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the expected and the actual proportion of exceptions is reduced, which improves
the unconditional performance of the model, as discussed previously. Furthermore,
and as is revealed from the back test analysis, this improvement is achieved without
generating clusters of patches of exceptions. In view of overwhelming statistical
support given by all the backtesting procedures and the quantile predictive regression,
we therefore must conclude that the VaR predictions from the risk models
extended with the variables that proxy for liquidity and, particularly, volume, are
able to track the dynamics followed by the true VaR process more closely than a
purely-returns based model.
3.3 B/M and size portfolios
It is interesting to analyze the quantile-predictability of other representative portfolios.
Fama and French (1992) found that B/M and size characteristics seem to
capture most of the cross-section of average stock returns. Also, it is usual that
investors consider risk pro…les based on growth/value and size to make …nancial
decisions. Hence, it deserves interest to analyze predictability at di¤erent quantiles
for B/M-sorted (Low30 and High30) and Size-sorted (Low30 and High30)
portfolios. The results based on predictive quantile regressions are similar to those
discussed previously. We therefore discuss the main results for the out-of-sample
analysis, since some meaningful di¤erences worthy of comment arise in this case.
[Insert Tables 6.1 and 6.2 around here]
Tables 6.1 and 6.2 report the main outcomes from the backtesting analysis of
the CAViaR models extended with trade- and order-related variables, respectively,
for the Low30 and High30 B/M portfolios given 2 11 . There are meaningful
di¤erences across these portfolios. The LR UC and LR CC tests tend to indicate
a correct performance for all the covariate-extended models for both portfolios.
However, the DQ and, particularly, the V QR test, show a much more conservative
picture about the overall predictability of the Low30 portfolio, particularly, for
spread-related variables. In contrast, the overall evidence of predictability for the
High30 B/M portfolio is much stronger. All the back tests, including V QR, suggest
that the spread-related variables (particularly, e¤ective and relative e¤ective
spread) tend to predict correctly the distribution of the return at the quantiles
analyzed. Therefore, the extent of empirical predictability varies attending to
growth/value pro…les of the portfolios involved, which in turn favours certain variables
over others. As in the case of the market portfolio, the V QR test rejects
the correct performance of all the returns-based risk models for all the quantiles
2 involved, both in the Low30 and High30 portfolios (results not reported
for saving space). Therefore, the overall evidence suggests that models including
state variables that capture di¤erent aspects of the trading process outperform
models based on returns exclusively.
[Insert Tables 7.1 and 7.2 around here]
11 The results of the independence test LR IND are not presented in this section in order to
save space, although these results are available upon request.
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