Question 1
Suppose that {Y_t} is stationary with autocovariance function ?_k.

a) Show that W_t=?Y_t=Y_t-Y_(t-1) is stationary by finding the mean and autocovariance function for {W_t} .

b) Show that U_t=?^2 Y_t=?[Y_t-Y_(t-1) ]= Y_t-2Y_(t-1)+Y_(t-2) is stationary. (you need not find the mean and autocovariance function for {U_t}.)

Question 2
Suppose Cov(X_t,X_(t-k))=?_k is free of t but that E(X_t )=3t.

a) Is {X_t} stationary?

b) Let Y_t=7-3t+X_t. Is {Y_t} stationary?

Question 3
Two processes {Z_t} and {Y_t} are said to be independent if for any time points t_1, t_2, ............t_m and
s_1,s_2, ..............s_nthe random variables {Z_(t_1 ),Z_(t_2 ),...............,Z_(t_m )} are independent of the random variables {Y_(s_1 ),Y_(s_2 ),...............,S_(s_n )}.

a) Show that if {Z_t} and {Y_t} are independent stationary processes, then W_t=Z_t+Y_t is stationary.

Solution Summary

The solution describes methods to verify whether a time series is stationary or not.

Statistics Time-Series Forecasting problems: 16.7, 16.13, 16.15, 16.47. ... Thus 84% variability in the charges can be predicted using the time series model. ...

... of change in statistical information over regular intervals of time. We project these patterns to arrive at estimate for the future. Thus times series helps us ...

... Calculations are done in MS Excel and Minitab. Time series analysis of Hurricane data. ... 2. Time Series Decomposition for all department stores. ...

... Thus 84.% variability in the charges can be predicted using the time series model. ... X'G times Inverse of X'X 0.12679 -0.000412302 -2.51538E-05. ...

...Time Series Forecasting ? ... Each time we toss the coin, the probability of either outcome is always 50 percent, no matter how many times the coin is tossed. ...

... 1 < i < n). Reliability of Series Systems of ... n components (and their failure times) are statisti ... Components 5. Denote system mission time T. Hence ...

...time. Then it will show only the PACF plot. First run the ACF command and copy the ACF plot and then run the PACF command. The ACF and PACF of the times series...