Xycoon logo
Time Series Analysis
Home    Site Map    Site Search    Xycoon College    Free Online Software    
horizontal divider
vertical whitespace

Time Series Analysis - ARIMA models - AR(1) process

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


b. The AR(1) process

The AR(1) process is defined as

Time Series Analysis - ARIMA models - AR(1) process

(V.I.1-83)

where Wt is a stationary time series, et is a white noise error term, and Ft is called the forecasting function. Now we derive the theoretical pattern of the ACF of an AR(1) process for identification purposes.

First, we note that (V.I.1-83) may be alternatively written in the form

(V.I.1-84)

Second, we multiply the AR(1) process in (V.I.1-83) by Wt-k in expectations form

(V.I.1-85)

Since we know that for k = 0 the RHS of eq. (V.I.1-85) may be rewritten as

(V.I.1-86)

and that for k > 0 the RHS of eq. (V.I.1-85) is

(V.I.1-87)

we may write the LHS of (V.I.1-85) as

(V.I.1-88)

From (V.I.1-88) we deduce

(V.I.1-89)

and

(V.I.1-90)

(figure V.I.1-1)

We can now easily observe how the theoretical ACF of an AR(1) process should look like. Note that we have already added the theoretical PACF of the AR(1) process since the first partial autocorrelation coefficient is exactly equivalent to the first autocorrelation coefficient.

In general, a linear filter process is stationary if the y(B) polynomial converges.

Remark that the AR(1) process is stationary if the solution for (1 - fB) = 0 is larger in absolute value than 1 (c.q. the roots of y(B) are, in absolute value, less than 1).

This solution is f-1. Hence, if the absolute value of the AR(1) parameter is less than 1, then model is stationary which can be illustrated by the fact that

(V.I.1-91)

For a general AR(p) model the solutions of

(V.I.1-92)

for which

(V.I.1-93)

must be satisfied in order to obtain stationarity.

vertical whitespace




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