Time Series Problem set page 40

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1. (a) Consider the quarterly earnings of Johnson & Johnson from 1960 to 1980 in the file q-earn-jnj.txt .
 
Perform a log transformation of the data, detrend and deseasonalize the data, and subtract the mean, in order to obtain a sequence of observations that appears to be stationary with zero mean. Plot the sample autocovariance or autocorrelation function of the obtained time series. Perform the Box-Ljung test for m=5 and m=10 and draw conclusions. Use some forecasting method built in in the software you are using to forecast 24 values and plot the original series together with the 24 predicted values. [Hint: This is fairly straightforward if you use the software ITSM - This will be demonstrated in class].
 
(b) Consider the accidental deaths between 1973 and 1978 in the file Deaths.txt . Repeat the tasks stated in (a) without the log transformation and forecast 36 values for the deaths time series. 
 
3. #1.8(a) see attachement p.40
 
and express the autocovariance function of the detrended and deseasonalized time series in terms of the stationary process {Y_t}. 
 
4. Suppose that {X_t} is a stationary time series with mean mu and ACF rho(.). Show that the best mean square predictor of X_{n+h} of the form  a X_n + b is obtained by choosing a = rho(h) and b = mu (1-rho(h)), where the best mean square predictor minimizes the mean square error (MSE)  E(X_{n+h} - predictor)^2. 
 
5. Find the ACVF of the time series  Y_t = Z_t - 1.2 Z_{t-1} - 1.6 Z_{t-2},  where {Z_t} ~ WN(0, 0.25).
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