7.9Consider the regression modelYi=b0+b1X1i+b2X2i+ui. Use approach 2

from Section 7 .3 to transform the regression so that you can use at-statistic to test

Approach 2: Transform the regression.If your statistical package cannot test the restriction

directly, the hypothesis in Equation (7 .16) can be tested using a trick in which the

original regression equation is rewritten to turn the restriction in Equation (7 .16) into a

restriction on a single regression coefficient. To be concrete, suppose there are only two

regressors,X1iandX2i, in the regression, so the population regression has the form

Yi=b0+b1X1i+b2X2i+ui. (7 .17)

Here is the trick: By subtracting and addingb2X1i, we have thatb1X1i+b2X2i=

b1X1i-b2X1i+b2X1i+b2X2i=1b1-b22X1i+b21X1i+X2i2=g1X1i+b2Vi,

whereg1=b1-b2andVi=X1i+X2i. Thus the population regression in Equation

(7 .17) can be rewritten as

Yi=b0+g1X1i+b2Vi+ui. (7 .18)

Because the coefficientg1in this equation isg1=b1-b2, under the null hypothesis

in Equation (7 .16)g1=0, while under the alternativeg1_0. Thus, by turning

Equation (7 .17) into Equation (7 .18), we have turned a restriction on two regression

coefficients into a restriction on a single regression coefficient.

a.b1=b2.

b.b1+2b2=0.

c.b1+b2=1. (Hint:You must redefine the dependent variable in the

regression.)

On the website http://www.pearsonhighered.com/stock_watson/, you will find the data file "Earnings_and_Height" that contains the data, to do the exercise.