Does Systemic Risk in the Financial Sector Predict Future Economic Downturns? Online Appendix The online appendix contains a more detailed presentation of our estimation methodologies, and some of our tables, figures, and discussions.
1. Generalized Pareto distribution (GPD) The generalized Pareto distribution of Pickands (1975) is utilized to model return distribution conditioning on extreme losses. Extremes are defined as the 10% left (lower) tail of the distribution of monthly returns for financial firms (SIC code 6000 and SIC code 6999) in excess of the onemonth Treasury bill rate. Let us call f (r) the probability density function (pdf) and F(r) the cumulative distribution function (cdf) of monthly excess stock returns r. First, we choose a low threshold l, so that all r i < l < 0 are defined to be in the negative tail of the distribution, where r 1,r 2,,r n are a sequence of excess stock returns. Then we denote the number of exceedances of l (or excess stock returns lower than l) by N l = card{i : i = 1,,n,r i < l}, (1) and the corresponding excesses by M 1,M 2,,M Nl. The excess distribution function of r is given by: F l (y) = P(r l y r < l) = P(M y r < l),y 0. (2) Using the threshold l, we now define the probabilities associated with r: P(r l) = F(l), (3) P(r l + y) = F(l + y), (4) where y < 0 is an exceedance of the threshold l. Finally, let F l (y) be given by F l (y) = F(l) F(l + y). (5) F(l) We thus obtain the F l (y), the conditional distribution of how extreme a r i is, given that it already qualifies as an extreme. Pickands (1975) shows that F l (y) is very close to the generalized Pareto distribution G min,ξ in equation (6): [ G min,ξ (M;µ,σ) = 1 + ξ ( µ M σ )] 1 ξ, (6) where µ, σ, and ξ are the location, scale, and shape parameters of the GPD, respectively. The shape parameter ξ, called the tail index, reflects the fatness of the distribution (i.e., the weight of the tails), whereas the parameters of scale σ and of location µ represent the dispersion and average of the extremes, respectively. 1 The GPD presented in equation (6) has a density function: g min (Φ;x) = ( )[ 1 1 + ξ σ ( µ M σ ( ) )] 1+ξ ξ. (7) 1 The generalized Pareto distribution presented in equation (6) nests the Pareto distribution, the uniform distribution, and the exponential distribution. The shape parameter ξ, determines the tail behavior of the distributions. For ξ > 0, the distribution has a polynomially decreasing tail (Pareto). For ξ = 0, the tail decreases exponentially (exponential). For ξ < 0, the distribution is short tailed (uniform). 1
The GPD parameters are estimated by maximizing the log-likelihood function of M i with respect to µ, σ, and ξ: ( ( )) 1 + ξ µ Mi LogL GDP = nln(σ) n. (8) ξ σ ) T ( ln 1 + ξ i=1 Bali (2003, 2007) shows that the GPD distribution yields a closed form solution for VaR: ( ) [ ( ) σ αn ξ ϑ GPD = µ + 1], (9) ξ n where n and N are the number of extremes and the number of total data points, respectively. Once the location µ, scale σ, and shape ξ parameters of the GPD distribution are estimated, one can find the VaR threshold ϑ GPD based on the choice of the loss probability level α. 2 In this paper, we first take the excess monthly returns on all financial firms from January 1973 to December 2009, and then for each month in our sample we define the extreme returns as the 10% left tail of the cross-sectional distribution of excess returns on financial firms. Assume that in month t we have 900 financial firms that yield 90 extreme return observations that are used to estimate the parameters of the generalized Pareto distribution. Once we have the location, scale, and shape parameters of the GPD, we estimate the 1% VaR measure using equation (9) with N = 900, n = 90, and α = 1%. This estimation process is repeated for each month using the extreme observations in the cross-section of excess returns on financial firms, and generates an aggregate 1% VaR measure of the U.S. financial system. 2. Skewed generalized error distribution (SGED) The skewed generalized error distribution (SGED) allows us to investigate the shape of the entire distribution of excess returns on financial firms in a given month, while providing flexibility of modeling tail-thickness and skewness. The probability density function for the SGED is f (r i ;µ,σ,κ,λ) = C ) ( σ exp 1 [1 + sign(r i µ + δσ)λ] κ θ κ σ κ r i µ + δσ κ, (10) where C = κ/(2θγ(1/κ)), θ = Γ(1/κ) 0.5 Γ(3/κ) 0.5 S(λ) 1, S(λ) = 1 + 3λ 2 4A 2 λ 2, A = Γ(2/κ)Γ(1/κ) 0.5 Γ(3/κ) 0.5, µ and σ are the mean and standard deviation of excess stock returns r, λ is a skewness parameter, sign is the sign function, and Γ(.) is the gamma function. The scaling parameters κ and λ obey the following constraints κ > 0 and 1 < λ < 1. The parameter κ controls the height and tails of the density function, and the skewness parameter λ controls the rate of descent of the density around the mode of r, where mode(r) = µ δσ. In the case of positive skewness ( λ > 0), the density function is skewed to the right. This is because for values of r < µ δσ, the return variable r is weighted by a greater value than unity and for values of r > µ δσ by a value less than unity. The opposite is true for negative λ. Note that λ and δ have the same sign, thus, in case of positive skewness 2 The original VaR values are negative since they are obtained from the left tail of the return distribution. We multiply all VaR values by 1, such that larger VaR measures are associated with more catastrophic losses. 2
(λ > 0), the mode(r) is less than the expected value of r. The parameter δ is Pearson s skewness (µ mode(r))/σ = δ. 3 The SGED parameters are estimated by maximizing the log-likelihood function of r i with respect to the parameters µ, σ, κ, and λ: LogL(r i ;µ,σ,κ,λ) = Nln(C) Nln(σ) 1 θ κ σ κ N i=1 ( r i µ + δσ κ ) (1 + sign(r i µ + δσ)λ) κ, (11) where C, θ, and δ are defined below equation (10), sign is the sign of the residuals (r i µ + δσ), and N is the sample size. To come up with an aggregate 1% VaR measure of the entire financial sector, for each month we use the cross-section of excess returns on financial firms and estimate the parameters of the SGED density. Given the estimates of the four parameters (µ,σ,κ,λ), we solve for the SGED VaR threshold ϑ SGED numerically by equalizing the area under the SGED density to the coverage probability at the given loss probability level α: ϑsged (α) f µ,σ,κ,λ (z)dz = α. (12) Numerical solution of equation (12) for each month from January 1973 to December 2009 yields monthly time-series of the 1% VaR measures from the SGED density. 3. Principal component analysis In this section, we use the principal component analysis (PCA) to extract the common component of catastrophic risk embedded in the three proxies in a parsimonious manner, while suppressing potential measurement error associated with the individual VaR measures. We first standardize each of the three measures before performing PCA. The Eigen values of the three components are 2.6715, 0.2161, and 0.1124, respectively. The first principal component explains about 90 percent of the corresponding sample variance. We, therefore, conclude that the first principal component amply captures the common variation among the three VaR measures. This leads us to measure the catastrophic risk in the financial system as of month t, denoted CAT FIN, as: CAT FIN t = 0.5710 ϑ ST GPD D D D + 0.5719 ϑst SGED + 0.5889 ϑst NP, (13) where ϑ ST GPD D, ϑst SGED D, and ϑst D NP correspond to the standardized VaR measures based on the GPD, the SGED, and the nonparametric methods, respectively. Equation (13) indicates that the CAT FIN loads almost equally on the three VaR measures. The Pearson correlation coefficients between CAT FIN and the three VaR measures are in the range of 0.9333 and 0.9626. Although the three VaR measures are significantly correlated with each other, they are not as highly correlated as with CAT FIN. This suggests that the first principal component sufficiently summarizes the common variation among the three VaR measures, while reducing the potential measurement error associated with the individual VaR measures. 3 The SGED reduces to the generalized error distribution of Subbotin (1923) for λ = 0, the Laplace distribution for λ = 0 and κ = 1, the normal distribution for λ = 0 and κ = 2, and the uniform distribution for λ = 0 and κ =. 3
Table A2 shows that after controlling for a wide variety of factors, the coefficient of CAT FIN is negative and highly significant, thereby predicting the CFNAI index up to seven months in advance. From the 1-month to 6-month ahead prediction of the CFNAI index, the coefficient estimates are found to be in the range of -0.099 and -0.123 and strongly significant with the Newey and West (1987) t- statistics ranging from -2.36 to -4.28. The slope coefficient of CAT FIN forecasting 7-month ahead CFNAI index is somewhat lower in absolute magnitude, but it is still significant with a t-statistic of -1.76. The adjusted R 2 values from the predictive regressions are economically significant in the range of 27% to 61% for 1-month to 7-month ahead predictability. 4. CAT FIN measure based on the daily time-series distribution We have so far estimated the catastrophic risk of financial and nonfinancial institutions using the crosssectional distribution of monthly excess returns. In this section, we introduce an alternative risk measure based on the time-series distribution of daily excess returns. For each month in our sample, we first determine the lowest daily excess returns on financial institutions over the past 1 to 6 months. The catastrophic risk of financial institutions, denoted VaR daily FIN, is then computed by taking the average of these lowest daily excess returns obtained from alternative measurement windows. The estimation windows are fixed at 1 to 6 months, and each fixed estimation window is updated on a monthly basis. Once we generate VaR daily FIN for the entire financial sector, we test its predictive power for forecasting future economic downturns with controlling for the aforementioned macroeconomic and financial variables: CFNAI t+n = α + γvar daily FIN,t + βx t + 12 i=1 λ i CFNAI t i+1 + ε t+n. (14) Table A5 shows that the γ coefficients of VaR daily FIN,t are negative and highly significant (at the 5% level or better), and forecast the CFNAI index up to 5 months in advance for all estimation windows used in computing the average nonparametric VaR measure. The statistical significance of the coefficient estimates are comparable to those obtained using the cross-sectional CAT FIN measure. We should also note that the tail of the cross-sectional distribution (VaR and Expected Shortfall) provides very similar estimates to the aggregate VaR measure obtained from the time-series distribution; the 1% VaR of the cross-sectional distribution and the aggregate catastrophic risk of the financial sector obtained from the time-series distribution are highly correlated through time. Depending on the estimation window, the correlation between the two measures is in the range of 70% and 81%. 4 Overall, we find that downside risk obtained from the cross-sectional distribution is closely related to the health of the banking sector as well as the downside risk of the aggregate financial sector. 4 Moreover, Figure A3 of the online appendix shows that the cross-sectional distribution of financial firm equity returns is closely correlated to the aggregate VaR measure obtained using the time-series distribution. 4
5. CAT FIN from the cross-sectional and time-series distribution In this section, we will first show that the volatility of the cross-sectional distribution can be viewed as an aggregate volatility of the financial sector. Then, we will demonstrate that the tail of the crosssectional distribution (VaR and Expected Shortfall) can be viewed as catastrophic risk of the aggregate financial sector. First, we compute the volatility of the cross-sectional distribution of financial firms returns. For each month in our sample, we calculate the standard deviation of monthly returns of financial firms. To present a link between the volatility of the cross-sectional distribution and the aggregate volatility of the financial sector, we generate a standard volatility benchmark. Specifically, we compute the monthly realized volatility of financial industry returns using daily returns in a month: σ 2 ind = r d = D t d=1 N i=1 r 2 d + 2 w i r i,d. (15) D t d=1 r d r d 1. (16) where r i,d is the excess return of financial firm i on day d, r d is the financial industry return on day d, computed as the value-weighted (equal-weighted) average of daily returns of financial firms. In equation (16), σ 2 ind is the monthly realized variance of the financial sector, D t is the number of trading days in month t. The second term on the right hand side of equation (16) adjusts for the autocorrelation in daily returns using the approach of French, Schwert, and Stambaugh (1987). As an alternative to monthly realized volatility, we use an AR(1)-GARCH(1,1) model to estimate the monthly conditional volatility of financial industry returns: R t = α + βr t 1 + ε t. (17) E(ε 2 t Ω t 1 ) = σ 2 t t 1 = γ 0 + γ 1 ε 2 t 1 + γ 2 σ 2 t 1. (18) where R t is the value-weighted (or equal-weighted) financial industry return for month t. µ t t 1 α + βr t 1 is the conditional mean of R t for month t based on the information set up to time t 1, denoted by Ω t 1. σt t 1 2 is the time-t expected conditional variance of the aggregate financial sector based on the information set up to time t 1. The conditional variance, σt t 1 2, in equation (18) follows a GARCH(1,1) process defined as a function of the last period s unexpected news (or information shocks), ε t 1, and the last period s variance, σt t 1 2. The monthly realized volatility of the financial industry returns in equation (16) and the monthly GARCH volatility of the financial industry returns in equation (18) can be viewed as an aggregate volatility benchmark for the entire financial sector. Figure A4 plots the volatility of the cross-sectional distribution and the monthly realized and monthly GARCH volatility of financial industry returns (calculated as the value-weighted and equalweighted returns of all financial firms). Clearly, they are highly correlated through time. Specifically, the sample correlation between the volatility of the cross-sectional distribution and the monthly realized volatility of the financial sector is 38.21% for the value-weighted returns and 47.43% for the equalweighted returns. The sample correlation between the volatility of the cross-sectional distribution and 5
the monthly GARCH volatility of the financial sector is 57.29% for the value-weighted returns and 43.44% for the equal-weighted returns. Overall, Figure A4 provides strong evidence that the volatility of the cross-sectional distribution is closely related to the volatility of the aggregate financial sector. In addition to using the cross-sectional distribution of monthly returns, we utilize the time-series distribution of daily returns to estimate VaR measure for each financial firm. For each month in our sample, we first determine the lowest daily returns on financial institutions over the past 1 month to 12 months. The catastrophic risk of the financial sector is then computed by taking the average of these lowest daily excess returns obtained from alternative measurement windows. The estimation windows are fixed at 1 to 6 months, and each fixed estimation window is updated on a monthly basis. 5 We will now show that the tail of the cross-sectional distribution (VaR and Expected Shortfall) provides very similar estimates to the aggregate VaR measure obtained from the time-series distribution. Figure A3 plots the 1% VaR of the cross-sectional distribution and the aggregate VaR measure of the financial sector obtained from the time-series distribution. Clearly, they are highly correlated through time. Depending on the estimation window, the sample correlation between the 1% VaR of the crosssectional distribution and the aggregate VaR measures are in the range of 70% and 81%. 6 Overall, Figure A3 provides strong evidence that downside risk obtained from the cross-sectional distribution is closely related to the downside risk of the aggregate financial sector. Hence, these results show that the tail of the cross-sectional distribution (VaR and Expected Shortfall) can be viewed as catastrophic risk of the financial sector. References Bali, T. G., 2003, An Extreme Value Approach to Estimating Volatility and Value at Risk, Journal of Business, 76, 83 108., 2007, A Generalized Extreme Value Approach to Financial Risk Measurement, Journal of Money, Credit, and Banking, 39, 1611 1647. French, K. R., G. W. Schwert, and R. F. Stambaugh, 1987, Expected stock returns and volatility, Journal of Financial Economics, 19(1), 3 29. Newey, W. K., and K. D. West, 1987, A simple, positive-definite, heteroscedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, 703 708. Pickands, J. I., 1975, Statistical Inference Using Extreme Order Statistics, Annals of Statistics, 3, 119 131. Subbotin, M. T., 1923, On the Law of Frequency Error, Matematicheskii Sbornik, 31, 296 301. 5 Assuming that we have 21 daily return observations in a month, the lowest daily return over the past 1, 2, 3, 4, 5, and 6 months, respectively, yield 4.76%, 2.38%, 1.59%, 1.19%, 0.95%, and 0.79% nonparametric VaR measures. 6 The correlation between the 1% VaR of the cross-sectional distribution and the aggregate VaR measure is 81.02% for 1-month estimation window; 77.23% for 2-month estimation window, 75.52% for 3-month estimation window; 72.90% for 4-month estimation window; 71.40% for 5-month estimation window; and 70.03% for 6-month estimation window. 6
Figure A1. This figure depicts the monthly number of financial firms (SICCD >= 6000 and SICCD <= 6999) that meet the data requirements in the period of January 1963 - December 2009. Figure A2. Market share of the big-firm group. This figure depicts the share of market cap and the number of firms in the big-firm group relative to the aggregate market cap and the total number of firms in the financial sector. The big-firm group includes all financial firms with market cap above the NYSE top size quintile breakpoint. The sample period is from January 1973 to December 2009. 7
Figure A3. 1% VaR from the cross-sectional and time-series distributions. This figure presents the time-series plots of the 1% VaR obtained from the cross-sectional return distribution and the 1% aggregate VaR obtained from the time-series distribution. The sample period is from January 1973 to December 2009. 8
Figure A4. Volatility of the cross-sectional distribution of financial firms returns vs. the realized and GARCH volatility of financial industry returns This figure presents the time-series plots of the standard deviation of monthly excess returns of financial firms returns (i.e. volatility of the crosssectional distribution) and the realized and GARCH volatility of financial industry returns. The sample period is from January 1973 to December 2009. 9
Table A1 Descriptive statistics on the monthly catastrophic risk measures in the financial sector Entries in Panel A report the Pearson correlation coefficients among the monthly VaR measures and the CATFIN. ϑ GPD, ϑ SGED, and ϑ NP denote the 1% VaR estimated from the GPD, the SGED, and the nonparametric method, respectively. CATFIN is the arithmetic average of the three VaR measures. Entries in Panel B report the descriptive statistics for the three VaR measures and the CATFIN. The sample period is from January 1973 to December 2009. Significance at the 10%, 5%, and 1% level is denoted by *, **, and ***, respectively. Panel A: Pearson correlation coefficients ϑ GPD ϑ SGED ϑ NP CATFIN 0.9374 0.9354 0.9576 ϑ GPD 0.7839 0.8594 ϑ SGED 0.8631 Panel B: Descriptive statistics Mean Median Std. dev. CATFIN 27.23% 25.34% 10.97% ϑ GPD 22.76% 20.27% 12.47% ϑ SGED 30.17% 27.84% 12.00% ϑ NP 28.75% 27.42% 10.44% 10
Table A2 Predictive ability of CATFIN computed from the PCA for the Chicago Fed National Activity Index (CFNAI) Entries report the coefficient estimates from the predictive regressions: CFNAIt+n = α + γcat FINt + βxt + 12 i=1 λ icfnait i+1 + εt+n, where CAT FINt is computed as the first principal component of the three VaR measures; Xt denotes a vector of control variables; CFNAIt+n denotes the n-month ahead CFNAI. Newey and West (1987) t-statistics are reported in parentheses. The sample period is from January 1973 to December 2009. Significance at the 10%, 5%, and 1% level is denoted by *, **, and ***, respectively. For expositional purposes, the slope coefficients of the 12 lags of the CFNAI are suppressed. They are available upon request. CFNAIt+n Intercept CATFIN DEF TERM RREL FIN RET FIN VOL FIN SKEW FIN BETA MKT RET MKT VOL CORR SIZE LEV Adj. R 2 n=1 0.521-0.109-33.419-1.729-129.018-0.345 0.618 0.019-0.082-0.377-1.127 0.612-0.173-10.009 60.64% (0.87) (-4.28) (-2.47) (-0.72) (-2.15) (-0.40) (0.21) (1.92) (-0.37) (-0.34) (-0.32) (1.35) (-2.23) (-1.23) n=2 0.505-0.093-3.705-8.000-151.477 0.652-3.828-0.006 0.096-0.426 0.335 0.900-0.181-11.062 56.99% (0.69) (-3.44) (-0.22) (-2.46) (-2.67) (0.54) (-1.04) (-0.69) (0.40) (-0.34) (0.07) (1.65) (-2.12) (-1.15) n=3 0.466-0.123 3.738-9.884-187.168 1.916-3.293-0.006 0.122-0.490 0.691 0.940-0.246-15.499 51.64% (0.51) (-3.92) (0.20) (-2.66) (-3.74) (1.09) (-1.00) (-0.53) (0.43) (-0.26) (0.18) (1.50) (-2.42) (-1.30) n=4 0.129-0.111 18.180-13.400-128.829 1.168-9.489-0.003 0.265-0.254 9.110 0.889-0.294-21.513 41.06% (0.12) (-3.16) (1.00) (-3.55) (-2.27) (0.68) (-1.78) (-0.28) (0.82) (-0.12) (1.54) (1.29) (-2.67) (-1.49) n=5 0.126-0.112 20.435-14.539-137.425 2.548-10.679 0.017 0.261-3.125 10.520 0.842-0.339-25.024 35.19% (0.11) (-2.94) (1.11) (-3.45) (-2.03) (1.67) (-1.91) (1.38) (0.71) (-1.62) (1.47) (1.12) (-2.75) (-1.64) n=6 0.440-0.099 30.389-15.267-113.855 1.926-12.121 0.001 0.222-0.995 12.427 0.376-0.359-24.475 32.62% (0.33) (-2.36) (1.66) (-3.67) (-1.36) (0.97) (-2.05) (0.06) (0.59) (-0.53) (1.63) (0.49) (-2.79) (-1.53) n=7 0.746-0.087 32.164-14.833-110.492 2.309-10.129 0.012 0.149-1.245 11.078 0.055-0.391-24.079 26.57% (0.51) (-1.76) (1.59) (-3.47) (-1.21) (1.08) (-2.47) (0.78) (0.38) (-0.62) (1.82) (0.07) (-2.68) (-1.40) n=8 0.732-0.073 45.091-15.843-73.457 1.327-13.167 0.011 0.232-0.884 15.171-0.393-0.385-23.463 24.70% (0.45) (-1.44) (2.05) (-3.39) (-0.87) (0.80) (-2.90) (0.66) (0.59) (-0.50) (2.46) (-0.47) (-2.71) (-1.39) n=9 1.025-0.071 37.838-14.229-115.216 1.410-11.406 0.020 0.140-1.144 12.600-0.525-0.390-21.636 21.36% (0.58) (-1.36) (1.80) (-3.08) (-1.22) (0.74) (-3.08) (1.09) (0.32) (-0.59) (2.42) (-0.59) (-2.44) (-1.30) n=10 1.438-0.083 26.133-12.795-167.919 0.200-10.624 0.012 0.079-0.114 12.939-0.642-0.411-19.534 19.81% (0.82) (-1.62) (1.28) (-2.79) (-1.69) (0.11) (-2.02) (0.67) (0.17) (-0.06) (2.21) (-0.70) (-2.52) (-1.26) n=11 1.453-0.100 28.621-11.835-129.961 1.038-7.824 0.031-0.044-0.275 11.228-1.052-0.464-23.616 18.56% (0.81) (-2.04) (1.50) (-2.59) (-1.36) (0.57) (-1.71) (1.97) (-0.09) (-0.15) (2.05) (-1.08) (-2.60) (-1.52) n=12 1.665-0.075 32.857-12.208-208.654 1.366-10.256 0.018-0.117-1.513 10.949-1.087-0.462-23.440 17.55% (0.94) (-1.44) (1.56) (-2.82) (-2.14) (0.68) (-1.66) (1.09) (-0.22) (-0.71) (1.65) (-1.07) (-2.60) (-1.49) 11
Table A3 Predictive ability of CATFIN for the Chicago Fed National Activity Index (CFNAI) Entries report the coefficient estimates from the predictive regressions: CFNAIt+n = α + γcat FINt + βxt + 12 i=1 λ icfnait i+1 + εt+n, where CAT FINt is computed as the average of the three VaR measures; Xt denotes a vector of control variables, defined in Section 3.1; CFNAIt+n denotes the n-month ahead CFNAI. Newey and West (1987) t-statistics are reported in parentheses. The sample period is from January 1973 to December 2009. Significance at the 10%, 5%, and 1% level is denoted by *, **, and ***, respectively. CFNAIt+n Intercept CATFIN DEF TERM RREL FIN RET FIN VOL FIN SKEW FIN BETA MKT RET MKT VOL CORR SIZE LEV Adj. R 2 n=1 0.973-1.627-33.127-1.789-128.589-0.358 0.584 0.019-0.081-0.370-1.112 0.612-0.173-9.958 60.64% (1.54) (-4.29) (-2.46) (-0.75) (-2.14) (-0.41) (0.20) (1.95) (-0.37) (-0.33) (-0.31) (1.35) (-2.23) (-1.22) n=2 0.888-1.378-3.439-8.054-151.109 0.642-3.863-0.006 0.097-0.420 0.350 0.900-0.181-11.010 56.98% (1.17) (-3.44) (-0.21) (-2.48) (-2.66) (0.53) (-1.05) (-0.67) (0.41) (-0.34) (0.07) (1.65) (-2.12) (-1.14) n=3 0.970-1.816 4.208-9.970-186.663 1.910-3.366-0.005 0.123-0.480 0.723 0.939-0.246-15.399 51.62% (1.02) (-3.88) (0.23) (-2.69) (-3.73) (1.08) (-1.02) (-0.50) (0.44) (-0.25) (0.18) (1.50) (-2.41) (-1.29) n=4 0.586-1.644 18.590-13.477-128.371 1.162-9.552-0.003 0.267-0.245 9.138 0.889-0.294-21.423 41.04% (0.53) (-3.13) (1.02) (-3.57) (-2.26) (0.68) (-1.79) (-0.26) (0.82) (-0.12) (1.55) (1.29) (-2.67) (-1.48) n=5 0.585-1.649 20.944-14.632-136.945 2.548-10.769 0.017 0.263-3.113 10.560 0.842-0.339-24.892 35.15% (0.49) (-2.90) (1.14) (-3.48) (-2.02) (1.67) (-1.92) (1.40) (0.72) (-1.62) (1.47) (1.12) (-2.75) (-1.63) n=6 0.841-1.441 30.982-15.374-113.374 1.936-12.242 0.001 0.226-0.980 12.482 0.376-0.357-24.289 32.57% (0.64) (-2.31) (1.70) (-3.69) (-1.36) (0.98) (-2.07) (0.08) (0.60) (-0.52) (1.64) (0.49) (-2.79) (-1.52) n=7 1.101-1.275 32.580-14.910-110.087 2.310-10.207 0.012 0.151-1.235 11.113 0.056-0.391-23.961 26.54% (0.74) (-1.74) (1.62) (-3.49) (-1.21) (1.08) (-2.49) (0.80) (0.38) (-0.62) (1.83) (0.07) (-2.68) (-1.39) n=8 1.026-1.057 45.511-15.920-73.101 1.332-13.252 0.011 0.235-0.874 15.210-0.393-0.384-23.331 24.67% (0.61) (-1.42) (2.07) (-3.41) (-0.87) (0.80) (-2.91) (0.68) (0.59) (-0.49) (2.46) (-0.47) (-2.71) (-1.38) n=9 1.311-1.030 38.230-14.300-114.868 1.414-11.483 0.020 0.143-1.135 12.635-0.525-0.389-21.517 21.33% (0.73) (-1.34) (1.82) (-3.10) (-1.22) (0.74) (-3.09) (1.10) (0.32) (-0.58) (2.42) (-0.59) (-2.44) (-1.29) n=10 1.775-1.213 26.539-12.871-167.550 0.200-10.705 0.013 0.081-0.103 12.977-0.642-0.410-19.415 19.78% (1.01) (-1.61) (1.30) (-2.81) (-1.69) (0.11) (-2.04) (0.68) (0.18) (-0.06) (2.21) (-0.70) (-2.51) (-1.25) n=11 1.860-1.463 29.185-11.938-129.478 1.042-7.938 0.031-0.041-0.261 11.279-1.052-0.463-23.445 18.51% (1.03) (-2.01) (1.53) (-2.62) (-1.36) (0.57) (-1.73) (1.99) (-0.08) (-0.14) (2.06) (-1.08) (-2.59) (-1.50) n=12 1.971-1.104 33.098-12.256-208.364 1.364-10.304 0.018-0.116-1.509 10.977-1.086-0.462-23.382 17.55% (1.11) (-1.45) (1.57) (-2.84) (-2.13) (0.68) (-1.68) (1.10) (-0.22) (-0.71) (1.66) (-1.07) (-2.60) (-1.48) 12
Table A3 (Continued) CFNAIt+n CFNAIt CFNAIt 1 CFNAIt 2 CFNAIt 3 CFNAIt 4 CFNAIt 5 CFNAIt 6 CFNAIt 7 CFNAIt 8 CFNAIt 9 CFNAIt 10 CFNAIt 11 n=1 0.267 0.301 0.164 0.012-0.102 0.035-0.034 0.058 0.025-0.044 0.013-0.117 (4.04) (6.67) (2.68) (0.24) (-2.20) (0.75) (-0.64) (1.15) (0.51) (-0.92) (0.22) (-1.95) n=2 0.366 0.252 0.066-0.089 0.013 0.001 0.065 0.053-0.025-0.029-0.078-0.011 (7.58) (4.35) (1.28) (-1.81) (0.29) (0.01) (1.33) (1.07) (-0.50) (-0.43) (-1.40) (-0.22) n=3 0.341 0.166-0.018 0.042-0.033 0.101 0.055 0.010-0.012-0.138 0.046-0.044 (5.94) (3.14) (-0.36) (0.76) (-0.66) (1.97) (1.24) (0.21) (-0.18) (-2.53) (0.95) (-0.83) n=4 0.242 0.079 0.093-0.035 0.054 0.089 0.026 0.035-0.130-0.061-0.034 0.063 (4.17) (1.42) (1.77) (-0.69) (0.96) (1.68) (0.53) (0.60) (-2.13) (-1.12) (-0.65) (0.92) n=5 0.113 0.170 0.003 0.071 0.058 0.048 0.032-0.091-0.060-0.104 0.110-0.016 (1.63) (2.99) (0.05) (1.27) (1.08) (0.95) (0.50) (-1.41) (-1.02) (-1.79) (1.83) (-0.21) n=6 0.168 0.041 0.094 0.075 0.031 0.059-0.089-0.030-0.112 0.045 0.054-0.013 (2.24) (0.77) (1.66) (1.43) (0.61) (0.91) (-1.43) (-0.54) (-1.91) (0.78) (0.85) (-0.19) n=7 0.070 0.155 0.104 0.049 0.040-0.070-0.023-0.076 0.038-0.026 0.044 0.011 (0.98) (2.70) (1.97) (0.96) (0.61) (-1.18) (-0.42) (-1.34) (0.58) (-0.36) (0.74) (0.14) n=8 0.154 0.130 0.057 0.050-0.079-0.004-0.079 0.060-0.022-0.014 0.072-0.010 (2.32) (2.28) (1.08) (0.80) (-1.30) (-0.06) (-1.42) (0.90) (-0.34) (-0.25) (1.05) (-0.16) n=9 0.142 0.112 0.079-0.056-0.014-0.062 0.056-0.008-0.017 0.019 0.056-0.027 (2.24) (2.25) (1.29) (-0.83) (-0.24) (-1.03) (0.86) (-0.12) (-0.32) (0.28) (0.87) (-0.48) n=10 0.108 0.118-0.018 0.021-0.066 0.074-0.012-0.012 0.027 0.031 0.067-0.114 (1.73) (2.20) (-0.25) (0.36) (-1.11) (1.14) (-0.19) (-0.23) (0.42) (0.50) (1.26) (-2.41) n=11 0.117 0.014 0.034-0.048 0.072 0.003-0.006 0.036 0.021 0.028-0.024-0.067 (1.75) (0.20) (0.58) (-0.81) (1.06) (0.04) (-0.11) (0.50) (0.35) (0.44) (-0.52) (-1.08) n=12-0.013 0.086-0.005 0.095 0.012 0.012 0.046 0.037 0.030-0.083-0.003-0.014 (-0.19) (1.43) (-0.08) (1.35) (0.17) (0.20) (0.63) (0.67) (0.50) (-1.58) (-0.07) (-0.22) 13
Table A4 Predictive ability of VaR measures for the CFNAI Entries report the coefficient estimates from the predictive regressions: CFNAIt+n = α + γvart + βxt + 12 i=1 λ icfnait i+1 + εt+n, where VaRt is one of the three VaR measures: ϑgpd, ϑsged, and ϑnp; Xt denotes a vector of control variables; CFNAIt+n denotes the n-month ahead CFNAI. Newey and West (1987) t-statistics are reported in parentheses. The sample period is from January 1973 to December 2009. Significance at the 10%, 5%, and 1% level is denoted by *, **, and ***, respectively. For expositional purposes, the slope coefficients on the control variables and the 12 lags of the CFNAI are suppressed. The are available upon request. CFNAIt+n ϑsged Adj. R 2 ϑnp Adj. R 2 ϑgpd Adj. R 2 n=1-1.374 60.51% -1.536 60.39% -1.066 60.19% (-3.68) (-3.76) (-3.87) n=2-1.478 57.46% -1.252 56.74% -0.684 56.36% (-3.94) (-3.00) (-2.50) n=3-1.639 51.68% -1.874 51.61% -1.011 50.68% (-4.03) (-3.92) (-2.77) n=4-1.569 41.29% -1.665 40.98% -0.868 40.18% (-3.23) (-3.26) (-2.30) n=5-1.886 36.17% -1.744 35.22% -0.581 33.9% (-3.76) (-3.06) (-1.42) n=6-1.169 32.37% -1.870 33.24% -0.685 31.82% (-2.06) (-3.02) (-1.39) n=7-1.314 26.85% -1.421 26.68% -0.522 25.87% (-2.10) (-1.88) (-1.01) n=8-0.924 24.65% -1.331 24.98% -0.473 24.25% (-1.34) (-1.71) (-0.98) n=9-1.014 21.47% -1.244 21.55% -0.405 20.88% (-1.55) (-1.53) (-0.74) n=10-1.106 19.83% -1.404 19.98% -0.577 19.26% (-1.68) (-1.72) (-1.14) n=11-1.198 18.35% -1.817 19.02% -0.726 17.79% (-1.90) (-2.37) (-1.41) n=12-1.244 17.97% -1.048 17.44% -0.472 17.07% (-2.14) (-1.23) (-0.85) 14
Table A5 Predictive ability of VaR daily FIN for the CFNAI Entries report the coefficient estimates on VaR daily FIN from the predictive regressions: CFNAI t+n = α + γvar daily FIN,t + βx t + 12 i=1 λ icfnait i+1 + εt+n, where CFNAIt+n denotes the n-month ahead CFNAI, and VaR daily FIN,t is the catastrophic risk measure in month t, is computed by first extracting the lowest daily excess returns on financial institutions over the past 1 to 6 months, and then taking the average of the lowest daily excess returns for the financial sector obtained from alternative measurement windows. The estimation windows are fixed at one to 6 months, and each fixed estimation window is updated on a monthly basis. Newey and West (1987) t-statistics are reported in parentheses. The sample period is from January 1973 to December 2009. Significance at the 10%, 5%, and 1% level is denoted by *, **, and ***, respectively. For expositional purposes, the slope coefficients on the control variables including 12 lags of the dependent variable are suppressed. They are available upon request. 1 month 2 months 3 months 4 months 5 months 6 months CFNAIt+n VaR daily FIN,t Adj. R 2 VaR daily FIN,t Adj. R 2 VaR daily FIN,t Adj. R 2 VaR daily FIN,t Adj. R 2 VaR daily FIN,t Adj. R 2 VaR daily FIN,t Adj. R 2 n=1-12.661 60.71% -10.999 61.21% -9.377 61.14% -7.059 60.51% -5.543 60.13% -5.079 60.09% (-4.40) (-4.75) (-4.62) (-3.48) (-2.99) (-2.85) n=2-13.178 57.61% -10.583 57.83% -7.172 57.1% -5.294 56.69% -4.845 56.66% -4.044 56.51% (-4.05) (-4.32) (-3.07) (-2.47) (-2.44) (-2.13) n=3-14.675 51.89% -9.540 51.34% -7.326 51% -6.827 51.05% -5.744 50.82% -4.909 50.63% (-4.13) (-3.20) (-2.96) (-3.07) (-2.71) (-2.49) n=4-10.714 40.65% -8.805 40.86% -7.944 40.98% -6.598 40.72% -5.448 40.46% -5.071 40.44% (-2.41) (-2.90) (-3.10) (-2.72) (-2.37) (-2.31) n=5-11.712 34.97% -10.093 35.39% -7.651 34.97% -6.449 34.76% -5.708 34.64% -5.215 34.57% (-2.29) (-2.75) (-2.38) (-2.12) (-1.94) (-1.89) n=6-7.538 31.96% -5.470 31.92% -4.427 31.85% -4.367 31.92% -3.757 31.84% -3.339 31.79% (-1.31) (-1.33) (-1.24) (-1.28) (-1.15) (-1.08) n=7-4.776 25.85% -5.075 26.07% -5.250 26.25% -4.981 26.29% -4.282 26.18% -3.332 26% (-0.85) (-1.27) (-1.46) (-1.46) (-1.31) (-1.10) n=8-2.866 24.13% -4.004 24.32% -3.622 24.34% -3.489 24.37% -2.293 24.2% -1.855 24.16% (-0.48) (-0.89) (-0.91) (-0.92) (-0.65) (-0.57) n=9-1.210 20.75% -2.415 20.83% -2.774 20.9% -2.038 20.84% -1.486 20.8% -0.935 20.76% (-0.19) (-0.52) (-0.67) (-0.53) (-0.41) (-0.27) n=10-2.901 19.04% -3.892 19.22% -3.126 19.18% -2.933 19.19% -2.196 19.1% -2.008 19.09% (-0.43) (-0.76) (-0.71) (-0.73) (-0.57) (-0.55) n=11-4.584 17.54% -3.400 17.53% -3.408 17.59% -3.152 17.59% -2.900 17.57% -2.450 17.52% (-0.75) (-0.75) (-0.88) (-0.86) (-0.83) (-0.75) n=12 1.364 16.89% -0.927 16.89% -1.186 16.9% -2.113 16.99% -1.829 16.97% -1.902 16.99% (0.21) (-0.21) (-0.30) (-0.57) (-0.52) (-0.57) 15
Table A6 Predictive ability of CAT FIN ES for the Chicago Fed National Activity Index (CFNAI) Entries report the coefficient estimates from the predictive regressions: CFNAIt+n = α+γcat FIN t ES +βxt + 12 i=1 λ icfnait i+1 +εt+n, where CAT FIN t ES is computed as the average of the three expected shortfall measures; Xt denotes a vector of control variables; CFNAIt+n denotes the n-month ahead CFNAI. Newey and West (1987) t-statistics are reported in parentheses. The sample period is from January 1973 to December 2009. Significance at the 10%, 5%, and 1% level is denoted by *, **, and ***, respectively. For expositional purposes, the slope coefficients on 12 lags of the CFNAI are suppressed. They are available upon request. CFNAIt+n Intercept CAT FIN ES DEF TERM RREL FIN RET FIN VOL FIN SKEW FIN BETA MKT RET MKT VOL CORR SIZE LEV Adj. R 2 n=1 0.976-1.096-32.373-2.034-135.404-0.037-0.329 0.020-0.071-0.263-0.900 0.586-0.160-9.360 60.35% (1.55) (-3.65) (-2.27) (-0.84) (-2.22) (-0.04) (-0.12) (2.05) (-0.32) (-0.24) (-0.26) (1.28) (-2.05) (-1.14) n=2 0.912-0.993-3.718-8.128-157.445 0.875-4.432-0.005 0.098-0.347 0.425 0.878-0.174-10.855 56.88% (1.20) (-3.24) (-0.22) (-2.45) (-2.76) (0.72) (-1.21) (-0.61) (0.41) (-0.28) (0.09) (1.60) (-2.03) (-1.12) n=3 0.978-1.229 4.468-10.234-194.361 2.242-4.345-0.004 0.140-0.348 0.929 0.929-0.230-14.568 51.26% (1.02) (-3.76) (0.24) (-2.72) (-3.83) (1.24) (-1.31) (-0.42) (0.49) (-0.18) (0.24) (1.47) (-2.26) (-1.21) n=4 0.625-1.213 17.492-13.508-136.241 1.406-10.129-0.002 0.269-0.158 9.172 0.876-0.286-21.256 40.98% (0.56) (-3.04) (0.92) (-3.54) (-2.39) (0.79) (-1.90) (-0.21) (0.82) (-0.08) (1.56) (1.27) (-2.60) (-1.49) n=5 0.706-1.446 16.708-14.165-147.155 2.613-10.609 0.017 0.232-3.052 10.218 0.816-0.346-26.070 35.68% (0.59) (-3.38) (0.87) (-3.36) (-2.18) (1.66) (-1.83) (1.37) (0.64) (-1.59) (1.41) (1.10) (-2.84) (-1.72) n=6 0.975-1.341 26.260-14.793-123.380 1.939-11.847 0.001 0.186-0.952 12.044 0.346-0.369-25.826 33.16% (0.73) (-2.88) (1.39) (-3.61) (-1.48) (0.96) (-1.90) (0.06) (0.51) (-0.51) (1.55) (0.46) (-2.95) (-1.64) n=7 1.217-1.175 28.555-14.409-118.986 2.321-9.883 0.012 0.116-1.206 10.737 0.027-0.401-25.278 26.97% (0.82) (-2.25) (1.43) (-3.40) (-1.31) (1.06) (-2.27) (0.78) (0.30) (-0.61) (1.74) (0.03) (-2.82) (-1.48) n=8 1.093-0.888 43.342-15.684-79.725 1.391-13.275 0.011 0.216-0.825 15.057-0.415-0.387-23.933 24.82% (0.65) (-1.60) (2.02) (-3.37) (-0.95) (0.83) (-2.84) (0.67) (0.55) (-0.47) (2.41) (-0.50) (-2.77) (-1.42) n=9 1.428-1.006 33.974-13.749-123.226 1.363-11.098 0.020 0.105-1.114 12.304-0.553-0.401-22.910 21.72% (0.79) (-1.76) (1.62) (-2.95) (-1.32) (0.72) (-2.94) (1.08) (0.24) (-0.58) (2.33) (-0.63) (-2.54) (-1.39) n=10 1.861-1.034 23.593-12.546-176.010 0.227-10.777 0.013 0.058-0.037 12.871-0.668-0.414-20.173 20% (1.05) (-1.83) (1.16) (-2.73) (-1.78) (0.12) (-2.10) (0.68) (0.13) (-0.02) (2.21) (-0.74) (-2.56) (-1.30) n=11 1.963-1.247 25.834-11.572-139.318 1.065-7.990 0.031-0.068-0.181 11.098-1.084-0.468-24.347 18.84% (1.09) (-2.32) (1.36) (-2.51) (-1.47) (0.58) (-1.82) (2.01) (-0.13) (-0.10) (2.06) (-1.12) (-2.63) (-1.56) n=12 2.110-1.095 28.743-11.680-216.833 1.222-9.671 0.017-0.160-1.443 10.387-1.120-0.477-24.981 18.04% (1.19) (-2.08) (1.35) (-2.73) (-2.22) (0.61) (-1.66) (1.09) (-0.30) (-0.69) (1.59) (-1.12) (-2.69) (-1.59) 16
Table A7 Predictive ability of CAT FIN and micro-level systemic risk measures This table reports the results of predictive regressions of the CFNAI on the CAT FIN, microlevel systemic risk measures, and the controls variables. The micro-level measures of systemic risk are the conditional tail risk (CT R), marginal expected shortfall (MES), and distance to default (DD). For expositional purposes, the slope coefficients on the control variables including 12 lags of the dependent variable are suppressed. They are available upon request. CFNAI t+n CATFIN CTR MES DD Adj. R 2 n=1-1.853 0.021-0.065 0.046 52.14% (-3.72) (0.17) (-0.79) (0.91) n=2-1.462-0.152-0.118 0.022 47.44% (-3.08) (-1.12) (-1.08) (0.44) n=3-1.641-0.135-0.153 0.018 42.16% (-2.97) (-0.85) (-1.23) (0.34) n=4-1.344-0.230-0.289 0.015 33.17% (-2.53) (-1.24) (-1.94) (0.27) n=5-1.540-0.200-0.368 0.004 30.82% (-2.72) (-0.92) (-2.45) (0.05) n=6-1.642-0.296-0.386 0.094 29.09% (-2.74) (-1.26) (-2.72) (1.21) 17
Table A8 Descriptive statistics on monthly catastrophic risk measures for the fake bank sample Entries report the descriptive statistics on the monthly VaR measures and the CATFIN for the fake bank sample. ϑ f ake GPD, ϑ f ake f ake SGED, and ϑnp denote the 1% VaR estimated from the GPD, the SGED, and the nonparametric method, respectively. CAT FIN f ake is the arithmetic average of the three VaR measures. The sample period is from January 1973 to December 2009. Mean Median Std. dev. CAT FIN f ake 32.42% 31.10% 10.67% ϑ f ake GPD 29.14% 26.48% 13.25% ϑ f ake SGED 35.69% 33.99% 11.84% ϑ f ake NP 32.42% 31.24% 9.84% 18
Table A9 Catastrophic risk of big and small financial firms For each month in our sample the NYSE top size quintile breakpoint is used to divide financial firms into two groups: big firms with market cap above the breakpoint and small firms with market cap below the breakpoint. CATFIN for big and small firms is denoted CATFINBIG and CATFINSML, respectively. Entries report the coefficient estimates from regressions of the n-month ahead CFNAI on CATFINBIG and CATFINSML after controlling for the large set of economic and financial variables and 12 lags of the dependent variable. The last two columns report the Wald statistics and the corresponding p-values from testing the equality of slope coefficients on CAT FINBIG and CAT FINSML. The Wald statistic is distributed as the Chi-square with one degree of freedom. Newey and West (1987) t-statistics are reported in parentheses. The sample period is from January 1973 to December 2009. Significance at the 10%, 5%, and 1% level is denoted by *, **, and ***, respectively. For expositional purposes, the slope coefficients on the control variables including 12 lags of the dependent variable are suppressed. They are available upon request. CFNAI t+n CATFINBIG CATFINSML Adj. R 2 CAT FINBIG CAT FINSML Wald p-value n=1-0.987-1.704 59.86% 0.717 0.88 0.35 (-1.84) (-4.07) n=2-1.567-1.185 55.05% -0.382 0.23 0.63 (-2.64) (-2.73) n=3-1.575-1.468 49.53% -0.107 0.01 0.91 (-2.49) (-2.76) n=4-1.881-1.446 36.62% -0.435 0.19 0.67 (-2.59) (-2.32) n=5-2.036-1.229 28.96% -0.807 0.48 0.49 (-2.67) (-1.63) n=6-2.157-1.128 27.85% -1.029 0.65 0.42 (-2.24) (-1.52) n=7-1.328-0.908 21.00% -0.420 0.10 0.75 (-1.52) (-1.00) n=8-1.404-0.698 17.83% -0.706 0.27 0.61 (-1.50) (-0.76) n=9-1.662-0.848 15.80% -0.814 0.32 0.57 (-1.77) (-0.88) n=10-1.754-0.966 14.22% -0.788 0.26 0.61 (-1.77) (-0.97) n=11-1.145-1.228 12.47% 0.083 0.00 0.95 (-1.28) (-1.33) n=12-1.653-1.193 11.47% -0.460 0.11 0.74 (-1.79) (-1.43) 19
Table A10 Predicting financial and economic uncertainty with CAT FIN This table reports coefficient estimates from the predictive regressions: U t+n = α + βcat FIN t + ε t+n, where U denotes one of the proxies for uncertainty: V IX, V XO, AV GVOL or PCAVOL. V IX and V XO are the Chicago Board Options Exchange (CBOE) s implied volatility indices and they provide investors with up-to-the-minute market estimates of expected future volatility by using real-time index option bid/ask quotes. AV GVOL and PCAVOL denote, respectively, the average and the first principal component of the volatility measures of a set of macroeconomic variables: (i) Default spread (DEF), (ii) Term spread (T ERM), (iii) Relative T-bill rate (RREL), (iv) Aggregate dividend yield (DIV ), (v) the monthly growth rate of the U.S. industrial production (INDP), (vi) the monthly inflation rate based on the U.S. consumer price index (INF), and (vii) the monthly excess return on the value-weighted CRSP index (MKT RET ). For each macroeconomic variable, we estimate the time-varying conditional volatility based on the standard AR(1)-GARCH(1,1) model. Newey and West (1987) t-statistics are reported in parentheses. Significance at the 10%, 5%, and 1% level is denoted by *, **, and ***, respectively. V IX t+n V XO t+n AV GVOL t+n PCAVOL t+n CAT FIN t Adj. R 2 CAT FIN t Adj. R 2 CAT FIN t Adj. R 2 CAT FIN t Adj. R 2 n=1 28.046 30.77% 29.487 22.61% 1.594 1.53% 4.389 1.15% (3.27) (3.30) (2.12) (2.31) n=2 29.344 25.78% 28.339 17.11% 1.850 1.12% 4.993 2.36% (3.79) (3.52) (2.72) (2.81) n=3 29.811 22.71% 29.088 15.94% 1.782 0.81% 4.895 2.33% (4.15) (3.93) (2.61) (2.74) n=4 29.671 18.79% 29.605 13.26% 1.939 1.30% 5.211 2.59% (4.14) (3.91) (2.84) (2.94) n=5 28.105 13.74% 28.953 8.17% 1.887 0.04% 5.199 1.77% (4.56) (4.19) (2.89) (2.96) n=6 29.467 9.01% 29.468 4.79% 1.730 0.48% 4.920 1.14% (5.46) (4.61) (2.73) (2.86) 20