I have the following econometric model which is known as the koyck distributed lag with an AR(1) error term.
$ (1) ~~ y_t = \beta(1-\rho) \sum_{i=0}^{\infty} \rho^{i}x_{t-i} + v_{t} $ where $v_{t} = \rho v_{t-1} + \epsilon_{t}$.
The model can be re-written as the lagged dependent variable model shown below:
$ (2)~~ y_{t} = \rho y_{t-1} + \beta(1-\rho) x_{t} + \epsilon_t $
Now, assuming the value of $\rho$ is known and fixed and that I have estimates, $\hat{\beta}$ and $\hat{\sigma}^2$, I think that can calculate the cumulative effect L periods ahead of given a new value of $x_{t}$.
Using (2), the cumulative effect L periods ahead, given new value of x_{t} is equal to the cumulative sum of the first term L periods ahead + the cumulative effect of $x_{t}$ L periods ahead. So, using (2) rather than (1) since the error term in (2) is i.i.d ( which makes things easier ), I think the cumulative forecast is just
\begin{equation} \left[ \rho \times y_{t-1} \sum_{i=0}^{L} \rho^{i}\right] + \left[x_{t} \times \hat{\beta}(1-\rho)\sum_{i=0}^{L} \rho^{i}\right] \end{equation}
So, I can calculate the forecast of total cumulative effect either by brute force or by using the formula for the finite sum of a geometric series.
But I don't know how to get the MSE of this forecast. I know the koyck distributed lag literature pretty well but I've never seen an expression for the MSE. I don't think the first term has any variance if $\rho$ is assumed known. But clearly one is assuming that future values of $x_{t}$ are zero which adds some variance and there is estimation error in $\hat{\beta}$. I'm also not even sure if the forecast is unbiased ? Thanks in advance for either a useful reference or maybe it can be derived ? I'm definitely willing to assume that the expected value of future $x_{t}$ values is zero with some variance but I still don't know how to calculate the MSE. Thanks again.
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@richard hardy: you explain things nicely in the link below. Distinguish between short run and long run effects. the only model differences are that I don't have an intercept , your $\beta$ is my $\rho$, and your $\gamma$ is my $\beta \times (1- \rho)$. but that's nitpicky stuff.
The question is different from what you explain because I don't want the infinite effect. I just want the effect for general L. Also, that effect that you describe ( the long run multiplier effect ) doesn't account for the fact that there already may be some "uumph" there already before x_t enters into the picture because $y_{t-1}$ might take on a value. So, that's why I used my poor terminology: forecast of cumulative effect. Finally, how could I calculate the MSE of the effect I'm talking about.
Also, I'm not clear on what wrong bracket you're referring to. My second term is the same as your long run multiplier expression but summed to L rather than to infinity. Thanks a lot.
