ARCH Models

ARCH Models

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ARCH (autoregressive conditional heteroskedasticity) models recognize the presence of successive periods of relative volatility and stability. The error variance, conditional on past information, evolves over time as a function of past errors. The model was introduced by Engle [1982]. Bollerslev [1986] proposed the GARCH (generalized ARCH) conditional variance specification that allows for a parsimonious parameterisation of the lag structure. Considerable interest has been in applications of ARCH/GARCH models to high frequency financial time series.

The HET command in SHAZAM provides features for maximum likelihood estimation of models with ARCH or GARCH errors.

Examples

The examples in this section use a data set of daily exchange rate changes for the Deutschemark/British pound. The data set is from Bollerslev and Ghysels [1996] and has been adopted as a benchmark data set by McCullough and Renfro [1999] (also see the discussion in McCullough and Vinod [1999]). The model of interest is:

      Y_t = 100 [ln(P_t) - ln(P_t-1)] = µ + _t

where P_t is the bilateral Deutschemark/British pound rate. The topics are:

• Testing for ARCH
• Estimation of a GARCH(1,1) model
• Benchmark comparisons of coefficients and standard errors

References

Baillie, R.T. and Bollerslev, T., "The Message in Daily Exchange Rates: A Conditional-Variance Tale", Journal of Business and Economic Statistics, Vol 7, 1989, pp. 297-305.

Bollerslev, T., "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, Vol. 31, 1986, pp. 307-327.

Bollerslev, T. and Ghysels, E., "Periodic Autoregressive Conditional Heteroscedasticity", Journal of Business and Economic Statistics, Vol 14, 1996, pp. 139-151.

Bollerslev, T. and Wooldridge, J.M., "Quasi Maximum Likelihood Estimation and Inference in Dynamic Models with Time Varying Covariances", Econometric Reviews, Vol. 11, 1992, pp. 143-172.

Engle, R.F., "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation", Econometrica, Vol. 50, 1982, pp. 987-1007.

Judge, G.G., Griffiths, W.E., Hill, R.C., Lutkepohl, H. and Lee, T., The Theory and Practice of Econometrics, Second Edition, Wiley, 1985.

McCullough, B.D. and Renfro, C.G., "Benchmarks and Software Standards: A Case Study of GARCH Procedures", Journal of Economic and Social Measurement, 1999, forthcoming.

McCullough, B.D. and Vinod, H.D., "The Numerical Reliability of Econometric Software", Journal of Economic Literature, Vol. 37, 1999, pp. 633-665.

Weiss, A.A., "Asymptotic Theory for ARCH Models: Estimation and Testing", Econometric Theory, Vol. 2, 1986, pp. 107-131.