"Frontmatter". In: Analysis of Financial Time Series

QUANTILE ESTIMATION 269ˆx 0.01 = p 2 − 0.01p 2 − p 1r (91) + 0.01 − p 1p 2 − p 1r (92)= 0.000010.0001(−3.658) +0.00011 0.00011 (−3.657)≈−3.657.The 1% 1-day horizon VaR of the long position is $365,709. Again, this amount islower than those obtained before by other methods.Discussion: Advantages of using the prior quantile method to VaR calculationinclude (a) simplicity, and (b) using no specific distributional assumption. However,the approach has several drawbacks. First, it assumes that the distribution of thereturn r t remains unchanged from the sample period to the prediction period. Giventhat VaR is concerned mainly with tail probability, this assumption implies that thepredicted loss cannot be greater than that of the historical loss. It is definitely not soin practice. Second, for extreme quantiles (i.e., when p is close to zero or unity), theempirical quantiles are not efficient estimates of the theoretical quantiles. Third, thedirect quantile estimation fails to take into account the effect of explanatory variablesthat are relevant to the portfolio under study. In real application, VaR obtained by theempirical quantile can serve as a lower bound for the actual VaR.7.4.2 Quantile RegressionIn real application, one often has explanatory variables available that are importantto the problem under study. For example, the action taken by Federal Reserve Bankson interest rates could have important impacts on the returns of U.S. stocks. It is thenmore appropriate to consider the distribution function r t+1 | F t ,whereF t includesthe explanatory variables. In other words, we are interested in the quantiles of thedistribution function of r t+1 given F t . Such a quantile is referred to as a regressionquantile in the literature; see Koenker and Bassett (1978).To understand regression quantile, it is helpful to cast the empirical quantile ofthe previous subsection as an estimation problem. For a given probability p, the pthquantile of {r t } is obtained bywhere w p (z) is defined byˆx p = argmin βw p (z) =n∑w p (r i − β),i=1{ pz if z ≥ 0(p − 1)z if z < 0.Regression quantile is a generalization of such an estimate.