Autocorrelation functions of covariance stationary series. While interviewing at a top investment bank, your interviewer is im- pressed by the fact that you have taken, a course on forecasting. She decides to test your knowledge of the autocovariance structure of covari- ance stationary series and lists five autocovariance functions:
a. γ(t, r) = α
b, γ(t, r) =e^-ar
d. γ(t, τ)=ατ ,
where α is a positive constant, which autocovariance function(s) are consistent with covariance stationarity, and which are not? Why?

Only the first autocovariance function  : γ(t,r)  is covariance stationary, the remaining are not covariance stationary

Explanation:

For a process to be covariance stationary/ weak stationary/ second order stationary it must satisfy these two conditions below:

In order words, {Xt} is said to be (weakly) stationary if :

1. it is independent of t, and

2. For each h, x(t + ћ, t) is independent of t.

In that case, we write:

 γX (h) = γX (h,0⇒)

Hence only the first autocovariance function  : γ(t,r)  is covariance stationary  since Autocorrelation function (ACF) is time independent.

The remaining are not covariance stationary because ACF is time dependent.

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