Alternative models and model representations
Given a set of K time series variables yt = (y1t,…, yKt)’, the basic VAR model is of the form
Vt – AiVt-i + • • • + AvVt-v + ut, (32.1)
where ut – (u1t,…, uKt)’ is an unobservable zero-mean independent white noise process with time invariant positive definite covariance matrix E(utu’) – £u and the Aj are (K x K) coefficient matrices. This model is often briefly referred to as a VAR(p) process because the number of lags is p.
The process is stable if
det(IK – A1z – … – Apzp) Ф 0 for | z | < 1. (32.2)
Assuming that it has been initiated in the infinite past, it generates stationary time series which have time invariant means, variances, and covariance structure. If the determinantal polynomial in (32.2) has roots for z – 1 (i. e. unit roots), then some or all of the variables are I(1) and they may also be cointegrated. Thus, the present model is general enough to accommodate variables with stochastic trends. On the other hand, it is not the most suitable type of model if interest centers on the cointegration relations because they do not appear explicitly in the VAR version (32.1). They are more easily analyzed by reparameterizing (32.1) to obtain the so-called vector error correction model (VECM):
AVt – nyt-1 + rjAvt-1 + … + Tp-jAyt-p+1 + ut, (32.3)
where П – -(IK – A1 – … – Ap) and 0 – -(Ai+1 + … + Ap) for i – 1,…, p – 1. This representation of the process is obtained from (32.1) by subtracting yt-1 from both sides and rearranging terms. Because Ayt does not contain stochastic trends by our assumption that all variables can be at most I(1), the term nyt-1 is the only one which includes I(1) variables. Hence, nyt-1 must also be I(0). Thus, it contains the cointegrating relations. The Г (j – 1,…, p – 1) are often referred to as the short-term or short-run parameters while nyt-1 is sometimes called long-run or long-term part. The model in (32.3) will be abbreviated as VECM( p) because p is the largest lag of the levels yt that appears in the model. To distinguish the VECM from the VAR model the latter is sometimes called the levels version. Of course, it is also possible to determine the Aj levels parameter matrices from the coefficients of the VECM as A1 – Г1 + П + IK, A i – Г – Г;-1 for i – 2,…, p – 1, and
A p – – rp -1.
If the VAR(p) process has unit roots, that is, det(IK – A1z – … – Apzp) – 0 for z – 1, the matrix П is singular. Suppose it has rank r, that is, rank(n) – r. Then it is well known that П can be written as a product П – aP’, where a and P are (K x r) matrices with rank(a) – rank(P) – r. Premultiplying an I(0) vector by some matrix results again in an I(0) process. Hence, premultiplying – aP’ yt-1 by (a’a) 1a’ shows that P’yt-1 is I(0) and, therefore, contains the cointegrating relations. Hence, there are r – шпЦП) linearly independent cointegrating relations among the components of yt. The matrices a and P are not unique so that there are many possible P matrices which contain the cointegrating relations or linear transformations of them. Consequently, cointegrating relations with economic content cannot be extracted purely from the observed time series. Some nonsample information is required to identify them uniquely.
Special cases included in (32.3) are I(0) processes for which r = K and systems that have a stable VAR representation in first differences. In the latter case, r = 0 and the term Пум disappears in (32.3). These boundary cases do not represent cointegrated systems in the usual sense. There are also other cases where no cointegration in the original sense is present although the model (32.3) has a cointegrating rank strictly between 0 and K. Still it is convenient to include these cases in the present framework because they can be accommodated easily as far as estimation and inference are concerned.
In practice the basic models (32.1) and (32.3) are usually too restrictive to represent the main characteristics of the data. In particular, deterministic terms such as an intercept, a linear trend term or seasonal dummy variables may be required for a proper representation of the data. There are two ways to include deterministic terms. The first possibility is to represent the observed variables yt as a sum of a deterministic term and a stochastic part,
yt = p t + x t, (32.4)
where pt is the deterministic part and xt is a stochastic process which may have a VAR or VECM representation as in (32.1) or (32.3), that is, xt = A1xt-1 +… + Apxt-p + ut or Axt = П xt-1 + Г1Аxt-1 + … + rp-1 Axt-p+1 + ut. In that case, if pt is a linear trend term, that is, pt = p0 + p1t, then yt has a VAR(p) representation of the form
yt = vo + vfi + A1yt-1 + … + АрУі-р + ut, (32.5)
where v0 = – Пр0 + (Z]p=1 ]AJ)p1 and v1 = – Пр1. In other words, v0 and v1 satisfy a set of restrictions. Note, however, that if (32.5) is regarded as the basic model without restrictions for v„ i = 0, 1, the model can in principle generate quadratic trends if I(1) variables are included, whereas in (32.4) with a deterministic term pt = p0 + p1t a linear trend term is permitted only. The fact that in (32.4) a clear partitioning of the process in a deterministic and a stochastic component is available is sometimes advantageous in theoretical derivations. Also, in practice, it may be possible to subtract the deterministic term first and then focus the analysis on the stochastic part which usually contains the behavioral relations. Therefore this part is often of primary interest in econometric analyses. Of course, a VECM( p) representation equivalent to (32.5) also exists.
In practice, these representations with possibly additional deterministic terms may still not be general enough. At times one may wish to include stochastic exogenous variables on top of the deterministic part. A fairly general VECM form which includes all these terms is
Ayt = ПУ^1 + Г1А yt-1 + … + Гр-1А yt-p+1 + CDt + Bzt + ut, (32.6)
where the zt are exogenous variables, Dt contains all regressors associated with deterministic terms, and C and B are parameter matrices.
All the models we have presented so far are reduced form models in that they do not include instantaneous relations between the endogenous variables yt. In practice it is often desirable to model the contemporaneous relations as well and therefore it is useful to consider a structural form
г = n*yM + Г *Лум + … + Г *r_1Ayt_r+1 + C*Dt + B*zt + vt, (32.7)
where vt is a (K x 1) zero mean white noise process with covariance matrix Xv and the П*, Г * (j = 0,…, p – 1), C* and B* are structural form parameter matrices. The reduced form corresponding to the structural model (32.7) is given in (32.6) with Г,- = (Г*0)-1Г* (j = 1,…, p – 1), C = (Г*0)-1C*, П = (Г*)-1П*, B = (Г *)-1B* and ut = (Г *)-1vt. Of course, a number of restrictions are usually imposed on the general forms of our models. These restrictions are important at the estimation stage which will be discussed next.
Because estimation of some of the special case models is particularly easy these cases will be considered in more detail in the following. We begin with the levels VAR representation (32.1) under the condition that no restrictions are imposed. Then estimation of the VECM (32.3) is treated and finally more general model variants are discussed.