Pooling of Time Series and Cross Section Data

Estimates of parameters from cross section data are often introduced into time series
regression as known with certainty, which leads to conditional estimates of the time series
regression. This paper develops a method of pooling cross section and time series data from
the Bayesian point of view, to estimate all the parameters simultaneously. It is shown that
the parameters which are common to both the regressions will have on the average sharper
posterior distributions. It is also demonstrated that the traditional method often leads to
underestimates of the standard errors of the time series estimates. The method is applied to
estimate a statistical demand function for the U.S. based on cross section and time series
data given in Tobin [18].
1. INTRODUCTION
INTHIS PAPER we consider the problem of pooling time series and cross section
data. Several interesting attempts have been made in the past to combine time
series and cross section data to estimate the parameters of a model-see Klein
[6], Marshack [S], Solow [12], Staehle [13], Stone [14], Tobin [IS], and Wold and
Jureen [19]. In all these studies, cross section data are used to obtain estimates of
some of the parameters of the model. These estimates are then introduced into
time series regression as known with certainty to estimate the other parameters
of the model. Thus the estimates obtained from the time series data are conditional
upon the estimates obtained from the cross section data, and hence the time series
regression yields only conditional estimates of the parameters. This fact is not
often mentioned in the literature. Tobin [IS], however, notes that a refinement of
this method is certainly necessary to avoid introducing the estimates of the parameters
as known with certainty. Kuh [7] has also pointed out that cross section
estimates may often be biased and therefore can contaminate the combined
estimates if they are introduced as point estimates. From the sampling theory point
of view, Durbin [3], Theil and Goldberger [16], and Theil [15] have suggested
methods of introducing extraneous information about some coefficients in a
regression model. It would be undoubtedly better and desirable to introduce
information from cross section data into the time series regression employing
these methods, rather than introducing them with certainty. But it is to be noted
that these estimation methods have only an asymptotic justification. In this paper,
a method for combining time series and cross section data is proposed from the
Bayesian point of view. With this method, the coefficients of the cross section and
time series regressions are simultaneously estimated, and exact finite sample results
are obtained. Also it is shown that the posterior distributions of the parameters
common to both time series and cross section regressions will on the average be
sharper than those obtained using the traditional method.