Econ 551b Econometrics II Problem Set 2

Prof. John Rust Due: February 17, 1999

Question 1

Let Y ∼ N (0, Σ) where Y is k ×1 random vector and Σ is k ×k positive definite matrix. Prove Y′Σ^{−1}Y ∼ χ^{2} (k). (Hint: Apply Jordan Decomposition on Σ.) Question 2

The Legendre Polynomials, denoted by P_{0}(x), P_{1}(x), P_{2}(x),..., can be obtained by finding the set of orthogonal vectors that span the same subspace in L^{2}([−1, 1]) as 1, x, x^{2}, x^{3},... Using the Gram-Schmidt Orthogonalization Process, find the first four Legendre Polynomials. Then normalize the Legendre Polynomials so that they are orthonormal. Question 3

Let β̂ = (β̂_{1}, β̂_{2})′ be the OLS estimate in the regression:

y = β_{1}x_{1} + β_{2}x_{2} + ϵ

And \overline{β} = (\overline{β}_{1}, \overline{β}_{2})′ be the estimate from stepwise regression. That is, first obtain \overline{β}_{1} as the OLS estimate from the regression:

y = β_{1}x_{1} + μ

Then obtain \overline{β}_{2} as the OLS estimate from the following regression:

y − P (y|x_{1}) = (x_{2} − P (x_{2}|x_{1}))β_{2} + η

Where P (y|x_{1}) and P (x_{2}|x_{1}) are projections of y and x_{2} on x_{1} respectively, i.e., they are the predicted values in the regression of y and x_{2} on x_{1} respectively. Show that ŷ = x_{1}β̂_{1} + x_{2}β̂_{2} and \overline{y} = x_{1}\overline{β}_{1} + (x_{2} − P (x_{2}|x_{1})\overline{β}_{2} are identical. Suggest when stepwise regression can be useful. Question 4

Suppose (Y, X) ∼ N(0, Σ), show that E(Y |X) = Xβ^{∗}, where β^{∗} = (E(X′X))^{−1}E(X′Y).