Nonparametric prediction in time series analysis 241The first differences ofXt(denoted byrtin the following) are then plotted in
Figure 3 where it is evident that the behaviour of the series changes considerably in
the time interval taken into account.
Following [10], which further assesses the nonlinearity of the data-generating
process ofrt, we examine the conditional mean ofrt, neglecting the conditional
heteroschedasticity that gives rise to the volatility clustering that can be clearly seen
in Figure 3.
Starting from these results, we firstly evaluate some features of the series using
the descriptive indexes presented in Table 4. In particular, the mean, the median, the
standard deviation, the skewness and kurtosis (given as third and fourth moment of
the standardised data respectively) ofrt, are computed. As expected, the distribution
ofrthas null median and shows negative skewness and heavy tails.
It is widely known that when the prediction of the mean level of asymmetric time
series needs to be generated, a parametric structure that can be properly applied is
the SETAR(2;p 1 ,p 2 ) model that treats positive and negative values ofrtdifferently.
This is the reason why the SETAR models have been widely applied to analyse and
forecast data related to financial markets (among others: [20] for a wide presentation
of the model and [12] for its application to financial data).
Ta b le 3 .Descriptive indexes ofrtMean Median S.D. Skewness Kurtosis
rt − 6 .224e-05 0 0. 2342 − 0. 5801 16. 6101Fig. 3.First differences of the weakly 90-day US T-bill rate (rt)