Three variables n, d, and y , all have zero sample means and unit sample variances. a fourth variable is c = n + d. in the regression of c on y , the slope is 0.8. in the regression of c on n, the slope is 0.5. in the regression of d on y the slope is 0.4. what is the error sum of squares in the regression of c on d? there are 21 observations.

Error sum of squares  SSE = 15

Step-by-step explanation:

Given:

C = N + D

Var C = Var N + Var D + 2 Cov(N,D)

          = 2(1+ Cov(N,D))

from 2 simple regreesion

however

Cov(c,y) = Cov(N +  D, y) = Cov(N,y) + Cov(D,y)

AND

Cov(C,N) =Cov(N +  D, N) =  Var N + Cov(D,N) = 0.5

Cov(D,N) = -0.5

therefore

Var(C) = 2(1-0.5) = 1

ALSO

Cov(C,D) = Cov( N+D,D)

               = Cov( N,D) +  Var(D)

               = -0.5 + 1 = 0.5

Slope of C on D = = 0.5

finaly we have

Error sum of square (SSE)  = SST -SSR

       = (n-1) Sc^2 - slope^2(n-1) Sd^2

       =20(1)^2 - 0.5^2(20)(1)

SSE = 15