which implies that with normally distributed disturbance, generalized l; east squares are also mle. as is the classical regression model the mle of gz is biased an unbiased estimator is 04-T-ECY-XBML)Q2-(Y-XBML 3 Estimation when n is Unknown If Q contains unknown parameters that must be estimated, then GLs is not feasible. But with an unrestricted Q, there are T(T+1)/2 additional parameters in o2Q2. This number is far too many to estimate with T observations. Obviousl some structures must be imposed on the model if we are to proceed 3.1 Feasible Generalized Least Squares The typical problem involves a small set parameter 0 such that Q2=S(0). For example, we may assume autocorrelated disturbance in the beginning of this apter as P1 Pr-1 1 a29=a then Q has only one additional unknown parameters, p. A model of heteroscedas- ticity that also has only one new parameters, a, is Definition If Q2 depends on a finite number of parameters, 81, 82,. Bp, and if S2 depends on consistent estimator, 01, 02,. 8,, the Q2 is called a consistent estimator of Q2 Definition: Let @2 be a consistent estimator of Q2. Then the feasible generalized least square estimator(FGLS) of B is B=(X9-1x)-x92-Y.which implies that with normally distributed disturbance, generalized l;east squares are also MLE. As is the classical regression model, the MLE of σ 2 is biased. An unbiased estimator is ˆσ 2 = 1 T − k (Y − XβˆML) 0Ω −1 (Y − XβˆML). 3 Estimation When Ω is Unknown If Ω contains unknown parameters that must be estimated, then GLS is not feasible. But with an unrestricted Ω, there are T(T + 1)/2 additional parameters in σ 2Ω. This number is far too many to estimate with T observations. Obviously, some structures must be imposed on the model if we are to proceed. 3.1 Feasible Generalized Least Squares The typical problem involves a small set parameter θ such that Ω = Ω(θ). For example, we may assume autocorrelated disturbance in the beginning of this chapter as σ 2Ω = σ 2         1 ρ1 . . . ρT −1 ρ1 1 . . . ρT −2 . . . . . . . . . . . . . . . . . . ρT −1 ρT −2 . . . 1         = σ 2         1 ρ 1 . . . ρ T −1 ρ 1 1 . . . ρ T −2 . . . . . . . . . . . . . . . . . . ρ T −1 ρ T −2 . . . 1         , then Ω has only one additional unknown parameters, ρ. A model of heteroscedas￾ticity that also has only one new parameters, α, is σ 2 t = σ 2x α 2t . Definition: If Ω depends on a finite number of parameters, θ1, θ2, ..., θp, and if Ωˆ depends on consistent estimator, ˆθ1, ˆθ2, ..., ˆθp, the Ωˆ is called a consistent estimator of Ω. Definition: Let Ωˆ be a consistent estimator of Ω. Then the feasible generalized least square estimator (FGLS) of β is βˇ = (X0Ωˆ −1X) −1X0Ωˆ −1Y. 7