Forecasting financial data with autoregressive and conditionally heteroscedastic processes has received a considerable attention in the literature. However, most of the models used in practice are stationary and have short memory properties which is not compatible with the slow decay of the empirical autocorrelations observed for these series. Then non stationary models with time-varying parameters have been introduced for dealing with financial returns. These models avoid the criticism made on stationary long memory processes which are always fitted over a large history of data. See for instance Mikosch and Stărică (2004) for a discussion about some nonstationarities and a theoretical basis of a possible explanation of the long memory properties usually found for financial time series. However, different types of nonstationary time series models have been introduced. For instance, Granger and Stărică (2005) consider a simple model of independent random variables with a time-varying unconditional variance, Engle and Rangel (2008) a multiplicative model with a time-varying unconditional variance and a GARCH component whereas Dahlhaus and Subba Rao (2006) or Fryzlewicz et al. (2008) consider an ARCH with time-varying parameters.
Deciding which model is the most compatible with a given data set is a problem of major importance because inference of time-varying parameters is a difficult task and too complex models can be hardly interpretable or unstable.
In this paper, we develop a variety of tools for testing parameters stability of ARCH processes and in particular for deciding between the three types of models discussed above. First, we show that non time-varying parameters in ARCH processes can be estimated at the usual root converge rate and we also study the asymptotic semiparametric efficiency of our procedure. Then we develop two statistical tests. The first one is used for testing if a subset of parameters is non time-varying and the second one for testing if the estimates of lag coefficients are significant. We also provide an information criterion for selecting the number of lags for time-varying ARCH processes.
Our tools are illustrated with three data sets. For the three series, we found that a time-varying intercept is clearly rejected over the period under study. The conclusion for the lags coefficients depends on the series. For instance, the daily exchange rates between the US Dollar and the Euro seems compatible with the model of Stărică and Granger (2005) with no second order dynamic. On the other hand, the absence of second order dynamic is rejected for the FTSE index, though nonstationarity leads to much smaller lag estimates with respect to the stationary case. In all the cases, a time-varying unconditional variance seems to have an important contribution to volatility and has to be taken into account.