Let $n=144$ observations from an $AR(1)$ model $$y_t=\phi y_{t-1}+\epsilon_t$$ where $\epsilon_t$ is White Noise with mean zero and variance $\sigma^2$. If $y_1=-1.7$, $y_{144}=-2.1$, $\sum_{t=2}^n y_t y_{t-1}=-128.6$ and $\sum_{t=1}^n y_t^2=246.4$.

Suppose $\sigma^2$ know, find $Cov(e_n(1),e_n(2))$, where $e_n(j)$ is the forecast error $j$ steps ahead.

This question is related with another that I posted Estimation of $\phi$ in $AR(1)$ process , but I preferred to separate them.

First I found the forecasts $y_t(1)$ and $y_t(2)$ with $t=144$.

$$y_t(1)=E[y_{t+1}|y_1,\dots,y_t]=E[\phi y_t|y_1,\dots,y_t]=\phi y_t$$ $$y_t(2)=E[y_{t+2}|y_1,\dots,y_t]=E[\phi y_{t+1}+\epsilon_{t+2}|y_1,\dots,y_t]=\phi^2 y_t$$

Then the forecast errors are $$e_n(1)=y_{t+1}-y_t(1)=\epsilon_{t+1}$$ $$e_n(2)=y_{t+2}-y_t(2)=\phi y_{t+1}+\epsilon_{t+2}-\phi^2 y_t$$

Finally the covariance is $$Cov(e_n(1),e_n(2))=Cov(\epsilon_{t+1},\phi y_{t+1}+\epsilon_{t+2}-\phi^2 y_t)$$ $$=\phi Cov(\epsilon_{t+1},\phi y_t+\epsilon_{t+1})=\phi Var(\epsilon_{t+1})=\phi \sigma^2$$

Is it right?

NOTE: Sorry if I poste multiple questions about this topic, but I'm learning alone and I don't have solutions manual.

  • 1
    $\begingroup$ The relation to the previous post is very similar. It is the exact same model with the exact same information on the data. The only difference I think is that you are trying to estimate a different parameter. Also you are formulating these problems in terms of checking your work. I don't think that is what this site is for. $\endgroup$ – Michael Chernick Dec 30 '16 at 19:18
  • 2
    $\begingroup$ @MichaelChernick has a point there: where self-study questions are concerned I think we need to draw a line between discussing specific doubts you may have about your approach & providing a general answer-checking service. There's some discussion at A difficulty with self-study-like questions of the form “Is this correct?”. $\endgroup$ – Scortchi Dec 31 '16 at 22:53

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.