Experimental Evidence on Combining Cross-Section and Time Series Information - Johnson, K. H.; Lyon, H. L. (The Review of Economics and Statistics, 1973)
IN recent years several research studies have used a data base consisting of a time series of cross-section samples. A primary reason is that panel data of this type are potentially richer in information than a single cross-section sample. To date, however, the question of how to best analyze data bases of this type has not been fully explored in any of the research. To illustrate, a number of prior research projects which have utilized this type of data base are briefly reviewed. Hoch (1962) used moving cross-section samples as the data base for estimating the parameters of a CobbDouglas production function by analysis of covariance. Specifically the data were collected on 63 Minnesota farms for the years 1946 to 1951. Hoch reported an observed difference between the least squares parameter estimates and covariance estimates and the elasticities developed from these estimates. In terms of method, the major conclusion was that the covariance model might produce less biased elasticities and marginal return estimates. Massy and Frank (1965) investigated the relationship between price changes and dealing activities on a -firm's market share for frequently purchased household and food products. Panel data covering a 101-week time period of family purchase history provided the data base for the study. However, the data were aggregated so no methodological insight could be inferred concerning the question of analyzing time series of cross-section data. Laughhunn and Lyon (1971) applied Bayesian regression in analyzing a time series of cross-section cigarette consumption data using the Tiao and Zellner (1964) approximation method. The primary methodological issue in this research was to observe differences that might exist between classical pooling and the Bayesian regression technique. Comparison of the two techniques revealed very little difference between either parameter estimates or standard errors. Schipper (1964) used covariance regression to analyze a series of cross-section samples (19541957) collected by the Survey Research Center, University of Michigan. The central focus of this study was to analyze consumer discretionary behavior particularly with respect to durable expenditures, short term debt, and discretionary saving. The major methodological finding was that several differences between the covariance regression model and individual cross-section regressions existed. Schipper suggested the individual cross-section analyses might be biased but could not prove this point since he did not use experimental data. Palda and Blair (1970) conducted an analysis of toothpaste demand by using multiple cross sections of data collected by MRCA during the period 1958-1962. One focus of their research was to investigate the potential cross-section specification bias, based on the rationale presented by Simon and Aigner (1970), that can exist because of omitted variables. An interpretation of the results led them to think that the covariance model may reduce the specification bias. This interpretation cannot be considered conclusive since the analysis was not conducted in an experimental framework. Since there is an interest on the part of economic and business researchers to use multiple cross-section sample data, this would appear to be a sufficient reason for evaluating the different methods available for combining and analyzing the samples. Earlier work in this area includes studies by Nerlove (1967, 1968). He assumed models of the form Yit = aYit-l + Uit and Yit = aYit-l + 1-Xit + Uit respectively with Uit = yi + Vit with yi and Vit uncorrelated where =o-2 = 2 +or2. The estimation methods used were OLS, generalized least squares utilizing known p (p = o-A2/o-X2) analysis-of-covariance estimates with cross-sectional effects only, two-round estimates based on an estimated value of p, and maximum likelihood estimates. Generally, Nerlove's findings indicated that generalized least squares (if p is known) produces good esti-