Statistik und Ökonometrie - Lehrstuhl für Empirische … · 2009-10-08 · 1 Lehrstuhl für...
Transcript of Statistik und Ökonometrie - Lehrstuhl für Empirische … · 2009-10-08 · 1 Lehrstuhl für...
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Chapter 1:
Introduction
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Contents:
I. Introduction ........................................................................................................................................ 3
I.1 Assets Returns ............................................................................................................................. 4
I.1.1 Definition of Asset Returns .................................................................................................... 5
I.1.2 Statistical Properties of Returns .......................................................................................... 11
I.1.3 Empirical Properties of Returns ........................................................................................... 20
I.2 Basics of Time Series Analysis .................................................................................................. 24
I.2.1 Stationarity........................................................................................................................... 24
I.2.2 Autocorrelation Function...................................................................................................... 28
I.2.3 Partial Autocorrelation Function (PACF) ............................................................................. 32
I.2.3 Transformation of Data ........................................................................................................ 34
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I. Introduction
Definition of Time Series
In time series analysis a time series is defined as a realisation of stochastic process where the time
index takes on a finite or countable infinite set of values. Denoted, e.g. {Yt | for all integers t}.
Time series models are all based on the assumption that the series to be forecasted has been
generated by a stochastic process. Therefore, we assume that each observed value Y1, Y2, …, YT in
the series is drawn randomly from a probability distribution.
A stochastic process exhibits a random process, denoted as {Yt}, which can take a value between -∞
and +∞. The observed value Yt at time t describes one realisation of these stochastic processes.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.1 Assets Returns
• return of an asset is a complete and scale-free summary of the investment opportunity
• return series are easier to handle than price series because former have more attractive
statistical properties e.g. stationarity
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.1.1 Definition of Asset Returns
a) One-Period Simple Return
Holding the asset for one period from date t-1 to date t would result in a simple gross return:
1 � R� � P�P���
P� � P���1 � R�
The corresponding one-period simple net return or simple return is
R� � P�P��� � 1 � P� � P���P���
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
b) Multi-period Simple Return Holding the asset for k periods between dates t - k and t gives a k-period simple gross return
1 � R��k� � P�P��� · P���P��� · … · P�����P���
1 � R��k� � P�P���
1 � R��k� � 1 � R� · 1 � R��� · … · 1 � R�����
1 � R��k� � ��1 � R����������
Thus, the k-period simple gross return is just the product of the k one-period simple gross return, which is called compound return.
R��k� � P� � P���P���
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Mostly, when we present descriptive statistics returns are annualized. If the asset was held for k years, then the annualized (average) return is defined as geometric mean of the k one-period simple gross returns:
�R��k�� � ���1 � R���������� �
�� � 1
We also can annualized the arithmetic mean of monthly average return by:
�R��k��� !"#$%& � �1 � '1k ( R��
��� )���
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
c) Continuously Compounded Return
r� � ln1 � R�
r� � ln P�P���
r� � ln P� � ln P��� r� � p� � p���
where pt = ln(Pt).
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Advantages of continuously compounding:
• Continuously multi-period return:
r��k� � ln1 � R��k� r��k� � ln�1 � R� · 1 � R��� · … · 1 � R������ r��k� � ln1 � R� � ln1 � R��� � . � ln1 � R����� r��k� � r� � r��� � . � r�����
where rt = ln(1+Rt).
• Statistical properties of log returns are more tractable
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
d) Relationship between Simple and Continuously Compounded Returns
The relationships between simple return Rt and continuously compounded return rt are:
r� � ln1 � R�
R� � e01 � 1
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.1.2 Statistical Properties of Returns
a) Moments of a Random Variable (RV)
The n-th moment of a continuous random variable Y is defined as
M � E�Y � � 5 y fy · dy�9�9 ,
where E stands for expectation and f(y) is the propability function of Y.
The first moment is called the mean or expectation of Y. It measures the central location of the
distribution. We denote the mean of Y by µY. The n-th central moment of Y is defined as
M � E�Y � µ; � � 5 y � µ; fy · dy�9�9 ,
provided that the integral exists.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Skewness:
Sy � E =y � µ;>σ;> @ Kurtosis:
Ky � E =y � µ;Bσ;B @
K(y) – 3 is called the excess kurtosis because K(y) = 3 for a normal distribution.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
The sample mean is
µC; � 1T ( y�,F���
the sample variance is
σG;� � 1T (y� � µC;�,F���
the sample skewness is
sCy � 1T ( y� � µC;>σG;>F
��� ,
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
the sample kurtosis is
kIy � 1T ( y� � µC;BσG;BF
��� .
Under normality assumption, KCL and MN L are distributed asymptotically as normal with zero mean
and variances 6/T and 24/T, respectively.
Financial data often exhibit leptokurtosis, i.e. a kurtosis higher than 3 or an excess kurtosis higher
than 0. We consider such return pattern especially for high frequency data, for example daily data.
For monthly, quarterly or yearly aggregated data the distribution turns more towards a normal
distribution.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Figure 1: Skewness and Excess Kurtosis
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Test for Skewness and Kurtosis:
Distribution:
sCy � 1T ( y� � µC;>σG;>F
��� ~N Q0, 6TT
kIy � 1T ( y� � µC;BσG;BF
��� ~N Q3, 24T T
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
For the coefficient of skewness the exact test of Urzua (1996) is defined as:
X Y√\X ] ^_
c � 6 · T � 2T � 1T � 3
The coefficient of kurtosis is significant different from 3, if
a kI � 3b24 T⁄ a ] ze
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
This test only works approximately for very large sample sizes due to the fact that the estimator for the kurtosis coefficient is biased. However, a test based on exact moments can be formulated as (Urzua 1996):
akI � a√b a ] ze
h � 3 · i � 1i � 1
b � 24 · TT � 2T � 3T � 1�T � 3T � 5
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
b) Test of Normality Jarque-Bera test statistic:
The test statistic measures the difference of the skewness and kurtosis of the series with those from
the normal distribution. The statistic is computed as:
JB � QT6T · QsC� � 14 kI � 3�T ~ χ�2
Under the null hypothesis of a normal distribution, the Jarque-Bera statistic is distributed as χ2 with 2
degrees of freedom.
1% ≈ 9,21
5% ≈ 5,99
The test is only adequate for large samples, whereas for small samples you have to interpret it
cautiously.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.1.3 Empirical Properties of Returns
Figure 2: DAX Total Return Index
0
2000
4000
6000
8000
65 70 75 80 85 90 95 00
DAXTR
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Figure 3: Simple discrete (monthly) returns of the DAX Total Return Index
-0,3
-0,2
-0,1
0
0,1
0,2
0,3
Jan
65
Jan
67
Jan
69
Jan
71
Jan
73
Jan
75
Jan
77
Jan
79
Jan
81
Jan
83
Jan
85
Jan
87
Jan
89
Jan
91
Jan
93
Jan
95
Jan
97
Jan
99
Jan
01
Jan
03
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Figure 4: Continously (monthly) returns of the DAX Total Return Index
-0,4
-0,3
-0,2
-0,1
0
0,1
0,2
0,3
Jan
65
Jan 6
7
Jan 6
9
Jan
71
Jan
73
Jan 7
5
Jan 7
7
Jan
79
Jan
81
Jan
83
Jan 8
5
Jan
87
Jan
89
Jan
91
Jan 9
3
Jan 9
5
Jan
97
Jan
99
Jan 0
1
Jan 0
3
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Figure 5: Descriptive Statistics of DAX Returns
0
20
40
60
80
100
-0.3 -0.2 -0.1 0.0 0.1 0.2
Series: DLNDAXTRSample 1965:01 2003:12Observations 468
Mean 0.004499Median 0.006365Maximum 0.193738Minimum -0.293327Std. Dev. 0.057752Skewness -0.645113Kurtosis 5.647069
Jarque-Bera 169.0973Probability 0.000000
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.2 Basics of Time Series Analysis
I.2.1 Stationarity
Strict Stationary:
Joint distribution: Y(t) = {Y(1), Y(2), …, Y(T)}
→ invariant under time shift
The random variables Y(t+1), … Y(t+n) have the same joint distribution as Y(t+1+c), …, Y(t+n+c),
with c as a arbitrary positive integer. This is a very strong condition that is hard to verify empirically.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Weak Stationarity:
Weak stationarity exists, when expected value, variance and covariance of the distribution random
variables are constant for all points of time.
a EYt � µ � constant, ∀ t, mean stationarity b VarYt � σt2� σ2 � constant, ∀ t variance stationarity, and c CovYt, Yt-j � σtj � σj � constant, ∀ t covariance stationarity,
Ewy� � µ�y��� � µ�x � Ewyy � µ�yy�� � µ�x for all t ≠ s.
The data of the underlying process are time invariant and neither the shape nor the parameters of
the distribution change over time. The covariance only depends on j, where j is an arbitrary integer.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Suppose that we have observed T data points {Ytt = 1, …, T}:
• weak stationarity implies that the time plot of the data would show that the T values fluctuate
with constant variation around a constant level.
• we assume that the first two moments of Yt are finite
• from definitions, if Yt is strictly stationary and its first two moments are finite, then Yt is also
weakly stationary, but the converse is not true in general
• however, if the time series Yt is normally distributed, then weak stationarity is equivalent to strict
stationarity.
27
Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
The covariance γj = Cov(Yt, Yt-j) is called the lag-j autocovariance of Yt with the following properties:
(a) γ0 = Var(Yt) and
(b) γ-j = γj.
Cov(Yt, Yt-(-j))= Cov(Yt-(-j), Yt) = Cov(Yt+j, Yt) = Cov(Yt1, Yt1-j), where t1 = t + j.
The statistical ratios to describe weakly stationary processes are
a) the autocovariance function for the direction of interrelation,
b) the autocorrelation function for the strength and direction of interrelation, and
c) the partial autocorrelation function for the contribution of adding an new regression coefficient.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.2.2 Autocorrelation Function Consider a weakly stationary series Yt. When the linear dependence between Yt and its past values
Yt-j is of interest, the concept of correlation is generalized to autocorrelation. The correlation
coefficient between Yt and Yt-j is called the lag-j autocorrelation of Yt and is commonly denoted by ρj,
which under the weak stationarity assumption is a function of j only. Specifically, we define
{| � }~��� , ���|b�h��� · �h����| � }~��� , ���|�h��� � �|��
where the property Var(Yt) = Var(Yt-j) for a weakly stationary series is used. From the definition, we
have ρ0 = 1, ρj = ρ-j, and -1 ≤ ρj ≤ 1. In addition, a weakly stationary series Yt is not serially correlated
if and only if ρj = 0 for all j > 0.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
For a given sample of returns T1tt }r{ = , let r be the sample mean, i.e. ∑
=
=T
1ttrT
1r
Then the first-order autocorrelation coefficient of rt is
ρC� � ∑ r� � r�r��� � r�F���∑ r� � r��F��� � ∑ r� � r�r��� � r�F�����∑ r� � r��F���
The standard error of the correlation coefficient is calculated as
��� � � 2√�
The autocorrelation coefficients of random data have a sampling distribution which is approximately
normally distributed with a mean of zero and a standard deviation of �√�. Under certain assumptions
this term can be viewed as a 95% confidence interval and empirical autocorrelation coefficients,
which lay outside of this interval, are significant different from null.
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Box and Pierce (1970) propose the Portmanteau statistic:
�� � i · ( {C|� ~ χ��|�� m � 1
H0: ρ1 = … = ρm = 0
H1: ρ1 ≠ 0 for i ∈ {1, …, m}.
→ rt ~ i.i.d. χ2(m)
Ljung-Box (1978) modify the Q(m) statistic to increase the power of the test in finite samples:
��� � ii � 2 · ( {C|�i � � ~ ��� � 1�|��
→ m ≈ ln(T)
→ number of autocorrelation coefficients should be around 20 % of the sample size
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Figure 6: Autocorrelation, Partial Autocorrelation and Ljung-Box (LB) Test
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Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.2.3 Partial Autocorrelation Function (PACF)
Assume an equation appear several variables, for example in the following model:
AR(1) : Yt = φ0,1 + φ1,1Yt-1 + u1t
AR(2) : Yt = φ0,2 + φ1,2Yt-1 + φ2,2Yt-2 + u2t
AR(3) : Yt = φ0,3 + φ1,3Yt-1 + φ2,3Yt-2 + φ3,3Yt-3 + u3t
…
AR(p): Yt = φ0,p + φ1,pYt-1 + φ2,pYt-2 + φ3,pYt-3 + … + φp,pYt-p + upt
What additional contribution is delivered from Yt-p, if the explanation is controlled by Yt-1 ... Yt-p-1?
33
Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Given {Yt} is a stationary process and the partial autocorrelation �I|,| = �I| (for j ≥ 2) is the partial
correlation of Yt-j and Yt under holding constant all random variables Yi, which lay between t-j < i < t:
• �I|,� describes the partial correlation between Yt-j and Yt-j+i
• �I� � 1
• �I� � {C� → autocorrelation coefficient of lag 1
• �I�| � �I| → the partial autocorrelation function is symmetric
• Generally the equation for the partial autocorrelation coefficients is defined as:
�� �
��������� ρ� for j � 1 ρ� � ( ����,#ρ��#
���#��
1 � ( ����,��������
��������������������� for j ] 1
�
34
Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.2.3 Transformation of Data
I.2.3.1 Differencing for Trends Elimination (Mean-Stationarity)
Basically, a non-constant mean of a time series arises from two different characters:
a) structural breaks with erratic changes of the means
b) continuous increase or decrease to a greater or lesser extent over time
35
Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Difference operator ∆
If Yt is the original series then it follows
∆Yt = Yt – Yt-1
the first differences of the time series Yt.
If a time series must be differenced twice we formulate
∆2Yt = ∆(∆Yt) = ∆(Yt – Yt-1) = ∆Yt - ∆Yt-1 = Yt - Yt-1 - Yt-1 + Yt-2 = Yt - 2Yt-1 + Yt-2
Consequently, a twice differencing corresponds to a filter which is applied to a series with the
weights of the filter (1,-2,1). If a time series is differenced d of times we can write ∆dYt.
36
Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
I.2.3.2 Box-Cox-Transformation for Stabilization of the Variance
If Y is the original and X the transformed series, then we can denote the approach in general as:
� � ¡ ��¢ � 1£log ��� ¥~�¥~� 0 ¦ £ ¦ 1£ � 0
To apply the transformation procedure we have to determine θ, which can approximately be chosen
that the variance of the series X is constant:
§� � \¨���¢
where σ is the standard deviation and µ is the mean of the time series Yt.
37
Lehrstuhl für Empirische Wirtschaftsforschung und ÖkonometrieDr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007
Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08
Divide the series into K sub-periods: s# � cm#��© with i = 1, …, K
We also can determine θ on the basis of the logarithmized equation:
ln(si) = ln c + (1 - θ) ln(mi)
As a special case of this transformation, for θ = 0, we receive the logarithmized transformation of the
time series:
Yt = log(Xt)