CHAPTER TEN: MULTICOLLINEARITY 375

is highly correlated with the rest of the X's. Thus, one may drop that Xi from the model, provided it does not lead to serious specification bias.

4. Detection of multicollinearity is half the battle. The other half is concerned with how to get rid of the problem. Again there are no sure methods, only a few rules of thumb. Some of these rules are as follows: (1) using extraneous or prior information, (2) combining cross-sectional and time series data, (3) omitting a highly collinear variable, (4) transforming data, and (5) obtaining additional or new data. Of course, which of these rules will work in practice will depend on the nature of the data and severity of the collinearity problem.

5. We noted the role of multicollinearity in prediction and pointed out that unless the collinearity structure continues in the future sample it is hazardous to use the estimated regression that has been plagued by multi-collinearity for the purpose of forecasting.

6. Although multicollinearity has received extensive (some would say excessive) attention in the literature, an equally important problem encountered in empirical research is that of micronumerosity, smallness of sample size. According to Goldberger, "When a research article complains about multicollinearity, readers ought to see whether the complaints would be convincing if "micronumerosity" were substituted for "multicollinearity."46 He suggests that the reader ought to decide how small n, the number of observations, is before deciding that one has a small-sample problem, just as one decides how high an R2 value is in an auxiliary regression before declaring that the collinearity problem is very severe.

10.1. In the k-variable linear regression model there are k normal equations to estimate the k unknowns. These normal equations are given in Appendix C. Assume that Xk is a perfect linear combination of the remaining X variables. How would you show that in this case it is impossible to estimate the k regression coefficients?

10.2. Consider the set of hypothetical data in Table 10.10. Suppose you want to fit the model

a. Can you estimate the three unknowns? Why or why not?

b. If not, what linear functions of these parameters, the estimable functions, can you estimate? Show the necessary calculations.

376 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL

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