# Calculations Regarding Time Series Analysis

Suppose Cov(X(t), X(t-k)) = y(k) is free of t but that E(X(t)) = 3t.

(a) Is {X(t)} stationary?

(b) Let Y(t) = 7 - 3 t + X(t). Is {Y(t)} stationary?

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2.11 Suppose C0v(X,,X,_ k) = yk is free oft but that E(X,) = 3:.

(a) Is {X,} stationary?

(b) Let Y, = 7 â€” 3! +X,. Is {Y,} stationary?

Solution

a) In order to show that {XI} is stationary, we have to show that its mean and autocovariance

functions are free of t.

It is given that the autocovariance function C01/(XI, X ,_k) = 7/k is free of t.

However, E (X I) = 3t , which varies with t.

Since E (X I) is not free of t, {Xt} is not stationary.

b) In order to show that {I/I} is stationary, we have to show that its mean and autocovariance

functions are free of t.

We have,

E(Y,) = E(7â€”3t+ XI)

: 7 â€” 3t + E (X I)

= 7â€”3t+3t (Since E(X,)=3t)

= 7, which is free of t.

Also,

C0v(Y,,Y,_k) = C0v(7 â€”3t+X,,7 â€”3(tâ€”k)+X,_k)

= C0v(X[,X,_k)

= 7/,6 , which is free of t.

Therefore, the mean and autocovariance functions of are free of t.

Thus, is stationary.

[Note that: If a and b are constants, then C01/(a + X ,b + Y) = Cot/(X ,Y) . Applying this result, we

get C01/(7 â€”3t+ X,,7 â€”3(tâ€”k)+ X,_k)= Cot/(XI, X,_k) since 7 â€”3t and 7 â€”3(tâ€”k) are

constants.]

https://brainmass.com/statistics/regression-analysis/calculations-regarding-time-series-analysis-522971