lecture 1: empirical properties of returnsfaculty.washington.edu/ezivot/econ589/empirical... ·...

1

Lecture 1: Empirical Properties of Returns

Econ 589Eric Zivot

© Eric Zivot 2008

Eric ZivotSpring 2011

Updated: March 29, 2011

Daily CC Returns on MSFT

0.1

r(t)

-0.2

-0.1

0.0

© Eric Zivot 2008

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

-0.3

2

Daily CC Returns on S&P 500

00.

05-0

.15

-0.1

0-0

.05

0.00

© Eric Zivot 2008

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

-0.2

0

60

Daily CC Returns on MSFTSample Quantiles:

min 1Q median 3Q max -0.3583 -0.01282 0 0.01554 0.1787

Distribution of Daily CC Returns on MSFT

20

30

40

50

Per

cent

of T

otal

Sample Moments: mean std skewness kurtosis

0.001278 0.02539 -0.7275 16.12

Number of Observations: 4365

© Eric Zivot 2008

0

10

-0.3 -0.2 -0.1 0.0 0.1 0.2

x

3

50

60Daily CC Returns on S&P 500 Sample Quantiles:

min 1Q median 3Q max -0.229 -0.004697 0.0004677 0.00581 0.08709

Sample Moments:

Distribution of Daily CC Returns on S&P 500

10

20

30

40

Per

cent

of T

otal

pmean std skewness kurtosis

0.0003276 0.01135 -2.083 45.18


© Eric Zivot 2008

0

10

-0.20 -0.15 -0.10 -0.05 0.0 0.05 0.10

x

0.2 Daily CC Returns on MSFT10/19/2000

Normal QQ-Plot

Test for Normality: Jarque-Bera

Null Hypothesis: data is normally

Test for Normality

-0.1

0.0

0.1Null Hypothesis: data is normally distributed

Test Statistics:MSFT

Test Stat 31685.64p.value 0.00

Dist. under Null: chi-square with 2 degrees of freedom

Total Observ.: 4365

© Eric Zivot 2008

-0.3

-0.2

-2 0 2

10/26/1987

10/19/1987

4

0.2


06/30/2003

QQ-Plot: Student-t with 4 degrees of freedom

-0.2

-0.1

0.0

0.1

03/17/1986

© Eric Zivot 2008

-0.3

-10 -5 0 5 10

03/17/1986

03/14/1986

Skew Normal Distribution

40.6

shape=5

40.6

shape=-5

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

x.vals

pdf

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

x.vals

pdf

0.3

0.4

shape=0

0.6

0.8

shape=1000

© Eric Zivot 2008

-3 -2 -1 0 1 2 3

0.0

0.1

0.2

x.vals

pdf

-3 -2 -1 0 1 2 3

0.0

0.2

0.4

x.vals

pdf

ξ = 0, ω = 1

5

Skew t Distribution

40.6

shape=5

StSn

40.6

shape=-5

StSn

-4 -2 0 2 4

0.0

0.2

0.4

x.vals

pdft

-4 -2 0 2 4

0.0

0.2

0.4

x.vals

pdft

30.

4

shape=0

StSn 6

0.8

shape=1000

StSn

© Eric Zivot 2008

-4 -2 0 2 4

0.0

0.1

0.2

0.3

x.vals

pdft

-4 -2 0 2 40.

00.2

0.4

0.6

x.vals

pdft

ξ = 0, ω = 1, ν = 5

QQ-Plot: MLE of Skew-t for MSFTlocation = -0.004, scale = 0.020, shape = 0.298, df = 4.973

0.2

-0.1

0.0

0.1

msft

© Eric Zivot 2008

-0.1 0.0 0.1 0.2

-0.3

-0.2

st quantiles

mle computed with R package sn, qqPlot() from R package car

6

0 05

Daily CC Returns on S&P 50010/21/1987


Null Hypothesis: data is normally

Normal QQ-Plot Test for Normality

-0.15

-0.10

-0.05

0.0

0.05

10/26/1987

Null Hypothesis: data is normally distributed

Test Statistics:SP500



© Eric Zivot 2008

-0.20

-2 0 2

10/19/1987

Total Observ.: 4365

0.1Daily CC Returns on S&P 500

06/30/2003

QQ-Plot: Student-t with 4 degrees of freedom

-0.1

0.0

03/17/1986

© Eric Zivot 2008

-0.2

-10 -5 0 5 10

03/14/1986

7

QQ-Plot: MLE of Skew-t for SP500location = 0.001, scale = 0.007, shape = -0.099, df = 3.329

0.10

-0.10

-0.05

0.00

0.05

sp50

0

© Eric Zivot 2008

-0.10 -0.05 0.00 0.05 0.10

-0.20

-0.15

-

st quantiles

mle computed with R package sn, qqPlot() from R package car

Monthly CC Returns on MSFT

0.3

0.4

r(t)

3-0

.2-0

.10.

00.

10.

2

© Eric Zivot 2008

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

-0.4

-0.3

8

Monthly CC Returns on S&P500

.05

0.10

-0.1

5-0

.10

-0.0

50.

000.

© Eric Zivot 2008

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003

-0.2

0

30

Monthly CC Returns on MSFTSample Quantiles:

min 1Q median 3Q max -0.3861 -0.03587 0.0336 0.09736 0.4384

Distribution for Monthly Returns on MSFT

10

20

Per

cent

of T

otal


0.03357 0.1145 0.1845 4.004


© Eric Zivot 2008

0

-0.4 -0.2 0.0 0.2 0.4

x

9

30

Monthly CC Returns on S&P500Sample Quantiles:

min 1Q median 3Q max

Distribution for Monthly Returns on S&P 500

10

15

20

25

Perc

ent o

f Tot

al

max -0.2066 -0.01921 0.0122 0.03875

0.125


0.008219 0.0459 -0.8377 5.186


© Eric Zivot 2008

0

5

-0.2 -0.1 0.0 0.1

x

0.401/01/2001

01/01/1987

Test for Normality: Shapiro-Wilks


Normal QQ-Plot Tests for Normality

0.0

0.2


Dist. under Null: normalTotal Observ.: 208



Test Stat 9.9223

© Eric Zivot 2008

-0.4

-0.2

-3 -2 -1 0 1 2 3

04/01/2000

p.value 0.0070


Total Observ.: 208

10

QQ-Plot: Student’s t with 10 df

0.405/01/2003

06/01/2003

0.0

0.2

05/01/2003

© Eric Zivot 2008

-0.4

-0.2

-2 0 2

03/01/1986

0.1


01/01/1987

Test for Normality: Shapiro-Wilks


Normal QQ-Plot Tests for Normality

-0.1

0.0


Dist. under Null: normalTotal Observ.: 208



Test Stat 65.7551

© Eric Zivot 2008

-0.2

-3 -2 -1 0 1 2 3

08/01/1998

10/01/1987

p.value 0.0000


Total Observ.: 208

11


06/01/2003

QQ-Plot: Student’s t with 7 df

-0.1

0.0

0.1

© Eric Zivot 2008

-0.2

-4 -2 0 2 4

04/01/1986

03/01/1986

Testing for Autocorrelation

AC

F.4

0.6

0.8

1.0


Test for Autocorrelation: Ljung-Box

Null Hypothesis: no autocorrelation



Lag0 10 20 30

0.0

0.2

0.6

0.8

1.0

Daily CC Returns on S&P 500


Total Observ.: 4365

Test for Autocorrelation: Ljung-Box

Null Hypothesis: no autocorrelation


© Eric Zivot 2008

Lag

AC

F

0 10 20 30

0.0

0.2

0.4

0 MSFT Test Stat 43.2323p.value 0.0019


Total Observ.: 4365

12

Stylized Facts of Daily Asset Returns

0.30

0.10

Microsoft Returns

0.20

0.05

S & P 500 Returns

1986 1990 1994 1998 2002

-0

1986 1990 1994 1998 2002

-0

1986 1990 1994 1998 2002

0.00

0.08

Microsoft Squared Returns

1986 1990 1994 1998 2002

0.00

00.

040

S & P 500 Squared ReturnsVolatility clustering

1986 1990 1994 1998 2002

0.00

0.30

Microsoft Absolute Returns

1986 1990 1994 1998 2002

0.00

0.30

S & P 500 Absolute Returns

Sample Autocorrelations of Daily Returns

L

AC

F

0 5 10 15 20

0.0

0.6

Microsoft Returns

L

AC

F

0 5 10 15 20

0.0

0.6

S&P 500 Returns

Lag Lag

Lag

AC

F

0 5 10 15 20

0.0

0.6


Lag

AC

F

0 5 10 15 20

0.0

0.6

S&P 500 Squared Returns

Microsoft Absolute Returns Microsoft Absolute Returns

Dependence in volatility

Lag

AC

F

0 5 10 15 20

0.0

0.6

Lag

AC

F

0 5 10 15 20

0.0

0.6

13

Stylized Facts for Monthly Asset Returns

1986 1990 1994 1998 2002

-0.3

0.4

Microsoft Returns

1986 1990 1994 1998 2002

-0.2

00.

10

S&P 500 Returns

1986 1990 1994 1998 2002

0.02

0.18


1986 1990 1994 1998 2002

0.00

5

S&P 500 Squared Returns

Microsoft Squared Returns S&P 500 Squared Returns

Less volatility clustering

Less volatility dependence

Lag

AC

F

0 5 10 15 20

0.0

0.6

q

Lag

AC

F

0 5 10 15 20

0.0

0.6

q

MSFT and S&P 500 Daily Returns

0.0

SP500

0.0

0.1

MSFT

-0.2

-0.1 Sample covariance matrix

MSFT SP500 MSFT 0.0006380948 0.0001700987

SP500 0.0001700987 0.0001262902

Sample correlation matrixMSFT SP500 MSFT 1.0000000 0.5992023SP500 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

14

EWMA Volatilities and Correlations

EWMA Conditional Correlation

SP500

EWMA Conditional Volatilities

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Correlations

60.

08

MSFT

0.01

0.02

0.03

0.04

SP500

0.1

0.2

0.3

1986 1988 1990 1992 1994 1996 1998 2000 20020.02

0.04

0.06

198619871988198919901991199219931994199519961997199819992000200120022003

lecture 1: empirical properties of returnsfaculty.washington.edu/ezivot/econ589/empirical... ·...

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