One widely used representation of an univariate time series is a ARMA model. To motivate the model, basically we can track two lines of thinking. First, a model depends on the level of the lagged observations. For example, if we observe a high realisation of GDP we would expect that the GDP in the next few periods are high as well. This way of thinking can be represented by an autoregressive model (AR model). An AR model of the order p can be written as:

_{}

where _{} and _{}.is a constant

In the second way of thinking, we can model that the observations of a random variable at time t are not only affected by the shock at time t, but also the shocks of prior periods. For example, if we observe a negative shock to the economy, say, 9/11, then we would expect that the negative effect affects the economy also for the near future. This way of thinking can be represented by a moving average model (MA model). A MA model of the order q can be written as:

_{}

If we combine both models we get a ARMA(p,q) model.

_{}.

A necessary condition of for ARMA models
are, that the ARMA equitation have a stationary solution. If the time
series is not stationary, we can transform it to a stationary time series
by differencing. ARMA models with differenced time series are called
ARIMA(p,d,q) (autoregressive integrated moving average) models, where
d is the number of differences to get a stationary time series.

**[notes]**

The parameter of an pure AR(p) model can be estimated
by OLS. Estimation of MA(q) or ARMA(p,q) models (with q>1) are
non linear. [web:reg] ARMA Add-In estimates this models using the
Levenberg-Marquardt algorithm. The derivates, which are needed for
the estimation and the covariance matrix, are computed with numeric
finite difference methods.

After estimation the Add-In displays the coefficient results (including
std.error, t-statistic, p-value), summary statistics (R², Adjusted
R², Standard Error of Regression, sum of squared residuals, log
likelihood, Durbin Watson, Akaike information criteria (AIC), Schwarz
criteria (SIC), inverted MA/AR roots, Impulse response function as
well as forecast evolution.

**[related links]**

All links will be open in a new window

wikipedia A description of the ARIMA models at wikipedia. (HTML)

Introduction to ARIMA models, by Barbara Bogacka. Lecture Notes. (PDF)

Levenberg
Marquard, by Manolis I. A. Lourakis. A Brief Description of the
Levenberg-Marquardt Algorithm Implemened

by levmar. (PDF)

Links to other sites from these pages are for information only and Kurt Annen accepts no responsibility or liability for access to, or the material on, any site which is linked from or to this site.

screenshot of [web-reg] arma add-in

**[download]**

for downloading click on the filename

File: **setup_arma.exe
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The [web:reg] ARMA Add-In was written by Kurt Annen.
This program is freeware. But I would highly appreciate if you could
give me credit for my work by providing me with information about possible
open positions as an economist. My focus as an economist is on econometrics
and dynamic macroeconomics. If you like the program, please send me
an email.

[arma excel add-in]

©
2005 Kurt Annen