16-10. What if instead of regressing a dependent variable Y on an explanatory variable X, we regress the Z-scores of Y on the Z-scores of X? From Yi = β0 + β1Xi + εi and Y = β0 + β1

X + ¯ε, Yi − Y

σY

= β1 Xi − X

σX

σX

σY

+ εi − ¯ε

σY or ZiY = β1σX

σY

ZiX + ui = β

∗

1ZiX + ui. (16.E.1)

Applying ordinary least squares to (16.E.1) renders

ˆZ iY = ˆ β

∗

1ZiX .

(What assumptions must ui satisfy?)

The quantity ˆ β

∗

1 is termed a standardized regression coefficient or simply a beta coefficient. So although the usual regression slope ˆ β1 represents the average rate of change in Y per unit change in X, we see that the beta coefficient ˆ β

∗

1 measures the said changes in standard deviation units; that is, when X increases by one standard deviation, Y changes by ˆ β

∗

1 standard deviations on the average. Hence the beta coefficient is independent of units. Note that we can easily recover the usual regression slope as ˆ β1 =

ˆ β

∗

1 (σY/σX). A glance at the structure of ˆ β

∗

1 reveals that it is simply the estimated coefficient of correlation between the variables X and Y.

For the following data set:

(a) Transform the X and Y variables to Z-scores.

(b) Use least squares to regress the Y Z-score on the X Z-score with the intercept suppressed.

(c) From ˆ β

∗

1 determine ˆ β1.

(d) Calculate the coefficient of correlation between X and Y directly and verify that it equals ˆ β

∗

1 .

X 1 2 3 4 5 6 7 8 Y 3 4 3 5 6 5 4 7