- Email: [email protected]
On a pooled estimator and its finite-sample moments
Journal
of Econometrics
48 (1991) 195-214.
North-Holland
On a pooled estimator and its finite-sample moments* William
M. Mikhail
The American Unkersity in Cairo, Cairo, Egypt
G.A. Ghazal Cairo University, Cairo, Egypt Received
May 1988, final version
received
February
1990
The present paper deals with the situation in which we have a single cross-section equation which could be estimated by ordinary least squares and a single time-series equation which is one of a complete system of simultaneous equations to which the application of OLS would lead to inconsistent estimates. A consistent pooled estimator is derived and the exact finite-sample moments of this estimator are evaluated on the assumption that lagged endogenous variables are not present. The exact moment function is expanded in terms of the inverse of the noncentrality parameter. The exact finite-sample moments of the two-stage least-squares estimator as derived by Richardson (1968) and Sawa (1972) could then be obtained as special cases, and the expansion sheds more light on further studies and comparisons.
1. Introduction As explained by Maddala (19711, there are basically two types of problems related to the pooling of cross-section and time-series data. First, there is the case of a time-series of cross-sections, for which many alternative approaches have been suggested by various authors. Second, there is the case of a single cross-section and a single time series, which has been analyzed by Tobin (1950), Chetty (1968), and Maddala (1971), but has probably not received as much attention as the first case. The present paper deals with this latter case, and does not address itself in any way to the case of temporal cross-sections. Tobin’s analysis of the problem has come to be referred to as the traditional method, and many authors followed in applying it to similar models. It consisted of estimating the parameters of the cross-section equation and introducing the estimates of the common parameters into the *The
authors
are grateful
0304.4076/91/$03.500
to the anonymous
1991-Elsevier
Science
referees
Publishers
for their most helpful
B.V. (North-Holland)
comments.
196
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
time-series equation as known with certainty. This of course led to conditional estimates of the parameters of the time-series equation and, consequently, to underestimates of the standard errors of the coefficients. However, even if the estimates of the common parameters obtained from the cross-section equation are substituted into the time-series equation taking the uncertainty into consideration, the resulting estimates, as shown by Durbin (1953) and Maddala (19711, will still be inefficient. Efficient estimates could only be obtained by estimating the parameters of the two equations simultaneously, as there is also some information about the common parameters in the time-series sample which should not be ignored. Chetty (1968) used the Bayesian approach to estimate all the parameters simultaneously. The method utilizes the information in both equations to derive the marginal posterior distribution of the common parameters and it is therefore likely, as argued by Chetty, to have smaller variances than those obtained from the traditional method. These parameters, as well as the other parameters in the two equations, will have, on the average, sharper posterior distributions when computed from the pooled samples than when computed from either sample. Chetty also claims that the method suggested by Durbin (1953) and simultaneous approaches which follow the lines of Theil and Goldberger (1961) and Theil (1963) will only have an asymptotic justification, while for his Bayesian method exact finite-sample results are obtained. Maddala (1971) obtained maximum-likelihood estimates equivalent to iterating the least-squares procedure in the simultaneous presentation of the two equations. Applying this to the data of the Tobin model, he found that the results agreed with those obtained by Chetty using Bayesian analysis. Some questions have been raised about the validity of the procedure of pooling time-series and cross-section data in general in providing meaningful estimates of the common parameters. It has been argued that the parameters which we try to combine are different and answer different questions and should therefore have different estimators, prior distributions, and other properties. In general, estimates depend on the other variables included in each of the two equations and the precise ceteris paribus assumptions made. However, it appears from careful examination of the literature that despite the skepticism of some researchers about the usefulness of pooling time-series and cross-section data, there has been no substantive evidence against such procedure. Even the empirical results obtained by Maddala on the Tobin model, which were mistaken by certain researchers to constitute an evidence against pooling, were indeed not conclusive. Maddala has found, by applying the likelihood-ratio test and comparing the relative likelihoods or the posterior distributions of the common parameters, that for the Tobin model data should not have been pooled. Maddala called the evidence unambiguous, with hardly any overlap between the two likelihood functions. This strong verdict against pooling, in a situation when there was substantial improvement in the accuracy of the combined estimator, was totally discouraging to
197
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
many people to consider pooling in similar situations. However, when Izan (1980) carried out the exercise all over again by allowing the disturbances in the time-series equation to follow a first-order autoregressive process, and also allowing for outliners found in Tobin’s cross-section data, she found that the likelihood-ratio test, after allowing for autocorrelation, resulted in acagainst cepting the pooling hypothesis. ’ This suggests that the decision pooling might have been a result of misspecification of the model in the form of the process generating the disturbance term. All authors however agree that before pooling is done, diagnostic checks should be performed to ensure the consistency of the two sets of data. This may be carried out by classical tests of significance or by a study of posterior distributions and posterior odds. The present paper deals with the situation in which the cross-section equation could be estimated by ordinary least squares, but the time-series equation is one of a complete system of simultaneous equations to which the application of OLS would lead to inconsistent estimates. The situation considered is one which often crops up in demand analysis where we may have a single cross-section equation based on family-budget data with observations corresponding to individual households at one point in time, and a single time-series equation with data corresponding to national-accounts statistics, and the two equations would be sharing one or more common parameters.2 A simple classical example is the estimation of the marginal propensity to consume food, where the time-series equation is an aggregate function for food consumption in which we regress food consumption on aggregate disposable income and the price of food relative to the cost of living, while the cross-section equation uses family-budget data to regress food consumption of households on disposable income and the size of the household. In the first equation, aggregate food consumption and disposable income should be treated in macroeconomic analysis as jointly dependent. The purpose for pooling in the previous example is to get more efficient estimates, in view of the fact that cross-sectional observations at one point in time might be affected by prior observations, and that time-series data, besides often suffering from collinearity among the explanatory variables, do not contain information on individual socio-demographic variables relevant to the study of variations in the dependent variable. Thus the pooling might ‘It should be noted that in Tobin’s aggregate time-series model.
example
there
is a lagged
endogenous
variable
in the
‘No aggregation of cross-sections over time is considered here, since observations correspond to individuals (or households) at one point in time. If aggregation is done over groups of individuals and if the coefficients differ among them, then this may lead to aggregation bias. In this case, we follow Zellner (1969) and assume that the group coefficient vector can be written as b, = b + E,, where E, is a random component with zero expectation, and the estimator may still be unbiased. Thus, if we assume that the differences are random and that it is the mean or expected value of the coefficients that is to be estimated, then no aggregation bias will result. However, aggregate error terms in this case will usually be heteroskedastic.
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W.M. Mikhail and G.A. Ghazal, On a pooled estimator
be thought of as extraneous information in the from of cross-sectional data for the time-series model or in the form of time-series data for the cross-sectional model. However, the pooling of time-series and cross-sectional data might be desirable in other situations for the purpose of achieving consistency. An example is the joined input-output and econometric approach to economy-wide modeling and projections of final demand. The estimation of the elements of the private consumption column in the I/O table is done using functions estimated from cross-sectional family-budget data, and its projected control total is also obtained from a function using the same cross-sectional data set. If, within the same modeling framework, nationalaccounts statistics are used to estimate econometric relationships and project economic aggregates, it might be desirable not to have different estimates of the elasticities within the same exercise, irrespective of the considerations of accuracy. To achieve consistency and obtain a projected column of final demands which agrees with both sets of data, the equality of the coefficients might be imposed in the context of pooling the time-series and cross-sectional data without, of course, neglecting the simultaneous nature of the econometric system. The method used in this paper constitutes an improvement over the other methods available for this type of problem in that it allows an analysis of more general time-series models. The consistent pooled estimator considered below takes into account the inherent features of the individual components of a simultaneous-equation model, and its superiority over the likelihood or iterated least-squares approach is assured by pre-weighting the equations as given below (rather than assigning a kind of averaging process) in a multidimensional form of distance minimizing. Other methods which neglect the simultaneous nature of the RHS jointly-dependent variables would in effect be treating them as exogenous, thus committing a specification error which may not only lead to widely different estimates of the common parameters but also result in erroneous values for the tests designed to answer the question of whether or not to pool. Such erroneously calculated values for these tests, based on inaccurate specifications, may indeed lead to wrong conclusions, as illustrated by Izan, and to the rejection of the hypothesis of pooling when in fact it should be accepted. The paper considers the case in which the time-series equation contains only two endogenous variables, and it is assumed that no lagged values of theseavariables would be present in the equation. This is done in order to facilitate the derivation of the exact finite-sample results given in the following sections and to enable comparisons with previous results on finite-sample properties, especially those obtained by Richardson (1968) and Sawa (1972). In the case of many endogenous variables and lagged values of them being present in the time-series equation, it would not be possible to derive exact finite-sample moments for the combined estimator, but an approximation to the distribution might be possible to obtain.
WM. Mikhail and G.A. Ghazal, On a pooled estimator
199
The organization of the paper is as follows. The model, assumptions, and the combined estimator are presented in section 2. In section 3, we evaluate the exact first and second moments of the estimator. In order to derive these moments, we used a method described as Lemma 1, which is an extension of a basic lemma given by Sawa. The main results of section 3 will be proved in section 4. To facilitate the investigation of the properties of the combined estimator, an asymptotic expansion of the bias and the mean square error based on the exact results will be given in section 5. Finally, in section 6, we derive the conditions under which the moments exist.
2. The model and the combined
estimator
Let us write a time-series simultaneous system of G linear equations relating G endogenous and K predetermined variables YB+ZT=U.
(2.1)
The first equation being estimated is assumed to contain nous variables and to be overidentified by zero restrictions coefficients. It may be written as YI = PY2 +
stochastic as
Z,Y, + UI?
only two endogeon the structural
(2.2)
where Y is a T x G matrix of T observations on G endogenous variables, y , and y, are the first and second columns of Y, respectively, Z is a TX K matrix of observations on K nonstochastic predetermined variables (not containing lagged endogenous variables) partitioned as Z = (Z,, Z,>, where Z, is a T x k, matrix of included predetermined variables and Z, is a TX k, matrix of excluded ones (K = k, + k,), and U is a T X G matrix of disturbances with first column u,. Also, let the cross-section equation be Y3=Px,+x2b+u
2,
(2.3)
where y, is an N x 1 vector of observations on the dependent variable, x, is an N ~‘1 vector of observations on the first nonstochastic independent variable which has the same coefficient as y, in the time-series equation, X, is an N x m matrix of the other nonstochastic independent variables with X=(x ,, X2>, and u2 is an N X 1 vector of disturbances. B, r, @, yl, and 6, are parameter matrices and vectors of conforming dimensions. The reduced form of the system (2.1) could be written as Y=-ZTB-‘+UB-‘=Zn+V
(2.4)
WM. Mikhail and G.A. Ghazal, On a pooled estimator
200
and
where (a,,~~) and (u,, u,) are the first two columns of II and I/, respectively, and ri (i = 1,2) are partitioned as r/ = CT/~, 7~:~) conformably with Z = (Z,, Z,). The traditional econometric assumptions include the specification that the disturbance vectors u1 and u2 have the following properties: 6)
Ecu,) = 0,
Ecu,)
= 0.
(ii) E(u,u;) = U~Z,, ECU&)
= a;~,,,,
E(u,u;)
= 0.
It should be noted here that the assumption of independence of U, in the time-series equation and u2 in the cross-section equation is not unrealistic. In the context of the Tobin model, referred to as an example in the introduction above, there is no reason to believe that the omitted variables and the errors of measurement in the time-series equation with its observations covering aggregate variables over a number of years to be correlated with omitted variables or errors of measurement in the cross-section equation where the data correspond to individuals or households at a single given .point in time.” Indeed, the independence of U, and u2 is one main reason why eqs. (2.2) and (2.3) are not analyzed together using a 3SLS or ML approach, besides the considerations of finite-sample properties mentioned elsewhere in this paper. It is also assumed that each row of CC,, L’~) is independently and identically distributed two-dimensional normal variate with zero mean and covariance matrix4 ?P = &, and that the N-dimensional random vector u2 is distributed as normal: u2
-
N(0,
a&).
To estimate /3 from the time-series equation only, we need to apply a consistent single-equation technique. The two-stage least squares seemed to be a good choice, first because of its simplicity and second because of the fact that the pooled estimator based on it possesses finite-sample moments, as “This was also assumed
by Chetty
(1968) and Maddala
(1971) in estimating
the Tobin model.
4There exists a set of transformations [see, for instance, Sawa (1972)] on the variables and parameters of the model which transforms it into one in which q = II. We shall here assume that the transformations have already been made.
201
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
will be seen below, that could be compared with the results Richardson (1968), Sawa (1972), and others. We then have
obtained
by
p^ =Y;pY,/Y;pY,,
(2.6)
P=Z(Z’Z)_‘Z’-Z,(Z{ZJ’z;
(2.7)
where
matrix of rank k,. from the cross-section
is an idempotent The estimator
equation
is
@=XiLYJX;LXr,
(2.8)
L=I,-X,(X;X*))‘x;
(2.9)
where
is an idempotent matrix of rank (N - m) = n, say. A consistent pooled estimator based on applying the 2SLS to the first equation would be obtained by applying the generalized least squares to the simultaneous presentation of the two equations after premultiplying the first of the matrix of all the predetermined equation by Z’, the transpose variables in the time-series model:
Writing
the disturbance Z'U] Cl=
[ its covariance
Applying
112
term in this equation
as
1 )
matrix will be
Aitken’s P*=(Y;PYl
GLS, we get the combined +XiLY,)/(Y;PY*+XiLX,).
estimator
p* of j3 as (2.10)
We see from (2.10) that the numerator of the combined estimator is the sum of the numerators in (2.6) and (2.81, and the denominator is the sum of the two denominators.
202
3. Evaluation
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
of the exact moments
Analytical complexities have been a serious impediment in deriving finite-sample properties of estimators, and consequently the results obtained so far are rather limited and cover only certain special cases. Exact densities and moments as well as approximate densities and moments of approximate distributions have been derived for 2SLS, k-class estimators, instrumentalvariable estimators, and maximum-likelihood estimators, as applied to simple econometric models. Many excellent summaries of such results are available [see, for instance, Judge et al. (198511, but we refer here briefly to those studies which the present paper either draws upon or compares with. Basmann’s pioneering work of 1961 was mainly concerned with the problem of nonexistence of finite moments. He derived the exact density when two exogenous variables are excluded, and also obtained the density for the case in which three exogenous variables were excluded but when one of the parameters of the density was identically equal to zero. He later showed that, for the first case, the estimator converges stochastically as one of the parameters of the density increases indefinitely, and derived expressions for the mean and the bias of the estimator. Kabe (1963) derived the same two density functions utilizing the noncentral Wishart distribution. Richardson (1968) extended the results of Basmann and Kabe to the more general case in which any number of exogenous variables may be excluded from the equation. Richardson adopted Kabe’s approach and derived the exact distribution of the estimator by making a change of variable in the noncentral Wishart distribution. He found that the number of excluded exogenous variables appears as a parameter in the density function but that the form of the density is not affected by the addition of more excluded exogenous variables. Sawa (1972) used a moment-generating function technique to evaluate the exact finite-sample moments of the k-class estimators for 0 I k I 1, and proved that for k > 1 the estimator does not possess even the first-order moment. He also expanded the exact moment functions in terms of the inverse of the noncentrality parameter. Both Richardson and Sawa have independently confirmed Basmann’s conjecture that the moments of 2SLS exist only if the order of the moment is less than the degree of identification of the equation. In evaluating the exact moments of the combined estimator, we need the following two lemmas. The first is an extension of a result given by Sawa (19721, and the second provides a result concerning the confluent hypergeometric functions. Lemma 1. Let X, be an almost everywhere positive random variable, C be a positive constant, and X2 and X3 be arbitrary random variables. If there also
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
exists a joint moment-generating function of X,, X,, @(t,,t2,t3)
203
and X,,
= E[~~I(~I+C)+‘ZXZ+~,~~],
(3.1)
for t, < E and 1t,j, (t, 1 < E, where E is some positice constant, then the rth order moment of (X, + X,>/( X, + C) is gicen by
x,+x,
Ep [ x, + c
1
r =_
1 r(r) i s= 0
1
(f;;t’&“-“’ s ;
dt,.
tz=o,r,=o.
(3.2) The lemma and results. Before presenting
the proof Lemma
7, = 7T/*z;(z
are straightforward
generalizations
2, we give the following
- z,(z;z,)~‘z;)z,~**/2,
l-/z =xiLx,.
of Sawa’s
definitions: (3.3) (3.4)
The parameter 7, given in (3.3) corresponds to 6 in Sawa’s results and to ~*/2 in Richardson’s results, where pu2 is the concentration parameter. The value of 7, is unknown. However, the relative bias of /3* could be computed exactly for given values of 7, and k,, from eq. (3.6) below, and the mean square error could be computed for given values of T,, k,, and p from eq. (3.7). Lemma 2 T(cr-r(y
1) ,Fl(
r(a+l)
a-r-l;a+l;T,)
r(cw-r) ,F,(a
+17,r(N+2) =
r(a-rr(a)
1)
-r;
Q + 2; 7,)
,F,(a-r-l;a;~,).
(3.5)
This result could be easily established if we start by applying the definition of the confluent hypergeometric function to the LHS of (3.5). We now present the two theorems on the exact moments.
WM. Mikhail and G.A. Ghazal, On a pooled estimator
204
Theorem 1. The exact first-order defined by (2.10) is given by E(P*)
moment
of the combined
estimator p*
5 [?lr
=Pe-vt
r=O
W2/2-r)
’ [”
r( k,/2
+ 2
W2/2-r-1)
2
IFl(k2/2
+ 1)
- r; b/2
,F,(k,/2-r-
r( k,/2)
+ 1;~~)
l;k&C~,)
1. (3.6)
Theorem 2. The exact second-order defined by (2.10) is given by
E(p**)
ca ( -v2/21rr(r+ 2)
= y
r =
X
moment of the combined estimator /3*
r!
0
W k, - 2)/2 r(k,/2)
r>,F,((k, - 2)/2-r;
k2/2;v1)
r((k2 -2)/2 -r) +%P2(l
+
2772)
+2)/z)
j-((k 2
r( k2/2 - r 1 +2v:B2r((k2 + 4),2) ’ F ’(k,/2 r((k, +772(1
+
- 4)~
-r;
(k, + 4)/z
771)
-r)
172P2) r(
k2/2)
x1F,((k2-4)/2-r;k2/2;r11)
. I
(3.7)
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
205
The exact first- and second-order moments of the combined estimator p *, as given in Theorem 1 and 2, depend on the cross-section data only through the parameter q2 defined by (3.4). Therefore, in case we are not incorporating cross-section data in the estimation procedure the exact first- and second-order moments of the 2SLS estimator can be derived from (3.6) and (3.71, respectively, by putting n2 = 0. Thus, we have the following: $orollaiy I. The exact first- and second-order moments of the 2SLS estimator p of p in (2.2) could be obtained from (3.6) and (3.7) as
E(i)
r( b/2)
=PT,ePVl
r(k,/2
em71 r( k,/2 - 1) = 2 ,F,(kz/2r( k,/2)
E(a’)
+‘I’
These (1972).
,F,(k,/2;K,/2+
+ 1)
results
,W,/2
(3.8)
l;~,),
l;k,/2;n,)
- 1)
Z-( k,/2
+ 1)
,F,(k,/2
- 1; k,/2
+ 1; 7,)
r( k,/2) + 2dP2 ,F,(k2/2;k,/2+2 r(k,/2 + 2)
;771).
agree
(1968)
with
those
given
by Richardson
I
(3.9)
and
Sawa
4. Proof of theorems The T-dimensional random vectors y, (a = 1,2) defined by (2.5) and the N-dimensional random vector y, defined by (2.3) are, by assumption, mutually independent and distributed normally with mean vectors pL, (a = 1,2,3), and covariance matrix I where, using notation as close as possible to Sawa’s,
Pa
=
CL3 =
z,rTT,,
Px,
+
Z2Tn2>
a
=
1,2,
(4.1) (4.2)
+X*62.
The joint moment-generating
function
of (Y;PY,
+x;Lx,),
YipY,, and X;LY,
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
206
is given by
If we restrict
the admissible
domains
of t, and t, to
t;+2t,<1, so that
(4.4)
D = I - 2t,P I R=
and the 2T x 2T matrix
- tiP2,
1
-t,P
-t,P
I - 2t,p
is positive definite, function becomes
it could
@(t,, t,, t3) =
1 IRl”2
then
(4.5)
be shown
that
the
moment-generating
exp{ (t, + t3P + +ti)v2
-~~CL;[I-(Z+tt2PP)D-1(Z+t2PP)]~2). There exist two orthogonal respectively, such that
matrices
H and
F of order
(4.6) TX T and
N X N,
(4.7)
which we can use to make the following
H'y,=A=
[;;I,
orthogonal
H’y,=O=
transformations:
[%‘I. (4.8)
F’x,=cp=
where
the vectors
[
1
cpl (p2 ,
A and
F’y,
=5=
8 have been
[
51 52
1’
partitioned
into
k,,
k,,
and
T-K
207
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
into m and
components, whereas the vectors ‘p and 5 have been partitioned n components, respectively. Then E(B,) = 0,
EC&) = 0, E(b) = Ps,,
E(5,)
E(e,) =q,,
where q, and qz are the second subvectors respectively, and ;4;9, = rl,>
(4.9)
= P%,
9$92 =
in
H’E(YI)
and
F’E(Yd
(4.10)
772.
The components of A, 19,and 5 are independently normally distributed with variance one, while the components of cp are nonstochastic. This transformation allows us to represent the moment-generating function as follows:
X
exp
_77 + 77&l+ M)* 1 1 - 2t, -t; I
+
772(
t,
+
t3P
+
3.:)
. I
(4.11) @(r,, t,, t3), we obtain the following for t I < i:
Differentiating
ev(-77, +77,/O - 2b) + w2) = 2877, (1 - 2t,)kz’2+’
exd-v, + v,/(l - 2tl) + t,r12) +
P772 (1
-
2t,)kz’2
(4.12)
208
WM. Mikhail and G.A. Ghazal, On a pooled estimator
and
1r,=o,r,=o
a2w,,t*J3) c,” (1 atsat2-s 2 3
[ s=O
exphl =
q2/2
+
+77,/U
-
%)
+
w2)
TIP*)
(1 - 2t,)kz’2+1
exp(h, +77,/U -W + %(1-t 27?1P2)
+fn2)
(1 - 2tl)k2'2+2
exd-17,+77,/U - 2t,) +t1172)
+ 471,77~P~
(1-
2tlp2+’
exd-17, +77,/U -2tJ fh71~) +
7?2(1
+
772P2)
(1 - 2t,p*
Using Lemma 1 together with (4.12) and expanding series, and also making the variate transformation 8 = l/(
1 - 2t,),
ose5
1
9
. exp(t,q2)
(4.13)
as a power
(4.14)
we find
E(P*) =Bep”’
cm 672/2Y r,
r=O
(4.15) Using the integral representation of the confluent hypergeometric function, we obtain the result given in Theorem 1. Similarly, using Lemma 1 together with (4.13), the variate transformation (4.14), and Lemma 2, we obtain the result given in Theorem 2.
5. Asymptotic
expansions
Using the asymptotic expansion of the confluent hypergeometric function5 in the analysis above, the following useful approximations to the bias and the mean square error could be easily obtained. 5See Levedev (1965, pp. 268-271) or Slater (1960, p. 60).
209
W.M. M&hail and G.A. Ghatal, On a pooled estimator
Theorem 3. The asymptotic expansion of the bias of the combined estimator p * of p up to the order 77~~ is gicen by
b(P*,P>=P{(l -W%;‘+
[U -k&)(2-k,/2)
-(772/2)(2-~2/2)17712}+0(7713).
(5.1)
Noting that the bias of the combined estimator p* is given by b(P*,p> = E(p*) - p, then substituting into (3.6) and rearranging, we can immediately establish the result. Theorem 4. The asymptotic expansion of the mean square error of the combined estimator /3 * of p up to order q ;2 is gicen by MSE(P*)
=E(p*
-@’
= i(1+
p=)n;’
-t/3=772(1
+ { [f(2
-
772/2)}77;=
- k,/2)
+
+ p=(2 - k,/2)‘]
(5.2)
0(x3).
To prove this result, we use the definition of the mean combined estimator, then substitute in (3.6) and (3.7).
square
error of the
Corollary 2. The asympiotic expansion of the bias and the mean square error of the 2SLS estimator p of B in (2.2) up to the order of q,-= are gicen, respectively, by
+
o( 77r3)7
(5.3)
MSE( ,!?) = E( p^ - p)2 =$(1+@‘)?7;‘+ + These
results
[3(2-k,/2)
+PZ(2-k2/2)‘]77;2
o(771-3).
are immediately
obtained
(5.4) by putting
n2 = 0 in (5.1) and (5.2).
WM. Mikhail and G.A. Ghazal, On a pooled estimator
210
6. Conditions
for the existence
of moments
The following two lemmas are existence of the moments of /3*.
needed
to find
the
conditions
for the
Lemma 3. If x - N(p, a’), for any A > 0, there exist positive numbers m( A, 0 and MA, 1) such that c’rn( A, 1) I E[lxl’]
_
for Ip/al
(6.1)
Lemma 4. Let X be an ecerywhere positive random cariable and C be Cl), a positive constant. Assume that the lth order moment of (l/(X+ E[(l/(X + C))‘], exists. Then
E[WW+Cl)‘]= j&;,(-‘)‘-%+&) dt, where @x+Jt)
(6.2)
is the moment generating function of (X + C>.
The proofs of the two lemmas are given in the appendix. Making the orthogonal transformations (4-81, the estimator
p*
is reduced
to 402 p*
+
5;(P2
T,
=
= w2
+
+
s,+
T2
(6.3)
say.
‘pi92
Making use of the expectations in (4.9) above and noting of 02,h2, and t2 are all equal to I, we then have E( T, + T,/O) var( T, + T2/0)
= E(h’#,
+ 5&2/o)
= var( hLe2 + 5;q2/O)
= Pq;e,
that the variances
+ Pq;cp, = PT_,
= S, + S, = a;.
(6.4) (6.5)
By assumption of normality, the numerator in (6.31 is conditionally distributed, given 0, according to the normal distribution with mean pr- and variance a;. It follows that E(lP*l’) To apply Lemma But
= E[E(l7,
+ 7-~li/~)(11/‘(~,
3 to the problem
at hand,
+S,)i’/~z)].
we need
(6.6)
to verify
/~~./arI
< 03.
(6.7)
211
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
Using write
the Cauchy-Schwarz
/&/a; Assuming that 4 is applicable.
5
inequality
[see Rao (1973, p. 54)], we can then
P2(q;q,+ q;q*) = P2P?, + 772).
n,,v2 < m, then The expectation
+WE(IV(& sE(lP*l’)
Ipr/aTl
+S2)l'/&)
df(AJ)E(ILW,
The existence of E(/l/(S, + S,)1’/0,). q;q, =2?l,
(6.8)
moments Since E(0,)
+&)l’/$). thus depends = q, and var(0,)
(6.9) on = I,
the
existence
of
=a;,z;[1-z,(z~Z,)-‘Z;]Z,rr2,.
S, = 8.$6, is distributed as noncentral x2 with k, degrees of freedom and the noncentrality parameter 7,) and S, = ‘p;‘p2 is a positive constant. Then, it follows from Lemma 4 that E[ll,‘(.S,
+S2)I’,‘B2] = (1/1.(1))/(’ (-4-‘@,,+,JWc -7j
(6.10)
where @s,+S,(t)= E[er(s,+s,q
= e~~sql
_ 2t)
-k/2
Using (6.11) in (6.10), and expanding the exponential the order of summation and integration, we get E(I
l/(S,
=
(1/n9)
+
e*77t~/(’
-20.
(6.11)
part erS2 and reversing
S2)1’/~2)
5
((
-s,W)
i-=0
x
Making
/_“,( _t)‘+r-
the variate @J= l/(1
‘(
1 _
2t)
-kZ/2 e*9,j/(l
-21)
dt.
(6.12)
transformation - 2t),
O<@Il,
(6.13)
212
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
we get
E(Il/(S, + %)I’/&)
= (eBq1/2’+‘T(I))r~)((
-S2/2)r/r!) (6.14)
The integration tion
in each term converges
T(kJ2-1-r)T(f+r)
for all r 2 0 and has the representa-
1Fdkz/2-l-r;
r( k/2)
k2/2;
T,),
if and only if I < k,/2. It has now been shown that E(Ip*l’> exists if and only if I < k,/2. Since the order condition for identifiability requires that k, 2 2, i.e., k, - 2 is the number of overidentifying restrictions in the time-series equation, it follows that Elp * 1 exists if the time-series equation is overidentified.
Appendix Proof of Lemma 3 Let 1 be even and y = (x - p)/a.
&/:
~[lxl'] =
(uy
m
Then
+p)‘e-Y2/2dy
(A.11 For odd 1, the integral
is zero. For even J,
&~~~“e~yz’2dy=
see Kendall
and Stuart
(I-J)! 2(/-5)/‘[([_J)/2]!’
(1977, p. 143).
(A.2)
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
213
Therefore,
(A-3)
The if A term If
summation term in < 00, and therefore is positive for A > 1 is odd, the above bY
+/-I’_<
lay
(A.3) depends on 1 and I~/(TI =A. It is clearly finite less than a positive number M(A, I>. Moreover, the 0, and so is greater than a positive number m(A, I). result still holds since
+pl ‘+I + 1,
which implies
E[lvy +/-A’] 5 E[IcY +/A’+‘] and since 1 + 1 is even when Accordingly, we can write a’m(A,I)
+ 1,
(A.4)
1 is odd, we have that E[ Ix/‘] is bounded
<,‘~(A,f).
(A.5)
Proof of Lemma 4
E[( A)‘] =c(
&)‘f(W
since in general
for a > 0 and d > 0,
T
/0 Making
epuxxyd-’ dx = apdT( d).
the variate
transformation 1 = -/” T(f)
which immediately
gives (6.2).
8 = - t, we find
-cc
for 1.
( -I,‘-‘[~~e’(x+c)f(x)
dx] dt,
214
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
References Basmann, R.L., 1961, A note on the exact finite sample frequency functions of generalized classical linear estimators in two leading overidentified cases, Journal of the American Statistical Association 56, 619-636. Chetty, V.K., 1968, Pooling of time series and cross section data, Econometrica, 279-290. Durbin, J., 1953, A note on regression when there is extraneous information about one of the coefficients, Journal of the American Statistical Association, 799-808. Hsiao, Chang, 1986, Analysis of panel data (Cambridge University Press, Cambridge). Izan, H.Y., 1980, To pool or not to pool? A reexamination of Tobin’s food demand problem, Journal of Econometrics 13, 391-402. Judge, G.G. et al., 1985, The theory and practice of econometrics (Wiley, New York, NY). Kabe, D.G., 1963, A note on the exact distributions of the CCL estimators in two leading overidentified cases, Journal of the American Statistical Association 58, 533-537. Kendall, M.G. and A. Stuart, 1977, The advanced theory of statistics, Vol. 1 (Hafner, New York, NY). Kloek, T., 1988, Macroeconomic models and econometrics, in: W. Driehuis, M.M.G. Fase, and H. den Hartog, eds., Challenges for macroeconomic modelling (Elsevier Science Publishers B.V., Amsterdam). Kuh, E., 1959, The validity of cross-sectionally estimated behaviour equation in time-series applications, Econometrica, 197-214. Levedev, N.N., 1965, Special functions and their applications (Prentice-Hall, Englewood Cliffs, NJ). Maddala, G.S., 1971, The likelihood approach to pooling cross-section and time-series data, Econometrica, 939-953. Rao. C.R., 1973, Linear statistical inference and its applications (Wiley, New York, NY). Richardson, D.H., 1968, The exact distribution of a structural coefficient estimator, Journal of the American Statistical Association, 1214-1226. Sawa, T., 1972, Finite-sample properties of the k-class estimators, Econometrica, 653-680. Slater, L.J., 1960, Confluent hypergeometric functions (Cambridge University Press, Cambridge). Theil, H., 1963, On the use of incomplete prior information in regression analysis, Journal of the American Statistical Association, 401-414. Theil, H. and A.S. Goldberger, 1961, On pure and mixed statistical estimation in economics, International Economic Review, 65-78. Tobin, J., 1950, A statistical demand function for food in the U.S.A., Journal of the Royal Statistical Society A, 113-141. Zellner, A., 1969, On the aggregation problem: A new approach to a troublesome problem, in: K.A. Fox et al., eds., Economic models, estimation and risk programming: Essays in honor of Gerhard Tintner (Springer-Verlag, New York, NY).
of Econometrics
48 (1991) 195-214.
North-Holland
On a pooled estimator and its finite-sample moments* William
M. Mikhail
The American Unkersity in Cairo, Cairo, Egypt
G.A. Ghazal Cairo University, Cairo, Egypt Received
May 1988, final version
received
February
1990
The present paper deals with the situation in which we have a single cross-section equation which could be estimated by ordinary least squares and a single time-series equation which is one of a complete system of simultaneous equations to which the application of OLS would lead to inconsistent estimates. A consistent pooled estimator is derived and the exact finite-sample moments of this estimator are evaluated on the assumption that lagged endogenous variables are not present. The exact moment function is expanded in terms of the inverse of the noncentrality parameter. The exact finite-sample moments of the two-stage least-squares estimator as derived by Richardson (1968) and Sawa (1972) could then be obtained as special cases, and the expansion sheds more light on further studies and comparisons.
1. Introduction As explained by Maddala (19711, there are basically two types of problems related to the pooling of cross-section and time-series data. First, there is the case of a time-series of cross-sections, for which many alternative approaches have been suggested by various authors. Second, there is the case of a single cross-section and a single time series, which has been analyzed by Tobin (1950), Chetty (1968), and Maddala (1971), but has probably not received as much attention as the first case. The present paper deals with this latter case, and does not address itself in any way to the case of temporal cross-sections. Tobin’s analysis of the problem has come to be referred to as the traditional method, and many authors followed in applying it to similar models. It consisted of estimating the parameters of the cross-section equation and introducing the estimates of the common parameters into the *The
authors
are grateful
0304.4076/91/$03.500
to the anonymous
1991-Elsevier
Science
referees
Publishers
for their most helpful
B.V. (North-Holland)
comments.
196
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
time-series equation as known with certainty. This of course led to conditional estimates of the parameters of the time-series equation and, consequently, to underestimates of the standard errors of the coefficients. However, even if the estimates of the common parameters obtained from the cross-section equation are substituted into the time-series equation taking the uncertainty into consideration, the resulting estimates, as shown by Durbin (1953) and Maddala (19711, will still be inefficient. Efficient estimates could only be obtained by estimating the parameters of the two equations simultaneously, as there is also some information about the common parameters in the time-series sample which should not be ignored. Chetty (1968) used the Bayesian approach to estimate all the parameters simultaneously. The method utilizes the information in both equations to derive the marginal posterior distribution of the common parameters and it is therefore likely, as argued by Chetty, to have smaller variances than those obtained from the traditional method. These parameters, as well as the other parameters in the two equations, will have, on the average, sharper posterior distributions when computed from the pooled samples than when computed from either sample. Chetty also claims that the method suggested by Durbin (1953) and simultaneous approaches which follow the lines of Theil and Goldberger (1961) and Theil (1963) will only have an asymptotic justification, while for his Bayesian method exact finite-sample results are obtained. Maddala (1971) obtained maximum-likelihood estimates equivalent to iterating the least-squares procedure in the simultaneous presentation of the two equations. Applying this to the data of the Tobin model, he found that the results agreed with those obtained by Chetty using Bayesian analysis. Some questions have been raised about the validity of the procedure of pooling time-series and cross-section data in general in providing meaningful estimates of the common parameters. It has been argued that the parameters which we try to combine are different and answer different questions and should therefore have different estimators, prior distributions, and other properties. In general, estimates depend on the other variables included in each of the two equations and the precise ceteris paribus assumptions made. However, it appears from careful examination of the literature that despite the skepticism of some researchers about the usefulness of pooling time-series and cross-section data, there has been no substantive evidence against such procedure. Even the empirical results obtained by Maddala on the Tobin model, which were mistaken by certain researchers to constitute an evidence against pooling, were indeed not conclusive. Maddala has found, by applying the likelihood-ratio test and comparing the relative likelihoods or the posterior distributions of the common parameters, that for the Tobin model data should not have been pooled. Maddala called the evidence unambiguous, with hardly any overlap between the two likelihood functions. This strong verdict against pooling, in a situation when there was substantial improvement in the accuracy of the combined estimator, was totally discouraging to
197
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
many people to consider pooling in similar situations. However, when Izan (1980) carried out the exercise all over again by allowing the disturbances in the time-series equation to follow a first-order autoregressive process, and also allowing for outliners found in Tobin’s cross-section data, she found that the likelihood-ratio test, after allowing for autocorrelation, resulted in acagainst cepting the pooling hypothesis. ’ This suggests that the decision pooling might have been a result of misspecification of the model in the form of the process generating the disturbance term. All authors however agree that before pooling is done, diagnostic checks should be performed to ensure the consistency of the two sets of data. This may be carried out by classical tests of significance or by a study of posterior distributions and posterior odds. The present paper deals with the situation in which the cross-section equation could be estimated by ordinary least squares, but the time-series equation is one of a complete system of simultaneous equations to which the application of OLS would lead to inconsistent estimates. The situation considered is one which often crops up in demand analysis where we may have a single cross-section equation based on family-budget data with observations corresponding to individual households at one point in time, and a single time-series equation with data corresponding to national-accounts statistics, and the two equations would be sharing one or more common parameters.2 A simple classical example is the estimation of the marginal propensity to consume food, where the time-series equation is an aggregate function for food consumption in which we regress food consumption on aggregate disposable income and the price of food relative to the cost of living, while the cross-section equation uses family-budget data to regress food consumption of households on disposable income and the size of the household. In the first equation, aggregate food consumption and disposable income should be treated in macroeconomic analysis as jointly dependent. The purpose for pooling in the previous example is to get more efficient estimates, in view of the fact that cross-sectional observations at one point in time might be affected by prior observations, and that time-series data, besides often suffering from collinearity among the explanatory variables, do not contain information on individual socio-demographic variables relevant to the study of variations in the dependent variable. Thus the pooling might ‘It should be noted that in Tobin’s aggregate time-series model.
example
there
is a lagged
endogenous
variable
in the
‘No aggregation of cross-sections over time is considered here, since observations correspond to individuals (or households) at one point in time. If aggregation is done over groups of individuals and if the coefficients differ among them, then this may lead to aggregation bias. In this case, we follow Zellner (1969) and assume that the group coefficient vector can be written as b, = b + E,, where E, is a random component with zero expectation, and the estimator may still be unbiased. Thus, if we assume that the differences are random and that it is the mean or expected value of the coefficients that is to be estimated, then no aggregation bias will result. However, aggregate error terms in this case will usually be heteroskedastic.
198
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
be thought of as extraneous information in the from of cross-sectional data for the time-series model or in the form of time-series data for the cross-sectional model. However, the pooling of time-series and cross-sectional data might be desirable in other situations for the purpose of achieving consistency. An example is the joined input-output and econometric approach to economy-wide modeling and projections of final demand. The estimation of the elements of the private consumption column in the I/O table is done using functions estimated from cross-sectional family-budget data, and its projected control total is also obtained from a function using the same cross-sectional data set. If, within the same modeling framework, nationalaccounts statistics are used to estimate econometric relationships and project economic aggregates, it might be desirable not to have different estimates of the elasticities within the same exercise, irrespective of the considerations of accuracy. To achieve consistency and obtain a projected column of final demands which agrees with both sets of data, the equality of the coefficients might be imposed in the context of pooling the time-series and cross-sectional data without, of course, neglecting the simultaneous nature of the econometric system. The method used in this paper constitutes an improvement over the other methods available for this type of problem in that it allows an analysis of more general time-series models. The consistent pooled estimator considered below takes into account the inherent features of the individual components of a simultaneous-equation model, and its superiority over the likelihood or iterated least-squares approach is assured by pre-weighting the equations as given below (rather than assigning a kind of averaging process) in a multidimensional form of distance minimizing. Other methods which neglect the simultaneous nature of the RHS jointly-dependent variables would in effect be treating them as exogenous, thus committing a specification error which may not only lead to widely different estimates of the common parameters but also result in erroneous values for the tests designed to answer the question of whether or not to pool. Such erroneously calculated values for these tests, based on inaccurate specifications, may indeed lead to wrong conclusions, as illustrated by Izan, and to the rejection of the hypothesis of pooling when in fact it should be accepted. The paper considers the case in which the time-series equation contains only two endogenous variables, and it is assumed that no lagged values of theseavariables would be present in the equation. This is done in order to facilitate the derivation of the exact finite-sample results given in the following sections and to enable comparisons with previous results on finite-sample properties, especially those obtained by Richardson (1968) and Sawa (1972). In the case of many endogenous variables and lagged values of them being present in the time-series equation, it would not be possible to derive exact finite-sample moments for the combined estimator, but an approximation to the distribution might be possible to obtain.
WM. Mikhail and G.A. Ghazal, On a pooled estimator
199
The organization of the paper is as follows. The model, assumptions, and the combined estimator are presented in section 2. In section 3, we evaluate the exact first and second moments of the estimator. In order to derive these moments, we used a method described as Lemma 1, which is an extension of a basic lemma given by Sawa. The main results of section 3 will be proved in section 4. To facilitate the investigation of the properties of the combined estimator, an asymptotic expansion of the bias and the mean square error based on the exact results will be given in section 5. Finally, in section 6, we derive the conditions under which the moments exist.
2. The model and the combined
estimator
Let us write a time-series simultaneous system of G linear equations relating G endogenous and K predetermined variables YB+ZT=U.
(2.1)
The first equation being estimated is assumed to contain nous variables and to be overidentified by zero restrictions coefficients. It may be written as YI = PY2 +
stochastic as
Z,Y, + UI?
only two endogeon the structural
(2.2)
where Y is a T x G matrix of T observations on G endogenous variables, y , and y, are the first and second columns of Y, respectively, Z is a TX K matrix of observations on K nonstochastic predetermined variables (not containing lagged endogenous variables) partitioned as Z = (Z,, Z,>, where Z, is a T x k, matrix of included predetermined variables and Z, is a TX k, matrix of excluded ones (K = k, + k,), and U is a T X G matrix of disturbances with first column u,. Also, let the cross-section equation be Y3=Px,+x2b+u
2,
(2.3)
where y, is an N x 1 vector of observations on the dependent variable, x, is an N ~‘1 vector of observations on the first nonstochastic independent variable which has the same coefficient as y, in the time-series equation, X, is an N x m matrix of the other nonstochastic independent variables with X=(x ,, X2>, and u2 is an N X 1 vector of disturbances. B, r, @, yl, and 6, are parameter matrices and vectors of conforming dimensions. The reduced form of the system (2.1) could be written as Y=-ZTB-‘+UB-‘=Zn+V
(2.4)
WM. Mikhail and G.A. Ghazal, On a pooled estimator
200
and
where (a,,~~) and (u,, u,) are the first two columns of II and I/, respectively, and ri (i = 1,2) are partitioned as r/ = CT/~, 7~:~) conformably with Z = (Z,, Z,). The traditional econometric assumptions include the specification that the disturbance vectors u1 and u2 have the following properties: 6)
Ecu,) = 0,
Ecu,)
= 0.
(ii) E(u,u;) = U~Z,, ECU&)
= a;~,,,,
E(u,u;)
= 0.
It should be noted here that the assumption of independence of U, in the time-series equation and u2 in the cross-section equation is not unrealistic. In the context of the Tobin model, referred to as an example in the introduction above, there is no reason to believe that the omitted variables and the errors of measurement in the time-series equation with its observations covering aggregate variables over a number of years to be correlated with omitted variables or errors of measurement in the cross-section equation where the data correspond to individuals or households at a single given .point in time.” Indeed, the independence of U, and u2 is one main reason why eqs. (2.2) and (2.3) are not analyzed together using a 3SLS or ML approach, besides the considerations of finite-sample properties mentioned elsewhere in this paper. It is also assumed that each row of CC,, L’~) is independently and identically distributed two-dimensional normal variate with zero mean and covariance matrix4 ?P = &, and that the N-dimensional random vector u2 is distributed as normal: u2
-
N(0,
a&).
To estimate /3 from the time-series equation only, we need to apply a consistent single-equation technique. The two-stage least squares seemed to be a good choice, first because of its simplicity and second because of the fact that the pooled estimator based on it possesses finite-sample moments, as “This was also assumed
by Chetty
(1968) and Maddala
(1971) in estimating
the Tobin model.
4There exists a set of transformations [see, for instance, Sawa (1972)] on the variables and parameters of the model which transforms it into one in which q = II. We shall here assume that the transformations have already been made.
201
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
will be seen below, that could be compared with the results Richardson (1968), Sawa (1972), and others. We then have
obtained
by
p^ =Y;pY,/Y;pY,,
(2.6)
P=Z(Z’Z)_‘Z’-Z,(Z{ZJ’z;
(2.7)
where
matrix of rank k,. from the cross-section
is an idempotent The estimator
equation
is
@=XiLYJX;LXr,
(2.8)
L=I,-X,(X;X*))‘x;
(2.9)
where
is an idempotent matrix of rank (N - m) = n, say. A consistent pooled estimator based on applying the 2SLS to the first equation would be obtained by applying the generalized least squares to the simultaneous presentation of the two equations after premultiplying the first of the matrix of all the predetermined equation by Z’, the transpose variables in the time-series model:
Writing
the disturbance Z'U] Cl=
[ its covariance
Applying
112
term in this equation
as
1 )
matrix will be
Aitken’s P*=(Y;PYl
GLS, we get the combined +XiLY,)/(Y;PY*+XiLX,).
estimator
p* of j3 as (2.10)
We see from (2.10) that the numerator of the combined estimator is the sum of the numerators in (2.6) and (2.81, and the denominator is the sum of the two denominators.
202
3. Evaluation
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
of the exact moments
Analytical complexities have been a serious impediment in deriving finite-sample properties of estimators, and consequently the results obtained so far are rather limited and cover only certain special cases. Exact densities and moments as well as approximate densities and moments of approximate distributions have been derived for 2SLS, k-class estimators, instrumentalvariable estimators, and maximum-likelihood estimators, as applied to simple econometric models. Many excellent summaries of such results are available [see, for instance, Judge et al. (198511, but we refer here briefly to those studies which the present paper either draws upon or compares with. Basmann’s pioneering work of 1961 was mainly concerned with the problem of nonexistence of finite moments. He derived the exact density when two exogenous variables are excluded, and also obtained the density for the case in which three exogenous variables were excluded but when one of the parameters of the density was identically equal to zero. He later showed that, for the first case, the estimator converges stochastically as one of the parameters of the density increases indefinitely, and derived expressions for the mean and the bias of the estimator. Kabe (1963) derived the same two density functions utilizing the noncentral Wishart distribution. Richardson (1968) extended the results of Basmann and Kabe to the more general case in which any number of exogenous variables may be excluded from the equation. Richardson adopted Kabe’s approach and derived the exact distribution of the estimator by making a change of variable in the noncentral Wishart distribution. He found that the number of excluded exogenous variables appears as a parameter in the density function but that the form of the density is not affected by the addition of more excluded exogenous variables. Sawa (1972) used a moment-generating function technique to evaluate the exact finite-sample moments of the k-class estimators for 0 I k I 1, and proved that for k > 1 the estimator does not possess even the first-order moment. He also expanded the exact moment functions in terms of the inverse of the noncentrality parameter. Both Richardson and Sawa have independently confirmed Basmann’s conjecture that the moments of 2SLS exist only if the order of the moment is less than the degree of identification of the equation. In evaluating the exact moments of the combined estimator, we need the following two lemmas. The first is an extension of a result given by Sawa (19721, and the second provides a result concerning the confluent hypergeometric functions. Lemma 1. Let X, be an almost everywhere positive random variable, C be a positive constant, and X2 and X3 be arbitrary random variables. If there also
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
exists a joint moment-generating function of X,, X,, @(t,,t2,t3)
203
and X,,
= E[~~I(~I+C)+‘ZXZ+~,~~],
(3.1)
for t, < E and 1t,j, (t, 1 < E, where E is some positice constant, then the rth order moment of (X, + X,>/( X, + C) is gicen by
x,+x,
Ep [ x, + c
1
r =_
1 r(r) i s= 0
1
(f;;t’&“-“’ s ;
dt,.
tz=o,r,=o.
(3.2) The lemma and results. Before presenting
the proof Lemma
7, = 7T/*z;(z
are straightforward
generalizations
2, we give the following
- z,(z;z,)~‘z;)z,~**/2,
l-/z =xiLx,.
of Sawa’s
definitions: (3.3) (3.4)
The parameter 7, given in (3.3) corresponds to 6 in Sawa’s results and to ~*/2 in Richardson’s results, where pu2 is the concentration parameter. The value of 7, is unknown. However, the relative bias of /3* could be computed exactly for given values of 7, and k,, from eq. (3.6) below, and the mean square error could be computed for given values of T,, k,, and p from eq. (3.7). Lemma 2 T(cr-r(y
1) ,Fl(
r(a+l)
a-r-l;a+l;T,)
r(cw-r) ,F,(a
+17,r(N+2) =
r(a-rr(a)
1)
-r;
Q + 2; 7,)
,F,(a-r-l;a;~,).
(3.5)
This result could be easily established if we start by applying the definition of the confluent hypergeometric function to the LHS of (3.5). We now present the two theorems on the exact moments.
WM. Mikhail and G.A. Ghazal, On a pooled estimator
204
Theorem 1. The exact first-order defined by (2.10) is given by E(P*)
moment
of the combined
estimator p*
5 [?lr
=Pe-vt
r=O
W2/2-r)
’ [”
r( k,/2
+ 2
W2/2-r-1)
2
IFl(k2/2
+ 1)
- r; b/2
,F,(k,/2-r-
r( k,/2)
+ 1;~~)
l;k&C~,)
1. (3.6)
Theorem 2. The exact second-order defined by (2.10) is given by
E(p**)
ca ( -v2/21rr(r+ 2)
= y
r =
X
moment of the combined estimator /3*
r!
0
W k, - 2)/2 r(k,/2)
r>,F,((k, - 2)/2-r;
k2/2;v1)
r((k2 -2)/2 -r) +%P2(l
+
2772)
+2)/z)
j-((k 2
r( k2/2 - r 1 +2v:B2r((k2 + 4),2) ’ F ’(k,/2 r((k, +772(1
+
- 4)~
-r;
(k, + 4)/z
771)
-r)
172P2) r(
k2/2)
x1F,((k2-4)/2-r;k2/2;r11)
. I
(3.7)
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
205
The exact first- and second-order moments of the combined estimator p *, as given in Theorem 1 and 2, depend on the cross-section data only through the parameter q2 defined by (3.4). Therefore, in case we are not incorporating cross-section data in the estimation procedure the exact first- and second-order moments of the 2SLS estimator can be derived from (3.6) and (3.71, respectively, by putting n2 = 0. Thus, we have the following: $orollaiy I. The exact first- and second-order moments of the 2SLS estimator p of p in (2.2) could be obtained from (3.6) and (3.7) as
E(i)
r( b/2)
=PT,ePVl
r(k,/2
em71 r( k,/2 - 1) = 2 ,F,(kz/2r( k,/2)
E(a’)
+‘I’
These (1972).
,F,(k,/2;K,/2+
+ 1)
results
,W,/2
(3.8)
l;~,),
l;k,/2;n,)
- 1)
Z-( k,/2
+ 1)
,F,(k,/2
- 1; k,/2
+ 1; 7,)
r( k,/2) + 2dP2 ,F,(k2/2;k,/2+2 r(k,/2 + 2)
;771).
agree
(1968)
with
those
given
by Richardson
I
(3.9)
and
Sawa
4. Proof of theorems The T-dimensional random vectors y, (a = 1,2) defined by (2.5) and the N-dimensional random vector y, defined by (2.3) are, by assumption, mutually independent and distributed normally with mean vectors pL, (a = 1,2,3), and covariance matrix I where, using notation as close as possible to Sawa’s,
Pa
=
CL3 =
z,rTT,,
Px,
+
Z2Tn2>
a
=
1,2,
(4.1) (4.2)
+X*62.
The joint moment-generating
function
of (Y;PY,
+x;Lx,),
YipY,, and X;LY,
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
206
is given by
If we restrict
the admissible
domains
of t, and t, to
t;+2t,<1, so that
(4.4)
D = I - 2t,P I R=
and the 2T x 2T matrix
- tiP2,
1
-t,P
-t,P
I - 2t,p
is positive definite, function becomes
it could
@(t,, t,, t3) =
1 IRl”2
then
(4.5)
be shown
that
the
moment-generating
exp{ (t, + t3P + +ti)v2
-~~CL;[I-(Z+tt2PP)D-1(Z+t2PP)]~2). There exist two orthogonal respectively, such that
matrices
H and
F of order
(4.6) TX T and
N X N,
(4.7)
which we can use to make the following
H'y,=A=
[;;I,
orthogonal
H’y,=O=
transformations:
[%‘I. (4.8)
F’x,=cp=
where
the vectors
[
1
cpl (p2 ,
A and
F’y,
=5=
8 have been
[
51 52
1’
partitioned
into
k,,
k,,
and
T-K
207
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
into m and
components, whereas the vectors ‘p and 5 have been partitioned n components, respectively. Then E(B,) = 0,
EC&) = 0, E(b) = Ps,,
E(5,)
E(e,) =q,,
where q, and qz are the second subvectors respectively, and ;4;9, = rl,>
(4.9)
= P%,
9$92 =
in
H’E(YI)
and
F’E(Yd
(4.10)
772.
The components of A, 19,and 5 are independently normally distributed with variance one, while the components of cp are nonstochastic. This transformation allows us to represent the moment-generating function as follows:
X
exp
_77 + 77&l+ M)* 1 1 - 2t, -t; I
+
772(
t,
+
t3P
+
3.:)
. I
(4.11) @(r,, t,, t3), we obtain the following for t I < i:
Differentiating
ev(-77, +77,/O - 2b) + w2) = 2877, (1 - 2t,)kz’2+’
exd-v, + v,/(l - 2tl) + t,r12) +
P772 (1
-
2t,)kz’2
(4.12)
208
WM. Mikhail and G.A. Ghazal, On a pooled estimator
and
1r,=o,r,=o
a2w,,t*J3) c,” (1 atsat2-s 2 3
[ s=O
exphl =
q2/2
+
+77,/U
-
%)
+
w2)
TIP*)
(1 - 2t,)kz’2+1
exp(h, +77,/U -W + %(1-t 27?1P2)
+fn2)
(1 - 2tl)k2'2+2
exd-17,+77,/U - 2t,) +t1172)
+ 471,77~P~
(1-
2tlp2+’
exd-17, +77,/U -2tJ fh71~) +
7?2(1
+
772P2)
(1 - 2t,p*
Using Lemma 1 together with (4.12) and expanding series, and also making the variate transformation 8 = l/(
1 - 2t,),
ose5
1
9
. exp(t,q2)
(4.13)
as a power
(4.14)
we find
E(P*) =Bep”’
cm 672/2Y r,
r=O
(4.15) Using the integral representation of the confluent hypergeometric function, we obtain the result given in Theorem 1. Similarly, using Lemma 1 together with (4.13), the variate transformation (4.14), and Lemma 2, we obtain the result given in Theorem 2.
5. Asymptotic
expansions
Using the asymptotic expansion of the confluent hypergeometric function5 in the analysis above, the following useful approximations to the bias and the mean square error could be easily obtained. 5See Levedev (1965, pp. 268-271) or Slater (1960, p. 60).
209
W.M. M&hail and G.A. Ghatal, On a pooled estimator
Theorem 3. The asymptotic expansion of the bias of the combined estimator p * of p up to the order 77~~ is gicen by
b(P*,P>=P{(l -W%;‘+
[U -k&)(2-k,/2)
-(772/2)(2-~2/2)17712}+0(7713).
(5.1)
Noting that the bias of the combined estimator p* is given by b(P*,p> = E(p*) - p, then substituting into (3.6) and rearranging, we can immediately establish the result. Theorem 4. The asymptotic expansion of the mean square error of the combined estimator /3 * of p up to order q ;2 is gicen by MSE(P*)
=E(p*
-@’
= i(1+
p=)n;’
-t/3=772(1
+ { [f(2
-
772/2)}77;=
- k,/2)
+
+ p=(2 - k,/2)‘]
(5.2)
0(x3).
To prove this result, we use the definition of the mean combined estimator, then substitute in (3.6) and (3.7).
square
error of the
Corollary 2. The asympiotic expansion of the bias and the mean square error of the 2SLS estimator p of B in (2.2) up to the order of q,-= are gicen, respectively, by
+
o( 77r3)7
(5.3)
MSE( ,!?) = E( p^ - p)2 =$(1+@‘)?7;‘+ + These
results
[3(2-k,/2)
+PZ(2-k2/2)‘]77;2
o(771-3).
are immediately
obtained
(5.4) by putting
n2 = 0 in (5.1) and (5.2).
WM. Mikhail and G.A. Ghazal, On a pooled estimator
210
6. Conditions
for the existence
of moments
The following two lemmas are existence of the moments of /3*.
needed
to find
the
conditions
for the
Lemma 3. If x - N(p, a’), for any A > 0, there exist positive numbers m( A, 0 and MA, 1) such that c’rn( A, 1) I E[lxl’]
_
for Ip/al
(6.1)
Lemma 4. Let X be an ecerywhere positive random cariable and C be Cl), a positive constant. Assume that the lth order moment of (l/(X+ E[(l/(X + C))‘], exists. Then
E[WW+Cl)‘]= j&;,(-‘)‘-%+&) dt, where @x+Jt)
(6.2)
is the moment generating function of (X + C>.
The proofs of the two lemmas are given in the appendix. Making the orthogonal transformations (4-81, the estimator
p*
is reduced
to 402 p*
+
5;(P2
T,
=
= w2
+
+
s,+
T2
(6.3)
say.
‘pi92
Making use of the expectations in (4.9) above and noting of 02,h2, and t2 are all equal to I, we then have E( T, + T,/O) var( T, + T2/0)
= E(h’#,
+ 5&2/o)
= var( hLe2 + 5;q2/O)
= Pq;e,
that the variances
+ Pq;cp, = PT_,
= S, + S, = a;.
(6.4) (6.5)
By assumption of normality, the numerator in (6.31 is conditionally distributed, given 0, according to the normal distribution with mean pr- and variance a;. It follows that E(lP*l’) To apply Lemma But
= E[E(l7,
+ 7-~li/~)(11/‘(~,
3 to the problem
at hand,
+S,)i’/~z)].
we need
(6.6)
to verify
/~~./arI
< 03.
(6.7)
211
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
Using write
the Cauchy-Schwarz
/&/a; Assuming that 4 is applicable.
5
inequality
[see Rao (1973, p. 54)], we can then
P2(q;q,+ q;q*) = P2P?, + 772).
n,,v2 < m, then The expectation
+WE(IV(& sE(lP*l’)
Ipr/aTl
+S2)l'/&)
df(AJ)E(ILW,
The existence of E(/l/(S, + S,)1’/0,). q;q, =2?l,
(6.8)
moments Since E(0,)
+&)l’/$). thus depends = q, and var(0,)
(6.9) on = I,
the
existence
of
=a;,z;[1-z,(z~Z,)-‘Z;]Z,rr2,.
S, = 8.$6, is distributed as noncentral x2 with k, degrees of freedom and the noncentrality parameter 7,) and S, = ‘p;‘p2 is a positive constant. Then, it follows from Lemma 4 that E[ll,‘(.S,
+S2)I’,‘B2] = (1/1.(1))/(’ (-4-‘@,,+,JWc -7j
(6.10)
where @s,+S,(t)= E[er(s,+s,q
= e~~sql
_ 2t)
-k/2
Using (6.11) in (6.10), and expanding the exponential the order of summation and integration, we get E(I
l/(S,
=
(1/n9)
+
e*77t~/(’
-20.
(6.11)
part erS2 and reversing
S2)1’/~2)
5
((
-s,W)
i-=0
x
Making
/_“,( _t)‘+r-
the variate @J= l/(1
‘(
1 _
2t)
-kZ/2 e*9,j/(l
-21)
dt.
(6.12)
transformation - 2t),
O<@Il,
(6.13)
212
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
we get
E(Il/(S, + %)I’/&)
= (eBq1/2’+‘T(I))r~)((
-S2/2)r/r!) (6.14)
The integration tion
in each term converges
T(kJ2-1-r)T(f+r)
for all r 2 0 and has the representa-
1Fdkz/2-l-r;
r( k/2)
k2/2;
T,),
if and only if I < k,/2. It has now been shown that E(Ip*l’> exists if and only if I < k,/2. Since the order condition for identifiability requires that k, 2 2, i.e., k, - 2 is the number of overidentifying restrictions in the time-series equation, it follows that Elp * 1 exists if the time-series equation is overidentified.
Appendix Proof of Lemma 3 Let 1 be even and y = (x - p)/a.
&/:
~[lxl'] =
(uy
m
Then
+p)‘e-Y2/2dy
(A.11 For odd 1, the integral
is zero. For even J,
&~~~“e~yz’2dy=
see Kendall
and Stuart
(I-J)! 2(/-5)/‘[([_J)/2]!’
(1977, p. 143).
(A.2)
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
213
Therefore,
(A-3)
The if A term If
summation term in < 00, and therefore is positive for A > 1 is odd, the above bY
+/-I’_<
lay
(A.3) depends on 1 and I~/(TI =A. It is clearly finite less than a positive number M(A, I>. Moreover, the 0, and so is greater than a positive number m(A, I). result still holds since
+pl ‘+I + 1,
which implies
E[lvy +/-A’] 5 E[IcY +/A’+‘] and since 1 + 1 is even when Accordingly, we can write a’m(A,I)
+ 1,
(A.4)
1 is odd, we have that E[ Ix/‘] is bounded
<,‘~(A,f).
(A.5)
Proof of Lemma 4
E[( A)‘] =c(
&)‘f(W
since in general
for a > 0 and d > 0,
T
/0 Making
epuxxyd-’ dx = apdT( d).
the variate
transformation 1 = -/” T(f)
which immediately
gives (6.2).
8 = - t, we find
-cc
for 1.
( -I,‘-‘[~~e’(x+c)f(x)
dx] dt,
214
W.M. Mikhail and G.A. Ghazal, On a pooled estimator
References Basmann, R.L., 1961, A note on the exact finite sample frequency functions of generalized classical linear estimators in two leading overidentified cases, Journal of the American Statistical Association 56, 619-636. Chetty, V.K., 1968, Pooling of time series and cross section data, Econometrica, 279-290. Durbin, J., 1953, A note on regression when there is extraneous information about one of the coefficients, Journal of the American Statistical Association, 799-808. Hsiao, Chang, 1986, Analysis of panel data (Cambridge University Press, Cambridge). Izan, H.Y., 1980, To pool or not to pool? A reexamination of Tobin’s food demand problem, Journal of Econometrics 13, 391-402. Judge, G.G. et al., 1985, The theory and practice of econometrics (Wiley, New York, NY). Kabe, D.G., 1963, A note on the exact distributions of the CCL estimators in two leading overidentified cases, Journal of the American Statistical Association 58, 533-537. Kendall, M.G. and A. Stuart, 1977, The advanced theory of statistics, Vol. 1 (Hafner, New York, NY). Kloek, T., 1988, Macroeconomic models and econometrics, in: W. Driehuis, M.M.G. Fase, and H. den Hartog, eds., Challenges for macroeconomic modelling (Elsevier Science Publishers B.V., Amsterdam). Kuh, E., 1959, The validity of cross-sectionally estimated behaviour equation in time-series applications, Econometrica, 197-214. Levedev, N.N., 1965, Special functions and their applications (Prentice-Hall, Englewood Cliffs, NJ). Maddala, G.S., 1971, The likelihood approach to pooling cross-section and time-series data, Econometrica, 939-953. Rao. C.R., 1973, Linear statistical inference and its applications (Wiley, New York, NY). Richardson, D.H., 1968, The exact distribution of a structural coefficient estimator, Journal of the American Statistical Association, 1214-1226. Sawa, T., 1972, Finite-sample properties of the k-class estimators, Econometrica, 653-680. Slater, L.J., 1960, Confluent hypergeometric functions (Cambridge University Press, Cambridge). Theil, H., 1963, On the use of incomplete prior information in regression analysis, Journal of the American Statistical Association, 401-414. Theil, H. and A.S. Goldberger, 1961, On pure and mixed statistical estimation in economics, International Economic Review, 65-78. Tobin, J., 1950, A statistical demand function for food in the U.S.A., Journal of the Royal Statistical Society A, 113-141. Zellner, A., 1969, On the aggregation problem: A new approach to a troublesome problem, in: K.A. Fox et al., eds., Economic models, estimation and risk programming: Essays in honor of Gerhard Tintner (Springer-Verlag, New York, NY).