Extreme Asymmetric Volatility, Leverage, Feedback and Asset Prices

Electronic copy available at: http://ssrn.com/abstract=1348563

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Electronic copy available at: http://ssrn.com/abstract=1348563

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Table 1: Descriptive Statistics

Descriptive statistics for unexpected return and volatility shocks, Yt and Zt. Ljung/Box Q(k)-statistics at lag k test for the null hypothesis of zero autocorrelation coefficients up to order k in each of the original as well as squared series. GARCH-f1 denotes maximum likelihood estimation results for model (2.2-3) with p = 4 and asymmetric specification f1. Results are derived under the assumption of Student-t distributed volatility

shocks Zt with an estimated number of degrees of freedom of 5.27 (standard error of 0.36) and a maximized log likelihood value of 7499.8. Given the model, Engle’s ARCH LM (k)-statistic at lag k = 1, 2 and 3 is computed from an auxiliary test regression and tests the null hypothesis of no ARCH effects in the shocks Zt up to order k. Sample period: January 2, 1990 to September 30, 2008.

Y Z Z: GARCH-f1 Mean 0.0265 0.0447 ω0 (t-value) 0.0000 (0.043) Median -0.00850 -0.0155 ω1 (t-value) -0.142 (-1.01) Maximum 3.52 10.04 ω2 (t-value) -0.241 (-1.19) Minimum -3.63 -4.34 γ0 (t-value) 0.0002** (6.15) Std. Dev. 0.74 1.01 γ1 (t-value) 0.120** (7.39) Skewness 0.0694 0.987 γ2 (t-value) -0.125** (-6.60) Kurtosis 3.79 9.01 γ3 (t-value) 0.864** (47.99) Q (1) original - (probability) Q (1) squared - (probability) 1.918 (0.17) 0.573 (0.45) 0.127 (0.72) 0.0920 (0.76) Engle LM (1) (probability) 0.0919 (0.76) Q (2) original - (probability) Q (2) squared - (probability) 5.367 (0.068) 5.859 (0.053) 0.190 (0.91) 0.426 (0.81) Engle LM (2) (probability) 0.428 (0.81) Q (3) original - (probability) Q (3) squared - (probability) 9.910* (0.019) 11.955** (0.008) 0.532 (0.91) 0.445 (0.93) Engle LM (3) (probability) 0.446 (0.93)

Table 2: Estimates of the Dynamic Conditional Correlation Model

Maximum likelihood parameter estimates of the Dynamic Conditional Correlation (DCC) between unexpected return and volatility shocks, Yt and Zt. The asymptotic mean test is based on the hypothesis that mean conditional correlation is equal to zero, the asymptotic variance test is based on the hypothesis that the variance of conditional correlation is equal to zero. The Jarque-Bera normality test is based on the null hypothesis that conditional correlation is normally distributed. Pearson's correlation test is based on the hypothesis that unconditional correlation is zero. Sample period: January 2, 1990 to September 30, 2008.

** denotes significance for a double-sided test at the 99% confidence level.

Dynamic Conditional Correlation ν1 ν2 _logL

Parameter estimate (standard error) 0.0158** (0.00188) 0.980** (0.00280) 3188.1

H0: Mean = 0 H0: Var = 0 Jarque-Bera

Test statistic (p-value) -418.3 (0.000) 51.9 (0.000) 480.8 (0.000)

Unconditional Correlation 95 % confidence interval

Parameter estimates (t-statistic)

-0.689 (-66.48)

Table 3: Threshold Choice

Guidance from three different threshold choice approaches for the upper and lower tail of the return and volatility shocks, Yt and Zt. MEF denotes graphical inspection of the mean excess plot and selection of a value beyond which the plot is approximately linear. The threshold plot method displays the resulting tail index estimates as a function of the chosen threshold. The given intervals denote the range of threshold choices for which the resulting tail estimates appear reasonably stable. Visual selection denotes a plausible threshold choice based on the results of the two given plot methods. Adaptive selection denotes the application of an automated method which aims at minimizing asymptotic mean squared error (see Beirlant et al. (2004), Section 4.7 ii). Sample period: January 2, 1990 to September 30, 2008.

Y −Y Z −Z

MEF plot 1.5 1.5 1.5 1.0

Threshold plot [0.2, 1.5] [1.0, 1.5] [1.5, 2.5] [0.7, 1.5]

Visual selection 1.5 1.2 1.5 1.5

Adaptive selection 0.7* 1.4* 1.6* 0.7*