126 A. D ́ıaz, F. Jare ̃no, and E. Navarro
0.060.070.080.090.10.110.120246810121416
term to maturity
NS(O) NS(G)more differences in the short- and long- termFig. 1.TSIR estimated byNSOandNSG(01.07.1994)In summary, we use four different estimation models: Nelson and Siegel [14],
NSG, and Vasicek and Fong [17],VFG, which take into account residuals weighted
by the reciprocal of maturity, andNSOandVFO, that is, with non-weighted residuals.
These alternative estimation procedures provide the input of the subsequent functional
principal component analysis.
3 GARCH models
VTS is an essential issue in finance, so it is important to have good volatility forecasts,
which are based on the fact that volatility is time-varying in high-frequency data. In
general, we can assume that there are several reasons to model and forecast volatility.
First of all, it is necessary to analyse the risk of holding an asset^6 and the value of
an option which depends crucially of the volatility of the underlying asset. Finally,
more efficient estimators can be obtained if heteroscedasticity in the errors is handled
properly.
In order to achieve these forecasts, extensive previousliterature has used autore-
gressive conditional heteroscedasticity (ARCH) models, as introduced by Engle [11]
and extended to generalized ARCH (GARCH) in Bollerslev [5]. These models nor-
mally improve the volatility estimates, to a large extent, compared with a constant
variance model and they provide good volatility forecasts, so they are widely used
in various branches of econometrics, especially in financial time series analysis. In
fact, it is usually assumed that interest rate volatility can beaccurately described by
GARCH models.
(^6) In fact, VaR estimates need as the main input the volatility of portfolio returns.