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.

a) both were commissioned explorers

c.

clarify customers’ doubts only when asked.