Question

Q6). Suppose that you want to fit two separate regression lines on the same data set - For the first least square fit, Y is the response variable and X is the predictor variable For the second least square fit, X is the response variable and Y is the predictor variable. (a). Show that the product of the slope estimates from the two regression lines is Show that the above two regression lines will never be perpendicular to each other (b).

Answer 1

a)

when Y is regressed on X

then slope =

Cov(X, Y) mi = Var(X

when X is regressed on Y

Cov(X, Y) m2 =-Var(Y)

product =

Cov(X, Y) Cov(X, Y) mmVarX) Var(Y

$Cov(X,Y) = \frac{\sum (X_i - \overline{X})(y_i - \overline{Y})}{n-1}$

Var(X)

Var(Y)-n

hence

m1 m2 is same as shown in image

b)

here m1m2 >= 0

for perpendicular

m1 m2 should be -1

hence two lines will never be perpendicular