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.

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
    [begin{align*}mathbb{E}[r_t]=mu=dfrac{phi_0}{1-phi_1}\text{Var}[r_t]=gamma_0=dfrac{sigma_a^2}{1-phi_1^2}end{align*}]
  • By using (phi_0=(1-phi_1)mu,) and repeated substitutions, the AR(1) model can be re-written as
    [begin{gather*}r_t-mu=a_t+phi_1a_{t-1}+phi_1^2a_{t-2}+cdots=displaystylesum_{i=0}^inftyphi_1^ia_{t-i}end{gather*}]
    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

[begin{align*}r_t=phi_0+phi_1r_{t-1}+cdots+phi_pr_{t-p}+a_t=phi_0+displaystylesum_{i=1}^pphi_ir_{t-i}+a_tend{align*}]

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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