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APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL 945

Applying (7.8.4) the adjusted coefficient of determination can be seen to be

Collecting our results thus far, we have

Y = 300.28625 + 0.74198X2i + 8.04356X3i-(78.31763) (0.04753) (2.98354)

t = (3.83421) (15.60956) (2.69598) R2 = 0.99761 R2 = 0.99722 df = 12

The interpretation of (C.10.14) is this: If both X2 and X3 are fixed at zero value, the average value of per capita personal consumption expenditure is estimated at about \$300. As usual, this mechanical interpretation of the intercept should be taken with a grain of salt. The partial regression coefficient of 0.74198 means that, holding all other variables constant, an increase in per capita income of, say, a dollar is accompanied by an increase in the mean per capita personal consumption expenditure of about 74 cents. In short, the marginal propensity to consume is estimated to be about 0.74, or 74 percent. Similarly, holding all other variables constant, the mean per capita personal consumption expenditure increased at the rate of about \$8 per year during the period of the study, 1956-1970. The R2 value of 0.9976 shows that the two explanatory variables accounted for over 99 percent of the variation in per capita consumption expenditure in the United States over the period 1956-1970. Although R2 dips slightly, it is still very high.

Turning to the statistical significance of the estimated coefficients, we see from (C.10.14) that each of the estimated coefficients is individually statistically significant at, say, the 5 percent level of significance: The ratios of the estimated coefficients to their standard errors (that is, t ratios) are 3.83421, 15.61077, and 2.69598, respectively. Using a two-tail t test at the 5 percent level of significance, we see that the critical t value for 12 df is 2.179. Each of the computed t values exceeds this critical value. Hence, individually we may reject the null hypothesis that the true population value of the relevant coefficient is zero.

As noted previously, we cannot apply the usual t test to test the hypothesis that \$2 = \$3 = 0 simultaneously because the t-test procedure assumes that an independent sample is drawn every time the t test is applied. If the same sample is used to test hypotheses about \$2 and \$3 simultaneously, it is likely that the estimators /32 and ¡33 are correlated, thus violating the assumption underlying the t-test procedure.9 As a matter of fact, a look at the variance-covariance matrix of (3 given in (C.10.9) shows that the estimators ji2 and /S3 are negatively correlated (the covariance between the two is —0.13705). Hence we cannot use the t test to test the null hypothesis that \$2 = \$3 = 0.

9See Sec. 8.5 for details.

946 APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL

But recall that a null hypothesis like 02 = p3 = 0, simultaneously, can be tested by the analysis of variance technique and the attendant F test, which were introduced in Chapter 8. For our problem, the analysis of variance table is Table C.5. Under the usual assumptions, we obtain