Appendix 1 Model Selection: GARCH Models. Parameter estimates and summary statistics for models of the form: 1 if ɛt i < 0 0 otherwise
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1 Appendix 1 Model Selection: GARCH Models Parameter estimates and summary statistics for models of the form: R t = µ + ɛ t ; ɛ t (0, h 2 t ) (1) h 2 t = α + 2 ( 2 ( 2 ( βi ht i) 2 + γi ɛt i) 2 + δi D t i ɛt i) 2 i=1 D t i = i=1 { 1 if ɛt i < 0 0 otherwise i=1 (2) (3) are found in Panel A for Tables A1.1 to A1.4 below. A model with p lags of h t, q lags of ɛ 2 t and r lags of D t ɛ 2 t is labeled GJR(p,q,r). Such a model excluding the asymmetric volatility term D t ɛ 2 t is labeled GARCH(p,q). These models were estimated on the period from January 4, 1988 to the end of September Presented in Panel B of the tables are the model Log Likelihood, Schwarz information criteria (BIC), and tests for residual autocorrelation (Wald AR), ARCH and Sign-ARCH. The test for autocorrelation is a χ 2 Wald test on 5 (3) lags of the daily (monthly) data residuals in the mean equation. The test for ARCH is a standard LM test, having us regress the current squared standardized residual on 24 (3) lags of the daily (monthly) data squared standardized residual, collecting the regression R 2 and forming z = nr 2, where n is the number of observations, and z is distributed χ 2 with 24 (3) degrees of freedom. The Sign-ARCH tests were proposed by Engle and Ng [1993] and test for stronger (asymmetric) volatility increases from negative shocks to the returns process. Our Sign-ARCH tests on daily data used 5 lags, 3 lags for monthly data. The R 2 for all the models below are 0 as the mean regression equation in each model contains only a constant, and so this statistic is not included. Of the models that remove significant evidence (at the 5% level) of time series dependence, including residual autocorrelation, ARCH and Sign-ARCH, the GARCH(1,1) specification has the best BIC as well as being the most simple model. The GJR(0,1,2) and GJR(0,1,1) models do very nearly as well on all criteria we investigate failing only one specification test at the 5% level, and slightly better by the BIC, but we will favor the more familiar GARCH(1,1) model. 1
2 Table A1.1 Daily Data, In-Sample 1988:1-1995:9 Volatility Models based on Equations (1)-(3) Parameter MLE MLE MLE MLE MLE MLE GARCH GARCH GARCH GARCH GARCH GARCH (1,0) (2,0) (1,1) (2,1) (1,2) (2,2) µ ( 0.019) ( 0.019) ( 0.016) ( 0.017) ( 0.017) ( 0.016) α ( 0.047) ( 0.054) ( 0.001) ( 0.001) ( 0.003) ( 0.003) β ( 0.008) ( 0.008) ( 1.168) ( 0.101) β ( 1.147) ( 0.100) γ ( 0.044) ( 0.044) ( 0.007) ( 0.034) ( 0.019) ( 0.012) γ ( 0.026) ( 0.034) ( 0.010) Log Likelihood BIC LM ARCH Wald AR Sign Test Negative Sign Test Positive Sign Test Joint Sign Test
3 Table A1.2 Daily Data, In-Sample 1988:1-1995:9 Volatility Models based on Equations (1)-(3) Parameter MLE MLE MLE MLE MLE MLE GJR GJR GJR GJR GJR GJR (1,0,1) (2,0,1) (1,1,1) (2,1,1) (1,2,1) (2,2,1) µ ( 0.019) ( 0.019) ( 0.017) ( 0.017) ( 0.017) ( 0.017) α ( 0.048) ( 0.055) ( 0.001) ( 0.001) ( 0.002) ( 0.002) β ( 0.008) ( 0.008) ( 0.999) ( 0.170) β ( 0.985) ( 0.171) γ ( 0.059) ( 0.059) ( 0.012) ( 0.030) ( 0.016) ( 0.016) γ ( 0.027) ( 0.033) ( 0.019) δ ( 0.089) ( 0.088) ( 0.014) ( 0.014) ( 0.026) ( 0.022) Log Likelihood BIC LM ARCH Wald AR Sign Test Negative Sign Test Positive Sign Test Joint Sign Test
4 Table A1.3 Daily Data, In-Sample 1988:1-1995:9 Volatility Models based on Equations (1)-(3) Parameter MLE MLE MLE MLE MLE MLE GJR GJR GJR GJR GJR GJR (1,0,2) (2,0,2) (1,1,2) (2,1,2) (1,2,2) (2,2,2) µ ( 0.019) ( 0.019) ( 0.017) ( 0.017) ( 0.000) ( 0.017) α ( 0.051) ( 0.055) ( 0.001) ( 0.001) ( 0.000) ( 0.002) β ( 0.007) ( 0.007) ( 0.000) ( 0.145) β ( 0.000) ( 0.143) γ ( 0.053) ( 0.052) ( 0.012) ( 0.022) ( 0.000) ( 0.017) γ ( 0.020) ( 0.026) ( 0.027) δ ( 0.083) ( 0.082) ( 0.055) ( 0.054) ( 0.000) ( 0.025) δ ( 0.055) ( 0.054) ( 0.053) ( 0.055) ( 0.000) ( 0.030) Log Likelihood BIC LM ARCH Wald AR Sign Test Negative Sign Test Positive Sign Test Joint Sign Test
5 Table A1.4 Daily Data, In-Sample 1988:1-1995:9 Volatility Models based on Equations (1)-(3) Parameter MLE MLE MLE MLE MLE MLE GJR GJR GJR GJR GJR GJR (0,0,1) (0,1,1) (0,2,1) (0,0,2) (0,1,2) (0,2,2) µ ( 0.019) ( 0.017) ( 0.017) ( 0.019) ( 0.017) ( 0.017) α ( 0.052) ( 0.001) ( 0.001) ( 0.056) ( 0.001) ( 0.002) β ( 0.004) ( 0.780) ( 0.004) ( 0.015) β ( 0.772) ( 0.012) δ ( 0.063) ( 0.007) ( 0.017) ( 0.058) ( 0.049) ( 0.011) δ ( 0.056) ( 0.050) ( 0.009) Log Likelihood BIC LM ARCH Wald AR Sign Test Negative Sign Test Positive Sign Test Joint Sign Test
6 Appendix 2 Additional Daily and Monthly Estimation Results Parameter estimates and summary statistics for models based on the equations below are found in Panel A for Tables A2.1 to A2.3. σ 2 t = α 0 + α 1 V t 1 + φ ARCH,0 ĥ 2 t 1 + φ ARCH,1 V t 1 σ 2 t 1 V t 1 = +φ Option,0 Ŝ 2 t 1 + φ Option,1 V t 1 Ŝ 2 t 1 { 1 if V olumet 1 1 ni=2 V olume (n 1) t i 0 otherwise R t = µ + ɛ t ; ɛ t (0, σ 2 t ) σ 2 t = α 0 + α 1 V t 1 + β 0 σ 2 t 1 + β 1 V t 1 σ 2 t 1 + γɛ 2 t 1 + φ Option,0 S 2 t 1 + φ Option,1 V t 1 S 2 t 1 Presented in Panel B of the tables are the model Log Likelihood, Schwarz information criteria (BIC), and tests for residual autocorrelation (Wald AR), ARCH and Sign-ARCH. The test for autocorrelation is a χ 2 Wald test on 5 (3) lags of the daily (monthly) data residuals in the mean equation. The test for ARCH is a standard LM test, having us regress the current squared standardized residual on 24 (3) lags of the daily (monthly) data squared standardized residual, collecting the regression R 2 and forming z = nr 2, where n is the number of observations, and z is distributed χ 2 with 24 (3) degrees of freedom. The Sign-ARCH tests were proposed by Engle and Ng [1993] and test for stronger (asymmetric) volatility increases from negative shocks to the returns process. Our Sign-ARCH tests on daily data used 5 lags, 3 lags for monthly data. The R 2 for all the models below are 0 as the mean regression equation in each model contains only a constant, and so this statistic is not included. These results are qualitatively identical to the results discussed in the paper. An outstanding difference is that residual Sign-ARCH effects for the daily data full sample are not expunged by any of the models, though standard ARCH and autocorrelation effects are not statistically significant for these models and data. Also, the full sample daily results are not as statistically significant and coefficient estimates of interest are not as large in magnitude as for the subsample In contrast, the monthly results are quite strong both in and out of sample, and change very little as the sample is extended to 2003 from Given that the option-implied volatility estimate is calibrated to make a one month volatility forecast, not a one day forecast, it is perhaps not surprising that the one-month horizon results are more stable. 6
7 TABLE A2.1 Monthly Data, In-Sample 1988:1-1995:9 Volatility Models Parameter Naive Options ARCH Combining Full MLE Name Only Model Model µ ( 0.364) ( 0.316) ( 0.328) ( 0.312) ( 0.290) α ( 2.155) ( 2.196) ( 0.372) ( 3.005) ( 1.339) β ( 0.047) ( 0.222) β ( 0.381) γ ( 0.016) ( 0.060) φ Option, ( 0.136) ( 0.163) ( 0.108) φ Option, ( 0.354) ( 0.219) φ ARCH, ( 0.584) φ ARCH, ( 0.774) Log Likelihood BIC Wald AR Test LM ARCH Test Sign Test Neg. Sign Test Pos. Sign Test Joint Sign Test
8 TABLE A2.2 Monthly Data, Full Sample 1988:1-2003:8 Volatility Models Parameter Naive Options ARCH Combining Full MLE Name Only Model Model µ ( 0.319) ( 0.264) ( 0.259) ( 0.263) ( 0.251) α ( 2.018) ( 1.561) ( 0.528) ( 1.736) ( 0.877) β β 1 ( 0.075) ( 0.189) ( 0.286) γ ( 0.056) ( 0.053) φ Option, ( 0.079) ( 0.132) ( 0.096) φ Option, φ ARCH,0 ( 0.246) ( 0.175) ( 0.330) φ ARCH, ( 0.465) Log Likelihood BIC Wald AR Test LM ARCH Test Sign Test Neg. Sign Test Pos. Sign Test Joint Sign Test
9 TABLE A2.3 Daily Data, Full Sample 1988:1-2003:8 Volatility Models Parameter Naive Options ARCH Combining Full MLE Name Only Model Model µ ( 0.018) ( 0.014) ( 0.014) ( 0.014) ( 0.014) α ( 0.048) ( 0.040) ( 0.002) ( 0.042) ( 0.038) β ( 0.008) ( 0.283) β ( 0.367) γ ( 0.009) ( 0.024) φ Option, ( 0.038) ( 0.076) ( 0.187) φ Option, ( 0.135) ( 0.234) φ ARCH,0 φ ARCH, ( 0.101) ( 0.176) Log Likelihood BIC Wald AR Test LM ARCH Test Sign Test Neg. Sign Test Pos. Sign Test Joint Sign Test
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