Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 218 / 5 / a / Solution

Use the stationary autoregressive process of order oneMultiplication by $X_{t-1}-\mu$ and expectation, using independence of the innovation from the past, gives the autocovariance equation $\gamma (1)=\varphi \gamma (0)$. With $\bar{X}=n^{-1}{\sum }_t{X}_t$, define the sample autocovariances using the same denominator:Substitution into that equation gives the Yule–Walker estimator for an autoregressive process of order oneThis is exactly the lag-one sample autocorrelation function under the divisor-$n$ convention, provided the sample has positive variance. Using different divisors for different lags would change the estimator.