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Let $y_1,y_2,\dots,y_{10}$ be a time series generated by $$y_t=\delta+\epsilon_t+\theta\epsilon_{t-2}$$ where $\epsilon_t$ is white noise with $E[\epsilon_t]=0$, $Var(\epsilon_t)=\sigma^2$ and $\delta$, $\theta$ are unknown parameters.

a) Propose a estimator for $\delta$ and find the variance of this estimator.

I don't know well how to do the estimation in moving average models (MA), but I think that the process converge to the mean. $$y_t\rightarrow \mu$$ $$y_t\rightarrow \delta$$ Then the estimator is $$\hat{\delta}=\frac{1}{n}\sum_{t=1}^n y_t$$

$$Var(\hat{\delta})=Var\Big(\frac{1}{n}\sum_{t=1}^n y_t\Big)=\frac{n \mathrm{\gamma}(0) + \sum_{t=1}^{n-1} 2(n-t) \mathrm{\gamma}(t)} {n^2}$$

where

$$\gamma(h)=Cov(y_t,y_{t-h})=Cov(\delta+\epsilon_t+\theta\epsilon_{t-2},\delta+\epsilon_{t-h}+\theta\epsilon_{t-h-2})$$ $$=Cov(\epsilon_t,\epsilon_{t-h})+\theta Cov(\epsilon_t,\epsilon_{t-h-2})+\theta Cov(\epsilon_{t-2},\epsilon_{t-h})+\theta^2Cov(\epsilon_{t-2},\epsilon_{t-h-h})$$ $$\gamma(h)=\begin{cases}\sigma^2(1+\theta^2) \qquad h=0\\ \theta\sigma^2 \qquad h=2 \\ 0 \qquad otherwise\end{cases}$$

Is it right? Is there a way to find explicity the mean and $\theta$ estimator at hand?

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    $\begingroup$ I think this is tricky: Note e_(t-1) is missing. So this is like a second order MA process with one parameter to be restricted to 0. There are issues relating to stationarity and finite moving averages have a autoregressive representation with an infinite number of parameters. So you can't rely on the AR(1) model to estimate the variance. Also in general there are restrictions on the parameters for stationarity and invertibility. Maybe you should look at Box and Jenkins classic book. $\endgroup$ – Michael Chernick Dec 23 '16 at 21:36
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    $\begingroup$ If you search google using box-jenkins method, the wikipedia article cites the first edition (1970) of Box and Jenkins book along with three other useful references. $\endgroup$ – Michael Chernick Dec 23 '16 at 21:42
  • $\begingroup$ @MichaelChernick Is necessary find the autoregressive representation to estimate the variance? I can't just use the covariance properties to find this variance, since that I have just $y_1,\dots,y_{10}$? $\endgroup$ – user72621 Dec 23 '16 at 23:04
  • $\begingroup$ You can't estimate infinitely many AR parameters based on an n=10. My point is conceptually that the AR(1) model is not the way to approach estimating the process variance. $\endgroup$ – Michael Chernick Dec 23 '16 at 23:17
  • $\begingroup$ @MichaelChernick It is not a general approach for any process? I mean, it's not just a variance propertie? $Var(\hat{\delta})=Var\Big(\frac{1}{n}\sum_{t=1}^n y_t\Big)=\frac{n \mathrm{\gamma}(0) + \sum_{t=1}^{n-1} 2(n-t) \mathrm{\gamma}(t)} {n^2}$ $\endgroup$ – user72621 Dec 23 '16 at 23:29

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