Statistik und Ökonometrie - Lehrstuhl für Empirische … · 2009-10-08 · 1 Lehrstuhl für...

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

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

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.

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

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��

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��

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��

�� )��

c) Continuously Compounded Return

r� � ln1 � R�

r� � ln P�P��

r� � ln P� � ln P�� r� � p� � p��

where pt = ln(Pt).

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

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

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.

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.

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

�� ,

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.

Figure 1: Skewness and Excess Kurtosis

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

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

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

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.

I.1.3 Empirical Properties of Returns

Figure 2: DAX Total Return Index

65 70 75 80 85 90 95 00

Figure 3: Simple discrete (monthly) returns of the DAX Total Return Index

Figure 4: Continously (monthly) returns of the DAX Total Return Index

Figure 5: Descriptive Statistics of DAX Returns

-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

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.

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.

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.

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.

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.

For a given sample of returns T1tt }r{ = , let r be the sample mean, i.e. ∑

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.

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

Figure 6: Autocorrelation, Partial Autocorrelation and Ljung-Box (LB) Test

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?

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 � ( ��,�ρ��

�� for j ] 1

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

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.

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.

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)

Statistik und Ökonometrie - Lehrstuhl für Empirische … · 2009-10-08 · 1 Lehrstuhl für...

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