series information and where each individual series may also be characterized by a nonstationary autoregressive process (Levin and Lin, 1993; Quah, 1994; lm et al., 1995; Tzavalis and Harris, 1996, among others). The purpose of this paper is to contribute to this literature by introducing a set of exact results for the bias of the OLS estimator derived from a dynamic panel data model with a unit root and investigate its behaviour when viewed as a function of the cross-section dimension. In the process we also obtain the joint moment generating function of the quadratic forms appearing in the numerator and denominator of the OLS estimator for both finite and asymptotic sample sizes, thus extending earlier results by White (1958) who considered the simple AR(1 ) model. The plan of the paper is as follows. Section 2 derives the relevant moment generating functions, Section 3 focuses on the behaviour of the exact bias of the OLS estimator and Section 4 concludes.
* Tel.: +44-118-9875123 ext.7470; fax: +44-118-9750236; e-mail: [email protected]. 0167-7152/98/$19.00 (~) 1998 Elsevier Science 13.V. All rights reserved PH S01 6 7 - 7 1 5 2 ( 9 7 ) 0 0 1 6 5 - X
146
Pitarakis / Statistics & Probability Letters 38 (1998) 145 150
J.-I'i
2. T h e m o m e n t generating function
Let Yit satisfy the following dynamic panel data model Yit = cPYit-I 4- git,
i = 1. . . . . N
and
t = 1. . . . . T,
(1)
where eit is a zero mean Gaussian process Vi, t with E(e 2) = ~ 2 Vi, t and where we let q5 E N. We will further assume that yio 0 Vi and with no loss of generality put a,:2 = 1. The OLS estimator o f the parameter ~b in (1) is given by =
~)~_ EiN1 ET= l YitYi,--I
(2)
zL, yL,
which for our purpose will be more conveniently written as the following ratio of quadratic forms in normal variables following the same notation as in White (1958)
Z'(IN ® A)z $ - 4) -- Z'(IN @ B ) z '
(3)
where z ' = (Y~lt.... , / X t ) is a 1 × N T vector with Y~t A and B are the following T x T matrices ~i
0
...
0
0
0
1
-q~
~1
...
0
0
0
o
I
-,~
...
o
0
0
--4'
1
0
-~
~
=
(Yil . . . . . Y i T ) , IN an N-dimensional identity matrix and
A= .
.
.
.
.
.
.
.
B=
0 0
.
.
l
.
0
I
½
0
...... 0)
1 .
.
......
.
.
... ...
0
.
°
i 0
0 0
The joint moment generating function o f the pair o f quadratic forms in the numerator and denominator of (3) is given by ~9T(U, V) = E[exp{z'(IN ® A)zu + Z'(IN ® B)zv}]
J . - Y . P i t a r a k i s I S t a t i s t i c s & P r o b a b i l i t y L e t t e r s 38 ( 1 9 9 8 )
145
150
147
where E[] denotes the expectation operator. Writing ~-}/N=l~t=l(Yit r -- dpyit-1 )2 = Z'(IN ® P)z with P given by the following T x T matrix (1 + ~2)
_~
0
......
0
-~ 0
(1 + ~2) -~
-~ (1 + ~2)
...... ......
o 0
o
o
o
...
(1+~2)
_~
o
0
0
..
-~
l
/
p=
the gaussianity assumption implies I t (IN G P)z}(dz) exp{(z'(Ix Q A)zu + Zt(IN Q B)zv - gz
tilT(U,V) = (2/~) -Nr/2
= (2/~) -NT/2 / exp{-IztO(u, v)z}(dz) = [D(u,v)1-1/2, where (dz) denotes the exterior product and D(u,v)=(Iu ® P ) - 2(IN @ A ) u - 2(IN ®B)v. The T x T matrix D(u, v) can be rewritten as D(u,v)= (IN ® M(u,v)), where M ( u , v ) = P - 2 A u - 2By. More specifically the T × T matrix M(u, v) is given by t 121 122
M(u,v)
122
0
...
...
O~
fll
122
...
,.,
0 0
0
122
121
...
.,.
0
0
0
...
12]
0
0
0
...
#2
=
122 1;
with 121(u,v) = 1 + ¢~2-F2~bu- 2v,
(4)
122(R,/)) = -(~b -{--bt). Theorem 2.1. The joint moment generatin# function of the pair [Z'(IN ® A)z,z'(IN ® B)z] is given by
!-{, ff/T(U,U):
~ 2 1
-}- 22 -- AI
2;] -N'2 j
where )q and ).2 are the distinct roots of 22 - 121(u,v)2 + p22(u,v) = O. Proof. See Appendix.
Note that the expression of the moment generating function in Theorem 2.1 is valid for ~bE ~ and VT and generalizes White's (1958) result to the dynamic panel data model in (1). It is also interesting to observe the strong similarities between the expression of the joint m.g.f, in Theorem 2.1 with the m.g.f, one would
148
J.-E
Pitarakis / Statistics & Probability Letters 38 (1998) 145-150
obtain by considering a seasonal AR(l) process (see Pitarakis, 1997). In what follows our analysis will concentrate on the case where all individual series are characterized by a unit root process by letting q5= 1. Under this assumption a proper normalization of ~ - 1 is obtained by dividing the numerator of (3) by T and its denominator by T 2 resulting in the Op(1) statistic T ( ~ - 1) (see Dickey and Fuller, 1979). The following corollary establishes the limiting moment generating function under this nonstationary specification and assuming that N is a fixed constant. Corollary 2.1. The moment 9eneratin# function of the limit&# distribution of T( q5 - 1) is" 9iven by A
~k(u,v) = e - ~
I
cosh -x/-L--~
u
sinh x/-S~
l
Proof. See Appendix. The expression of ~b(u, v) obtained in Corollary 2.1 represents the moment generating function of the asymptotic distribution of the normalized OLS coefficient estimated from (1). At this stage it is worth comparing its shape with that obtained from a simple AR(1) with a unit root. From Dickey and Fuller (1979) and Phillips (1986) the following distributional results hold for the numerator and denominator of the OLS estimator obtained form (1) under N = 1
~'~Tt=lXt-I'Y't d~ ~001 W ( r ) d W ( r ) , T
2 t=l TT2
~
fo' W2(r)dr,
where W(r) is a standard Wiener process and d denotes convergence in distribution. Note that the above convergence results also hold under a set of weaker assumptions on the error process {et} driving the system in (1). From the expression of our estimator in (2) and the use of the continuous mapping theorem it is then straightforward to obtain the following expression for the asymptotic distribution of T(~b - 1)
2,5, £ Wi(r)dW,(r) 2iL, Y0W, r)dr Thus the expression of ~,(u, v) in Corollary 2.1 can also be viewed as the joint moment generating function of the pair
In the recent unit root literature (N = 1) most of the relevant distributions have been tabulated via the use of Monte Carlo simulations since the above stochastic integrals involving Wiener processes are not directly usable for analytic derivations. Our results in Theorem 2.1 and Corollary 2.1 however open the way to further analytical results in the context of panel data models since closed form expressions for the exact density and distribution functions of T(~ - 1) can be obtained using the well known inversion theorem for characteristic functions along the lines of Abadir (1993).
J.-E Pitarakis / Statistics & Probability Letters 38 (1998) 145 150
149
3. Exact moments and biases
If we let br = E[q~- 1] denote the bias of q~ under the unit root model, Tbr will converge to the mean of the asymptotic distribution of T(q~- 1) as T goes to infinity. Since Corollary 2.1 provides us with the expression of the moment generating function of the limiting distribution of T(q~ - 1) we can obtain its moments using existing results on expectations of ratios of quadratic forms (see Sawa, 1978). Letting Q1 and Q2 denote two quadratic forms and O(u, v) their joint moment generating function, the mth order moment (assuming it exists) of their ratio is given by
L\ Qz J
[
8u"
],=0 dr.
(5)
Letting i~i denote the ith order moment of the limiting distribution of T(q~ - 1 ) and using (5) and the m.g.f. in Corollary 2.1 we obtain the following exact expressions for the first and second moments
#l _
X
t
2
( c o s h t) N/2
N2 f0"~
It2 = 8
N ~o~
dt + ~
sinht
(cosh t) (N+2)/2 dt,
N2f0
t3 (cosh t) N/2 d t - ~ -
t2 sinht
( c o s h t) (N+2)/2
dt +
N(N + 2) f 0 ~ t(sinh t) 2 8 (cosh t)( N+4)/2 dt.
Note that the above integrals are convergent and can be solved analytically by writing the hyperbolic functions in terms of exponentials and integrating termwise. The Table 1 provides the exact mean and variance for specific magnitudes of the cross-section dimension. The magnitudes corresponding to N = 1 represent the mean and variance of the well known Dickey and Fuller distribution since for N = 1 the model collapses to a simple AR(1) with a unit root. As it can clearly be seen from the first column of Table 1, the distribution shifts rightwards as N the number of individuals increases, with its mean^shifting from -1.78143 under N = 1 to -0.11807 under N = 12. This also implies that the asymptotic bias in 4 decreases in absolute value as we move from N = 1 to N = 4 , 8 or 12 since we can write it as br = (1/T)E[T(~- 1)]. It is important to stress that our results are operating under the assumption that N is a fixed constant and that only the time dimension T is allowed to grow towards infinity. From the expressions of ¢~t and p2 above it is clear that if we let N tend to infinity the normalized OLS coefficient (i.e. T(q~- 1)) will collapse to a point mass at zero (also suggesting that the bias in q~ is a decreasing function of N). This can obviously be remedied by using an alternative normalization which would take into account both the cross-section and time dimensions.
Table 1 Exact mean and variance of the asymptotic distribution of T(~ - I) for various values of the cross-section dimension N N
Mean
Variance
1 4 8 12
-1.7814 -0.3863 -0.1817 -0.1181
10.1121 0.9760 0.3615 0.2285
150
J . - E Pitarakis / Statistics & Probability Letters 38 (1998) 145-150
4. Conclusion Using a simple nonstationary dynamic panel data model this paper has shown analytically that the asymptotic bias of the OLS estimator is a decreasing function of the cross-section dimension. In the process we have also derived a convenient expression for the joint moment generating function of the quadratic forms appearing in the numerator and denominator of the OLS estimator, valid for all sample sizes. Our results highlight the potential benefits of pooling time series and cross-section data when the underlying individual processes are characterized as having a unit root in their AR representation. Although beyond the scope of this note, our expression of the limiting moment generating function in Corollary 2.1 can also be used to derive distributional results under the assumption that both T and N grow towards infinity.
Appendix Proof of Theorem 2.1. From the properties of the Kronecker product we have ID(u, v)[-1/2 1Ig ®M(u, v)l-1/2 = IM(u,v)l -N/2. Following White (1958) the determinant of M can be evaluated by noting that it can be written in the form of a second order homogeneous difference equation. Indeed, expanding the determinant of M by the elements of its first column and letting M(T) denote the determinant of M we have M ( T ) = lqM ( T - 1 ) - I ~ M ( T - 2) and the result in Theorem 2.1 follows immediately. =
Proof of Corollary 2.1. Letting ~b= 1 in (4) and solving 22 - #12 +/~22 = 0 for 2 leads to ),j= 1 + u - v ± iv/~(1 + u - v/2) 1/2 for j---1,2. Replacing u by u/T and v by v/T 2 and expanding we obtain 2j = 1 + [(u + i v / ~ ) / T ] + O(T-2). It then follows that (1 - 2j)/(21 - 22)= - (u ± i v / ~ ) / ( 2 i v / ~ ) where it is understood that j = 1 corresponds to the plus sign and j = 2 to the minus sign. Finally using (1 + (Z/T))r--~ ez we have 2f ~ exp(u± i x / ~ ) and the result follows from l i m r ~ ~r(u/T,v/T 2) where ~T(.,.) is the moment generating function in Theorem 2.1.
References Abadir, K.M., 1993. The limiting distribution of the autocorrelation coefficient under a unit root. Ann. Statist. 21, 1058-1070. Dickey, D., Fuller, W., 1979. Distribution of the estimators for autoregressive time series with a unit root. J. Am. Statist. Assoc. 74, 427-432. Gradhshteyn, I.S., Rhyzhik, I.M., 1980. Tables of Integrals, Series and Products, Academic Press, New-York. Ira, K., Pesaran, H., Shin, Y., 1995. Testing for unit roots in heterogeneous panels. DAE Working Paper No. 9526, Cambridge University. Levin, A., Lin, C.F., 1993. Unit root tests in panel data: asymptotic and finite sample properties. Unpublished Manuscript, University of California, San Diego. Phillips, P.C.B., 1986. Understanding spurious regressions in econometrics. J. Econom. 33, 311 340. Pitarakis, J.Y., 1997. Moment generating functions and further exact results for seasonal autoregressions. Discussion Paper, Reading University. Quah, D., 1994. Exploiting cross section variation for unit root inference in dynamic data. Econom. Lett. 44, 9 19. Sawa, T., 1978. The exact moments of the least squares estimator for the autoregressive model. ,I. Econom. 8, 159-172. Tzavalis, E., Harris, R., 1996. Testing for unit roots in dynamic panel data models. Discussion Paper, Exeter University. White, J., 1958. The limiting distribution of the serial correlation coefficient in the explosive case. Ann. Math. Statist. 29, 1188-1197.
1/5
STATISTICS& PROBABILrrY LETTERS ELSEVIER
Statistics & Probability Letters 38 (1998) 145-150
On the bias of the OLS estimator in a nonstationary dynamic panel data model I Jean-Yves
Pitarakis *
Department of Eeonomics, University of Readiny, PO Box 218, Whitekniyhts, RG6 6AA, Readin 9, UK Received 1 April 1997; received in revised form l July 1997
Abstract
This paper derives the joint moment generating function of two quadratic forms appearing in the OLS estimator from a dynamic panel data model. The result is then used to investigate the effect of the cross-section dimension on the asymptotic bias. (~) 1998 Elsevier Science B.V. All rights reserved
Keywords: Panel data; Moment generating function; Unit roots
1. Introduction
Since the early 1980s a considerable body of research in the time series econometrics literature has focused on the properties of estimators and test statistics arising from models containing variables having unit roots in their autoregressive representation. Most of the theoretical work in this area has concentrated mainly on aggregate economic time series. More recently, the growing availability of highly disaggregate data has also directed research interests towards panel data models combining both individual and time series information and where each individual series may also be characterized by a nonstationary autoregressive process (Levin and Lin, 1993; Quah, 1994; lm et al., 1995; Tzavalis and Harris, 1996, among others). The purpose of this paper is to contribute to this literature by introducing a set of exact results for the bias of the OLS estimator derived from a dynamic panel data model with a unit root and investigate its behaviour when viewed as a function of the cross-section dimension. In the process we also obtain the joint moment generating function of the quadratic forms appearing in the numerator and denominator of the OLS estimator for both finite and asymptotic sample sizes, thus extending earlier results by White (1958) who considered the simple AR(1 ) model. The plan of the paper is as follows. Section 2 derives the relevant moment generating functions, Section 3 focuses on the behaviour of the exact bias of the OLS estimator and Section 4 concludes.
* Tel.: +44-118-9875123 ext.7470; fax: +44-118-9750236; e-mail: [email protected]. 0167-7152/98/$19.00 (~) 1998 Elsevier Science 13.V. All rights reserved PH S01 6 7 - 7 1 5 2 ( 9 7 ) 0 0 1 6 5 - X
146
Pitarakis / Statistics & Probability Letters 38 (1998) 145 150
J.-I'i
2. T h e m o m e n t generating function
Let Yit satisfy the following dynamic panel data model Yit = cPYit-I 4- git,
i = 1. . . . . N
and
t = 1. . . . . T,
(1)
where eit is a zero mean Gaussian process Vi, t with E(e 2) = ~ 2 Vi, t and where we let q5 E N. We will further assume that yio 0 Vi and with no loss of generality put a,:2 = 1. The OLS estimator o f the parameter ~b in (1) is given by =
~)~_ EiN1 ET= l YitYi,--I
(2)
zL, yL,
which for our purpose will be more conveniently written as the following ratio of quadratic forms in normal variables following the same notation as in White (1958)
Z'(IN ® A)z $ - 4) -- Z'(IN @ B ) z '
(3)
where z ' = (Y~lt.... , / X t ) is a 1 × N T vector with Y~t A and B are the following T x T matrices ~i
0
...
0
0
0
1
-q~
~1
...
0
0
0
o
I
-,~
...
o
0
0
--4'
1
0
-~
~
=
(Yil . . . . . Y i T ) , IN an N-dimensional identity matrix and
A= .
.
.
.
.
.
.
.
B=
0 0
.
.
l
.
0
I
½
0
...... 0)
1 .
.
......
.
.
... ...
0
.
°
i 0
0 0
The joint moment generating function o f the pair o f quadratic forms in the numerator and denominator of (3) is given by ~9T(U, V) = E[exp{z'(IN ® A)zu + Z'(IN ® B)zv}]
J . - Y . P i t a r a k i s I S t a t i s t i c s & P r o b a b i l i t y L e t t e r s 38 ( 1 9 9 8 )
145
150
147
where E[] denotes the expectation operator. Writing ~-}/N=l~t=l(Yit r -- dpyit-1 )2 = Z'(IN ® P)z with P given by the following T x T matrix (1 + ~2)
_~
0
......
0
-~ 0
(1 + ~2) -~
-~ (1 + ~2)
...... ......
o 0
o
o
o
...
(1+~2)
_~
o
0
0
..
-~
l
/
p=
the gaussianity assumption implies I t (IN G P)z}(dz) exp{(z'(Ix Q A)zu + Zt(IN Q B)zv - gz
tilT(U,V) = (2/~) -Nr/2
= (2/~) -NT/2 / exp{-IztO(u, v)z}(dz) = [D(u,v)1-1/2, where (dz) denotes the exterior product and D(u,v)=(Iu ® P ) - 2(IN @ A ) u - 2(IN ®B)v. The T x T matrix D(u, v) can be rewritten as D(u,v)= (IN ® M(u,v)), where M ( u , v ) = P - 2 A u - 2By. More specifically the T × T matrix M(u, v) is given by t 121 122
M(u,v)
122
0
...
...
O~
fll
122
...
,.,
0 0
0
122
121
...
.,.
0
0
0
...
12]
0
0
0
...
#2
=
122 1;
with 121(u,v) = 1 + ¢~2-F2~bu- 2v,
(4)
122(R,/)) = -(~b -{--bt). Theorem 2.1. The joint moment generatin# function of the pair [Z'(IN ® A)z,z'(IN ® B)z] is given by
!-{, ff/T(U,U):
~ 2 1
-}- 22 -- AI
2;] -N'2 j
where )q and ).2 are the distinct roots of 22 - 121(u,v)2 + p22(u,v) = O. Proof. See Appendix.
Note that the expression of the moment generating function in Theorem 2.1 is valid for ~bE ~ and VT and generalizes White's (1958) result to the dynamic panel data model in (1). It is also interesting to observe the strong similarities between the expression of the joint m.g.f, in Theorem 2.1 with the m.g.f, one would
148
J.-E
Pitarakis / Statistics & Probability Letters 38 (1998) 145-150
obtain by considering a seasonal AR(l) process (see Pitarakis, 1997). In what follows our analysis will concentrate on the case where all individual series are characterized by a unit root process by letting q5= 1. Under this assumption a proper normalization of ~ - 1 is obtained by dividing the numerator of (3) by T and its denominator by T 2 resulting in the Op(1) statistic T ( ~ - 1) (see Dickey and Fuller, 1979). The following corollary establishes the limiting moment generating function under this nonstationary specification and assuming that N is a fixed constant. Corollary 2.1. The moment 9eneratin# function of the limit&# distribution of T( q5 - 1) is" 9iven by A
~k(u,v) = e - ~
I
cosh -x/-L--~
u
sinh x/-S~
l
Proof. See Appendix. The expression of ~b(u, v) obtained in Corollary 2.1 represents the moment generating function of the asymptotic distribution of the normalized OLS coefficient estimated from (1). At this stage it is worth comparing its shape with that obtained from a simple AR(1) with a unit root. From Dickey and Fuller (1979) and Phillips (1986) the following distributional results hold for the numerator and denominator of the OLS estimator obtained form (1) under N = 1
~'~Tt=lXt-I'Y't d~ ~001 W ( r ) d W ( r ) , T
2 t=l TT2
~
fo' W2(r)dr,
where W(r) is a standard Wiener process and d denotes convergence in distribution. Note that the above convergence results also hold under a set of weaker assumptions on the error process {et} driving the system in (1). From the expression of our estimator in (2) and the use of the continuous mapping theorem it is then straightforward to obtain the following expression for the asymptotic distribution of T(~b - 1)
2,5, £ Wi(r)dW,(r) 2iL, Y0W, r)dr Thus the expression of ~,(u, v) in Corollary 2.1 can also be viewed as the joint moment generating function of the pair
In the recent unit root literature (N = 1) most of the relevant distributions have been tabulated via the use of Monte Carlo simulations since the above stochastic integrals involving Wiener processes are not directly usable for analytic derivations. Our results in Theorem 2.1 and Corollary 2.1 however open the way to further analytical results in the context of panel data models since closed form expressions for the exact density and distribution functions of T(~ - 1) can be obtained using the well known inversion theorem for characteristic functions along the lines of Abadir (1993).
J.-E Pitarakis / Statistics & Probability Letters 38 (1998) 145 150
149
3. Exact moments and biases
If we let br = E[q~- 1] denote the bias of q~ under the unit root model, Tbr will converge to the mean of the asymptotic distribution of T(q~- 1) as T goes to infinity. Since Corollary 2.1 provides us with the expression of the moment generating function of the limiting distribution of T(q~ - 1) we can obtain its moments using existing results on expectations of ratios of quadratic forms (see Sawa, 1978). Letting Q1 and Q2 denote two quadratic forms and O(u, v) their joint moment generating function, the mth order moment (assuming it exists) of their ratio is given by
L\ Qz J
[
8u"
],=0 dr.
(5)
Letting i~i denote the ith order moment of the limiting distribution of T(q~ - 1 ) and using (5) and the m.g.f. in Corollary 2.1 we obtain the following exact expressions for the first and second moments
#l _
X
t
2
( c o s h t) N/2
N2 f0"~
It2 = 8
N ~o~
dt + ~
sinht
(cosh t) (N+2)/2 dt,
N2f0
t3 (cosh t) N/2 d t - ~ -
t2 sinht
( c o s h t) (N+2)/2
dt +
N(N + 2) f 0 ~ t(sinh t) 2 8 (cosh t)( N+4)/2 dt.
Note that the above integrals are convergent and can be solved analytically by writing the hyperbolic functions in terms of exponentials and integrating termwise. The Table 1 provides the exact mean and variance for specific magnitudes of the cross-section dimension. The magnitudes corresponding to N = 1 represent the mean and variance of the well known Dickey and Fuller distribution since for N = 1 the model collapses to a simple AR(1) with a unit root. As it can clearly be seen from the first column of Table 1, the distribution shifts rightwards as N the number of individuals increases, with its mean^shifting from -1.78143 under N = 1 to -0.11807 under N = 12. This also implies that the asymptotic bias in 4 decreases in absolute value as we move from N = 1 to N = 4 , 8 or 12 since we can write it as br = (1/T)E[T(~- 1)]. It is important to stress that our results are operating under the assumption that N is a fixed constant and that only the time dimension T is allowed to grow towards infinity. From the expressions of ¢~t and p2 above it is clear that if we let N tend to infinity the normalized OLS coefficient (i.e. T(q~- 1)) will collapse to a point mass at zero (also suggesting that the bias in q~ is a decreasing function of N). This can obviously be remedied by using an alternative normalization which would take into account both the cross-section and time dimensions.
Table 1 Exact mean and variance of the asymptotic distribution of T(~ - I) for various values of the cross-section dimension N N
Mean
Variance
1 4 8 12
-1.7814 -0.3863 -0.1817 -0.1181
10.1121 0.9760 0.3615 0.2285
150
J . - E Pitarakis / Statistics & Probability Letters 38 (1998) 145-150
4. Conclusion Using a simple nonstationary dynamic panel data model this paper has shown analytically that the asymptotic bias of the OLS estimator is a decreasing function of the cross-section dimension. In the process we have also derived a convenient expression for the joint moment generating function of the quadratic forms appearing in the numerator and denominator of the OLS estimator, valid for all sample sizes. Our results highlight the potential benefits of pooling time series and cross-section data when the underlying individual processes are characterized as having a unit root in their AR representation. Although beyond the scope of this note, our expression of the limiting moment generating function in Corollary 2.1 can also be used to derive distributional results under the assumption that both T and N grow towards infinity.
Appendix Proof of Theorem 2.1. From the properties of the Kronecker product we have ID(u, v)[-1/2 1Ig ®M(u, v)l-1/2 = IM(u,v)l -N/2. Following White (1958) the determinant of M can be evaluated by noting that it can be written in the form of a second order homogeneous difference equation. Indeed, expanding the determinant of M by the elements of its first column and letting M(T) denote the determinant of M we have M ( T ) = lqM ( T - 1 ) - I ~ M ( T - 2) and the result in Theorem 2.1 follows immediately. =
Proof of Corollary 2.1. Letting ~b= 1 in (4) and solving 22 - #12 +/~22 = 0 for 2 leads to ),j= 1 + u - v ± iv/~(1 + u - v/2) 1/2 for j---1,2. Replacing u by u/T and v by v/T 2 and expanding we obtain 2j = 1 + [(u + i v / ~ ) / T ] + O(T-2). It then follows that (1 - 2j)/(21 - 22)= - (u ± i v / ~ ) / ( 2 i v / ~ ) where it is understood that j = 1 corresponds to the plus sign and j = 2 to the minus sign. Finally using (1 + (Z/T))r--~ ez we have 2f ~ exp(u± i x / ~ ) and the result follows from l i m r ~ ~r(u/T,v/T 2) where ~T(.,.) is the moment generating function in Theorem 2.1.
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