Estimates a vector auto-regressive time series model with optional moving average components.

The type double function is imsls_d_vector_autoregression.

An array containing the estimated coefficients. The array has length n_cols×(trend+n_xvars+n_cols×(p+q+1)) if a non-trivial A0 is included in the model. Otherwise, the array has length n_cols×(trend+n_xvars+n_cols×(p+q)). trend = 1 if IMSLS_TREND is specified, and 0 otherwise.

The array elements occur in this order:

This arrangement corresponds to the vectorized coefficient matrix,

or

See also optional argument IMSLS_RETURN_USER for a different form of these output values. To release this space, use imsls_free. If no value can be computed, or IMSLS_RETURN_USER is supplied, returns NULL.

This function estimates a vector-autoregression moving average model using one and two-stage multivariate least squares regressions.

The general model can be written in operator notation as

where yt is a K-dimensional real-valued time series with stationary mean μ , ut is a K-dimensional white noise series with a non-singular covariance matrix, Σu, and xt respresents a matrix of deterministic components such as trend or seasonal variables. For autoregression (ar) lag parameter, p ≥ 0 , and moving average (ma) lag parameter, q ≥ 0, the operators are defined as

where L is the lag or backshift operator, defined as

and the Aj, Mj are K×K matrices of coefficients. That is,

The model has many equivalent forms, such as

A pure vector autoregression with q = 0 is often denoted VAR(p) and the autoregressive moving-average is VARMA(p,q).

The estimation procedures for the most part assume that the underlying time series is stable and invertible. These conditions are satisfied when

In other words, the conditions for a stable and invertible process are that the roots of the determinantal polynomial are outside the (complex) unit circle, or equivalently, that the eigenvalues of the determinantal matrix have modulus less than 1. To provide for unit-root tests, this function returns the eigenvalues of the estimated model via the optional argument IMSLS_UNIT_ROOT. In the case of unit-roots, when some fall on the unit circle (|z| = 1) there may still be stationary combinations of the variables. This behavior, known as cointegration, is discussed below.

Another form of expressing the VARMA(p,q) model is the error-correcting form:

where

The error-correcting or error-correction model (ECM) was developed for economic variables that tend to track close to each other due to common trends. The concept is closely aligned with that of cointegration. A K-dimensional process is said to be integrated of order 1, denoted as yt ~ I(1), when yt is non-stationary while the first-differenced series, Δyt, is stationary. In this case, there may exist a stationary linear combination in the levels (non-differenced versions). That is, there may exist a

where at least one βit ≠ 0 such that zt is stationary.

The interpretation is that cointegrated variables have common trends that cause them to move together in some sense. Consider the price of a commodity in two different locations. At their individual levels, they are non-stationary, unpredictable processes, but market forces keep the two prices from being too far apart so the difference in their prices is stationary.

We obtain an estimate of each coefficient matrix in the ECM form of the model. Then we are interested in the decomposition of Π = αβ ′ where α,β are K×irank (rank irank coefficient matrices). From the ECM form it can be deduced that zt = β ′yt is the cointegrating relationship under the given assumptions. However, the cointegrating rank r is not known in practice. As discussed in Lütkepohl, 2007, Ch. 7, the most common tests for the correct cointegrating rank are

and

The likelihood ratio test statistic for testing the first hypothesis is known as the trace test, and the second is known as the maximum eigenvalue test (Johansen 1988, 1995). In particular,

where λ1 ≥ … ≥ λK are eigenvalues of a particular matrix associated with the likelihood function. To allow for various tests, this function returns the eigenvalues λ1 ≥ … ≥ λK through the optional argument IMSLS_VECM_EIGENVALUES.

Partition the K- dimensional time series as yt’ = (zt,xt)’, where zt is of length m and xt is of length K-m. In general, xt is said to Granger-cause zt when predictions for zt can be improved by taking xt into account, and vice-versa. In the context of a stable and invertible VAR(p) or VARMA(p,q) process, a test for Granger causality amounts to testing for certain 0 restrictions on the AR or MA coefficients. Given a partition yt’ = (zt,xt)’, the test of

is performed using a version of the Wald statistic which has an approximate F-distribution,

where N is the number of 0 restrictions induced by the test. This function returns the test statistic and associated p-value when a test is specified through the optional argument IMSLS_CAUSALITY.

In this example we use a small data set with 2 time series. A VARMA(1,0) model is requested.

In this example, we fit the unrestricted VARMA(1,1) model and request 1 and 2 step ahead forecasts for each observation.

In this example we fit the following VARMA(1,1) restricted model on simulated data of two dimensions:

In this specification there are 7 free parameters:

The first stage and the second stage results are shown. From the second stage results, we see that (to two decimals) the estimate is