Autoregressive–moving-average model

The acronym ARMA ( autoregressive moving average ) and the ajar because artificial words ARMAX and ARIMA denote linear models for stationary, discrete-time stochastic processes. They are used for time series analysis in the measurement technique in statistics and there especially used in econometrics. Since they play a central role in the Box-Jenkins method, they are known there under the name Box-Jenkins models. The prediction models of economic institutions and banks are composed usually of ARMA models. One can view these models as linear difference equations or differential equation systems. They define the processes are collectively referred to also as a linear stochastic processes.

  • 2.1 ARMAX and ARIMA
  • 2.2 Interpretation of the moving average part

Mathematical definition of an ARMA process

In the model noise terms and weighted past values ​​of the time series are included linearly. ARMA models are one of the main tools for the prediction of observed stochastic signals. Are the signals to be modeled is not stationary, then you have to differentiate, if necessary, before modeling to remove the trend.

MA model

The signal consists of a by a moving average ( moving average ) of length m smoothed signal (not directly measurable ) other time series and a noise term (j = 0) together.

AR model

The signal consists of a smoothed signal of its n previous values ​​and a noise term.

ARMA model

This model is ( n m) also referred to as ARMA model, where n and m is the order of the process mean.

With the help of the so-called shift operator L (from lag = time difference):

To write shorter also:

Where θ and both finite polynomials ( of degree n and m) represent:

Substantive interpretation: What is MA and what is AR?

Moving Average

Is a so-called white noise, a random variable that is (usually distributed for all pairwise independent and identically Gaussian ) with mean and variance. The dependence is restricted in MA -terms on the expectation value: Is

Then, the expected value of Y with each time step, only shifted. itself is determined stochastically.

Auto- regression

Unlike the auto- regression part: Here is deterministic from the past dependent. in

Subject just then a fault, if a fault is subject.

ARMA models in the statistics

Regression models play a major role in the statistics. In econometrics often several time series of the form must be brought into relation, the so-called economic indicators, eg interest rates, unemployment, investment, etc. A distinction is made between endogenous time-dependent variables Y ( t) (which are thus explained by the model ) and exogenous variables X (t), which are defined from outside. With them you can the general linear system of equations

Formulate. B, Y ​​, A and X are matrices with as many rows as observations and as many columns as variables of the corresponding type. Each time counts as an observation. Is one and the same variable at different times ( that is, as Y (t ), Y ( t-1) etc. ) into the equation system, it is counted as a number of variables. So the equation has three variables. This is crucial for the ARMA models. is a vector with as many rows as observations.


If the regressor X, this is described by ARMAX - models. Follow only the differences of Y into the model so that after the model predictions must be re- "integrated", we speak of ARIMA models, the I stands for " Integrated". The differences are here produce a stationary and adjusted for non- seasonal trends model for the calculation.

All models of the ARMA family have this system of linear equations for the foundation. Many systems can be estimated with a simple linear regression. The prerequisite is that the standard errors of the estimates are unbiased, that is not auto-correlating the error terms of Y. The correlation of errors among themselves while not distorting the estimator itself, but the associated standard error (usually he is greatly underestimated ). If autoregression terms of the form

Present, is usually before such autocorrelation of the error terms.

Interpretation of the moving average part

Said linear system of equations is a term of the form

Therefore extended by a autoregression of the error terms. Practically play mainly extensions of order 1:

A role. This is a Markov process.

The term "MA" for such a purely stochastic processes is rather misleading. ARMA models are therefore simultaneous models for deterministic context models ( AR- share, corresponding regression model) and stochastic processes (MA component).

ARMA models (also ARMAX, ARIMA ) are estimated by nonlinear regression method.