# Vector autoregression analysis

## Introduction

The vector autoregression analysis estimates the linear dependencies among a number of time series. The analysis can produce fitted values and forecasts for those series.

In addition to estimating a given system, you can also automatically test different models and let the analysis pick the best one based on an information criteria.

There is a report that contains the estimated parameters of the system as well as a number of statistics that can be used as a test of the system's validity and stability.

The estimation is made using all common valid observations for the model series in the selected estimation sample.

## Estimation model

A vector autoregression can be thought of as a system of linear regressions, but the emphasis is on using lagged values of the dependent variables to model a set of variables. There is an equation for each variable that explains its evolution based on its own lags and the lags of other variables in the model.

The dependent variables are called *endogenous* variables. There may also be *exogenous* variables. Such variables are only explanatory not modeled in the system.

A model may be denoted as being of order *p*, called VAR(p), containing *K* endogenous variables. If there are 2 variables in a VAR(1) model, the system of equations can be written as:

The expression can be written in expanded form as:

$\left(\begin{array}{c}{y}_{1\text{,}t}\\ {y}_{2\text{,}t}\end{array}\right)=\left(\begin{array}{c}{v}_{1}\\ {v}_{2}\end{array}\right)+\left(\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right)\left(\begin{array}{c}{y}_{1\text{,}t-1}\\ {y}_{2\text{,}t-1}\end{array}\right)+\left(\begin{array}{c}{u}_{1\text{,}t}\\ {u}_{2\text{,}t}\end{array}\right)$The equations can thus be explicitly written as:

${y}_{1\text{,}t}={v}_{1}+{a}_{11}{y}_{1\text{,}t-1}+{a}_{12}{y}_{2\text{,}t-1}+{u}_{1\text{,}t}$${y}_{2\text{,}t}={v}_{2}+{a}_{21}{y}_{1\text{,}t-1}+{a}_{22}{y}_{2\text{,}t-1}+{u}_{2\text{,}t}$

The present value of y depends on an *intercept v*, the lagged value of itself and the other variable, and then there is an error term *u*. Each error term is supposed to be uncorrelated with all lags of itself and lags of the other error terms.

An arbitrary number of successive forecasts can be calculated and you must specify an end date for the forecast calculation.

When a system contains exogenous variables, assume that these are included in the vector *x* together with their lags and possibly including lag 0 (contemporaneous variables) so that *x* contains s elements. The system of equations for a model called VARX(p, s) can then be written as:

When there are exogenous variables, forecasts can only be calculated as long as there is data available for all the exogenous variables. You might want to add forecasts to these variables before they are passed on to the VAR analysis.

For a *symmetric system*, where each equation contains the same explanatory variables and lags, OLS (ordinary least squares) is used as the estimation method. For *asymmetric systems*, GLS (generalized least squares) is used, which requires an iterative procedure. This is more computationally intense and the system might not converge fast enough to find a solution for large systems.

## Settings

#### Estimation sample range

Specify the limits of the estimation sample range. The default range will be the largest range where there is data for all the series.

#### Output residuals

When this option is selected a time series containing the residuals will be calculated.

#### Output the endogenous series

Select this option in order to include the endogenous series in the output.

#### Output the exogenous series

Select this option in order to include the exogenous series in the output.

#### Calculate impulse response

Select this option in order to calculate the impulse response of the specified length. Select in what equation the impulse should be applied and what variable.

(Available in version 1.17 and later.)

#### Confidence band

Confidence bands for forecasts of each equation are computed using the VAR estimator covariance matrix. Since the VAR is ideally a stable linear dynamic system, the forecasted values are dynamically generated. This means that they converge toward some mean (zero if normalized). Therefore, the error bands must also converge to a constant value, an upper and lower bound respectively. Because not much is known about the small sample properties about the Feasible Generalized Least Squares estimator used in the VAR, only asymptotic errors are computed. This makes the this makes the estimated error terms less reliable for estimations from short time series. It can be shown that the estimator variance of the FGLS is lower than or at least equal to that of standard OLS. [Added in version 1.16]

#### Autocorrelation test lags

Select this option in order to include a Portmanteau autocorrelation test in the report. Specify the number of lags to include. The number of lags should be larger than the highest number of lags of the endogenous or exogenous variables.

#### Max endogenous lags

Specify the maximum number of lags to include for the endogenous variables. You can further refine which lags to include in the model on the “Lag settings for endogenous variables in the equations” tab.

#### Max exogenous lags

Specify the maximum number of lags to include for the exogenous variables. You can further refine which lags to include in the model on the “Lag settings for exogenous variables in the equations” tab.

#### Find best model based on max endogenous lags for information criteria

Select this option in order to let the system automatically test what combination of symmetric lags are optimal based on the selected information criteria.

You can select the minimum and maximum number of lags of the endogenous variables to test and also the minimum and maximum of different lags (regressors) to include in each round of tests.

Select the setting “Require stable process” in order to disqualify any model where the roots of the characteristic equation indicate that the model is not stable.

#### Type

Select if a series should be included in the model as a endogenous variable, exogenous variable or not at all.

#### Diff

Use first order differences of this variable in the model.

(Available in version 1.17 and later.)

#### Intercept

Select if the intercept should be included in the model for endogenous variables.

#### Equation name

Optionally specify the name of the equation to be used in the report.

#### Variable name

Optionally specify the name of the equation to be used in the report.

## Bibliography

Lütkepohl, H. *New Introduction to Multiple Time Series Analysis.* Berlin: Springer, 2005