A Brief Introduction to Vector Autoregressions

Introduction

The basic premise is that we can use the interaction among several variables to improve our forecast of each individual variable. At time t the forecast of a variable y is a function of its own past values as well as the present and past values of other variables in the system. The basic premise is that structural systems models of the economy have done a poor job of forecasting. As an alternative, some modelers have proposed using atheoretic vector autoregressive (VAR) models. A VAR does not come with the set of exclusion restrictions necessary to identify and estimate a structural model. As a consequence, both Keynesians and Monetarists could use the same VAR to forecast GDP.

As in univariate analysis of the last chapter, the simplest model is a first order process, VAR(1). Since this is the easiest specification to deal with, we'll use it almost exclusively. Most of the results stated here can be generalized to higher order VAR processes. We'll write the first order process as

The matrix A is an nxn matrix of unknown coefficients. The error terms have expectation zero, are independent over time, but may be contemporaneously correlated, so the nxn error covariance matrix need not be diagonal. A little thought probably suggests to you that the AR system is just a reduced form for some unstated structural model.

If y_t is a stationary random vector then an equivalent moving average representation can be derived. The process y_t is stationary if and only if the roots of the determinantal equation

where L is an n-dimensional lag operator, lie outside the unit circle. The infinite order MA representation is

where B = (I-AL).

Conversely, if y_t is a finite MA and the roots of |B|=0 are outside the unit circle, then the process is invertible and can be rewritten as a possibly infinite order autoregression.

To pursue the different representations a bit further, the process might be a mixed AR(p) and MA(q). Hence

.

If the AR part is stationary then the model can be rewritten as a MA, possibly of infinite order. Similarly if the MA part is invertible then the process can be written as an AR, possibly of infinite order.

To avoid the problem of determining p and q in the vector ARMA, we often write the model as a VAR of arbitrary finite order. Since the right hand side variables will all be past realizations of y:(nx1), it is possible to apply OLS to estimate the unknown coefficients in the matrix A.

Alternative Representations

To proceed we'll consider the two variable VAR(1) given by

According to the model, an increase in u_t will cause x_t to increase by one unit and will cause y_t to increase by r units as well. Because of the contemporaneous covariance between u and v, a change in v cannot be attributed to a pure innovation in y.

The moving average representation of the system can be derived by recursive substitution and written as

Notice that the MA representation of x does not include the current value of v, nor does the MA representation of y include the current value of u. Is this contrary to the earlier assertion that a one unit innovation in x_t will cause y_t to increase by r units? No, since the contemporaneous correlation between u and v is r. A one unit change in u produces an r unit change in v, which does enter the MA rpresentation of y. To pursue this, we proceed by first writing the model in matrix form:

.. 1

The innovations in this system of equations are given by

The information set for x_t is x_t-1 and y_t-1. Similarly for y_t it is x_t-1 and y_t-1.

The Cholesky Decompostion, the lower triangular square root such that , of the error covariance matrix is

Multiply both sides of the VAR by H to get the recursive model

.. 2

The new error terms are given by

and they have zero correlation. The innovations in this second model are given by

The information set for x and y is now different than that for the first representation.

In model ..2 there are no endogenous variables in x_t, but the current value of x_t appears in y_t. Hence an innovation in x in the first equation has a direct effect on y. In representation 2 a change in u_t has no effect on y_t by construction.

The moving average representation of the system has the same sort of characteristic. Derive the MA model from ..2

.. 3

The current value of e₂ does not appear in x_t, but the current value of e₁ does appear in y_t. We have a second triangularization of the same system. Now a change in e_1t by one unit produces a change of r units in y, although e₁ and e₂ are still uncorrelated.

Finally, these three triangularizations are not unique. Their representations depend on the ordering of the variables in the model. If we were to put y first, then we would get a different set of relationships. The only circumstance under which ordering doesn't matter is when u and v are uncorrelated.

Exogeneity

With all of these representations, an obvious question is whether x or y or vice are exogenous. In this usage we interpret the word 'expgenous' in the sense of a variable being determined outside the system. There are several different notions of exogeneity. In order to understand them all we'll first have to consider the general vector ARMA model, then look at our specific two variable case.

.. 4

The nx1 dimensional random vector y has been partitioned into y₁:n₁x1 and y₂:n₂x1. The error vector has been partitioned conformably. y₁ is strictly exogenous with respect to y₂ when it is not affected either directly or indirectly by lagged values of y₂. If c₁₂(L)=0, then lagged values of y₂ do not affect y₁. If d₁₂(L)=0 and d₂₁(L)=0 then lagged values of y₂ do not enter the y₁ equation. These three restriction matrices are necessary and sufficient. Let's impose the restrictions and multiply through by the inverse of the MA coefficient matrix to get the VAR representation:

This gives us the necessary and sufficient condition for the exogeneity of y₁ with respect to y₂. In the VAR representation, y₁ is exogenous with respect to y₂ if and only if the coefficients on y₂ are all zero.

Now impose the exogeneity restrictions and multiply the ARMA in ..4 by the inverse of the AR coefficient matrix to get the MA representation of the system.

Hence, in the MA representation of the system, y₁ is exogenous with respect to y₂ if and only if the coefficients on are all zero.

Sometimes you will see reference to the notion of weak exogeneity. The idea is similar to that of sufficient statistics. A statistic is sufficient if the likelihood function can be factored into a part that depends on the statistic and the unknown parameter and a part that depends only on the data. The weak exogeneity of y1 with respect y2 requires knowledge of the joint likelihood function for y₁ and y₂, say . If this can be factored into the product of two likelihood functions such that can take any values in its parameter space given a specific admissable value for .

There is also the notion of a variable being predetermined. The variable y₂ is said to be predetermined if it is unaffected by present and future values of but is affected by past values of .

To cement these notions we again consider a simple model

.. 5 .. 6

The disturbances u_t and v_t are independent and serially uncorrelated. Since x_t is correlated with the error term in eqn 5, and similarly for y_t in eqn 6, we need to solve for the reduced forms.

Note that these reduced forms are actually two AR(1)'s that are related through their error terms and are a special case of the alternative representations that we considered earlier. If then the reduced form for x_t becomes . x_t is now no longer contemporaneously correlated with the error u_t, although it will be affected by past values of u through y_t-1. We say that x_t is predetermined.

Suppose that and so that x_t = v_t. Now x_t is independent of past, present and future values of u_t, and it is said to be strictly exogenous.

Suppose that all we really needed was a consistent and asymptotically efficient of in equation 5. If in equation 6 then x_t and u_t are contemporaneously uncorrelated so an OLS estimate of will be consistent and asymptotically efficient. In this case xt is said to be weakly exogenous for . If u and v had a joint normal distribution you could prove this using the factorization suggested above.

Suppose we now wish to use our estimate of and equation 5 to forecast y_t+1 for a given x_{t+1 .} If , then y_t tells us what x_t+1 must be and the forecasted y_t+1 tells us what x_t+2 via equation 6. As a result of this feedback we are not able to pick any value we wish for x_{t+2 .} We cannot use equation 5 in isolation from equation 6 for forecasting purposes. If , then we are able to use equation 5 in isolation for forecasting. In this instance x_t is said to be strongly exogenous in equation 5.

Finally, suppose x_t in equation 5 is a policy variable. If is invariant to policy changes then x_t is said to be super-exogenous.

Causality

To examine the notion of Wiener - Granger causality in a two variable system we introduce two summary tables. The first lays out the notation and models, the second presents the results. Each pair of rows is a model. Hence there are four models.

Table 1 The Models
Var(ujt)=Tj	Var(wjt)=Gj	Cov(u2t, w2t)=C

The third model, the 5^th and 6^th rows are found by pre-multiplying the second model, the 3^rd and 4^th rows, by .

Table 2 Wiener Granger Causality
(1) z causes x if and only if
for all s	or	G1 > G2 and T3 > T4
(2) x causes z if and only if
for all s	or	T1 > T2 and G3 > G4
(3) z does not cause x if and only if
for all s	or	G1 = G2 and T3 = T4
(4) x does not cause z if and only if
for all s	or	T1 = T2 and G3 = G4
(5) There is instantaneous causality between x and z if and only if
for all s	or	T2 > T3 and G2 > G3

Exogeneity and causality are not the same thing. Tests for the absence of a causal ordering can be used to refute strict exogeneity in a given specification, but such tests cannot be used to eatablish it. Furthermore, unidirectional causality is neither necessary nor sufficient for inference to proceed on a subset of variables, all we require is that the variables be weakly exogenous.