This blog piece intends to introduce a new addin (i.e. SIMULUGARCH) that extends the current capability of EViews’ available features for the forecasting of univariate GARCH models.
Table of Contents
 Introduction
 Forecasting with Simulation or Bootstrap
 Application to price of Bitcoin
 Files
 References
Introduction
Estimation of conditional volatility is not an easy task as it is an unobserved phenomenon and therefore certain assumptions need to be made for that purpose. Once the model parameters are identified, it is relatively straightforward to produce forecasts. However, unlike the regular mean models (e.g. OLS, ARIMA etc.), generating a confidence interval around the forecast of conditional volatility requires an additional effort.Forecasting with Simulation or Bootstrap
Suppose that we prefer a GARCH(1,1) model to explain the volatility dynamics of the logarithmic return of a financial asset:\begin{align*} \Delta \log(P_t) &= r_t = \bar{r} + e_t\\ e_t &= \epsilon_t \sigma_t\\ \sigma_t^2 &= \omega + \alpha_1 e_{t  1}^2 + \beta_1\sigma_{t  1}^2 \end{align*} where $ \epsilon_t \sim IID(0,1) $. As shown by Enders (2014), hstepahead forecast of the conditional variance is as follows:
\begin{align*} \sigma_{t + h}^2 &= \omega + \alpha_1 e_{t + h 1}^2 + \beta_1\sigma_{t + h  1}^2\\ E(\sigma_{t+h}^2) &= \omega + \alpha_1 E(e_{t + h  1}^2) + \beta_1 E(\sigma_{t + h  1}^2)\\ E(e_{t+h}^2) &= E(e_{t + h}^2\sigma_{t + h}^2) = E(\sigma_{t + h}^2)\\ E(\sigma_{t + h}^2) &= \omega + (\alpha_1 + \beta_1)E(\sigma_{t + h  1}^2) \end{align*} If $ (\alpha_1 + \beta_1) < 1 $, then it implies that forecasts of conditional variance will converge to a longrun value of $ E(\sigma_t^2) = \omega/(1  \alpha_1  \beta_1) $.
Median of conditional variance would be a useful gauge as a central tendency since the variance is a squared value and therefore has a skewed distribution towards larger values. In order to compute the median value along with the associated confidence interval, we need different realizations of forecasted values of conditional variance. One can either simulate or bootstrap the values of innovations (i.e. $ \epsilon_t $) to do so. Simulation generates random samples of innovations from the theoretical distribution assumed in the estimation of model. Bootstrap, on the other hand, does resampling (with replacement) of innovations and is therefore mimics the sampling process successfully as long as the observed distribution of sample resembles the distribution of population.
Application to price of Bitcoin
Bitcoin has emerged as the newest and wellknown kid on the block (of investment products) and its value has been quite volatile so far (XBTUSD.WF1).Simple visual inspections of price level and log returns show us the explosive dynamics and large fluctuations during the analysis period of 20112021 (SIMULUGARCH_EXAMPLE.PRG).










Median scenario for volatility is a gradual increase over the coming month (i.e. 22 business days). This should be expected as the longrun value (i.e. unconditional variance) is calculated to be around 156. However, please keep in mind that median value is always smaller than the mean in rightskewed distributions.




Files
References
 Enders, W. (2014), Applied Economic Time Series, Fourth Edition", John Wiley & Sons.
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ReplyDeleteVeery nice piece!
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