This question concerns the model $$X(t) = 0.5X(t-1) + Z(t) - Z(t-1) + Z(t-2)$$ in which $Z(t)$ is white noise whith mean 0 and variance $\sigma^2$. It is assumed that $E(X(t)) = 0$ for all t.
Part (a) Is this model stationary?

A. Impossible to determine.
B. No.
C. Yes.


Part (b) Show the following: $$E(X(t)Z(t)) = \sigma^2$$ $$E(X(t)Z(t-1)) = -0.5\sigma^2$$ $$E(X(t)Z(t-2)) = 0.75\sigma^2$$


Part (c) With $\gamma(k) = E( X(t)X(t-k) ),$ show that $$\gamma(0) = 0.5\gamma(1) + 2.25 \sigma^2,$$ $$\gamma(1) = 0.5\gamma(0) - 1.5\sigma^2.$$


Part (d) Find the variance of $X(t)$ in terms of $\sigma^2$, and the autocorrelation function of $X(t)$ at lag 1.


Part (e) Writing the model in the form $$X(t) = \sum_{j=1}^{\infty} \psi_j Z(t-j)$$ find $\psi_0, \psi_1, \psi_2.$




Part (f) After de-trending the data, a environmental scientist fits the model to the daily noon temparatures (in Celsius) in Whitehorse over sixty consecutive days. A portion of the data $x(t)$ and the fitted values $\hat{x}(t)$ are given below (to 2 decimal places):
Use the model and the above information to forecast the next two values of the time series, showing your working clearly.



Part (g) Given that the estimate of from the fitted model is 1.01, provide approximate 95% confidence intervals for each of your forecasts in (f).