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1 Supplementary Appendix Measuring crisis risk using conditional copulas: An empirical analysis of the 2008 shipping crisis Sebastian Opitz, Henry Seidel and Alexander Szimayer

2 Model specification Table S.1: Breusch-Godfrey LM test for serial correlation BDI BY Lag order i R 2 (T i) p-value R 2 (T i) p-value This table presents the Breusch-Godfrey LM statistics for serial correlation of Breusch-Godfrey test for different lag orders using standardized residuals of the quasi maximum likelihood estimation of Equations (1) and (4), see Breusch (1978) and Godfrey (1978). Table S.2: ARCH LM test of standardized residuals of risk factors BDI BY Lag order i ARCH LM p-value ARCH LM p-value This table presents the ARCH LM statistics of ARCH test (see Engle, 1982) for different lag orders using standardized residuals of the quasi maximum likelihood estimation of Equations (1) and (4). The null hypothesis is no ARCH up to the selected lag. * indicates the rejection of H 0 at the 5% level. 1

3 Table S.3: Distribution tests of standardized residuals BDI BY KS test AD test KS test AD test 5%-level test statistic %-level critical value p-value This table presents the test statistics of Kolmogorov-Smirnov test (KS) as well as Anderson-Darling test (AD) using standardized residuals of the quasi maximum likelihood estimation of Equations (1) and (4). The null hypothesis is that the data is t-distributed Quantiles of t(6.277) Quantiles of t(4.465) Quantiles of BDI Quantiles of BY (a) QQ-plot of BDI residual quantiles versus Student s t quantiles (b) QQ-plot of BY residual quantiles versus Student s t quantiles Figure S.1: QQ-plots of standardized residuals t(6.277) empirical cdf 0.9 t(4.465) empirical cdf (a) Empirical vs. theoretical distribution: BDI (b) Empirical vs. theoretical distribution: BY Figure S.2: Marginal distribution plots: empirical vs. theoretical distribution 2

4 Model estimates Table S.4: ML-estimates Model (1) (2) (3) (4) Conditioning factors unconditional 3 OF R,t 3 3 MSCI,t 3 3 OF R,t 3 & 3 MSCI,t 3 Parameter Estimate SE Estimate SE Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,S β BY,D BDI σbdi, σbdi, σbdi, BY σby, σby, σby, σby, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL This table presents the maximum-likelihood model estimates specified in Equations (1), (4), (7), and (8) over the sample period from 05/1997 to 12/2014. LL is the log-likelihood. *, **, and *** denote statistical significance at the 10%, 5%, and 1% level, respectively. Figures in [ ] are standard errors. 3

5 Robustness results Table S.5: ML-estimates for out-of-sample analysis Model 05/ / / / / /2006 Parameter Estimate SE Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,S β BY,D BDI σ 2 BDI BY σ 2 BY BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL

6 λ λ estimation period I 05/ /2005 out-of-sample 01/ /2014 Figure S.3: Out-of-sample estimation of shipping crisis risk starting in 01/ estimation period III 05/ /2006 out-of-sample 01/ /2014 Figure S.4: Out-of-sample estimation of shipping crisis risk starting in 01/2007 5

7 Table S.6: ML-estimates for alternative copula model I Model Full mtf-copula Mirrored Frank copula Mirrored Gumbel copula Parameter Estimate SE Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,S β BY,D BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci κ θ, κ θ,of R κ θ,msci LL

8 Table S.7: ML-estimates for alternative copula model II Model Mirrored Clayton copula Mirrored t-copula Parameter Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,OF R β BDI,MSCI BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,OF R β BY,MSCI BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci κ η, κ η,of R κ η,msci κ ρ, κ ρ,of R κ ρ,msci LL

9 Table S.8: ML-estimates for VAR(0)-model VAR(0) Parameter Estimate SE Mean equation BDI β BDI, β BDI,S β BDI,D BY β BY, β BY,S β BY,D BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL

10 Table S.9: ML-estimates with lag 1 for different window widths Window 1 month 2 months 3 months 6 months Parameter Estimate SE Estimate SE Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,S β BY,D BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL

11 Table S.10: ML-estimates with lag 2 for different window widths Window 1 month 2 months 3 months 6 months Parameter Estimate SE Estimate SE Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,S β BY,D BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL

12 Table S.11: ML-estimates with lag 3 for different window widths Window 1 month 2 months 3 months 6 months Parameter Estimate SE Estimate SE Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,S β BY,D BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL

13 Table S.12: ML-estimates with lag 6 for different window widths Window 1 month 2 months 3 months 6 months Parameter Estimate SE Estimate SE Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,S β BY,D BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL

14 30% 25% 20% 15% 10% 5% 0% BofA Merrill Lynch US Corporate B Index BofA Merrill Lynch US Corporate Shipping Index Figure S.5: Effective yields of US corporate bond (B) index and US high-yield shipping bond index 100% 80% 60% 40% 20% 0% -20% -40% % 30% 25% 20% 15% 10% 5% 0% Orderbook-to-fleet ratio (left axis) Share of fleet > 20 years in total fleet (right axis) Orderbook-to-fleet residual (left axis) Figure S.6: Adjustment of orderbook-to-fleet ratio 13

15 Table S.13: ML-estimates using shipping bond yield as risk factor for cost of capital Model (1) (2) (3) (4) Conditioning factors unconditional 3 OF R,t 3 3 MSCI,t 3 3 OF R,t 3 & 3 MSCI,t 3 Parameter Estimate SE Estimate SE Estimate SE Estimate SE BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β SY, β SY 1, β SY 1, β SY 1, β SY 1, β SY 2, β SY 2, β SY 2, β SY 2, β SY,S β SY,D BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 SY, σ 2 SY, σ 2 SY, σ 2 SY, σ 2 SY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL

16 Table S.14: ML-estimates for alternative conditioning variables Conditioning factors 3 OF R,t 3 & 3 EX,t 2 3 η OF R,t 3 & 3 MSCI,t 3 BDI β BDI, β BDI1, β BDI1, β BDI1, β BDI1, β BDI2, β BDI2, β BDI2, β BDI2, β BDI,S β BDI,D BY β BY, β BY 1, β BY 1, β BY 1, β BY 1, β BY 2, β BY 2, β BY 2, β BY 2, β BY,S β BY,D BDI σ 2 BDI, σ 2 BDI, σ 2 BDI, BY σ 2 BY, σ 2 BY, σ 2 BY, σ 2 BY, BDI ν BDI BY ν BY κ λ, κ λ,of R κ λ,msci θ LL Sample period 03/ /

17 References Breusch, T. S. (1978). Testing for autocorrelation in dynamic linear models. Australian Economic Papers 17(31): Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation. Econometrica: Journal of the Econometric Society 50(4): Godfrey, L. G. (1978). Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables. Econometrica: Journal of the Econometric Society 46(6):

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