Vector autoregression

Vectorautoregressive (short VAR models ) are very widely used econometric models for simultaneous estimation of several equations. They are the multi-dimensional analogue of the autoregressive model. They belong to the upper class model of VARMA models. In this type of time series models, the endogenous variables are determined both by its own past values ​​( random shock), as well as by the past values ​​of the other endogenous variables. The variables are therefore also referred to as delayed exogenous. So there is a feedback between the variables when the covariance matrix is non- diagonal.

Motivation

A simple two-dimensional VAR model contains two time series and which are explained and yet another time series that is used to explain. The model equations are then

The values ​​of the two time series and the time thus depend on

• The past two time series, in the VAR (1 ) model only on the values ​​of the previous period, and,
• In the VAR (p ) model can further lags are involved (from English " time shifting" ),

In the model, the model parameters must

• ,
• And

Iteratively estimated from the data.

Demarcation to transfer function models

There are similarities between the VAR models and transfer function models. A VAR ( 1 ) model can not be regarded as a causal transfer function model. Reason is the respective contemporaneous correlation of the shock variables. By orthogonalization of the shock variables ( the variance- covariance matrix diagonalization ), a VAR ( 1 ) model still be converted to a transfer function causal model.

The structure of a VAR estimation

Where the vector of endogenous variables, the vector of exogenous variables and the error term respectively.

To select the optimal number of lags (optimal lag order ) the Akaike or the Schwarz criterion can be used. It should be noted that any added delay taken while bringing additional explanatory power, but at the same time eliminated degrees of freedom from which the estimation will be removed from the empiricism and on. The Akaike criterion overestimates the number of delays, the black - criterion is to be economical.

It is important that the time series of detrended ( Hodrick -Prescott about with filters) and must be stationary. This is the case when all the roots of the auxiliary equation are outside the unit circle.

The calculation of the covariance matrix is recursive. Helmut Lütkepohl developed a simple method for calculating the Startmatrizen.

Criticism of VAR

VAR models are not well-founded theory, they give a purely statistical relationship again. The choice of the time series, the period and the pulse sequence (see Cholesky decomposition ) are free and can therefore lead to a lack of robustness of the results. In part, have VAR models according to relationships which can not be caused by the mutual influence of the variables, but also by a common reaction to third, unacceptable taken, variables. Also provides an addition of already represents a significant complication of the model, only one variable, depending on the lag order, so VARs remain limited to relatively small models. The results of VAR estimates should therefore be closely examined.