Problem set 3 Problem set 3 Buan 6340 Linear algebra 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....

The linear algebra question is to be done on paper and the python programming file should be a .py file not a .pynb. Deadline: 7th March 2020


Problem set 3 Problem set 3 Buan 6340 Linear algebra 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 e′Σ−1e and obtain the optimal estimator. It may be helpful to take the square root Σ, i.e., Σ = Σ1/2Σ1/2. Programming 2. OLS estimation of AR(1): Consider the following AR(1) model: xt = α+ ρxt−1 + et Generate data from this model. Then estimate the model using ordinary least squares. The estimated ρ̂ 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. 1 Linear algebra Programming