Reference no: EM132793750

Problem (a) Define the ML estimator. What are its asymptotic properties? What do you understand by the invariance principle?

Problem (b) Consider the multiple linear regression model y=Xβ0+u, with u∼N(0,σ^2I).
Given a sample of dimension T, derive the MLE of β0 and σ0. Clearly state any assumption you are making.

Problem (c) Derive the asymptotic covariance matrix of the MLE β,σˆ . Write down the asymptotic variance of σˆ2 and describe how can one use it to construct a confidence interval for σ0 ^2.

Problem (d) Does it turn out that the asymptotic covariance matrix of the MLE β ,σˆ2 is diagonal? What is the main implication ?

Problem (e) Compare the MLE σˆ2 with the OLS estimator s2 = e′e/(T - k) where e = y - Xβ, T is sample size and k is the rank of X. What are the differences and analogies ?