Financial Econometrics Slides-01: RETURN PROPERTIES Part II
Stylized Facts of Returns
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School of Economics Financial Econometrics
Stylized Facts of Returns
Financial Econometrics
Slides-01: RETURN PROPERTIES Part II
School of Economics1
1Copyright University of Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Shape Characteristics: Population
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Let Xt be a random variable with pdf f(x)
= E[Xt] : center
= var(Xt) = E[(Xt )2] : spread
skewness(Xt) = S(X) = E
: symmetry
kustosis(Xt) = K(X) = E
: tail thickness
K(X) 3 : Excess kurtosis
Note: The kth moment and central moment of Xt are:
mk = E[(Xt )k]
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Shape Characteristics of Random Variable
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Why are the mean and variance of returns important?
They are concerned with long-term return and risk, respectively.
Why is return symmetry of interest in financial study?
Symmetry has important implications in holding short or long financial
positions and in risk management.
Why is kurtosis important?
Related to volatility forecasting, efficiency in estimation and tests, etc.
High kurtosis implies heavy (or long) tails in distribution.
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Examle: Normal Random Variable
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Normal Distribution
( 2)
var() = 2
skew() = 0
kurt() = 3
= 0 for odd
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Shape Characteristics: Sample moments
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Sample moments
Let { } denote a random sample of size where is a realization
of the random variable
( )2 = 2
dskew = 3
dkurt = 4
( )
Note: we divide by 1 to get unbiased estimates. Check software to see
how moments are computed.
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Shape Characteristics: Visually
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Topic 1. Features of Some Financial Time Series
UNSW Business School,
Slides-01, Financial Econometrics 20
Often is reported as a deviation from Normal K=3:
Topic 1. Features of Some Financial Time Series
UNSW Business School,
Slides-01, Financial Econometrics 21
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Testing for normality
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QQ-plot: plot standardized empirical quantiles vs. theoretical quantiles
from specified distribution. Note: Shapiro-Wilks (SW) test for normality:
correlation coefficient between values used in QQ-plot
Jarque-Bera (JB) test for normality
( kurt 3)2
Note: if rt is N(,
T skew N(0, 6), and
T ( kurt 3) N(0, 24)
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Shape Characterirtics: Normality test
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The null hypothesis:
H0 : Data (the return) Xt are Normally distributed.
1 Skewness test: Zsk =
Reject H0 if |zsk| is too large (> 1.96, at 5%).
2 Kurtosis test: Zkt =
Reject H0 if |zkt| is too large (> 1.96, at 5%).
3 Jaque-Bera test: JB = Z2ks + Z
Reject JB is too large (> 5.99 at 5%)
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Example: Descriptive Statistics
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Topic 1. Features of Some Financial Time Series
Descriptive statistics
eg. NYSE index prices: (19950103-20020830)
Composite, Industrial,
Trans, Utility, Finance.
Descriptive statistics of log returns.
Correlations of log returns
250 500 750 1000 1250 1500 1750
Composite Industrial Trans Utility Finance
Mean 0.035 0.034 0.031 0.007 0.052
Std. Dev. 1.006 1.009 1.320 1.087 1.310
Skewness -0.316 -0.386 -1.044 -0.275 -0.042
Kurtosis 7.224 7.755 18.103 5.637 5.772
Composite Industrial Trans Utility Finance
Composite 1
Industrial 0.983 1
Trans 0.731 0.708 1
Utility 0.769 0.711 0.505 1
Finance 0.885 0.800 0.668 0.623 1
Portfolio variance and
diversification:
+ 2Cov(*, @)]
UNSW Business School,
Slides-01, Financial Econometrics 23
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Example: Descriptive Statistics
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Topic 1. Features of Some Financial Time Series
Descriptive statistics
Normality test
eg. Comp. index log return
time series plot
-6 -4 -2 0 2 4
Series: RC
Sample 1 1931
Observations 1930
Mean 0.035300
Median 0.052285
Maximum5.178704
Minimum -6.791142
Std. Dev. 1.006207
Skewness-0.315728
Kurtosis 7.224376
Jarque-Bera1467.129
Probability0.000000
250 500 750 1000 1250 1500 1750
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Slides-01, Financial Econometrics 25
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Stylized Fact: Large kurtosis
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Topic 1. Features of Some Financial Time Series
Descriptive statistics
Some stylised facts about index return series
concentration around zero with a few large outliers
large standard deviations (volatile)
negative skewness (longer tail at the negative side)
large kurtosis (tail probabilities larger than normal)
large variation followed by large ones (clustering)
-6 -4 -2 0 2 4
Series: RC
Sample 1 1931
Observations 1930
Mean 0.035300
Median 0.052285
Maximum5.178704
Minimum -6.791142
Std. Dev. 1.006207
Skewness-0.315728
Kurtosis 7.224376
Jarque-Bera1467.129
Probability0.000000
250 500 750 1000 1250 1500 1750
leptokurtic
Histogram of RC
-6 -4 -2 0 2 4
UNSW Business School,
Slides-01, Financial Econometrics 27
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Descriptive statistics: Autocorrelation
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Predictability
We say Xt+1 is predictable if information at t , eg. {Xt, Xt1. , }, helps
to improve our prediction of Xt+1.
In particular, Xt+1 is predictable if Xt+1 is correlated with Xtj for some
j > 0 (ie. Cov(Xt+1, Xtj) 6= 0).
Autocorrelation Function (ACF)
Autocovariance: j = Cov(Xt, Xtj) = Cov(Xt, Xt+j)
Sample autocovariance: j =
t=j+1(Xt X)(Xtj X)
Autocorrelation: j =
Sample Autocorrelation: j =
Partial autocorrelation (PAC)
PAC pj is a measure of the direct relation between Xt and Xtj for
j = 1, 2,
pj is the correlation between Xt and Xtj after controlling for the effects
of Xt and Xt1 Xtj+1
p1 = 11 in Xt = 10 + 11Xt1 + e1t
p2 = 11 in Xt = 20 + 21Xt1 + 22Xt2 + e2t,
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Test for autocorrelation
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The null hypothesis: H0: There is no autocorrelation (White noise process)
1 Autocorrelation test:
T j N(0, 1) under the null hypothesis
Reject if |j | is too large (> 1.96/
T , at 5% significance level)
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Joint Hypothesis Tests
We can also test the joint hypothesis that all m of the k correlation
coefficients are simultaneously equal to zero using the Q-statistic
developed by Box and Pierce:
where T=sample size, m=maximum lag length
The Q-statistic is asymptotically distributed as a 2m.
However, the Box Pierce test has poor small sample properties, so a
variant has been developed, called the Ljung-Box statistic:
= T (T + 2)
This statistic is very useful as a portmanteau (general) test of linear
dependence in time series.
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
An ACF Example
Question:
Suppose that a researcher had estimated the first 5 autocorrelation
coefficients using a series of length 100 observations, and found them to
be (from 1 to 5): 0.207, -0.013, 0.086, 0.005, -0.022.
Test each of the individual coefficient for significance, and use both the
Box-Pierce and Ljung-Box tests to establish whether they are jointly
significant.
A coefficient would be significant if it lies outside (0.196,+0.196) at the
5% level, so only the first autocorrelation coefficient is significant.
Q = 5.09 and Q = 5.26
Compared with a tabulated 2(5)=11.1 at the 5% level, so the 5
coefficients are jointly insignificant.
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Example: ACF/PACF
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Topic 1. Features of Some Financial Time Series
Descriptive statistics
eg. NYSE composite return
AC test at 5% level:
1.96/ d = 0.04462,
rs is rejected at
= 1,2,5,12
LB test at 5% level:
rs is rejected for
all , as all p-values
are less than 0.05.
UNSW Business School,
Slides-01, Financial Econometrics 32
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Example: ACF/PACF of squared Returns
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Topic 1. Features of Some Financial Time Series
Descriptive statistics
What about squared returns?
Usually strongly correlated.
Why squared returns?
5 G()
eg. NYSE Composite
return squared
UNSW Business School,
Slides-01, Financial Econometrics 33
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Summary of stylized Facts
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KEY stylised facts about financial return series
1 the returns have small, often non-significant autocorrelations (no linear
return predictability)
2 the squared returns have strong positive autocorrelations (predictability in
volatility, volatility clustering)
3 large kurtosis (heavy tails, tail probabilities larger than normal)
School of Economics Financial Econometrics
Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation
Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material.
Characterizing Financial time series:
asset price and returns
stylised facts about index return series
Normality tests Zks, Zkt, JB
Predictability in returns
Autocovariance and autocorrelation
Tests for autocorrelation: AC test and Qm
Next week: Application of linear regression in Finance (asset pricing)
School of Economics Financial Econometrics
Stylized Facts of Returns
Shape Characteristics
Testing for Normality
Autocorrelation
CS: assignmentchef QQ: 1823890830 Email: [email protected]
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