Estimation of Elasticity of Substitution in American Manufacturing Industry from Pooled Cross-Section and Time-Series Observations - Blair, Roger D.; Kraft, John (The Review of Economics and Statistics, 1974)
IN 1961, Arrow, Chenery, Minhas, and Solow (ACMS) (1961) introduced their now familiar production function V y[8 K-P + (1-8)L-P]-1/P (1.) where V is value added per man-year, K is capital, L is man-years of labor, and y, 8, and p are the efficiency, distribution, and substitution parameters, respectively. It is well known that the elasticity of substitution, 1/(1 + p), can be estimated by estimating b in the profit maximizing conditi'on log (V/L) _log a + b log w + u (2) where w is the annual wage rate of production workers. fact, ACMS obtained very good results by using international data from 19 countries for various census years between 1949 and 1955. These data represented up to 24 ISIC industries at the three-digit level. 1963, C. E. Ferguson (1963) used U.S. Census of Manfactures data to fit the regression equation (2). Whereas ACMS obtained good results, Ferguson was disappointed in his: In the entire list of 129 items, R2 is significant at P < .05 in only 50%o of the cases. The bcoefficient is significant 70% of the time . . . But in more than half of these, b was not found to be significantly different from one (1963, p. 306). Ferguson recognized a possible reason for such results. The requires different relative factor prices for different observations. With only a little variation in the wage rate, the regression coefficients will have large standard errors. Unfortunately, when Ferguson's paper appeared there was no way to correct or improve the sample. Now, it is possible to pool time-series and cross-sectional data and to recognize the possibility of cross-sectional heteroscedasticity and time-wise autoregression of the disturbance terms. Jan Kmenta has termed this a cross-sectionally heteroscedastic and time-wise autoregressive model (1971, p. 509). We shall use this to estimate the elasticity of substitution, b, in a modification of regression equation (2). We expect that the increased variability of the independent variable will improve the results. This system of production functions for various indtustries provides a classic example of a case where the method of seemingly unrelated regressions may be applied. Thus, we also obtain two-stage Aitken estimates of the elasticity of substitution' using the pooled data. Estimation of the elasticity of substitution by pooling time-series and cross-section data requires a modification of regression equation (2). As the regression stands, there is an implicit assumption of no technological progress over time. This assumption is removed by specifying the as log (V/L)_ log a + b logw + c2T2 + C3T3 + C4T4 + u (3) where T2, T3, and T4 are dummy variables representing the years 1958, 1963, and 1967, respectively.' The introduction of the dummy variables into equation (3) allows for the possibility of technological progress in each of the cross-section years of 1958, 1963, and 1967. Although this results in a loss of three degrees of freedom, it does allow our to capture the influence of technological progress. The Received for publication August 8, 1973. Revision received for publication October 12, 1973. * We have benefited greatly from the comments offered by our colleagues,. David Denslow and Frank Sloan. The encouragement and helpful suggestions of the late C. E. Ferguson, Jan Kmenta, John Moroney, and an anonymous referee are gratefully acknowledged. Jerry R. Jackson provided invaluable assistance in the computations. Of course, we must exonerate everyone but ourselves of all blame for what follows. An earlier version of this paper was presented at the annual meetings of the Econometric Society in December 197 1. 1 We are indebted to an anonymous referee for suggesting this means of accounting for technological progress.