Exercises

C.1. For the illustrative example discussed in Section C.10 the X'X and X'y using the data in the deviation form are as follows:

1,103,111.333 16,984

955,099.333 14,854.000

16,984 280

a. Estimate and 03.

b. How would you estimate ?

c. Obtain the variance of ji2 and ji3 and their covariances.

d. Obtain R2 and R2.

e. Comparing your results with those given in Section C.10, what do you find are the advantages of the deviation form?

C.2. Refer to exercise 22.23. Using the data given therein, set up the appropriate (X'X) matrix and the X'y vector and estimate the parameter vector p and its variance-covariance matrix. Also obtain R2. How would you test

10In these days of high-speed computers there may not be need for the deviation form. But it simplifies formulas and therefore calculations if one is working with a desk calculator and dealing with large numbers.

950 APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL

the hypothesis that the elasticities of Ml with respect to GDP and interest rate R are numerically the same? C.3. Testing the equality of two regression coefficients. Suppose that you are given the following regression model:

Yi = ft + p2 X2i + ft X3i + u and you want to test the hypothesis that p2 = p3. If we assume that the ui are normally distributed, it can be shown that t =

follows the t distribution with n — 3 df (see Section 8.6). (In general, for the k-variable case the df are n — k.) Therefore, the preceding t test can be used to test the null hypothesis ft = ft •

Apply the preceding t test to test the hypothesis that the true values of P2 and f)3 in the regression (C.10.14) are identical. Hint: Use the var-cov matrix of p given in (C.10.9).

C.4. Expressing higher-order correlations in terms of lower-order correlations. Correlation coefficients of order p can be expressed in terms of correlation coefficients of order p — 1 by the following reduction formula:

r1 1.345... (p-1) — [r1 p.345...(p-1)r2p.3 45...(p-1)]

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