Multiple Time Series Estimation

V.III.2 Multiple Time Series Estimation

If the VAR(p) model is written in structural form as

Y = BZ + E with

(V.III.2-1)

then the following LS estimator can be obtained

(V.III.2-2)

An alternate estimator can be shown to be

(V.III.2-3)

which is much easier to implement into a computer program since it makes use of the SUR estimator (when all exogenous variables are present in all equations). Note that the vec operator transforms a matrix into a vector where the first column is followed by the second, the third, etc...

It can be shown that the LS estimator is normal, and consistent

(V.III.2-4)

Note that in the case of a VAR(p) model with cointegration, an unrestricted LS estimator (as in (V.III.2-2) or (V.III.2-3)) can be shown to be asymptotically consistent (w.r.t. the parameters, and the error covariance matrix). An MLE for the VAR(p) with cointegrated variables model does also exist (LUETKEPOHL 1991). The MLE allows to explicitly constrain for the cointegration relationships. Of course, the cointegrated variables will not be differenced to induce stationarity (due to the definition of cointegration).

A restricted (stable) VAR model (RVAR)

(V.III.2-5)

(V.III.2-6)

which is equivalent to MLE under Gaussian errors. If the rank of R is equal to M, and if the VAR model is stable with white noise errors it can be shown that

(V.III.2-7)

This result is still valid if an EGLS is used in stead of GLS, because RVAR-EGLS is asymptotically equivalent to the RVAR-GLS. Note, that the (unrestricted) LS estimator of the error covariance is consistent.

The estimation of the parameters of a stable and invertible VARMA(p,q) model is almost always done with a MLE (or a FIMLE). A special problem however, pops up in this context. If the VARMA model is written somewhat more general

(V.III.2-8)

then this model is called to be nonunique. The property of nonuniqueness can be reformulated in the context of the compact matrix representation. The general VARMA model can be written as YG = BZ + EW, where G contains the "contemporaneous effects" among endogenous variables, whereas W represents the "contemporaneous effects" of the i-th equation error on the j-th endogenous variable. This problem is analogous to the identification problem in econometrics (see chapter III.III).

It can be shown, however, that the echelon form of a VARMA model has a unique parameterization. The echelon form can be defined (on using the LUETKEPOHL (1991) notation) as follows

(V.III.2-9)

(V.III.2-10)

The MLE can be shown to be asymptotically normal, and consistent (for the parameters (even with constant), and the error-covariance matrix).