Volatility of original vs. first-differenced series in ARIMA+GARCH in R

Question

I want to do ARIMA-GARCH modelling on the daily prices of crude. Since ARIMA-GARCH model cannot be applied to the time series in R, I first differenced the series to convert it into a stationary series and applied ARMA-GARCH to the differenced series. When I forecasted using this model, I got a two series of "Series" (forecast of the mean model, I hope I am right) and "Sigma" (forecast of the standard deviations) of the DIFFERENCED SERIES.

Now, I want to know whether the standard deviation of the differenced series is the same as that of the original series. If not, how do I get the forecast of the standard deviation (variance or volatility) of the original series?

Answer 1

The conditional standard deviation of a differenced series $\Delta x_t$ as modelled with an ARMA(p,q)-GARCH(s,r) model will be the same as the conditional standard deviation of the original series $x_t$ as modelled with an ARIMA(p,1,q)-GARCH(s,r). So if you cannot run an ARIMA(p,1,q)-GARCH(s,r) for $x_t$ because of technical reasons, you can use ARMA(p,q)-GARCH(s,r) for $\Delta x_t$ and still get the conditional standard deviation you are interested in.

Now if $x_t$ is integrated, the unconditional standard deviation is undefined. Meanwhile, the unconditional standard deviation of $\Delta x_t$ may still be well defined.

Answer 2

Your question seems to be closely related to the question that has been answered here. GARCH models have some requirements to fulfill before it makes sense using them.

Here the model building approach from Tsay's "Analysis of Financial Timeseries", p. 113:

Building a volatility model for an asset return series consists of four steps:

1. Specify a mean equation by testing for serial dependance in the data an, if necessary, building an econometric model (e. g., an ARMA model) for the teturn series to remove any linear dependence.
2. Use the residuals of the mean equation to test for ARCH effects.
3. Specify a volatility model if ARCH effects are statistically significant, and perform a joint estimation of the mean and volatility equations.
4. Check the fitted model carefully and refine if necessary.

These are namely no remaining autocorrelation in the first order (Step 1) and conditional heteroscedasticity, also called ARCH-effect, in the second order (Step 2).

I assume you checked these before you applied the ARIMA model, decided these were relevant and specified your mean-equation order(s).

So given these prerequisites the conditional volatility from your GARCH model only makes sense if you used them on the differenced series with your ARMA-GARCH specification.

The unconditional volatility probably won't be the same for the two series, but in general one is interested in the conditional volatility, which is the Sigma from your ARMA-GARCH estimation, when applying a GARCH model.

As to what exactly the "Series"-output is I can't help you without a concrete code example or a more detailled desciption of what package / function you used for your estimation.