# Time Series

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.

Also,

=

=

=

= , 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.