The use of ARCH models, which have enjoyed increasing popularity during the last decade, solves important statistical problems in the estimation of the volatility, as well as in the goodness of these models in forecast intervals. Apart from their simplicity, the main reason for the success of ARCH models is that they take into account non-linearities and changes in the forecasting of future values (a comparative forecasting approach between traditional and bootstrap methodology can be seen in Olave and Miguel (2001)). Moreover, the joint estimation of the mean and the variance can be easily made.
In the following example of financial markets, we will apply different statistical tools in order to fit the best model to the data.
First, we analyse whether there is any type of dependence in the
mean of this series. To that end, we analyse graphically the
existence of ARMA structure in the mean. The autocorrelation plots
obtained by functions
acfplot
and
pacfplot
of
reveal no presence of statistically
significant autocorrelations or partial autocorrelations, at least
in the first lags. In the light of these results (see figures
6.13 and 6.14), we will not fit a usual MA(1),
which is very often the model in this financial series, where the
``leverage'' effect in the mean appears as a moving average effect
in the residuals.
In a second stage, we would determine the type of heteroscedasticity by means of the squared residuals (in our particular example the squared data). Note that the ARCH model can be seen as an ARMA on squared residuals. Figures 6.15 and 6.16 show a slow decreasing of significative lags up to a high order, indicating the convenience of selecting a GARCH(1,1) model. In this case we can say that the ``leverage'' effect is present in the conditional variance.
The usual ARCH test confirms us the presence of ARCH effects. So, finally we use the function garchest to obtain the estimations of the model. The order of the model fitted to the data depends on the autocorrelation and partial autocorrelation structure.
The results of the GARCH(1,1) model fitted to the data can be seen in table 6.3.
We can see that the sum of
and
both parameters are significant, indicating a high persistence of
volatility for this data in this period of time.
This time series only shows a moderate significant autocorrelation coefficient in lag 5, but we do not take it into account by filtering the data. The squared data show a clearly significant pattern in the ACF and PACF functions, meaning again a strong ARCH effect in this time series. In figures 6.17 and 6.18 we see the corresponding autocorrelations of the squared centred time series.
The main information given by the autocorrelation plots is
confirmed by the LM test of the ARCH effect, and similarly by the
"TR2" test, as you can see if you execute
XEGarch16
.
At the sighting of the results, we opt for fitting a GARCH(1,1)
model to the data in
XEGarch17
.
Figure 6.19 represents the original time series plus the two volatility bands around it. We see that the strong moving average does not allow very high variations in the estimated conditional variance
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