Introduction to Autoregressive models

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

1  AR(1) model

1.1  Model definition


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


  • 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.

1.2  Properties of AR(1) model

  • It can be shown that the sufficient and necessary condition for an AR(1) model to be weakly conditional is (|phi_1|lt 1).
  • For a weakly stationary AR(1) model, the unconditional mean and variance can be derived as following
  • By using (phi_0=(1-phi_1)mu,) and repeated substitutions, the AR(1) model can be re-written as
    which is a linear combination of past innovations, therefore AR(1) is a linear time series model.
  • It can be shwon that, for a weakly stationary AR(1) time series, the autocovariance is
    [begin{gather*}gamma_0=dfrac{sigma_a^2}{1-phi_1^2}gamma_l=phi_1cdotgamma_{l-1},quadforall,,lgt 0end{gather*}]
    and the autocoefficient function (ACF) is
    [begin{gather*}rho_0=1rho_l=phi_1cdotrho_{l-1},quadforall,,lgt 0Longrightarrow,quad rho_l=phi_1^l,quadforall,,lgt 0end{gather*}]
    Therefore, the ACF of a weakly stationary AR(1) series decays exponentially with rate (phi_1,) and starting value (rho_0=1). In more details,

1.3  Example of simulated AR(1) series

Two simulated AR(1) series


2  AR(2) model


3  AR(p) model

3.1  Model definition


This model says that p variables ({r_{t-i}}), (i=1,cdots,p), jointly determine the conditional expectation of (r_t,) given the past data.

4  Identify AR models in practice

4.1  Partial autocorrelation function (PACF)

4.2  Information criteris

4.3  Selection rule

4.4 Model check



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