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##### 实证金融研究：Empirical Finance：ARMA models for stationary stochastic processes

Empirical Finance
Lecture 5: ARMA models for stationary stochastic processes
stuart.fraser@wbs.ac.uk
Room D1.18 (Social Studies)Warwick Business School 2
Introduction

1. Different types of time series model:
AR, MA and ARMA models.
2. The ACFsand PACFsof different types of time series models.
3. Time series model selection using ACFsand PACFs.
4. Time series model selection using information criteria.Warwick Business School 3
Introduction
ARMA models provide predictions of a time series using past values of the series and/or innovations (error terms).
ARMA models are usually atheoretical/purely statistical models (not normally based on economic/finance theory).
The principle use of ARMA models is for forecasting a series (not policy).
ARMA models often provide better out of sample forecasts than structural (i.e., theory motivated) models:
⇒Seminar 4: Forecast comparisons of GMM CIP (structural) model versus an ARMA model for the forward premium.Warwick Business School 4
White noise process
A white noise process is a basic ‘building block’for time-series models.
In essence, white noise is a process with no temporal structure –it’s purely ‘random’.
()()().0 ,0,covvar0:process noisewhitemean)(zero a of Properties2≠===−kEkttttεεσεε
Is white noise a stationary
or non-stationary process?Warwick Business School 5
WoldDecomposition Theorem
Any weakly stationary process can be decomposed into the sum of a: 1. Purely deterministic component plus2. A linear combination of white noise processesIf the number of weights is infinite we need to assume that the weights are absolutely summablefor the series to be convergent/stationary. For example, if the weights decay geometrically to zero then the series is convergent/stationary (see below)..1 ,...002211=+=++++=Σ∞=−−−ψεψμεψεψεμjjtjttttyψΣ∞=∞<0jjψWarwick Business School 6
WoldDecomposition Theorem
The Wolddecomposition forms the basis for ARMA modeling.
Different patterns of ψweights give rise to different types of ARMA model.
Also the ‘memory’of a time-series process depends on the Woldform of the model.
There is a one to one correspondence between the pattern of the ψweights in the Woldform of a series and its autocorrelation function.
Without loss of generality we’ll assume the deterministic component/mean μ=0 in the remainder of today’s analysis.Warwick Business School 7
Autoregressive (AR) processes
Suppose
Or
Where L is the ‘lag operator’:
()tttttttttjjyyεφφεεφεεφφεεφψ+=+++=+++==−−−−−121221......()ttyLεφ=−1mttmmttmttyyLyyLyLy+−−−===1
First-order AR process: AR(1)Warwick Business School 8
Sums of geometric series (useful results for later)
The sum to nterms of a geometric series is given byThereforeAccordinglyIf then The sum of an infinite geometric series is therefore 12...−++++=nnarararaS()()nnnnraarararararararS−=−−−−++++=−−1......1212()rraSnn−−=110lim=∞→nnrraS−=∞11<rWarwick Business School 9
Sums of geometric series: AR(1) model
The Woldform of an AR(1) model is an infinite geometric series:The term in brackets is an infinite geometric series with Therefore()tttttLLyεφφεφφεε...1...22221+++=+++=−−Lraφ== 论文英语论文网提供整理，提供论文代写英语论文代写代写论文代写英语论文代写留学生论文代写英文论文留学生论文代写相关核心关键词搜索。

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