Efficient Inference in a Random Coefficient Regression Model
P. A. V. B. Swamy
Econometrica, Vol. 38, No. 2. (Mar., 1970), pp. 311-323.
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Econornetrica, Vol. 38, No. 2 (March, 1970)
EFFICIENT INFERENCE IN A RANDOM COEFFICIENT REGRESSION MODEL In this paper an attempt is made to estimate a regression equation using a time series
of cross sections. It is assumed that the coefficient vector is distributed across units with the
same mean and the same variance-covariance matrix. The distribution of the coefficient
vector is assumed to be invariant to translations along the time axis. A consistent and an
asymptotically efficient estimator for the mean vector and an unbiased estimator for the
variance-covariance matrix of the coefficient vector have been suggested. Some asymptotic
procedures for testing linear hypotheses on the means and the variances of coefficients
have been described. Finally, the estimation procedure is applied in the analysis of annual
investment data, 1935-54, for eleven firms.
1.
INTRODUCTION
As OBSERVED BY Klein [8, pp. 216-181, it is unlikely that interindividual differences
observed in a cross section sample can be explained by a simple regression equation with a few independent variables. In such situations the coefficient vector of a
regression model can be treated as random to account for interindividual heterogeneity. In deriving the production function, the supply function for factors, and
the demand function for a product, Nerlove [lo, pp. 34-35 and Ch. 43 found it
appropriate to treat the elasticities of output with respect to inputs and of factor
supplies and product demand as random variables differing from firm to firm.
Kuh [9] and Nerlove [20, pp. 157-1871 treated the intercept as random and the
slopes as fixed parameters in estimating a relationship from a time series of cross
sections. In a recent note, Zellner [17] applied a regression model with random
coefficients to the aggregation problem and showed that for such models and for a
certain range of specifying assumptions, there would be no aggregation bias in the
least squares estimates of coefficients in a macro equation obtained by aggregating
a micro equation over all micro units. Thus the value of specifying a random
coefficient regression (RCR) model may be substantial in econometric work.
Under certain assumptions, we will study RCR models which treat both intercept
and slope coefficients as random variables and will develop appropriate statistical
inference procedures.
For empirical implementation of an RCR model specified in this paper, panel
data provide the base. By panel data we mean temporal cross section data obtained
by assembling cross sections of several years, with the same cross section units
appearing in all years. The cross section units could be households or firms or
any economic units.
The plan of the paper is as follows. In Section 2 we specify the model that we
are analyzing and describe the proposed estimation procedure. For conditions
' This paper forms a part of the author's doctoral dissertation submitted to the University of
Wisconsin. This research was supported by the National Science Foundation under Grant GS-1350.
The author would like to express his indebtedness to Professor A. S. Goldberger, under whose guidance
this research was carried out, and to Professor A. Zellner for helpful comments and correction.
312
P. A. V. B. SWAMY
generally encountered, we propose an estimation procedure that yields consistent
and asymptotically efficient estimators for the parameters of the model. In Section 3
we indicate the procedure of constructing forecast intervals for the predictions of
coefficients. Section 4 is devoted to establishing the properties of estimators constructed in Section 2. In Section 5 we turn to consideration of two asymptotic tests
of linear hypotheses on the parameters of RCR models. In Section 6 we describe an
asymptotic test of equality between coefficient vectors in N relations with heteroskedastic disturbances. Then the estimation and testing procedures presented in
Sections 3-5 are applied in Section 7. Lastly, we present some concluding remarks
in Section 8.
2.
EFFICIENT ESTIMATORS OF RCR EQUATIONS USING PANEL DATA
Consider the following model :
(i
=
1,2, . . . , N).
There are T observations on each of the N individual units. Observed are y i
and Xi for i = 1,2, . . . , N. The matrix Xi (i = 1,2, . . . , N) is of rank A, containing
observations on A nonstochastic regressors, xit,(t = 1,2,. . . T ; ;I. = 1,2,. . . A ) ;
6, and u iare unobserved random vectors.
We assume that for i, j = 1,2,. . . N :
(2.2d)
Piand _uj are independent ;
pi and -pj for i # j are independent
pi = fl + di (i = 1,2,. . . N) where di is a
(2.2e)
-
A x 1 vector of random elements.
Let
~ s s u m ~ t k (2.2b)
n s and (2.2~)
are equivalent to saying that Edi = 0 and
respectively. Assumption (2.2e) implies that di and dj for i # j are independent.
We can write the equation in (2.1) as
RANDOM COEFFICIENT REGRESSION MODEL
or more compactly as
(2.4a)
y=
Xb
+ Dd + _u
where _y = [_y;,_y;,
. . .y;]', X = [ X ; , X i , . . . X&]', d = [d;,di,. . . d&]', _u = [ui,
. . . _u&I1,
the 0's are-T x A null matrices, and D denotes the block-diagonal
matrix on the r.h.s. of (2.4). Under the assumptions (2.2aH2.2e), the N T x 1
disturbance vector Dh: + _u has the covariance matrix
XIAX;
ollI
0
. ..
0
_u;,
(2.5) V(@=
I
+
0
X2AXi + 0
~ 2 1
...
4
0
0
. . . XNAX&+ oNNI
0
where the 0's are T x T null matrices. The matrix V(8) is symmetric of dimension
N T x N T and is a function of the Xi's and an unknown f[A(A + 1) + 2N]element parametric vector 8 containing all the distinct elements of A and aii (i =
1,2,. . . ,N) arranged in an order. We assume that each element of V(8) has continuous first order derivatives. In some :[A(A + 1) + 2N]-dimensional interval
A that contains Q o , the true value of 8, as an interior point, V ( 8 )is positi;e definite
for all 8 E A.
The model (2.1) represents a temporal cross section situation with the subscript
t representing time and the subscript i representing a micro unit. Assumption (a)
implies constant variances and zero covariances from period to period as well as
the absence of any auto or serial correlation of the disturbance terms. The aiiis the
variance of the ith micro unit's disturbance term for any time period. In other
words the disturbances are heteroskedastic and they have different variances for
different micro units. Furthermore, they are structurally and temporally uncorrelated. Assumptions (b) and (c) imply that the regression coefficient vectors Pi
are random and uncorrelated across micro units, but follow the same distribution
with mean fl and variance-covariance matrix A. This distribution is assumed to
be stable over time. Specification of a RCR model is appealing in the analysis of
cross section data since it permits corresponding coefficients to be different for
different individual units. A regression model with random intercept is a particular case of the specification (2.1).
Wald [16] and Hildreth and Houck [7] considered the problem of estimating
parameters in a random coefficient model and suggested some estimators that
require only a single sample of T observations on each of the variables in the
model. Their specifications are different from the specification in (2.2) and in
Swamy [13]. Hildreth and Houck have attempted to estimate 2A parameters with
Tobservations whereas in this paper we attempt to estimate:[A(A + I)] + A + N
parameters with N T observations. If we treat the coefficient vector Pi as fixed
but different for different individual units, then the number of coefficient vectors
to be estimated grows with the number of units in the sample. Data on each individual unit should be used to estimate its own coefficient vector. This difficulty
will not be present if we can make assumptions (b) and (c) in (2.2).
314
P. A.
v.
B. SWAMY
We now regard (2.4a) as a single equation regression model and try to obtain
estimators with desirable properties for the parameters of the model. The parameters to be estimated are the elements of ,8 and 6.
To estimate ,8 we apply Aitken's generalized least squares to (2.4a). Thus a
minimum variance linear unbiased estimator of - is
(2.6)
h ( 8 ) = ( ~ ' ~ ( 8 ) - ~ ~ ) -- ~ ~ ' ~ ( 8 ) - ~ y
Applying the matrix result in Rao [12, p. 29, (2.9)] to (2.6) we have
where
W? =
[il+
(A
o,(X;Xj)-')-'
I-'
(A
+ CTii(XjXi)- '} -
and hi = (XiXi)-'XPi.
The variance-covariance matrix of the estimator h(0) is easily shown to be
(XIV(Q)-'X)-' or
The estimator (2.7) will be recognized as the weighted average of the estimators
= 1,2, . . . ,N) with weights inversely proportional to their covariance matrices.
It is to be noted that (2.7) is equal to the estimator suggested by Rao [ll]if X I =
X2 = . . . = X N .
We shall now assume specific distributions for the variables involved.
hi (i
Under these additional assumptions h(8) is a maximum likelihood estimator
of j. In addition to this property h(8)is also a minimum variance estimator not only
within the class of linear unbiased estimators but also within the class of all
unbiased estimators because the variance of h(8) can easily be shown to be equal
to the Cramer-Rao lower bound (cf. [12, p. 2651).
Despite these optimal properties it is impossible to use (2.7) in practice since
A and oii (i = 1,2,. . . , N) are usually unknown. What we propose to do is to
employ estimators of A and the oii's in constructing the Aitken estimator.
RANDOM COEFFICIENT REGRESSION MODEL
An unbiased estimator of oiiis given by
where M i
=
I - X i ( X i X i ) - ' X i . An unbiased estimator of A is given by
where
Given that we have the estimates sii (i = 1,2,. . . , N ) and d^ we can form the
estimator
where
3.
CONSTRUCTION OF FORECAST INTERVALS
Under the set of assumptions in (2.2)and (2.10),
According to Rao's Lemma 2 [ll], the function hi in (2.7) is the best linear
unbiased predictor of Pi. The prediction error (hi - Pi) = ( X i x i ) -' X i u i , which is a
yi i = u:Mi_uibecause
linear function of ui ,isindependent of the quadratic form y l ~ ( X i x i ) - ' X i M i = 0. It follows from the above argument that
where t is a ( A x 1) vector of arbitrary elements.
- with (1 - E ) probability is given by
A forecast interval for /'pi
where tta is the upper :a point of the t distribution with (T - A ) d.f.
The estimator d^is not a maximum likelihood estimator of A and in some numerical problems it will
yield negative estimates for the variances of coefficients. An instructive numerical example on this
problem is discussed in Swamy [14].
316
P. A .
V. B.
SWAMY
If we set up the the forecast interval (3.3) for each element in Ji, or for different
linear combinations, trJi, we will get many intervals covering their respective
elements of Pi or linear combinations, /'Pi. But these say little about the probability
of the forecast intervals simultaneousl~covering their respective elements of Ji
or linear combinations t t P i . For this we have to construct simultaneous forecast
regions. First recall the familiar formula for the probability that the elements of an
- are contained in an ellipsoid in the A-dimensional -Pi
observational vector on Ji
space. That is,
where Fa is the upper a point of the F distribution with A, T - A d.f.
Using the Cauchy-Schwartz inequality we can derive from (3.4)the simultaneous
- with probability 1 - a :
forecast intervals which cover all linear combinations of Pi
(3.5)
Pr (['Pi
E tlbi
+ ( s ~ ~ A F ~ ~ ' ( x ; x , ) -for' ~ )all+ e) = 1 - a .
For any particular linear combination, d'Ji,
-
so that the confidence coefficient is greater than (1 - a) (cf. [12, p. 1981).
4.
ASYMPTOTIC PROPERTIES OF
E(8)
We assume that
(4.1)
lim T-'XlX,
=
Ci
T-+ m
exists and is nonsingular.
4.1. Consistency of
h(8)
From equations (4.1) and (2.9):
(4.2)
limT+,var [h(B)] = 0.
N+m
Hence plim h(8) = 8.
Following the sime argument as in Goldberger [3, pp. 269-721, we can easily
show that
where
N
*
N
N
RANDOM COEFFICIENT REGRESSION MODEL
and that
(4.5)
Ihi - Pil
plim,,,
=
0,
and
(4.6)
plim,, ,1&8)- mpl = 0
where mp = (l/N)C?= Pi.
Hence, by the invariance property of consistency [3, p. 1301
(4.7)
plim,,
,h(4) =
N- m
a.
Thus, h(8) is a consistent estimator of p. From (4.6) it follows that the asymptotic
distribution of h(8) is the same as the distribution of mB (cf. [12, p. 1011). Since the
variance of the latter is equal to A/N, we can treat &N as the estimated asymptotic
variance-covariance matrix of h(8).
4.2. Asymptotic EfJiciency of h(8)
The consistent estimator h(8) is said to be efficient if
in probability, where P(bl, b2, . . . , hNl_P,A , oii) is the joint probability density of
the random vectors h , . . . hN, and I ( P ) is Fisher's information measure on the
parameter (cf. [12, p. 2851).
That h(8)does indeed satisfy the condition (4.9) can be easily shown as follows.
Under the additional assumptions (2.10a) and (2.10b) we have
(4.10)
1 d log P(h, . . . hNIP,A, oii(i = 1,2,. . .
N)) N[l([)]N
aa
-
and
(4.11)
plim
T+
m
f i (h(8) - a) - (b(0)-
=
l
=
h(0) -
P
0.
Notice here that the definition of asymptotic efficiency given in (4.9) can be
adopted only ifwe know the form of the distribution of the hi's. Without making the
assumptions (2.10a) and (2.10b) we can still show, using (4.2), (4.3), and (4.4), that
(4.12)
plim 1 h(8) - @t))J
T* rn
=
0.
According to a convergence theorem in [12, p. 1011 the asymptotic distribution
of h(8) is the same as that of ti(@),because the difference of these two quantities has
zero probability limit. Since it is known that under general conditions the asymptotic distribution of f i ( h ( 0 ) - 8) is normal, the asymptotic distribution of
fi(h(8) - is also normal. since the variance of the limiting distribution of
p)
318
P. A. V . B. SWAMY
h(8) is equal to AlN, which is the reciprocal of Fisher's information measure on P
in a sample of size N from the population of pi,
- h(8) is the best asymptotical~
normal estimator (cf. [12, p. 2841).
5.
ASYMPTOTIC TESTS OF LINEAR HYPOTHESES ON THE PARAMETERS OF
RCR MODELS
We now require assumptions (2.10a) and (2.10b) to derive the following asymptotic tests.
5.1. Asymptotic Tests of Hypotheses on the Mean of Coeficient Vectors
In this section we suggest a criterion of testing the hypothesis, -j = -P o (a preassigned value), under the assumptions (a) to (e) in (2.2), and the assumptions (a)
and (b) in (2.10).
Using the results in (4.3), (4.4),and ( 4 4 , we can show that
in probability as T -+ co and N is fixed. According to the limit theorem in [12,
p. 1011 the asymptotic distribution of [h(@ - j ] ' ~ ( ~ ) - ' [ h ( 8-) j ] is the same as
the distribution of N(N - l)(mD- j)'SF1(aD [). The distribution of the latter
can be derived following the derivation of Hotelling's distribution in Rao [12,
p. 4581.
Under the assumptions in (2.2) and (2.10) the asymptotic distribution of
--
(5'2)
N-A
A(N - 1)
- [I'~(Q)- ' [ h ( ~-) 01
is F with A, N - A d.f. Since the exact finite sample distribution of (5.2) is not
known, the power ofa test based on (5.2)in small samples cannot be discussed here.
However, asymptotically the test of Ho : j = Po based on (5.2) is equivalent to the
test ofthe same hypothesis based on ~oteiling'sT 2statistic. The optimal properties
of the latter are well known (cf. Anderson [I, pp. 115-1 181).
5.2. Asymptotic Tests of Linear Hypotheses on the Variances of Coeficients
It follows from (4.3) that
in probability as T + cc and N is fixed, where L is a (A x 1)vector offixed elements.
Since SDis distributed as Wishart with parameters N - 1 and A, G'S& is x2a:
distributed with N - 1 d.f., where o f is equal to G'AL (cf. [12, p. 4521). Therefore,
according to the limit theorem in [12, p. 1011, the asymptotic distribution of
RANDOM COEFFICIENT REGRESSION MODEL
(N - l ) t t i f t is x2a: with N - 1 d.f. The statistic
can be used to test the hypothesis that A
6.
=
A , , not all elements of which are zero.
ASYMPTOTIC TESTS OF EQUALITY BETWEEN FIXED COEFFICIENT VECTORS IN
N RELATIONS WITH HETEROSKEDASTIC DISTURBANCES
Before attempting to estimate any model under the set of assumptions in (2.2),
it is better to test whether the coefficient vectors pi (i = 1, 2,. . . N) are fixed and
are all equal. Depending on the outcome of this preliminary test we can decide
whether assumptions (2.2b) to (2.2e) are plausible for the situation under study.
Consider the following hypothesis :
The above hypothesis states that the coefficient vectors, pi, are fixed and the
sample units under study are homogeneous in so far as the Pi's are concerned.
Further if (6.1) is valid, there will be no aggregation bias involved in simple aggregation of linear relations of (2.1) type (cf. Theil [15]). If hypothesis (6.1) is not true,
then the data on all units cannot be pooled to estimate a single relationship
between variables. One way of getting around this difficulty is to make assumptions
(b)to (e)in (2.2),if they are reasonable, and pool the data to estimate a relationship
of the type (2.1).
If the hypothesis (6.1) is true, then the hi (i = 1,2,. . . N) in (2.7) are N unbiased
and independent estimators of the same parametric vector b. The distribution of
(hi - p) is assumed to be NA(O,aii(XiXi)-I).Consider the homogeneity statistic
where
If the hi's do not estimate the same parametric vector, then the differences are
reflected in the statistic HI, and therefore it may be used to test & in (6.1). The
asymptotic distribution of Ho is x2 with A(N - 1) d.f. as T + rn and N is fixed
(cf. [12, pp. 323-241). The asymptotic distribution of Ho/A(N - 1) can also be
approximated by F with A(N - I), N ( T - A) d.f. (cf. Zellner [18]).
7.
APPLICATION OF METHODS IN THE ANALYSIS OF DATA
To illustrate application of the methods discussed above, we utilize the investment equation developed by Grunfeld [6] and described by Boot and de Wit [2],
and Griliches and Wallace [5]. Grunfeld's investment function involves a firm's
320
P. A. V. B. SWAMY
current gross investment, yti, being dependent on the value of firm's outstanding
shares at the beginning ofthe year,
and on its beginning-of-year capital
stock, x ~ l,i.
~ That
- is, the micro investment function for a firm is
with the subscripts i and t denoting an observation for the ith firm in the tth year.
Herein we present estimates of (7.1) for eleven corporations3 in the U.S. by the
method described above. The annual data, 1935-54, for each corporation are
taken from [2].
On the assumption that the same equation of the type (7.1) is applicable to all
corporations in our cross section sample of size eleven, we have a system of eleven
equations, each of which relates to a single corporation. We write the set of eleven
equations in matrix notation as
where -y i is a 20 x 1 vector of observations on yti, X i is a 20 x 2 matrix of observations on xlt-l,i and x ~ ~ -Pi~is, a~ 2, x 1 vector of random coefficients and ui
is a 20 x 1 vector of additive disturbances. The intercept term Boi is eliminated
by expressing the observations on each variable as deviations from their respective
firm means.
Before adopting the random coefficient approach to estimate the model (7.2),
we conduct a preliminary test of the following hypothesis using (6.2):
Hb:pl = -P2 = . . . = PN = P,
(7.3)
where Bi, for every i, is a 2 x 1 vector of fixed coefficients. The value of the statistic
in (6.2)on the basis of our sample data is 14.4521. This value is well above the
five per cent value of F with 20 and 187 d.f. so that the data do not support the
hypothesis that the coefficient vectors in (7.2) are the same for all corporations.
Under these conditions it may be reasonable to assume that the coefficient
vector in (7.2) is random across units. We now make assumptions (a) to (e) in (2.2)
for i, j = 1,2,. . . , 11. Assumption (b) in (2.2) implies that the data on all the 11
corporations contain some information on p, the means of coefficients, and they
can be utilized in estimating P. Adopting the formula (2.13) we have
The figures given in the parentheses are the estimated asymptotic standard
errors of 6,(8) and b2(8). They are obtained by taking the square root of the
diagonal elements of 2/11, where 2 is the estimate of the variance-covariance
2 is obtained by using the formula (2.12):
matrix of the pi's.
The names of these corporations are (1) General Motors, (2) U.S. Steel, (3) General Electric, (4)
Chrysler, (5) Atlantic Refineries, (6) IBM, (7) Union Oil, (8) Westinghouse, (9) Goodyear, (10) Diamond
Match, (11) American Steel Foundries.
RANDOM COEFFICIENT REGRESSION MODEL
321
O n the basis of the asymptotic distribution of d'&, which is derived in (5.4), we
obtain the following interval estimates for the variances of the marginal distributions of pli and pZi:
(7.6) (0.0005 <
< 0.0034)
(0.0091 < a;,, < 0.0575)
where a;, is the variance of the marginal distribution of PIiand
is the variance
of the marginal distribution of PZi.
If the above interval estimates based on a single sample cover the true values of
the variances of coefficients, then they indicate that the true value of a;, is small,
lying between 0.0005 and 0.0034.
We now turn to an application of (5.2) to test the hypothesis
In the present application the value of the statistic (5.2) is 30.8720 and this falls
well above the five per cent value of F with 2 and 9 d.f., so that the data cannot be
regarded as consistent with the hypothesis (7.7).
Aggregating over i we can write the estimated aggregate relationship as
where
and
The above equation indicates that at the aggregate level the value of the firm's
outstanding shares is an important variable in explaining the investment. By
following different approaches Griliches and Wallace [5] and Gould [4] criticized
the theory underlying the model (7.1).
Assuming that the coefficients in a regression equation are random across units
in a temporal cross section analysis we have presented an efficient method of
estimating the mean and the variance-covariance matrix of a distribution which
the coefficient vector follows. We have suggested a consistent and an asymptotically
efficient estimator for the mean, and an unbiased and a consistent estimator for
the variance-covariance matrix. We have developed two asymptotic procedures,
one to test a linear hypothesis on the means of coefficients and another to test a
322 P. A. V. B. SWAMY
linear hypothesis on the variances of coefficients. We have also described an
asymptotic test of equality between fixed coefficient vectors in N relations with
heteroskedastic disturbances.
There is a close connection between RCR models as treated in this paper and a
Bayesian approach to fixed coefficient models. For estimation purposes we treated
the coefficient vector pi as a random variable without specifying the form of its
distribution and built a n estimator for the mean proceeding conditionally upon
a given sample estimate of the variance-covariance matrix v@),
a fact that appears
to explain why our results in the main are large sample results. A Bayesian, while
analyzing a fixed coefficient model, would also assume that the actual value
of the coefficient vector in the sampled population was determined by a random
experiment. In such cases pi is a random variable having a certain a priori distribution. He would put a
pdf on the coefficient vector and combine it with the
likelihood of a sample via Bayes' theorem. He would be able to integrate out the
nuisance parameters and obtain a posterior density for the coefficient vector.
In closing, we note that many, if not all, micro units are heterogeneous with
regard to the regression coefficient vector in a model. If we proceed blithely with
cross section analysis ignoring such heterogeneity, we may be led to erroneous
inferences. Application of the random coefficient approach to situations that satisfy
the specifying assumptions introduced above leads to correct results.
This last conclusion should not be generalized to all situations. Obviously, it
is not hard to imagine situations that do not satisfy the above specifying assumptions. As usual, each situation has to be analyzed carefully.
State University of New York, Buffalo
Manuscript received November, 1967; revision received August, 1968.
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Efficient Inference in a Random Coefficient Regression Model
P. A. V. B. Swamy
Econometrica, Vol. 38, No. 2. (Mar., 1970), pp. 311-323.
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Investment Demand: An Empirical Contribution to the Aggregation Problem
J. C. G. Boot; G. M. de Wit
International Economic Review, Vol. 1, No. 1. (Jan., 1960), pp. 3-30.
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Market Value and the Theory of Investment of the Firm
John P. Gould
The American Economic Review, Vol. 57, No. 4. (Sep., 1967), pp. 910-913.
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Some Estimators for a Linear Model with Random Coefficients
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The Validity of Cross-Sectionally Estimated Behavior Equations in Time Series Applications
Edwin Kuh
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A Note on Regression Analysis
Abraham Wald
The Annals of Mathematical Statistics, Vol. 18, No. 4. (Dec., 1947), pp. 586-589.
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