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.