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    Time Series

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    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.

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    Solution Preview

    PlSolution 1

    a) We have,

    = 0, since is stationary.

    = , which is free of t.
    Therefore, is stationary.
    b) From part (a) we have shown ...

    Solution Summary

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