Question

PLZ HELP!

2. In the simple regression model, show that Bo is consistent under the as- sumption SLR 4.

Answer 1

Assumption SLR.4 (Zero Conditional Mean )

E(u|x) = 0

Cov(x,e) = E(Xu) - E(X)E(u) = E(Xu) =E(E(XU|x))= E (xE(U|x)) = 0

We'll look at β0-y--81z first. The law of large numbers says that y converges to E(y) Ao + gE(z) and if βι-+ A then βι x converges to g E(z). This means βο will be consistent ifa is. Now looking at A and assuming all variances and covariances are finite and well defined we have Cov(y, x) Var(x) Var(x) Cov(E,) Var(x) which equals A so long as Cov(c, z) To prove the stronger claim that the estimators are consistent in mean square we can start with the 2 (XTX)-1 . Here X is the data matrix and variance covariance matrix for (β0, β1) which equals σ for simple linear regression this is just [1; z] where 1 is a vector of ones and z -(xi, x2,..., "n,) is the predictor set. If we go through the linear algebra we get and the denominator 2 long as Ση-1 (2,-2)2-> oo as n-> oo every element of this matrix goes to zero, including the r-nx is nothing but the sum of squares for c. This means that as diagonal elements