16-10. What if instead of regressing a dependent variable Y on an explanatory variable X, we regress
Question:
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
Step by Step Answer:
Advanced Statistics From An Elementary Point Of View
ISBN: 9780120884940
1st Edition
Authors: Michael J Panik