Solved – GARCH or TARCH model when assymetric volatility but GARCH(1,1) already performs well

garchrvolatility-forecasting

I have the following question: I am analyzing Brent Oil returns and I have found that there's a significant negative sign bias. So first, I have tried with a GARCH(1,1) and it turns out that completly cleans the ACF, so there are no significant autocorrelations left, and adjusts quite decently to the data.

However, due to the presence of negative sign bias I've decided to perform a TARCH(1,1) but it turns out that performs dramatically worse than GARCH(1,1), as ACF is not cleant and residuals still have lot of noise.

Any suggestion of how I should tackle this issue?

Thank you very much!

Best Answer

Citing the vignette of "rugarch" package in R,

The sign bias test of Engle and Ng (1993) <...> tests the presence of leverage effects in the standardized residuals (to capture possible misspecification of the GARCH model), by regressing the squared standardized residuals on lagged negative and positive shocks as follows:

$$ \hat z_t^2 = c_0 + c_1 I\{\hat\varepsilon_{t-1}<0\} + c_2 > I\{\hat\varepsilon_{t-1}<0\} \cdot \varepsilon_{t-1} + c_3 > I\{\hat\varepsilon_{t-1}\geqslant 0\} \cdot \varepsilon_{t-1} + u_t $$

where $\hat\varepsilon_t$ are the estimated residuals from the GARCH process. The Null Hypotheses are $H_0: \ c_i = 0$ (for $i = 1, 2, 3$), and that jointly $H_0: \ c_1 = c_2 = c_3 = 0$. <...> [If rejected,] using instead a model such as the apARCH would likely alleviate these effects.

If one, several or all the hypotheses are rejected, the idea is to use a model that allows for asymmetric effects such as GJR-GARCH, APARCH, TARCH. (Hence, you may try the ones you have not tried yet.)

However, if you are unable to obtain a satisfactory model that allows for asymmetry, perhaps using a relatively good model without assymetry will be a lesser evil than using a poor asymmetric model. The model choice is essentially an empirical question. You could try doing out-of-sample performance evaluation to select the model that suits your needs best.

References

Engle, Robert F., and Victor K. Ng. "Measuring and testing the impact of news on volatility." The Journal of Finance 48.5 (1993): 1749-1778.
Ghalanos, Alexios. "Introduction to the rugarch package." Version 1.3-1 (2015).

Related Solutions

Solved – Fitting a GARCH(1,1) model

Depends on the coefficients. Simple Monte-Carlo analysis suggests that a lot, about 1000, which is quite surprising.

N <- 1000
n <- 1000+N
a <- c(0.2, 0.3, 0.4)  # GARCH(1,1) coefficients
e <- rnorm(n)  
x <- double(n)
s <-double(n)
x[1] <- rnorm(1) 
s[1] <- 0
for(i in 2:n)  # Generate GARCH(1,1) process
{
  s[i] <- a[1]+a[3]*s[i-1]+a[2]*x[i-1]^2    
  x[i] <- e[i]*sqrt(s[i])
}
x <- ts(x[1000+1:N])
x.garch <- garchFit(data=x)  # Fit GARCH(1,1) 
summary(x.garch)

I modified example code from garch from tseries package, but I used garchFit from fGarch package, since it seemed that it gave better results. I used 1000 values for burn-in.

Solved – Modelling time varying volatility when GARCH(1,1) coefficients sum to value greater one

First, you have not told us what this series represents. Subject-matter knowledge is generally useful when trying to model the data.

the volatility of the series is increasing <...> The log returns exhibit this even more clearly

It would be more accurate to say that volatility has been very low in the beginning, high for a while and then quite low again. There is no linear trend in the magnitude of volatility -- see your second figure.

Which alternative model could I use to model this behavior of the volatility of the series <...> ?

You may want to reconsider the model for the conditional mean first. Perhaps you could model the apparent trend in the original series and the dip during approx. 2009 by including some other explanatory variables and turning the AR model into an ARX model. Also, you could consider ARCH-in-mean specification, as the volatility seems to have been the highest when the series was going down.
Now to answer the question, there is a great variety of GARCH models suited for different features of the data. One survey (not necessarily the best) can be found here: Terasvirta "An introduction to univariate GARCH models" (2009). You could take a look at the stylized facts and the characteristics of the different models and see if there is one that suits you.
You could also have a GARCH model with exogenous variables. E.g. a GARCH model with a time trend could suit a case where conditional variance is increasing linearly (which does not seem to be the case here, though). Or you could have an IGARCH model where the conditional variance is an integrated process. That also answers your second question:

When modelling the conditional mean of the series the classic approach would be to integrate the series. Is there a similar approach considerable for the volatility?

Sorry for being a bit vague about the choice of the model, but I think it would be careless to recommend a particular model just by having seen a graph of the data.

I would be also very happy about any hint to a particular Stata or R command that seem promising to model the volatility of this time series.

In R I may recommend using "rugarch" package with main functions ugarchspec (for specifying a GARCH model) and ugarchfit (for estimating the specified model for a given data series). There is a nice vignette to consult.

P.S.

the estimated coefficients sum to a value greater than one <...> implying that volatility is growing without bounds which is clearly undesirable

It is not only undesirable, but it also does not seem to fit the data -- I do not see gradually exploding volatility in the second figure.

Best Answer

Related Solutions

Solved – Fitting a GARCH(1,1) model

Solved – Modelling time varying volatility when GARCH(1,1) coefficients sum to value greater one

Related Question