Introduction to Autoregressive models

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.

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R in Time Series: Autoregressive model

This tutorial gives brief introduction about the autoregressive model in time series.

1   Definition

A time series \(\{x_t\}\,\) is an autoregressive process of order p, denoted by AR(p), if

\[x_t=\alpha_1x_{t-1}+\alpha_2x_{t-2}+\cdots+\alpha_px_{t-p}+w_t=\sum_{i=1}^p\alpha_ix_{t-i}+w_t\] Read more R in Time Series: Autoregressive model