Reference no: EM132435641

Assume that the population regression function is

Y_i = βX_i + ε_i.

where X and Y and ε are defined above. This is a regression through the origin (no intercept).

A. Under the homoskedastic normal regression assumptions, the t-statistic will have a Student t distribution with n-1 degrees of freedom (not n-2 degrees of freedom). Explain.

B. Will the residuals sum to zero in this case? Explain and show your derivations. (Extra 10 points: show this - and explain - with real or simulated data.)

Consider the following simple regression model
y_t = β₁ + β₂t + ε_t;  t = 1, 2, 3, 4, 5, 6

where E[ε_t] = 0, E[ε_t²] = σ²t², and E[ε_t ε_s]=0 for t different than s, and let ε = (ε₁, ε₂, ε₄, ε₅, ε₆), and E[εε'] = σ²ψ.

A. Specify the covariance ψ.
B. Determine ψ^-1.
C. Determine the covariance matrix of the Least Squares estimator for β' = (β₁ , β₂).
D. Determine the covariance matrix of the Generalized Least Squares (GLS) estimator for β.
E. Calculate E[σ²] where σ² = (y - Xb)'(Y - Xb)/3 , and b is the LS estimator for β.