12-FinEcon-Model Sel Page 339 Wednesday, February 4, 2004 12:59 PM
Financial Econometrics: Model Selection, Estimation, and Testing 339VAR models can be written in equivalent forms that will be useful in
the next section. In particular, a VAR model can be written in terms of
the differences ∆xt in the following error-correction form:∆xt = (ΦΦΦΦ 1 L + ΦΦΦΦ 2 L^2 + ... + ΦΦΦΦn – 1 Ln –^1 )∆xt + ΠΠΠΠLn –^1 xt + Dst + εεεεtwhere the first n – 1 terms are in first differences and the last term is in levels.
The multivariate random walk model of log prices is the simplest
VAR model:xt = xt + m + εεεεt∆xt = m + εεεεtNote that in this model log prices are autoregressive while returns (that
is, the first differences) are simply correlated multivariate white noise
plus a constant term.
As we know from our discussion on ARMA models (see Chapter
11), the stationarity and stability properties of a VAR model depend on
the roots of the polynomial matrixA N
1 z + A 2 z(^2) + ... + Az
n
In particular, if all the roots of the above polynomial are strictly outside
the unit circle, then the VAR process is stationary. In this case, the VAR
process can be inverted and rewritten as an infinite moving average of a
white-noise process. If all the roots are outside the unit circle with the
exception of some root which is on the unit circle, then the VAR process
is integrated. In this case it cannot be inverted as an infinite moving
average. If some of the roots are inside the unit circle, then the process is
explosive. If the VAR process starts at some initial point characterized
by initial values or distributions, then the process cannot be stationary.
However, if all the roots are outside the unit circle, the process is
asymptotically stationary. If some root is equal to 1, then the process
can be differentiated to obtain an asymptotically stationary process.
COINTEGRATION
Let’s now look at the problem of representation of multivariate time
series from a different angle. Recall that a variable is integrated of order