Efficiency of FGLS (Feasible GLS) estimator

I hava the following model

$$y=X\beta+u$$

where y and u are $$(n\times 1)$$ a matrix with full column rank and $$\beta$$ is an unknown $$(k\times 1)$$ vector of parameters.

$$E(u)=0$$ and $$var(u)=\sigma^2 V$$ where V is $$(n\times n)$$ symmetric and positive definite non diagonal matrix

And $$u_t=\phi u_{t-1}+v_t$$ and $$u_t=v_t+\theta v_{t-1}$$ where $$\theta$$ is an unknown parameter with $$|\theta |<1$$

I want to discuss the efficiency of the estimator obtained as follows

For that, use Feasible GLS.

(i) I run the regression $$y=X\beta+u$$ to obtain $$\hat{u_t}$$

(ii)by using $$\hat{u_t}$$, I run the auxiliary regression $$u_t=\phi u_{t-1}+v_t$$ to obtain $$\hat{\phi}$$ and $$\hat{v_t}$$

(iii) by using $$\hat{\phi}$$ and $$\hat{v_t}$$, find the variance covariance matrix $$\omega = \sigma^2 \hat{V}$$ and

apply GLS to get $$\hat{\beta_{FGLS}}=(X’\hat{V}^{-1}X)^{-1}X’\hat{V}^{-1}y$$

Please share your ideas about the efficiency of this estimator. I think that we ignore $$u_t=v_t+\theta v_{t-1}$$. Thus, it is not efficient. But how can explain this in correct way?

• Your question suggests that $u_t$ simultaneously is an AR(1) and an MA(1) process, which is impossible. Can you clarify? Do you mean to say that an AR(1) is fitted to the residuals while the true process is MA(1) (or vice versa)? Commented Nov 7, 2022 at 10:02