## VAR Model Estimation

### Preparing VAR Models for Fitting

To create a model of multiple time series data, decide on a VAR model form, and fit parameters to the data. When you have a fitted model, check if the model fits the data adequately.

To fit a model to data, you must have:

Time series data, as described in Multivariate Time Series Data

At least one

`varm`

model object configured for estimation, as described in Vector Autoregression (VAR) Model Creation

There are several Econometrics Toolbox™ functions that aid these tasks, including:

`estimate`

, which fits VARX models.`summarize`

, which displays and returns parameter estimates and other summary statistics from fitting the model.`lratiotest`

and`aicbic`

, which can help determine the number of lags to include in a model.`infer`

, which infers model residuals for diagnostic checking.`forecast`

, which creates forecasts that can be used to check the adequacy of the fit, as described in VAR Model Forecasting, Simulation, and Analysis

### Fitting Models to Data

`estimate`

performs parameter estimation for VAR and VARX models only. For definitions of these terms and other model definitions, see Types of Stationary Multivariate Time Series Models. For an example of fitting a VAR model to data, see Fit VAR Model of CPI and Unemployment Rate.

#### How `estimate`

Works

Before fitting the model to data, `estimate`

requires at least

presample
observations to initialize the model, where * Mdl*.P

*is a*

`Mdl`

`varm`

model object and `P`

is the
property storing the model degree. You can specify your own presample
observations using the `Y0`

or `Presample`

name-value argument. Or, by default, `estimate`

removes the
first `Mdl`

.P

observations from the
estimation sample. Therefore, if you let `estimate`

take
requisite presample observations from the input estimation sample data, then the
effective sample size decreases.`estimate`

finds maximum likelihood estimates of the
parameters present in the model. Specifically, `estimate`

estimates the parameters corresponding to these `varm`

model
properties: `Constant`

, `AR`

,
`Trend`

, `Beta`

, and
`Covariance`

. For VAR models, `estimate`

uses a direct solution algorithm that requires no iterations. For VARX models,
`estimate`

optimizes the likelihood using the
expectation-conditional-maximization (ECM) algorithm. The iterations usually
converge quickly, unless two or more exogenous data streams are proportional to
each other. In that case, there is no unique maximum likelihood estimator, and
the iterations might not converge. You can set the maximum number of iterations
with the `MaxIterations`

name-value argument of
`estimate`

, which has a default value of
`1000`

.

For numeric array data inputs, `estimate`

removes entire
observations from the data containing at least one missing value
(`NaN`

). For table or timetable data inputs,
`estimate`

issues an error when any observation is missing.
For more details, see `estimate`

.

`estimate`

calculates the loglikelihood of the data, giving
it as an output of the fitted model. Use this output in testing the quality of
the model. For example, see Select Appropriate Lag Order and Examining the Stability of a Fitted Model.

### Examining the Stability of a Fitted Model

When you enter the name of a fitted model at the command line, you obtain a object summary. In the `Description`

row of the summary, `varm`

indicates whether the VAR model is stable or stationary.

Another way to determine stationarity of the VAR model is to create a lag operator polynomial object using the estimated autoregression coefficients (see `LagOP`

), and then passing the lag operator to `isStable`

. For example, suppose * EstMdl* is an estimated VAR model. The following shows how to determine the model stability using lag operator polynomial objects. Observe that

`LagOp`

requires the coefficient of lag `0`

.ar = [{eye(3)} ar]; % Include the lag 0 coefficient. Mdl = LagOp(ar); Mdl = reflect(Mdl); % Negate all lags > 0 isStable(Mdl)

If the VAR model is stable, then `isStable`

returns a Boolean value of `1`

, and `0`

otherwise. Regression components can destabilize an otherwise stable VAR model. However, you can use the process to determine the stability of the VAR polynomial in the model.

Stable models yield reliable results, while unstable ones might not.

Stability and invertibility are equivalent to all eigenvalues of the associated lag operators having modulus less than 1. In fact, `isStable`

evaluates these quantities by calculating eigenvalues. For more information, see `isStable`

or Hamilton [102].