# 2 X2it 3 X3it

8Dielman, op. cit., Sayrs, op. cit., Jan Kmenta, Elements of Econometrics, 2d ed., Macmillan, New York, 1986, Chap. 12.

648 PART THREE: TOPICS IN ECONOMETRICS

where

The composite error term wit consists of two components, si, which is the cross-section, or individual-specific, error component, and uit, which is the combined time series and cross-section error component. The term error components model derives its name because the composite error term wit consists of two (or more) error components.

The usual assumptions made by ECM are that

E(siUit) = 0 E(si Sj) = 0 (i = j) E(uituis) = E(uitujt) = E(uitujs) = 0 (i = j; t = s).

that is, the individual error components are not correlated with each other and are not autocorrelated across both cross-section and time series units.

Notice carefully the difference between FEM and ECM. In FEM each cross-sectional unit has its own (fixed) intercept value, in all N such values for N cross-sectional units. In ECM, on the other hand, the intercept ft represents the mean value of all the (cross-sectional) intercepts and the error component Si represents the (random) deviation of individual intercept from this mean value. However, keep in mind that Si is not directly observable; it is what is known as an unobservable, or latent, variable.

As a result of the assumptions stated in (16.4.5), it follows that

Now if af = 0, there is no difference between models (16.2.1) and (16.4.3), in which case we can simply pool all the (cross-sectional and time series) observations and just run the pooled regression, as we did in (16.3.1).

As (16.4.7) shows, the error term wit is homoscedastic. However, it can be shown that wit and wis (t = s) are correlated; that is, the error terms of a given cross-sectional unit at two different points in time are correlated. The correlation coefficient, corr (wit, wis), is as follows:

Notice two special features of the preceding correlation coefficient. First, for any given cross-sectional unit, the value of the correlation between error terms at two different times remains the same no matter how far apart the

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two time periods are, as is clear from (16.4.8). This is in strong contrast to the first-order [AR(1)] scheme that we discussed in Chapter 12, where we found that the correlation between time periods declines over time. Second, the correlation structure given in (16.4.8) remains the same for all cross-sectional units; that is, it is identical for all individuals.

If we do not take this correlation structure into account, and estimate (16.4.3) by OLS, the resulting estimators will be inefficient. The most appropriate method here is the method of generalized least squares (GLS).

We will not discuss the mathematics of GLS in the present context because of its complexity.10 Since most modern statistical software packages now have routines to estimate ECM (as well as FEM), we will only present the results for our investment example. But before we do that, it may be noted that we can easily extend (16.4.4) to allow for a random error component to take into account variation over time (see exercise 16.6).

The results of ECM estimation of the Grunfeld investment function are presented in Table 16.3. Several aspects of this regression should be noted. First, if you sum the random effect values given for the four companies, it will be zero, as it should (why?). Second, the mean value of the random error component, ei, is the common intercept value of -73.0353. The random effect value of GE of —169.9282 tells us by how much the random error component of GE differs from the common intercept value. Similar interpretation applies to the other three values of the random effects. Third, the R2 value is obtained from the transformed GLS regression.

If you compare the results of the ECM model given in Table 16.3 with those obtained from FEM, you will see that generally the coefficient values of the two X variables do not seem to differ much, except for those given in Table 16.2, where we allowed the slope coefficients of the two variables to differ across cross-sectional units.