Estimate the model using ordinary least squares

Question

Assignment Part 1: General Least Squares:

Suppose we are estimating a linear model:

y = Xβ + e

However instead of the V ar(e) = σ²In, we know that V ar(e) = Σ. Derive the optimal OLS estimator given this condition. That is, minimize erΣ-1e and obtain the optimal estimator. It may be helpful to take the square root Σ, i.e., Σ = Σ^1/2Σ^1/2.

Programming

Assignment Part 2: OLS estimation of AR(1):

Consider the following AR(1) model:

x_t = α + ρx_t-1 + e_t

Generate data from this model. Then estimate the model using ordinary least squares. The estimated ρˆ that you will find will be biased. Write a simulation to study this bias. See if you can find the functional form of the bias via simulation.

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