3. Consider the following relationship between the amount of money spent by a st

Question

3. Consider the following relationship between the amount of money spent by a state on welfare programs (y) and the states revenue (x): where D is a dummy variable that takes the value of 1 if the state legislature is controlled by Democrats and 0 otherwise. Your friend however defines another dummy variable R, which takes the value of 1 if the legislature is controlled by Republicans and 0 otherwise, and then estimates the following model instead: (a) Describe step by step how you can obtain estimates of the 's from those of the 's without running another regression. That is, you should derive expressions for the 's in terms of the 's. (HINT: Recognize that R-1-D. Plug this into the second equation and rearrange.) (b) How do the R-squared's compare across the two models?

Explanation / Answer

Here,

y = 0+ 1D + 2x + 3(D x x) + u ....(i)

D = 0 or 1

for republican

so y = 0+ 1 R + 2x + 3(R x x) + u ...(ii)

Putting

R = 1 - D

y = 0+ 1(1- D) + 2x + 3 [(1 -D) x x) + u

y = 0+ 1 - 1 D + 2x +  3 [ x - x X D] + u ....(iii)

comparing (i) and (iii)

0+ 1D + 2 x + 3(D x x) + u = 0+ 1 - 1 D + 2x +  3 [ x - D x x] + u

0= 0+ 1

1  =  - 1

2=  2+ 3

3 = - 3

(b) R -squared compare across the two models will be the same as the independent variables are same here and collinear.

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