A.      What is ã₀, the variance of Y_t?

B.      What is ã₁, the covariance of Y_t and Y_t-1?

C.      What is ñ₁, the autocorrelation of Y_t and Y_t-1?

D.     What is ñ₂, the correlation of Y_t and Y_t-2?

Part A: See Cryer and Chan, chapter 4, top of page 56:

ã₀ = ó² / (1 – ö²) = 4 / (1 – 0.16) = 4.762

Later modules refer to this process as AR(1), an autoregressive process of order 1. Final exam problems say: an AR(1) process with ö = 0.4 and ó²_ε = 4.

Part B: See Cryer and Chan, chapter 4, middle of page 56:

ã₁ = ö × ó² / (1 – ö²) = 0.4 × 4 / (1 – 0.16) = 1.905

Part C: See Cryer and Chan, chapter 4, equation 4.1.3 at the bottom of page 56:

ñ₁ = ö = 0.4

Part D: See Cryer and Chan, chapter 4, equation 4.1.3 at the bottom of page 56:

ã₂ = ö² = 0.4² = 0.160

This exercise is simple. Final exam problems are more complex. The autoregressive process may be of an order higher than 1, the ö parameters may be positive or negative, the process may have a moving average part, and the parameters may be estimated from the observed sample autocorrelations. The logic is the same for all the scenarios. This exercise is a good starting point for stationary time series.