Question 2

Information:

The expected change in Y,ΔY, associated with the change in X1,ΔX1, holding X2,...,Xk constant, is the difference between the value of the population regression function before and after changing X1, holding X2, . . . , Xk constant.

That is, the expected change in Y is the difference:

ΔY =f(X1 +ΔX1,X2,...,Xk)−f(X1,X2,...,Xk)

The estimator of this unknown population difference is the difference between the predicted

ˆ

values for these two cases. Let f (X1, X2, . . . , Xk) be the predicted value of Y based on the

ˆ

estimator f of the population regression function. Then the predicted change in Y is

ˆˆˆ

ΔY =f(X1 +ΔX1,X2,...,Xk)−f(X1,X2,...,Xk)

Question Starts here:

Consider the regression model Yi = β0 + β1X1i + β2X2i + β3 (X1i × X2i) + ui. Using the given information above, show:

a. ΔY/ΔX1 = β1 + β3X2 ( effect of change in X1, holding X2 constant).

b. ΔY/ΔX2 = β2 + β3X1 ( effect of change in X2, holding X1 constant).

c. If X1 changes by ΔX1 and X2 changes by ΔX2, then ΔY = (β1 + β3X2) ΔX1 + (β2 + β3X1) ΔX2 + β3ΔX1ΔX2 Write down everything clearly, show your work in detail.

Data