Eplain how to obtain the weighted least squares wls

Question

Consider the linear regression model

Y_t = β₀ + β1X_1t + β₂X_2t + u_t   t =1,2,...,n,

where Xit and X_2t are non-stochastic stochastic explanatory variables and u_t is a random disturbance such that E(u_t) = 0 for all t.

(a)  Suppose that E(u_tu_s) = 0 for all t ≠ s and E(u_t²) = σ² exp(X_1t + X_2t), where σ² is an unknown positive constant. What are the statistical properties of the OLS estimator of β = ( β₀, β₁, β₂)' in this model? Explain how to obtain the weighted least squares (WLS) estimate of p. What are the statistical properties of WLS estimates?

(b)  Suppose that u_t = Φu_t-4 + ε_t, where Φ is a known parameter and ε_t are random variables such that E(ε_t) = 0, E(ε_t²) = σ_ε² > 0, and E(ε_tε_s) = 0 for all t ≠ s. Explain how to obtain an efficient estimate of β.

(c)  Suppose that the u_t's are homoskedastic and serially uncorrelated, and that X_1t and X_2t are stochastic with E(X_1tu_t) = 0 for all t and E(X_2tu_t)≠ 0. Assuming that three valid instruments (W_1t, W_2t, W_3t) are available, explain how to obtain the instrumental-variables (IV) estimate of β. Would you recommend the use of OLS or IV in this case?