Xycoon logo
Wold Decomposition
Home    Site Map    Site Search    Xycoon College    Free Online Software    
horizontal divider
vertical whitespace

Time Series Analysis - ARIMA Models - Wold decomposition theorem

[Home] [Up] [Basics] [AR(1) process] [AR(2) process] [AR(p) process] [MA(1) process] [MA(2) process] [MA(q) process] [ARMA(1,1) process] [ARMA(p,q) process] [Non stationarity] [Differencing] [Behavior] [Inverse Autocorr.] [Unit Root Tests] [Wold's decomp.]

Wold's decomposition theorem

The most fundamental justification for time series analysis (as described in this text) is due to Wold's decomposition theorem, where it is explicitly proved that any (stationary) time series can be decomposed into two different parts. The first (deterministic) part can be exactly described by a linear combination of its own past, the second part is a MA component of a finite order.

A slightly adapted version of Wold's decomposition theorem states that any real-valued stationary process Yt can be written as

Time Series Analysis - ARIMA Models - Wold decomposition theorem


where yt and zt are not correlated.


where yt is deterministic




(zt has an uncorrelated error with zero mean).

Because of its importance for time series analysis in general, and in practice, we will discuss the proof of Wold's decomp. theorem shortly.

Since there seems to be much confusion in literature about this theorem, we will only discuss the proof of another version (than that described above) which can be proved quite easily. In order to make a distinction with the previous description, we will explicitly use other symbols.

Denote a stationary time series Wt with zero mean and finite variance.

In order to forecast the time series by means of a linear combination of its own past


a criterion is used to optimize the parameter values. This criterion is


which is called the sum of squared residuals (SSR).

The normal equations (c.q. eq. (V.I.1-176) differentiated w.r.t. the parameters) is easily found to be


In matrix notation eq. (V.I.1-177) becomes




which is symmetric about both diagonals due to the stationary of Wt and with


On adding an error component et,n to eq. (V.I.1-175) it can be shown that


The first part of (V.I.1-181) is almost trivial


The second part of (V.I.1-181) is


with a RHS equal to zero since the parameters satisfy (V.I.1-177) (Q.E.D.).

On repeating the previous procedure we obtain


Remark that the error components (V.I.1-181) and (V.I.1-184) are uncorrelated due to (V.I.1-181), from which we obviously find


On substituting (V.I.1-184) into (V.I.1-175) it is obvious that


where xt,n(1) depends on the past of Wt-1

Also it is obvious from (V.I.1-183) that


and from (II.II.1-27) and the fact that et,n is independent from the other regressors of (V.I.1-186), that


On repeating the step described above it is easy to obtain




On applying (V.I.1-189) and (V.I.1-190) on Wt-i we obtain


where xt-i,n always depends on it's own past only, and where evidently


vertical whitespace

AR(1) process
AR(2) process
AR(p) process
MA(1) process
MA(2) process
MA(q) process
ARMA(1,1) process
ARMA(p,q) process
Non stationarity
Inverse Autocorr.
Unit Root Tests
Wold's decomp.
horizontal divider
No news at the moment...
horizontal divider

© 2000-2012 All rights reserved. All Photographs (jpg files) are the property of Corel Corporation, Microsoft and their licensors. We acquired a non-transferable license to use these pictures in this website.
The free use of the scientific content in this website is granted for non commercial use only. In any case, the source (url) should always be clearly displayed. Under no circumstances are you allowed to reproduce, copy or redistribute the design, layout, or any content of this website (for commercial use) including any materials contained herein without the express written permission.

Information provided on this web site is provided "AS IS" without warranty of any kind, either express or implied, including, without limitation, warranties of merchantability, fitness for a particular purpose, and noninfringement. We use reasonable efforts to include accurate and timely information and periodically updates the information without notice. However, we make no warranties or representations as to the accuracy or completeness of such information, and it assumes no liability or responsibility for errors or omissions in the content of this web site. Your use of this web site is AT YOUR OWN RISK. Under no circumstances and under no legal theory shall we be liable to you or any other person for any direct, indirect, special, incidental, exemplary, or consequential damages arising from your access to, or use of, this web site.

Contributions and Scientific Research: Prof. Dr. E. Borghers, Prof. Dr. P. Wessa
Please, cite this website when used in publications: Xycoon (or Authors), Statistics - Econometrics - Forecasting (Title), Office for Research Development and Education (Publisher), http://www.xycoon.com/ (URL), (access or printout date).

Comments, Feedback, Bugs, Errors | Privacy Policy Web Awards