Higher-order Dependence in the General Power ARCH Process and the Role of Power Parameter

In a recent paper, Ding, Granger and Engle (1993) introduced a class of autoregressive conditional heteroskedastic models called Asymmetric Power Autoregressive Conditional Heteroskedastic (A-PARCH) models. The authors showed that this class contains as special cases a large number of well-known ARCH and GARCH models. The A-PARCH model contains a particular power parameter that makes the conditional variance equation nonlinear in parameters. Among other things, Ding, Granger and Engle showed that by letting the power parameter approach zero, the A-PARCH family of models also includes the log-arithmic GARCH model as a special case. Hentschel (1995) defined a slightly extended A-PARCH model and showed that after this extension, the A-PARCH model also contains the exponential GARCH (EGARCH) model of Nelson (1991) as a special case as the power parameter approaches zero. Allowing this to happen in a general A-PARCH model forms a starting-point for our investigation. Applications of the A-PARCH model to return series of stocks and exchange rates have revealed some regularities in the estimated values of the power parameter; see Ding, Granger and Engle (1993), Brooks, Faff, McKenzie and Mitchell (2000) and McKenzie and Mitchell (2002).We add to these results by fitting symmetric first-order PARCH models to return series of 30 most actively traded stocks of the Stockholm Stock Index. Our results agree with the previous ones and suggest that the power parameter lowers the autocorrelations of squared observations compared to the corresponding autocorrelations implied, other things equal, by the standard first-order GARCH model. In the present situation this means estimating the autocorrelation function of the squared observations from the data and comparing that with the corresponding values obtained by plugging the parameter estimates into the theoretical expressions of the autocorrelations. Another example can be found in He and Teräsvirta (1999d). The plan of the paper is as follows. Section 2 defines the class of models of interest and introduces notation. The main theoretical results appear in Section 3. Section 4 contains a comparison of autocorrelation functions of squared observations for different models and Section 5 a discussion of empirical examples. Finally, conclusions appear in Section 6. All proofs can be found in Appendix.