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3.2 Market portfolio analysis
Throughout the following subsections we analyze the ability of the predictive variables
to forecast day-ahead VaR for the returns of the volume-weighted market
portfolio, both through a predictive regression that involves the whole sample
and an out-of-sample predictive analysis in a rolling-window approach. The focus
on day-ahead estimation is consistent with the holding period considered for
internal risk control by most …nancial …rms (Taylor, 2008). Owing to their empirical
relevance in risk management, we are particularly interested in the quantiles
2 f0:01; 0:05g ; but shall also analyze a wider range of probabilities given by
= f0:01; 0:025; 0:05; 0:075g to characterize predictability along the left tail of
the distribution. It should be mentioned that in this section we also considered the
analysis on the returns of the value-weighted market portfolio, which are not presented
for saving space as the qualitative evidence is completely analogous to that
discussed below. The complete analysis is available from authors upon request.
The daily returns of the volume-weighted market portfolio over the total period
analyzed (see Figure 1 below) include di¤erent episodes in terms of activity and
volatility. The beginning of the sample corresponds to the period that followed the
market crash in October 1987. After the extraordinary crash, the volatility of the
market decreased progressively and reverted to normal levels. In 1998, Long-Term
Capital Management failure in the hedge-fund industry led to a massive bailout
by other major banks and investment houses that, in turn, generated an excess
of volatility in the market and which preceded the burst of the dot-com …rms
in 2000. Finally, the data from 2000 onwards show the large excess of volatility
that characterized the market after the burst of the technological bubble. This
particular period, depicted by the grey line in Figure 1, will be analyzed in detail
in the out-of-sample analysis (back test) later on.
[Insert Figure 1 around here]
3.2.1 Predictive regression analysis
We …rst analyze predictability through predictive-like quantile regressions. More
speci…cally, we estimate the unrestricted CAViaR model
V aR ;t = ;0 + ;1 V aR ;t 1 + ;2 jr t 1 j + j log (X t 1 ) j (6)
for any of the conditioning variables X t described in Section 3.1, using the entire
sample, from 01-04-1988 to 12-31-2002, with 3; 782 observations. Our main
interest is to test the suitability of these unrestricted models against a restricted
CAViaR speci…cations based solely on returns. Clearly, a rejection of the null hypothesis
H 0 : = 0 for a speci…c model and quantile provides formal evidence
on the suitability of the posited variable as a predictor of the conditional distribution.
The objective function in the quantile regression minimization problem is
optimized through the Simulated Annealing optimization algorithm (Go¤e, Ferier
and Rogers, 1994), while the asymptotic covariance matrix is estimated using a
robust sandwich-type estimator based on a k-nearest neighbor kernel, as described
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