Does Tail Dependence Make A Difference In the ... - Boston College

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where T is the total number of observation in the sub-sample It+1 i = 1 when financial institutions are in distress, and N is the number of violations for CoVaR which satisfy It+1 i × I m|i t+1 = 1, that is, when both financial firms and market are in distress. Therefore N/T is the empirical hit ratio for the CoV aR in the sub-sample It+1 i = 1. 4.3.2 Conditional Coverage Testing If the forecast of CoVaR is precise, we would not expect that the violation of CoVaR is clustered. In other words, the hit sequence I m|i t+1 should be independent over time. Christoffersen (1998) suggested the following likelihood approach for the independence test LR cc = 2log (1 − π 01) n00 π n01 01 (1 − π 11) n10 π n11 11 (1 − N/T ) T −N (N/T ) N ∼ χ(1) under H o where π ij i, j = 0, 1 is the transition probability. For example, conditional on today being a non-violation (I m|i t = 0), the probability of tomorrow being a violation(I m|i t+1 = 1) is π 0,1. and n ij i, j = 0, 1 is the total number of days that state j occurred after state i occurred the previous days. Therefore, the transition probability can be calculated as π 0,1 = n0,1 n 0,0+n 0,1 and π 1,1 = n1,1 n 1,0+n 1,1 . An approximate estimates of CoVaR should satisfy at least these two backtests. 5 Data and Empirical Results 5.1 Data The recent financial crisis provides ample evidence of a risk spill-over effect from individual financial institutions to the whole financial industry. The sample studied in this paper is common and widely used in the recent literatures. We consider daily holding period returns for almost the same panel of financial institutions studied by Acharya et al.(2010), Brownlees and Engel (2011) and Sylvain Benoit et al. (2012) between January 3, 2000 and December 30, 2011. The sample contains almost all U.S. financial institutions with equity market capitalization in excess of 5bln USD as of end of June 2007 (92 firms in total), among which, there are about 64 companies that have been trading [ Insert Table 1 Here ] continuously during the whole sample period in which each institution has 3018 observations. Following Brownless and Engle (2011), the sample of institutions can be split into four groups based on their two-digit SIC classification codes (Depositories, Insurance, Broker-Dealers and Others). Table 1 presents the full list of financial institutions studied in the paper. The return on the Dow Jones U.S. Financials Index (DJUSFN) is used as a proxy for the market return of the financial system. CoVaR and MES are computed at the most common confidence interval τ = 5%. We obtain our data from Bloomsburg terminal. [ Insert Table 3 Here ] Table 3 gives some descriptive statistics for the unfiltered return series over the whole sample. We can observe that the broker-dealers companies have been the riskiest groups: high volatility, large kurtosis, large VaR and Expected Shortfall (ES). The Jarque-Bera test confirms that all time series of daily returns are not normally distributed with significant excess kurtosis. The Ljung-Box test can be rejected for the market index (DowJones), 24
Depositories and Broker-Dealer companies,which indicates the presence of autocorrelation for these groups of financial institutions. As the characteristics of the extremes of the distribution show, the asymmetry of the distribution toward right tail is mildly pronounced for all categories of financial firms. [ Insert Table 4 Here ] Table 4 provides summary statistics on the parameter estimates (median across firms) for the institutions in various industry categories. As we can see, the difference between all categories are trivial. The individual volatility estimation displays the same persistence across all groups of financial firms. The univariate distributions have fat tails as expected. The degrees of freedom ν of skewed t distribution ranges between 3.8 to 4.1. The asymmetry parameter λ is found to be close to zero, indicating the univariate distribution being quite symmetrical. The estimation of the dynamic student t copula model shows that all financial firms are highly correlated with the market index return, with the median linear correlation ρ ranging from 0.64 to 0.77, which is consistent with the fact that the dependence between financial institutions and market is strong. Figure 9: The upper panel of figure displays the median across firms of conditional correlation for the dynamic student t copula. The lower panel of figure presents the median across firms of lower(upper) tail dependence for mixture Copula C+G+N. The acronyms C+G+N refers to the mixture copula: Clayton+Gumbel+Normal. The parameters of mixture copula model are estimation every 1 month in the Rolling window of 24 months. Figure 9 displays the median across firms of conditional correlation ρ for dynamic student t copula estimated by ’GAS’ model discussed in Section 4.2, The lower panel presents the median across firms of lower (upper) tail dependence for mixture Copula model Clayton+Gumbel+Normal. It is clear that the conditional correlation is higher during the financial crisis than that prior to crises, which is consistent with the well-documented empirical results that conditional correlations increase during an economics downturn. However, the rolling window estima- 25
Page 1 and 2: Does Tail Dependence Make A Differe
Page 3 and 4: Figure 1: The upper panel graphs di
Page 5 and 6: more related to the “too big to f
Page 7 and 8: CoV aR (hereafter called CoV aR )
Page 9 and 10: Theorem 1 (Sklar)(1959) Let G be a
Page 11 and 12: Therefore the systemic risk measure
Page 13 and 14: Figure 3: This figure displays 8000
Page 15 and 16: (lower dependence with the market).
Page 17 and 18: higher value of systemic risk. Figu
Page 19 and 20: oth estimated based on the joint di
Page 21 and 22: Figure 8: This figure displays the
Page 23: the Goodness of Fit (GoF) of copula
Page 27 and 28: 5.2 Empirical Results 5.2.1 Estimat
Page 29 and 30: [ Insert Table 6 Here ] Table 6 pre
Page 31 and 32: isk measure (last panel of Table 9)
Page 33 and 34: References Acharya, V. V., Pedersen
Page 35 and 36: Appendix Proof on the relationship
Page 37 and 38: Therefore v = F Y (y) = Φ( y −
Page 39 and 40: In the case of continuous random va
Page 41 and 42: Table 2: Bivariate Copula Functions
Page 43 and 44: Table 5: AIC for the Different Copu
Page 45: Table 8: Systemic Risk Rankings:

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