L48 ARCH LM Test & GARCH Models

The ARCH Model

Testing for ARCH Effect

Lagrange Multiplier (LM) Test (ARCH Test)

• a.k.a. Engle's ARCH test
• Steps
  1. Estimate the mean equation (Fit a usual TS model)
  2. Square the residuals from the mean equation to get a proxy for the variance at each \(t\)
  3. Regress the squared residuals on lags (check if variance depends on past residuals)
     • In particular for ARCH(1) models
  4. Test for significance
     • \(H_{0}:\) there are no ARCH effects (no conditional heteroskedasticity)

Picking optimal ARCH model

• Order = \(q\), number of lagged squared residuals to model conditional variance
• How?
  • Perform an ARCH effect test
  • choose initial order \(q\)
  • Use information criteria of fitted ARCH models different orders and pick one with the least AIC.
  • Examine PACF of squared residuals
    • Peaks in the PACF \(\implies\) significant lags corresponding to ARCH order

The GARCH Model

Generalized Autoregressive Conditional Heteroskedasticity

• Adds an AR component to the conditional variance
• ARCH model = more flexibility and efficiency
• GARCH model captures past variances and past squared residuals

\(GARCH(p,q)\) model specifies

$$
\sigma_{t}^{2} = \omega + \sum_{i=1}^{q} \alpha_{i} \epsilon_{t-i}^{2} + \sum_{j=1}^{p} \beta_{j} \sigma^{2}_{t-j}
$$
where,

• \(\epsilon_{t}\) residuals
• \(\sigma_{t}^{2}\) is the conditional variance at time \(t\)
• \(\omega > 0\) is constant
• \(\alpha_{i}:\) ARCH terms (squared residuals)
• \(\beta_{j}:\) GARCH terms (past variances)
• \(p,q:\) orders

Key Features

• Volatility Clustering: models periods of high and low volatility commonly observed in financial data.
• Mean Reversion: \(\sum \alpha_{i} +\sum\beta_{j} \lt 1\) \(\implies\) volatility eventually reverts to a long-term level.
• Leverage Effects: Standard GARCH cannot handle these asymmetric effects of negative shocks affecting volatility more than positive shocks.
  • EGARCH or GJR-GARCH

Steps to Fit a GARCH Model

1. Perform an ARCH effect test to confirm presence of conditional heteroskedasticity. (Heteroskedascity)
2. Use model selection criteria like AIC or BIC (\(p\) and \(q\))
3. Use MLE to fit GARCH model. (Estimation)
4. Check diagnostics (Residuals should be homoscedastic)
5. Use the fitted GARCH model to forecast future variances (Forecasting)

\(GARCH(1,1)\) model

• Most commonly used models for volatility modelling

• Incorporates lagged conditional variance to ARCH model's lagged squared residuals

• Mean

  • \(y_{t} = \mu + \epsilon_{t}\)
• Variance
  • \(\sigma_{t}^{2} = \omega + \alpha \epsilon_{t-1}^{2} + \beta \sigma_{t-1}^{2}\)

Key Properties

• Volatility Clustering
• Mean Reversion
• Stationarity: \(\alpha + \beta < 1\) \(\implies\) model is stationary. \(\alpha+\beta \approx 1\) \(\implies\) volatility exhibits long term memory/persistance.

Practical Applications

1. Financial Markets: Asset Price Modelling
2. Financial Markets: Option Pricing
   • Compute implied volatility
3. Risk Management: Value-at-Risk (VaR)
   • risk of portfolio loss over a certain time horizon
4. Risk Management: Stress Testing
   • Simulate worst-case scenarios by considering periods of high volatility