Timo Teräsvirta

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.

The topic of this chapter is forecasting with nonlinear models. First, a number of well-known nonlinear models are introduced and their properties discussed. These include the smooth transition regression model, the switching regression model whose univariate counterpart is called threshold autoregressive model, the Markov-switching or hidden Markov regression model, the artificial neural network model, and a couple of other models.
Many of these nonlinear models nest a linear model. For this reason, it is advisable to test linearity before estimating the nonlinear model one thinks will fit the data. A number of linearity tests are discussed. These form a part of model specification: the remaining steps of nonlinear model building are parameter estimation and evaluation that are also briefly considered.
There are two possibilities of generating forecasts from nonlinear models. Sometimes it is possible to use analytical formulas as in linear models. In many other cases, however, forecasts more than one periods ahead have to be generated numerically. Methods for doing that are presented and compared.
The accuracy of point forecasts can be compared using various criteria and statistical tests. Some of these tests have the property that they are not applicable when one of the two models under comparison nests the other one. Tests that have been developed in order to work in this situation are described.
The chapter also contains a simulation study showing how, in some situations, forecasts from a correctly specified nonlinear model may be inferior to ones from a certain linear model.
There exist relatively large studies in which the forecasting performance of nonlinear models is compared with that of linear models using actual macroeconomic series. Main features of some such studies are briefly presented and lessons from them described. In general, no dominant nonlinear (or linear) model has emerged.

This chapter contains section titled:
INTRODUCTION
STAR MODEL
FORECASTING WITH STAR MODELS
EXAMPLES
CONCLUSIONS