This tutorial presents introduction of autoregressive models, and theoir implementation in R.

### 1 AR(1) model

#### 1.1 Model definition

[r_t=phi_0+phi_1r_{t-1}+a_t]

where ({a_t},) is a white noise series of mean zero and variance (sigma_a^2).

Notes:

- AR(1) model is widely used not only for returns, as shown with (r_t,) here, but also for volatility with (r_t,) replaced with (sigma_t).
- Conditional on past return (r_{t-1}), we have
**conditional**mean and variance as following[begin{gather*}mathbb{E}[r_t|r_{t-1}]=phi_0+phi_1r_{t-1}\text{Var}[r_t|r_{t-1}]=sigma_a^2end{gather*}]This is a Markov property in that, conditional on (r_{t-1}), the return (r_t,) is not correlated with (r_{t-i},) for*i > 1*.