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Extensions of the GARCH Model

  • Specific variance of GARCH model
    • tailored to capture different characteristics in TS, particularly volatility dynamics

GJR-GARCH

  • Glosten, Jagannathan and Runkle GARCH
  • Asymmetric effects in volatility
\[ \sigma_{t} ^{2} = \omega + \alpha \epsilon_{t-1}^{2} + \gamma\epsilon_{t-1}^{2}I(\epsilon_{t-1}\lt 0) + \beta \sigma_{t-1}^{2} \]
  • \(I(x <0)=\begin{cases}1, & x \lt 0 \\ 0, & ow\end{cases}\)
  • \(\gamma \gt 0\) captures impact of negative shocks
  • Applications: Financial markets (bad news tends to increase volatility more than good news)

EGARCH

  • Handles Leverage effects without requiring non-negativity constraints on the parameters ensures that conditional variance is always positive.
\[ \log(\sigma_{t}^{2}) = \omega + \beta \log(\sigma_{t-1}^{2}) + \alpha \dfrac{\epsilon_{t-1}}{\alpha_{t-1}} + \gamma\left(\lvert \dfrac{\epsilon_{t-1}}{\sigma_{t-1}} \rvert - E\left[\lvert \dfrac{\epsilon_{t-1}}{\sigma_{t-1}} \rvert \right] \right) \]
  • log transformation ensures \(\sigma_{t}^{2}\) stays positive, while

  • \(\gamma \gt 0\) captures the impact of negative shocks.

  • Application: financial assets with pronounced

TGARCH

  • Threshold GARCH is similar to GJR-GARCH
  • Difference = standard deviation instead of variance
\[ \sigma_{t} ^{2} = \omega + \alpha \epsilon_{t-1} + \gamma\epsilon_{t-1}I(\epsilon_{t-1}\lt 0) + \beta \sigma_{t-1}^{2} \]

ARARCH

  • Asymmetric Power ARCH
\[ \sigma_{t}^\delta = \omega + \alpha(|\epsilon_{t-1}| - \gamma e_{t-1})^{\delta} + \beta \sigma_{t-1}^{\delta} \]
  • where
    • \(\delta\) = power parameter, determines transformation of conditional volatility
    • \(\gamma\) = asymmetry parameter
  • Markets with non-linear and asymmetric volatility patterns: energy/commodity markets

FIGARCH

  • Fractionally Integrated GARCH
  • Long memory property of volatility, capturing persistence that decays slowly over time
\[ \phi(L)(1-L)^{d} \sigma_{t}^{2} = \omega + (1- \beta(L))\epsilon_{t}^{2} \]
  • \((1-L)^{d}\) is the fractional differencing operator with \(0 \lt d \lt 1\)
  • Application: persistence in bond yields and exchange rates

MGARCH

  • Multivariate GARCH
  • Multiple time series capturing co-movements in volatilities
  • Popular Specifications
    • VARCH Model. Directly models varcov matrix but overparameterization
    • BEKK Model: Imposes structure to varcov for more parsimonious modeling
    • DCC-GARCH (Dynamic Conditional Correlation) → Models time-dependent correlations alongside volatilities.

NGARCH

  • Nonlinear GARCH: capture complex non-linearity
\[ \sigma_{t}^{2} = \omega + \alpha\epsilon_{t-1}^{2} + \beta \sigma_{t-1}^{2} + \lambda\epsilon_{t-1}\sigma_{t-1} \]
  • \(\lambda:\) Captures non-linear interactions by multiplication term \(\epsilon_{t-1}\sigma_{t-1}\)
  • Application: High-frequency financial data with complex volatility behavior.

HARCH

  • Heterogenous ARCH explains volatility using returns aggregated over multiple time horizons
\[ \sigma_{t}^{2} = \omega + \sum_{k=1}^K \alpha_{k} \left( \dfrac{1}{k}\sum_{i=1}^{k} \epsilon_{t-i}^{2} \right) \]
  • Captures the idea that market participants operate over heterogenous time scales
  • Application: Markets with participants reacting to news over different horizons

EGARCH-X/GARCH-X

  • Extends GARCH by incorporating exogenous covariate \(X_{t}\), affecting volatility
\[ \sigma_{t}^{2} = \omega + \sum_{i=1}^{q} \alpha_{i}\epsilon_{t-i}^{2} + \sum_{j=1}^{p} \beta_{j}\sigma_{t-j}^{2} X_{t} \]
  • Application: Modeling volatility with macroeconomic indicators, news sentiment or financial stress indices

ARMA + GARCH Models

  • ARMA for the mean (serial correlation)

  • GARCH for modelling conditional variance (time varying volatility)

  • Mean: \(ARMA(p,q)\)

    • Get residuals from this equation

  • Variance: \(GARCH(r,s)\)

    • Use them to model variance

  • Application

    • Stock returns
    • VaR
    • GDP growth and interest rates
    • Commodity prices: oil/electricity