Standardized regression coefficients:74 (a) *Show that the standardized coefficients can be computed as b* R(1 XX rXy, where RXX is the correlation matrix of

Standardized regression coefficients:74

(a) *Show that the standardized coefficients can be computed as b* ¼ R(1 XX rXy, where RXX is the correlation matrix of the explanatory variables, and rXy is the vector of correlations between the explanatory variables and the response variable. [Hints: Let ZX

ðn · kÞ

[ fðXij ( XjÞ=Sjg contain the standardized explanatory variables, and let zy

ðn · 1Þ

[ fðYi ( YÞ=SY g contain the standardized response variable. The regression equation for the standardized variables in matrix form is zy ¼ ZX b* þ e*. Multiply both sides of this equation by Z0 X =ðn ( 1Þ.]

(b) The correlation matrix in Table 9.2 is taken from Blau and Duncan’s (1967) work on social stratification. Using these correlations, along with the results in part (a), find the standardized coefficients for the regression of current occupational status on father’s education, father’s occupational status, respondent’s education, and the status of the respondent’s first job. Why is the slope for father’s education so small? Is it reasonable to conclude that father’s education is unimportant as a cause of the respondent’s occupational status (recall Section 6.3)?

(c) *Prove that the squared multiple correlation for the regression of Y on X1; ... ; Xk can be written as R2 ¼ B*

1rr1 þ###þ B*

k rrk ¼ r0 yX b*

[Hint: Multiply zy ¼ ZX b* þ e* through by z0 y=ðn ( 1Þ.] Use this result to calculate the multiple correlation for Blau and Duncan’s regression.