GARCH Modelling

GARCH stands for Generalised Auto-Regressive Conditional Heteroscedasticity. It is a very simple technique for estimating and forecasting time-varying or conditional volatility. 

To put it simply, when higher (lower) volatility is followed by higher (lower) volatility the return series does have ARCH effect or volatility clustering. If there exists a first-order serial correlation in squared returns (Use Ljung-Box test), it indicates that volatility is clustered and ARCH effect exists. Hence, volatility is not constant but time-varying (conditional).

ARCH models were introduced by Engle (1982), and Generalized ARCH or GARCH were proposed by Bollerslev (1986). ARCH process is not used nowadays because GARCH can be used in place of ARCH with fewer parameters. GARCH (1,1) would be sufficient to handle most of the financial problems as it can capture the stylizes facts of the past volatility patterns. The Equation for GARCH (1,1) is as follows:

Therefore, while estimating GARCH models we need three things:

1. Mean equation
2. Variance equation
3. Distributional assumption

GARCH is a very complex model, but it could be estimated easily in R, Eviews, or STATA.

The coefficient α indicates the reaction of volatility to the unexpected return or shocks, whereas, the coefficient β shows the persistence of the volatility, i.e. how long the volatility would take to revert back to long-run volatility {ω / (1 – α – β)}. 

Most of the time (α + β) < 1. If (α + β) > 1, we cannot use simple GARCH models and we would use Integrated GARCH (IGARCH) models in that case.

Equation (1) represents the simplest vanilla symmetric GARCH model. But sometimes the conditional volatility is not symmetric, that means the impact of positive and negative unexpected returns (or news) is not the same on the volatility. 

Generally, it is observed that the impact of the negative news is more on the volatility that the impact of the positive news. Because of the bad news the leverage (Debt/Equity ratio changes), which makes the company more susceptible to bankruptcy. If equity capital declines due to the negative news, Debt/Equity ratio increases which is termed as leverage effect, which can be captured by an Asymmetric GARCH (AGARCH) model. There are many asymmetric GARCH model, but Threshold GARCH (TGARCH or GJR GARCH) is used most often to capture such asymmetric conditional volatility.

Prerequisite for GARCH:

The data should be stationary

Advantages:

1. Better understand the volatility patterns of a stationary time-series
2. Volatility Forecasting based on the GARCH models

  

The procedure is as follows to fit GARCH models:

1. Make a time-series plot of the squared returns. If you can see clustering in the squared returns, ARCH effect is there.
2. Use Ljung-Box test at lag 1. If you can reject the null hypothesis of no serial correlation, ARCH effect exists.
3. Fit the appropriate GARCH model (GARCH, TGARCH, or EGARCH)
4. Find the residuals and do diagnostic tests. If the residuals are IID, then model is good. Otherwise repeat the process.
5. Use the fitted model for the forecasting purpose.

Do it in R as shown in the video below:

Do it in Eviews as shown in the video below:

References

Alexander, C. (2001), Market Models: A Guide to Financial Data Analysis, John Wiley & Sons, available at: http://sro.sussex.ac.uk/40646/ (accessed 26 June 2014).

Bollerslev, T. (1986), “Generalized autoregressive conditional heteroskedasticity”, Journal of Econometrics, Vol. 31 No. 3, pp. 307–327.

 Engle, R.F. (1982), “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation”, Econometrica, Vol. 50 No. 4, pp. 987–1007.