Xycoon logo
MA(2) process
Home    Site Map    Site Search    Free Online Software    
horizontal divider
vertical whitespace

Time Series Analysis - ARIMA Models - MA(2) process

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

f. The MA(2) process

By definition the MA(2) process is

Time Series Analysis - ARIMA Models - MA(2) process


which can be rewritten on using (V.I.1-139)


where Wt is a stationary time series, et is a white noise error component, and Ft is the forecasting function.

into eq. (V.I.1-46) and (V.I.1-45) we obtain


Hence the theoretical ACF can be deduced


The invertibility conditions can be shown to be


(compare with the stationarity conditions of the AR(2) process).

The deduction of the theoretical PACF is rather complicated but can be shown to be dominated by the sum of two exponentials (in case of real roots), or by decreasing sine waves (in case the roots are complex).

These two possible cases are shown in figures (V.I.1-5) and (V.I.1-6).

(figure V.I.1-5)

(figure V.I.1-6)

vertical whitespace

AR(1) process
AR(2) process
AR(p) process
MA(1) process
MA(q) process
ARMA(1,1) process
ARMA(p,q) process
Wold's decomp.
Non stationarity
Inverse Autocorr.
Unit Root Tests
MA(2) process
horizontal divider
horizontal divider

© 2000-2022 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