# Statistik und £â€“konometrie - Lehrstuhl f£¼r Empirische 2009-10-08¢  1 Lehrstuhl f£¼r Empirische Wirtschaftsforschung

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Lehrstuhl für Empirische Wirtschaftsforschung und Ökonometrie Dr. 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 Ökonometrie Dr. 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 Ökonometrie Dr. 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 Ökonometrie Dr. 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 Ökonometrie Dr. 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 Ökonometrie Dr. 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 Ökonometrie Dr. 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 Ökonometrie Dr. 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� � ln 1 � 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 Ökonometrie Dr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007

Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data Analysis ● Winter Term 2007/08

• Continuously multi-period return:

r��k� � ln 1 � R��k� r��k� � ln� 1 � R� · 1 � R��� · … · 1 � R����� � r��k� � ln 1 � R� � ln 1 � R��� � . � ln 1 � R����� r��k� � r� � r��� � . � r����� where rt = ln(1+Rt).

• Statistical properties of log returns are more tractable

• 10

Lehrstuhl für Empirische Wirtschaftsforschung und Ökonometrie Dr. 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� � ln 1 � R� R� � e01 � 1

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Lehrstuhl für Empirische Wirtschaftsforschung und Ökonometrie Dr. 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 f y · 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; f y · dy�9�9 , provided that the integral exists.

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Lehrstuhl für Empirische Wirtschaftsforschung und Ökonometrie Dr. 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:

S y � E = y � µ; >σ;> @ Kurtosis:

K y � E = y � µ; Bσ;B @

K(y) – 3 is called the excess kurtosis because K(y) = 3 for a normal distribution.

• 13

Lehrstuhl für Empirische Wirtschaftsforschung und Ökonometrie Dr. Roland Füss ? Statistik II: Schließende Statistik ? SS 2007

Department of Empirical Research and Econometrics Dr. Roland Füss ● Financial Data A

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