I was trying to find the OLS estimator for the model:
$Y$ = $\beta_0$ + $\beta_1X_{1t}$ + $\beta_2X_{2t}$ +.......+ $\beta_5X_{5t}$ + $e$
t = 1,2,3 ......, 50 time ordered observations
X is a full ordered rank matrix with 5 columns
and error term is distributed as:
$e_t = \lambda + \gamma*e_{t-1} + \rho*x_{3t} + u_t$
$u_t = N(0, \sigma^2$)
This is how I was doing it:
$Var(e_t) = Var(\lambda + \gamma*e_{t-1} + \rho*x_{3t} + u_t)=$
$Var(\gamma*e_{t-1} + \rho*x_{3t} + u_t) =$
$\gamma^2*Var(e_{t-1}) + \rho^2*Var(x_{3t}) + \sigma^2 + 2*\gamma*cov(e_{t-1}, u_t) + 2*\rho*\gamma*cov(e_{t-1},x_{3t}) + 2*\rho*cov(x_{3t}, u_t)$
$Cov(e_{t-1}, u_t) = 0$
$Cov(x_{3t}, u_t) = 0$
How do I proceed further?
Since once we have the sample, can we simply say that $X_{3t}$ will be known and hence a constant
so
$Var(x_{3t}) = 0$
$cov(e_{t-1},x_{3t}) = 0$
So, the variance equation is reduced to :
$Var(e_t) =\gamma^2*Var(e_{t-1}) + \sigma^2$ and we just proceed similar to AR(1)
Is this correct or I am awfully wrong here? Please help me out!
Also, how do we correct for mean since the conditional mean independence assumption is violated here. I am not sure how to correct for it