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646 PART THREE: TOPICS IN ECONOMETRICS

All in all, it seems that the investment functions of the four companies are different. This might suggest that the data of the four companies are not "poolable," in which case one can estimate the investment functions for each company separately. (See exercise 16.13.) This is a reminder that panel data regression models may not be appropriate in each situation, despite the availability of both time series and cross-sectional data.

A Caution on the Use of the Fixed Effects, or LSDV, Model. Although easy to use, the LSDV model has some problems that need to be borne in mind.

First, if you introduce too many dummy variables, as in the case of model (16.3.7), you will run up against the degrees of freedom problem. In the case of (16.3.7), we have 80 observations, but only 55 degrees of freedom—we lose 3 df for the three company dummies, 19 df for the 19 year dummies, 2 for the two slope coefficients, and 1 for the common intercept.

Second, with so many variables in the model, there is always the possibility of multicollinearity, which might make precise estimation of one or more parameters difficult.

Third, suppose in the FEM (16.3.1) we also include variables such as sex, color, and ethnicity, which are time invariant too because an individual's sex color, or ethnicity does not change over time. Hence, the LSDV approach may not be able to identify the impact of such time-invariant variables.

Fourth, we have to think carefully about the error term uit. All the results we have presented so far are based on the assumption that the error term follows the classical assumptions, namely, uit ~ N(0, a2). Since the i index refers to cross-sectional observations and t to time series observations, the classical assumption for uit may have to be modified. There are several possibilities.

1. We can assume that the error variance is the same for all cross-section units or we can assume that the error variance is heteroscedastic.

2. For each individual we can assume that there is no autocorrelation over time. Thus, for example, we can assume that the error term of the investment function for General Motors is nonautocorrelated. Or we could assume that it is autocorrelated, say, of the AR(1) type.

3. For a given time, it is possible that the error term for General Motors is correlated with the error term for, say, U.S. Steel or both U.S. Steel and Westinghouse.7 Or, we could assume that there is no such correlation.

4. We can think of other permutations and combinations of the error term. As you will quickly realize, allowing for one or more of these possibilities will make the analysis that much more complicated. Space and mathematical demands preclude us from considering all the possibilities. A somewhat accessible discussion of the various possibilities can be found in

7This leads to the so-called seemingly unrelated regression (SURE) modeling, originally proposed by Arnold Zellner. For a discussion of this model, see Terry E. Dielman, op. cit.

CHAPTER SIXTEEN: PANEL DATA REGRESSION MODELS 647

Dielman, Sayrs, and Kmenta.8 However, some of the problems may be alleviated if we resort to the so-called random effects model, which we discuss next.

Although straightforward to apply, fixed effects, or LSDV, modeling can be expensive in terms of degrees of freedom if we have several cross-sectional units. Besides, as Kmenta notes:

An obvious question in connection with the co variance [i.e., LSDV] model is whether the inclusion of the dummy variables—and the consequent loss of the number of degrees of freedom—is really necessary. The reasoning underlying the covariance model is that in specifying the regression model we have failed to include relevant explanatory variables that do not change over time (and possibly others that do change over time but have the same value for all cross-sectional units), and that the inclusion of dummy variables is a cover up of our ignorance [emphasis added].9

If the dummy variables do in fact represent a lack of knowledge about the (true) model, why not express this ignorance through the disturbance term uit? This is precisely the approach suggested by the proponents of the so-called error components model (ECM) or random effects model (REM). The basic idea is to start with (16.3.2):

Instead of treating 01i as fixed, we assume that it is a random variable with a mean value of 01 (no subscript i here). And the intercept value for an individual company can be expressed as where ei is a random error term with a mean value of zero and variance of a¡2.

What we are essentially saying is that the four firms included in our sample are a drawing from a much larger universe of such companies and that they have a common mean value for the intercept ( = 01) and the individual differences in the intercept values of each company are reflected in the error term ei.

16.4 ESTIMATION OF PANEL DATA REGRESSION MODELS: THE RANDOM EFFECTS APPROACH