CA Foundation · Quantitative Aptitude · Correlation and Regression
If every value of X is multiplied by 5 and then increased by 3, while Y is left unchanged, what happens to the regression coefficient of Y on X?
The regression coefficient of Y on X becomes one-fifth of its original value. Adding 3 is a change of origin and has no effect, but multiplying X by 5 is a change of scale, which multiplies the covariance by 5 and the variance of X by 25.
- AIt becomes one-fifth of its original valueCorrect
- BIt becomes five times its original value
- CIt remains unchanged
- DIt increases by 3
Explanation
Regression coefficients are independent of change of origin but not of scale. With u = 5x + 3, byu = Cov(y,u)/Var(u) = 5Cov(y,x)/(25 Var(x)) = byx/5. Thus the coefficient becomes one-fifth. Saying it is unchanged ignores the scale change.
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