VaR Forecasting in Times of Increased Volatility

The paper evaluates several hundred one-day-ahead VaR forecasting models in the time period between the years 2004 and 2009 on data from six world stock indices - DJI, GSPC, IXIC, FTSE, GDAXI and N225. The models model mean using the ARMA processes with up to two lags and variance with one of GARCH, EGARCH or TARCH processes with up to two lags. The models are estimated on the data from the in-sample period and their forecasting accuracy is evaluated on the out-of-sample data, which are more volatile. The main aim of the paper is to test whether a model estimated on data with lower volatility can be used in periods with higher volatility. The evaluation is based on the conditional coverage test and is performed on each stock index separately. The primary result of the paper is that the volatility is best modelled using a GARCH process and that an ARMA process pattern cannot be found in analyzed time series.

Authors:

• Ivo Jánský
• Milan Rippel

Keywords:

• VaR
• risk analysis
• conditional volatility
• garch
• egarch
• tarch
• moving average process
• autoregressive process

References:

[1] Akgiray, V. (1989): "Conditional Heteroscedasticity in Time Series of
Stock Returns: Evidence and Forecasts. The Journal of Business, 62, 55-
80.
[2] Angelidis, T., Benos, A., & Deggianakis, S. (2003). The Use of GARCH
Models in VaR Estimation.
[3] Black, F. (1976). Studies of stock market volatility changes. Proceedings
of the American Statistical Association, Business and Economic
Statistics Section , 177-181.
[4] Bollerslev, T. (1986). Generalized Autoregressive Conditional
Heteroskedasticity. Journal of Econometrics, 31, 307-327.
[5] Brooks, C., & Persand, G. (2003). The effect of asymmetries on stock
index return Value-at-Risk estimates. The Journal of Risk Finance, 4,
29-42.
[6] Christoffersen, P. (1998). Evaluation interval forecasts. International
Economic Review, 39, 841-862.
[7] Costello, A., Asem, E., & Gardner, E. (2008). Comparison of
historically simulated VaR: Evidence from oil prices. Energy
economics, 30, 2154-1266.
[8] Diebold, F., & Mariano, R. S. (2002). Comparing predictive accuracy.
Journal of Business and Economic Statistics, 20 (1), 134-144.
[9] Engle, R. F. (1982). Autogregressive Conditional Heteroscedasticity
with Estimates of the Variance of United Kingdom Inflation.
Econometrica, 50, 987-1008.
[10] JánskÛ, I. (2011). Value-at-risk forecasting with the ARMA-GARCH
family of models during the recent financial crisis. Prague: Institute of
Economic Studies.
[11] Mandelbrot, B. (1963). The Variation of Certain Speculative Prices. The
Journal of Business, 36, 394-419.
[12] Mandelbrot, B. (1967). The Variation of Some Other Speculative Prices.
The Journal of Business, 40, 393-413.
[13] Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A
new approach. Econometrica, 59, 347-370.
[14] Pagan, A. R., & Schwert, G. W. (1990). Alternative models for
conditional stock volatility. Journal of Econometrics, 45 (1-2), 267-290.
[15] Rabemananjara, R., & Zakoïan, J. M. (1993). Threshold ARCH models
and asymmetries in volatility. Journal of Applied Econometrics, 8, 31-
49.
[16] Zakoïan, J. M. (1994). Threshold heteroskedastic models. Journal of
Economic Dynamics and Control, 18, 931 - 955.