Three approaches to the estimation of production functions
The macroeconomic production functions are frequently estimated from the time series of aggregated data. Unfortunately, this approach has not worked very well for the East European countries, because the number of observations is still small and multicollinearity high. For these reasons it is virtually impossible to separate the contribution of technical change (time trend) from the contribution of capital and
The attempts to estimate production functions from the cross-sectional data gave frequently much better results. However, if the cross-section data are available for sectors or industrial branches, but not for individual firms, different kind of objections may be raised. It can be argued that fitting the single production function to the observations related to different sectors or branches implies the assumption that all the sectors or branches have identical production function. This is, however, not so. It is possible to assume that each sector or branch has its own individual parameters of the production function, and that only the mean values of these parameters are estimated.
Let us take, for example, the model CDNRS-NTP-LO and suppose that its parameters are generally different in each sector:
lgY = ai + bilgK + gilgL + ei (I = 1, 2, …n)
The index i here indicates sector. Now, suppose that we have only one observation for each sector. Clearly, in such a case it is impossible to estimate all 3 n parameters a i, b i and g i . It is possible, however, to use the sectoral observations to estimate the mean values a, b and g.
a = (1/n) S ai, b = (Sbi lgK i )/S lgK i, g = (Sgi lgL i )/S lgL i
by applying the least squares method to the regression equation
lgY i = a + blgK i + glgL i + ui
|3 || |
u i = (a i - a) + (bi - b)lgK i + (gi - g)lgL i + ei
If E(ei) = 0, then it follows from (3.2) and (3.4) that E(u i) = 0. For a moderately large number of sectors the distribution of u i may very well be close to normal.
The cross-section approach has one obvious drawback. All the observations relate to the same period of time and, therefore, it is impossible to estimate the change of the total factor productivity. However, if combined with time series analysis, the cross-sectional estimates may help to improve the reliability of the estimated rates of technical progress. Two such combinations are possible:
(1) Parameters of F(K, L) can be estimated first from the cross-section data for some selected year and then inserted into the aggregate production function, so that only parameters of the term A(t) need to be estimated from time series.4 This may help to by step multicollinearity and also to moderately increase the number of observations used in regressions. However, a serious question can be raised: are the parameters estimated from the cross-sectional observations likely to be the same as the "true" parameters of the aggregate production function?
It was shown in (3.2) that the cross-sectional estimates may be interpreted as mean values of individual (sectoral) parameters. The parameters of the aggregate (time series) production function are also a sort of "mean values", but it follows from (3.1) to (3.4) that a, b, g would coincide with true parameters of the aggregate production function only if the aggregation was done by summing the logarithms of observed variables. Because this is not the usual way of aggregation, the discussed way of combining cross-section and time series estimates is inconsistent and should be rejected.
(2) The second way of combination is to estimate simultaneously all the parameters, by running regressions on the time series of crosssectionally disaggregated data. In this case the full matrix of n x T observations - where n is the number of sectors and T is the number of years - must be available. The advantage of such an approach are obvious: The problem of multicollinearity is eliminated, the number of observations is considerably increased, and all the parameters are estimated from the disaggregated data so that the inconsistency is avoided. Unfortunately, the usual assumptions about the random disturbance are not valid in this ease. The residuals from the regressions using this type of combination approach are almost always time-wise autocorrelated and crosssectionally heteroseedastic. This "price" is 'however' not very high, as both autocorrelation and heteroscedasticity are less serious obstacles than multicollinearity.
Let us now formulate more exactly the three possible approaches to the estimation of production functions. Suppose we have all necessary data organized into matrices
Each row of these matrices represents one sector and each column represents one year, so that Yit, Kit , and Lit are output, capital, and labor in the sector i and the year t respectively.
(1) The pure cross-section approach will mean to run separate regressions for each year, i.e. for each column of matrices of observations. For example if the model CDNRS-NTP-LG is to be estimated, then T separate regressions
lgY it = at + bt lgK it + gt lgL it + eit
one for each t will be run. It is assumed, that for each t the distribution of eit is approximately normal and
E(eit) = 0; E(e2it) = s2t; E(eit ejt) = 0 for i ¹ j.
The pure time series approach will mean to run separate regressions for each sector, i.e. for each row of matrices of observations. If the model CDNRSCRTP-LG is to be estimated, n separate regressions
lgY it = ai + bi lgK it + gi lgL it + ri t + eit
one for each i will be run. It is assumed, that for each i eit is approximately normally distributed and
E(eit) = 0; E(e2it) = s2i; E(eit eis) = 0 for s ¹ t.
It is apparent that the pure cross-section approach results in time series of estimated parameters at, bt and gt each of them representing a cross-sectional mean for a given year. The pure time series approach provides in this case sectoral, rather than aggregate production functions. It will result in n separate sets of estimated parameters ai, bi, gi and ri which are supposed to be constant over the whole observed period.
Having these results we can check whether the time series of cross-sectionally estimated parameters of the production function support the assumption of constancy in time. Secondly, we should check, whether the crosssectionally estimated parameters are really mean values of sectoral parameters. Only after such a consistency check we should step forward to the combination.
The combination of cross-section and time series analyses will mean to run one regression over all the observed data. Many variants of the combination approach are possible. In the most simple case only one set of parameters a, b, g and r (model CDNRS-CRTP-LG) is estimated from the regression equation
lgY it = a + b lgK it + g lgL it + r t + eit
In this case a, b, g and r are uniform for all sectors and all years. It will be assumed that
eit = a i + e*it E(e*it) = 0; E(e2it) = s2i; E(eit eis) = 0
The random disturbance is assumed to be cross-sectionally heteroscedastic. It is also likely to be time-wise autocorrelated. In this paper we shall attempt to eliminate the sectoral heteroscedasticity by removing the components a i from residuals.5 This will be done by applying the ordinary least squares method with sectoral dummy variables. In this case a, b, g and r are uniform for all sectors and all years. It will be assumed that
eit = a i + e*it E(e*it) = 0; E(e2it) = s2i; E(eit eis) = 0
The random disturbance is assumed to be cross-sectionally heteroscedastic. It is also likely to be time-wise autocorrelated. In this paper we shall attempt to eliminate the sectoral heteroscedasticity by removing the components a i from residuals.5 This will be done by applying the ordinary least squares method with sectoral dummy variables.