ARCH Model

Describing an ARCH Model

• An autoregressive conditionally heteroscedastic (or ARCH) model is a time series model of variance
• In other words, an ARCH model is used to model the conditional variance when the conditional variance follows a pattern
• On the other hand, an ARMA model is used to model the conditional mean when the conditional mean follows a pattern
• Said another way, ARCH models are used to describe a changing, possibly volatile variance
• Although an ARCH model could possibly be used to describe a gradually increasing variance over time, most often it is used in situations in which there may be short periods of increased variation
• Essentially, an ARCH model could be used for any series that has periods of increased or decreased variance

Representing the ARCH Model

• We can model any trending volatility by adjusting how we model the errors

• Specifically, the ARCH(1) model represents errors terms as the following:

  $$\epsilon_{t} = \omega_{t} + \sqrt{\alpha_{0} + \alpha_{1}\epsilon_{t-1}^{2}}$$
  • Where $\omega_{t}$ is a white noise term representing some random, unpredictable component
  • Where $\epsilon_{t}$ represents the volatility in the current time period
  • Where $\alpha_{0}$ and $\alpha_{1}$ represent some coefficients for their respective time periods
  • Where $\epsilon_{t-1}^{2}$ represents the volatility in the previous time period

• Specifically, the ARCH(2) model represents error terms as the following:

  $$\epsilon_{t} = \omega_{t} + \sqrt{\alpha_{0} + \alpha_{1}\epsilon_{t-1}^{2} + \alpha_{2}\epsilon_{t-2}^{2}}$$
  • Where $\omega_{t}$ is a white noise term representing some random, unpredictable component
  • Where $\epsilon_{t}$ represents the volatility in the current time period
  • Where $\alpha_{0}, \alpha_{1},$ and $\alpha_{2}$ represent some coefficients for their respective time periods
  • Where $\epsilon_{t-1}^{2}$ represents the volatility in the previous time period
  • Where $\epsilon_{t-2}^{2}$ represents the volatility in the two previous time periods

Testing for ARCH Models

1. Fit our best possible ARCH model
2. Consider how the model fits against the residuals graphically

3. Create a correlogram to to choose the best number of lags to include in the ARCH model

   • A correlogram is an autocorrelation plot

References

• ARCH Model Video
• Defining ARCH Models
• Examples of ARCH Models