# Time Series Problem set page 40

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.40and express the autocovariance function of the detrended and deseasonalized time series in terms of the stationary process {Yt}. 4. Suppose that {Xt} 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 Xn + 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 Yt = Zt 1.2 Z{t-1} 1.6 Z{t-2}, where {Zt} ~ WN(0, 0.25).