Question

2. Consider a simple linear regression model for a response variable Y, a single predictor variable xi, i-1,..., n, and having Gaussian (i.e. normally distributed) errors: -Bai +Ej, Eii.i.d. N(0, σ2) This model is often called "regression through the origin" since EĢ-0 if xi 0. (a) Write down the likelihood function for the parameters β and σ2. (b) Find the MLEs for β and σ2, explicitly showing that they are unique maximizers of the likelihood function (Hint: The function g(x) -log(x) +1- r for r > 0 takes only nonpositive values.) (c) Compute the mean and variance of β, the MLE for β. In your own words, what does the Gauss-Markov theorem tell us about this variance? (d) Let Y.-ßng be the fitted values and ei-Y-Y the residuals. Compute 2 (e) Propose an unbiased estimator for ơ2 based on your answer to part d)

Hi all,

I need help with these questions. Here is my work so far and in b am having trouble showing it is a "unique" maximizer for variance. I would also appreciate it if someone with a good heard can also do the rest of the problems.

Thank you in advance.

Answer 1

8 2 Σ Xi