124 A. D ́ıaz, F. Jare ̃no, and E. Navarro
second or higher moments of the zero coupon rates. Nevertheless, in this paper we
focus on the second moment of interest rates derived from alternative term structure
methods. So, the aim of this paper is to analyse if there are significant differences
between the estimates of the VTS depending on the model used for estimating the
term structure of interest rates (TSIR).
In this study we compare Nelson and Siegel [14],NSO, Vasicek and Fong [17],
VFO, and both models using two alternative hypotheses about the error variance. First
we assume homoscedasticity in the bond price errors and so does the term structure
as estimated by OLS. Alternatively, a heteroscedastic error structure is employed
estimating by GLS weighting pricing errors by the inverse of its duration,NSGand
VFG.
In the literature, to minimise errors in prices is usual in order to optimise any
model for estimating the TSIR. Nevertheless, this procedure tends to misestimate
short-term interest rates. This is because an error in short-termbond prices induces
an error in the estimation of short-term interest rates greater than the error in long-term
interest rates produced by the same error in long-term bond prices. In order to solve
this problem, it is usual to weight pricing errors by the reciprocal of bond Macaulay’s
duration.^1
Once estimates of TSIR are obtained, we proceed to estimate interest rate volatil-
ities using conditional volatility models (GARCH models).
In addition, we try to identify the three main components in the representation of
the VTS for each model. Some researchers have studied this subject, finding that a
small number of factors are able to represent the behaviour of the TSIR [3, 13, 15].
Nevertheless, this analysis has not been applied, to a large extent, to the VTS (except,
e.g., [1]).
We apply our methodology to the VTS from estimates of the Spanish TSIR. The
data used in this empirical analysis are the Spanish Treasury bill and bond prices of
actual transactions from January 1994 to December2006.
We show statistically significant differences between estimates of the term struc-
ture of interest rate volatilities depending on the model used to estimate the term
structure and the heteroscedasticity structure of errors (NSO,NSG,VFOandVFG),
mainly in the short-term (less than one year) and in the long-term (more than ten years)
volatility. This inspection could have significant consequences for a lot of issues re-
lated to risk management in fixed income markets. On the other hand, we find three
principal components (PCs) that can be interpreted as level, slope and curvature and
they are not significantly different among our eight proposed models.
The rest of our paper is organised as follows. The next section describes the
data used in this paper and the methodologies employed to estimate the TSIR: the
Nelson and Siegel [14], NS, and Vasicek and Fong [17], VF, models. The third section
describes the model used to estimate the term structure of volatilities. The fourth
section analyses the differences in the VTS from our eight different models. Finally,
the last two sections include a principal component analysis of VTS and, finally,
summary and conclusions.
(^1) This correction is usual in official estimations of the central banks [2].