Forecasting with GARCH

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Forecasting with an ARCH/GARCH model is done by continuing the iteration of the model beyond the data that we have. If we set t=T+1t=T+1 for an GARCH(1,1) we get the equation σ^T+1∣T2=E (yT+12∣FT)=ω^+α^1 yT2+β^1 σ^T2\widehat\sigma_{T+1|T}^2 =\mathbb E\,( y_{T+1}^2|\mathcal F_T) =\widehat\omega+\widehat\alpha_1\,y_{T}^2 +\widehat\beta_1\, \widehat\sigma_{T}^2 where ω^,α^1,β^1\widehat\omega,\widehat\alpha_1,\widehat\beta_1 are estimated parameters, yT2y_T^2 is observed at time TT and σ^T2=ω^∑j=0T−1β^1j+α^1∑j=1T−1β^1j−1 yT−j2+β^1T σ^12,\widehat\sigma_{T}^2 = \widehat\omega\sum_{j=0}^{T-1} \widehat\beta_1^j+\widehat\alpha_1\sum_{j=1}^{T-1}\widehat\beta_1^{j-1}\,y_{T-j}^2 +\widehat\beta_1^T\, \widehat\sigma_{1}^2, and σ^12=ω^/(1−α^1−β^1)\widehat\sigma_1^2 = \widehat\omega/(1-\widehat\alpha_1-\widehat\beta_1). If β^1\widehat\beta_1 is small and TT large, the initial value for σ^12\widehat\sigma_1^2 will not matter much.

Multistep forecast is achieved by noticing that σ^T+h∣T2=E (σT+h2∣FT)=E (yT+h2∣FT)\widehat\sigma_{T+h|T}^2= \mathbb E\,(\sigma_{T+h}^2|\mathcal F_T)=\mathbb E\,(y_{T+h}^2|\mathcal F_T), such that σ^T+h∣T2=ω+α1E(yT+h−12∣FT)+β1E (σT+h−1∣FT)=ω+α1 σ^T+h−1∣T2+β1 σ^T+h−1∣T2=ω+(α1+β1)σ^T+h−1∣T2\begin{equation*} \begin{split} \widehat\sigma_{T+h|T}^2 &= \omega+\alpha_1\mathbb E(y_{T+h-1}^2|\mathcal F_T) +\beta_1\mathbb E\,(\sigma_{T+h-1}|\mathcal F_T)\\&=\omega+\alpha_1\,\widehat\sigma_{T+h-1|T}^2 +\beta_1\,\widehat\sigma_{T+h-1|T}^2\\ &=\omega+(\alpha_1+\beta_1)\widehat\sigma_{T+h-1|T}^2 \end{split} \end{equation*} If we simply iterate this formula backwards until we reach something that is known at time TT. To simplify notation, let γ=α1+β1\gamma = \alpha_1+\beta_1 and we also skip the hats for the time being. σ^T+h∣T2=ω+γσ^T+h−1∣T2=ω+γ(ω+γ σ^T+h−2∣T2)=ω+ωγ+γ2 σ^T+h−2∣T2=ω+ωγ+ωγ2+γ3 σ^T+h−3∣T2⋮=ω∑j=0h−2γj+γh−1 σ^T+1∣T2=ω∑j=0h−2γj+γh−1 (ω+α1yT2+β1σ^T2)=ω∑j=0h−1γj+γh−1 (α1yT2+β1σ^T2)\begin{equation*} \begin{split} \widehat\sigma_{T+h|T}^2 &=\omega+\gamma \widehat\sigma_{T+h-1|T}^2 \\ &= \omega+\gamma(\omega+\gamma\, \widehat\sigma_{T+h-2|T}^2) \\ &= \omega+\omega\gamma+\gamma^2\, \widehat\sigma_{T+h-2|T}^2 \\ &= \omega+\omega\gamma+\omega\gamma^2 + \gamma^3\, \widehat\sigma_{T+h-3|T}^2 \\ &\quad\vdots\\ &= \omega\sum_{j=0}^{h-2}\gamma^j + \gamma^{h-1}\,\widehat\sigma_{T+1|T}^2\\ &= \omega\sum_{j=0}^{h-2}\gamma^j + \gamma^{h-1}\,(\omega+\alpha_1 y_T^2+\beta_1\widehat\sigma_T^2)\\ &= \omega\sum_{j=0}^{h-1}\gamma^j + \gamma^{h-1}\,(\alpha_1 y_T^2+\beta_1\widehat\sigma_T^2)\\ \end{split} \end{equation*} Thus, inserting the α1+β1\alpha_1+\beta_1 for γ\gamma we have that σ^T+h∣T2=ω^∑j=0h−1(α^1+β^1)j+(α^1+β^1)h−1 (α^1yT2+β^1σ^T2).\widehat\sigma_{T+h|T}^2 = \widehat\omega\sum_{j=0}^{h-1}(\widehat\alpha_1+\widehat\beta_1)^j + (\widehat\alpha_1+\widehat\beta_1)^{h-1}\,(\widehat\alpha_1 y_T^2+\widehat\beta_1\widehat\sigma_T^2). Assuming α^1+β^1<1\widehat\alpha_1+\widehat\beta_1<1, we get that when h→∞h\to\infty, lim⁡h→∞σ^T+h∣T=ω^1−α^1−β^1,\lim_{h\to\infty}\widehat\sigma_{T+h|T} = \frac{\widehat\omega}{1-\widehat\alpha_1-\widehat\beta_1}, i.e. the forecast will approach the unconditional variance, which intuitively makes sense.

For the conditional variance we are typically not so interested in creating predictions intervals for the volatility, but rather use the point forecast for the volatility to make prediction intervals for the variable of interest (i.e. the stock returns). We forecast the varying forecast variance and use that instead of the fixed one we have seen used in other settings. Since the expectation of a GARCH model is zero, a 100(1−α)%100(1-\alpha)\% prediction interval (under Gaussian assumptions) is given as ±zα/2 σ^T+h∣T.\pm z_{\alpha/2}\,\widehat \sigma_{T+h|T}. We can also use volatility forecasting to calculate risk measures, such as Value At Risk (VaR) or Expected Shortfall (ES) (see McNeil et al, 2005, page 161).

References

  • Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the econometric society, 987-1007.
  • McNeil, A. J., Frey, R., & Embrechts, P. (2005). Quantitative risk management: concepts, techniques and tools-revised edition. Princeton university press.