The probability level α is assumed to be uniformly distributed between 0 and 1. When α follows other distributions, the difference between implied correlations and linear correlations is assigned different weights at different probability levels. Ang and Chen (2002), for example, chose weights proportional to the number of observations.

Under the null hypothesis that returns follow multivariate normal distribution, the linear correlation is appropriate and the theoretical implied correlations equal the theoretical linear correlation. Since the sample mean, the empirical estimates of VaR and ES are consistent estimators of their corresponding theoretical values, the estimated implied correlations converge to the theoretical implied correlations, which equal the linear correlation under _H_ 0. The alternative hypothesis is that the linear correlation underestimates (overestimates) correlations during market downturns (upturns). The null hypothesis is rejected when the test statistic is too large.

I use bootstrap to test the significance of the test statistics. The bootstrap technique provides a simple way to test statistics whose distribution is unknown, but can be simulated.<sup>6</sup> Take the _H_ statistics for example. I use the following procedure to make statistical inference. Given two assets with sample size _n_ , Step A: Calculate the test statistic _H_<sup>ˆ</sup> , sample means, standard deviations of each individual asset, and their correlation using the empirical data.

Step B: Draw _n_ pairs of data from the bivariate normal distribution using the sample mean, standard deviation, and correlation calculated in Step A.

Step C: Compute the test statistic, named _H_<sup>(1)</sup> for _H_ statistic from the simulated data.

Step D: Repeat step B and step C M times and get a sequence of test statistics, _H_<sup>(1)</sup> ,..., _H_<sup>(</sup><sup>_M_)</sup> . ˆ <u>1</u> _M_ Step E: Calculate the _p_ -values, _pH_ = _M_ +1 � _m_ =1<sup>(</sup><sup>_I_(</sup><sup>_H_(</sup><sup>_m_)≥</sup><sup>_H_ˆ) + 1),where</sup><sup>_I_(·)isknownasthe</sup> indicator function. The hypothesis is rejected at level α if the _p_ -value is less than or equal to α.

Notice that the _H_ statistics in this paper differ from the statistic in Ang and Chen (2002) in several ways. First, the thresholds considered in the paper are continuous, while the thresholds in the paper of Ang and Chen (2002) are a number of discrete points. Second, while Ang and Chen (2002) used the quadratic deviation and the sum of deviation between the linear correlation and the exceedance correlation, this paper measures the maximum and the average of the deviation of the linear correlation from implied correlations. It is common to construct statistics using maximum. See Hansen (1996), and Davies (1977, 1987) for example. Third, this paper uses the risk measureimplied correlations, which are free of conditioning bias, to measure correlation asymmetries, while Ang and Chen (2002) have to adjust conditioning bias of the exceedance correlation. Fourth, the significance of the test statistic is examined by bootstrap in this paper, while Ang and Chen (2002) used the generalized method of moments (GMM) and the delta method to get the standard deviation of test statistic first and then calculate the p-value.

> 6The asymptotic distribution of the test statistics is very complex. See Appendix A.

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