Autoregressive Time Series

We have the autoregressive timer series of order 1

$$Z_n=\varphi Z_{n-1}+{ϵ}_n$$

We assume the $ϵ$ are independent random variables with $\mathbb{E}[X]=0$ or mean of 0.

We say that this sequence of $Z$s is a discreteMarkov Process. This is because knowledge of $Z_{n-2}$ and earlier does not provide any information about $Z_n$ if $Z_{n-1}$ is known.

We say that AR(1) or Autoregressive 1 is a Markov Process.

AR(2) would simply be the autoregressive time series with a two step lag

$$Z_n=\varphi Z_{n-1}+{\varphi }_2{Z}_{n-2}+{ϵ}_n$$

We also refer to this as the second order Markov Process.

This generalizes to AR(p) where we would have $p$ lags.