Lecture 1: Empirical Properties of Returns Econ 589 Eric Zivot Spring 2011 Updated: March 29, 2011 Daily CC Returns on MSFT -0.3 r(t) -0.2-0.1 0.1 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 1
Daily CC Returns on S&P 500-0.20 0 5-0.15-0.10-5 0 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Distribution of Daily CC Returns on MSFT Percent of Total 60 50 40 30 20 Daily CC Returns on MSFT Sample Quantiles: min 1Q median 3Q max -0.3583-1282 0 1554 0.1787 Sample Moments: mean std skewness kurtosis 01278 2539-0.7275 16.12 Number of Observations: 4365 10 0-0.3-0.2-0.1 0.1 0.2 x 2
Distribution of Daily CC Returns on S&P 500 Percent of Total 60 50 40 30 20 Daily CC Returns on S&P 500 Sample Quantiles: min 1Q median 3Q max -0.229-04697 004677 0581 8709 Sample Moments: mean std skewness kurtosis 003276 1135-2.083 45.18 Number of Observations: 4365 10 0-0.20-0.15-0.10-5 5 0.10 x Normal QQ-Plot Test for Normality 0.2 0.1-0.1 10/26/1987-0.2 Daily CC Returns on MSFT 10/19/2000 Test for Normality: Jarque-Bera Null Hypothesis: data is normally distributed MSFT Test Stat 31685.64 p.value 0 Dist. under Null: chi-square with 2 degrees of freedom Total Observ.: 4365-0.3 10/19/1987-2 0 2 3
QQ-Plot: Student-t with 4 degrees of freedom Daily CC Returns on MSFT 0.2 06/30/2003 0.1-0.1-0.2 03/17/1986-0.3 03/14/1986-10 -5 0 5 10 Skew Normal Distribution shape=5 shape=-5 pdf 0.2 0.4 0.6 pdf 0.2 0.4 0.6-3 -2-1 0 1 2 3-3 -2-1 0 1 2 3 shape=0 shape=1000 pdf 0.1 0.2 pdf 0.2 0.4 0.3 0.4 0.6 0.8-3 -2-1 0 1 2 3-3 -2-1 0 1 2 3 ξ = 0, ω = 1 4
Skew t Distribution shape=5 shape=-5 St St pdft 0.2 0.4 0.6 Sn pdft 0.2 0.4 0.6 Sn -4-2 0 2 4-4 -2 0 2 4 shape=0 shape=1000 pdft 0.1 0.2 0.3 0.4 St Sn pdft 0.2 0.4 0.6 0.8 St Sn -4-2 0 2 4-4 -2 0 2 4 ξ = 0, ω = 1, ν = 5 QQ-Plot: MLE of Skew-t for MSFT location = -04, scale = 20, shape = 0.298, df = 4.973 msft -0.3-0.2-0.1 0.1 0.2 mle computed with R package sn, qqplot() from R package car -0.1 0.1 0.2 st quantiles 5
Normal QQ-Plot Test for Normality 005 5 Daily CC Returns on S&P 500 10/21/1987 Test for Normality: Jarque-Bera Null Hypothesis: data is normally distributed -5 10/26/1987-0.10-0.15-0.20 SP500 Test Stat 326705.3 p.value Dist. under Null: chi-square with 2 degrees of freedom Total Observ.: 4365 10/19/1987-2 0 2 QQ-Plot: Student-t with 4 degrees of freedom 0.1 Daily CC Returns on S&P 500 06/30/2003-0.1 03/17/1986-0.2 03/14/1986-10 -5 0 5 10 6
QQ-Plot: MLE of Skew-t for SP500 location = 01, scale = 07, shape = -99, df = 3.329 sp500-0.10-5 0 5 0.10-0.20-0.15 - mle computed with R package sn, qqplot() from R package car -0.10-5 0 5 0.10 st quantiles Monthly CC Returns on MSFT r(t) -0.4-0.3 3-0.2-0.1 0.1 0.2 0.3 0.4 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 7
Monthly CC Returns on S&P500-0.20-0.15-0.10-5 0 0..05 0.10 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 Distribution for Monthly Returns on MSFT 30 Monthly CC Returns on MSFT Sample Quantiles: min 1Q median 3Q max -0.3861-3587 336 9736 0.4384 Percent of Total 20 Sample Moments: mean std skewness kurtosis 3357 0.1145 0.1845 4.004 Number of Observations: 208 10 0-0.4-0.2 0.2 0.4 x 8
Distribution for Monthly Returns on S&P 500 30 25 Monthly CC Returns on S&P500 Sample Quantiles: min 1Q median 3Q max -0.2066-1921 122 3875 0.125 Percent of Total 20 15 10 Sample Moments: mean std skewness kurtosis 08219 459-0.8377 5.186 Number of Observations: 208 5 0-0.2-0.1 0.1 x Normal QQ-Plot Tests for Normality 0.4 0.2-0.2 01/01/1987 01/01/2001 Test for Normality: Shapiro-Wilks MSFT Test Stat 0.9906 p.value 0.9558 Dist. under Null: normal Total Observ.: 208 Test for Normality: Jarque-Bera MSFT Test Stat 9.9223 p.value 070 04/01/2000-0.4-3 -2-1 0 1 2 3 Dist. under Null: chi-square with 2 degrees of freedom Total Observ.: 208 9
QQ-Plot: Student s t with 10 df 0.4 06/01/2003 05/01/2003 0.2-0.2 03/01/1986-0.4-2 0 2 Normal QQ-Plot Tests for Normality 0.1 Monthly CC Returns on S&P500 01/01/1987 Test for Normality: Shapiro-Wilks SP500 Test Stat 0.9699 p.value 154 Dist. under Null: normal Total Observ.: 208 Test for Normality: Jarque-Bera -0.1 08/01/1998-0.2 10/01/1987-3 -2-1 0 1 2 3 SP500 Test Stat 65.7551 p.value 000 Dist. under Null: chi-square with 2 degrees of freedom Total Observ.: 208 10
QQ-Plot: Student s t with 7 df Monthly CC Returns on S&P500 0.1 06/01/2003-0.1 04/01/1986 03/01/1986-0.2-4 -2 0 2 4 Testing for Autocorrelation Test for Autocorrelation: Ljung-Box Daily CC Returns on MSFT Null Hypothesis: no autocorrelation.4 0.6 0.8 1.0 0.2 0.6 0.8 1.0 0.2 0.4 0 0 10 20 30 Daily CC Returns on S&P 500 0 10 20 30 SP500 Test Stat 35.1892 p.value 191 Dist. under Null: chi-square with 20 degrees of freedom Total Observ.: 4365 Test for Autocorrelation: Ljung-Box Null Hypothesis: no autocorrelation MSFT Test Stat 43.2323 p.value 019 Dist. under Null: chi-square with 20 degrees of freedom Total Observ.: 4365 11
Stylized Facts of Daily Asset Returns Microsoft Returns S & P 500 Returns 0.30 0.10-0 Microsoft Squared Returns S & P 500 Squared Returns -0 0.20 5 Volatility clustering 0 8 00 40 Microsoft Absolute Returns S & P 500 Absolute Returns 0 0.30 0 0.30 Sample Autocorrelations of Daily Returns Microsoft Returns S&P 500 Returns 0.6 0.6 Microsoft Squared Returns Dependence in volatility S&P 500 Squared Returns 0.6 0.6 Microsoft Absolute Returns Microsoft Absolute Returns 0.6 0.6 12
Stylized Facts for Monthly Asset Returns Microsoft Returns S&P 500 Returns -0.3 0.4-0.20 0.10 Microsoft Squared Returns Less volatility clustering S&P 500 Squared Returns 2 0.18 05 Microsoft Squared Returns Less volatility dependence S&P 500 Squared Returns 0.6 0.6 MSFT and S&P 500 Daily Returns SP500 0.1-0.2-0.1 Sample covariance matrix MSFT SP500 MSFT 006380948 001700987 SP500 001700987 001262902 MSFT Sample correlation matrix MSFT SP500 MSFT 1.0000000 0.5992023 SP500 0.5992023 1.0000000-0.3-0.2-0.1 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 13
EWMA Volatilities and Correlations EWMA Conditional Volatilities EWMA Conditional Correlation SP500 1 2 3 4 6 8 2 4 6 MSFT 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Correlations 1986 1988 1990 1992 1994 1996 1998 2000 2002 198619871988198919901991199219931994199519961997199819992000200120022003 14