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.

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)

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