RDP 8604: Leading Indexes – Do They? 2. The VAR Methodology

In general we will be concerned with a (n×1) vector of n endogenous variables yt containing a leading index and (n−1) variables representing the business cycle, whether these be indexes of the business cycle (e.g., a coincident index) or variables that might be expected to move with the business cycle (e.g., employment). We assume that Yt is generated by the following mth order vector-autoregression

where Dt is a (n×1) vector representing the deterministic component of Yt (generally a polynomial in time), Bj are (n×n) matrices and εt is a (n×1) vector of multivariate white noise residuals (or innovations).

Equation (1) is specified and estimated as an “unrestricted reduced form”. As is the hallmark of VARs, there are no exclusion restrictions within the Bj. matrices. Rather, the Bj's are uniquely determined under the orthogonality conditions E(εt ) = 0 and E(Yt-jεt) = 0, j=1, …, m, and are estimated by ordinary least squares. Since, in this paper, it is relatively straightforward to decide what variables should be in Yt, the only pretesting involved with the fitting of equation (1) is in choosing the appropriate lag length m. In general we choose the smallest m such that εt is indistinguishable from a multivariate white noise process.[4]

Tests which are commonly applied to the VAR are tests for Granger-causality which test whether a variable, say Y1t is useful in forecasting another variable, say Y2t. Y1t is said to be useful in forecasting Y2t if the inclusion of lags of Y1t. in the equation for Y2t significantly reduces the forecast variance. Thus it tests whether lags of Y1t. contain any additional information on Y2t which is not already contained in the lags of Y2t itself.

The model presented in equation (1) is difficult to describe in terms of the Bj coefficients. The best descriptive devices are the innovation accounting techniques suggested in Sims (1980, p.21) and described by Litterman (1979, pp.74–85). The first of these techniques of innovation accounting are the impulse response functions which describe the dynamic response of variables in the VAR to an impulse in one of the variables. To understand these impulse response functions, consider the moving average representation of equation (1), obtained by repeated back substitution for Yt−j

where Mj is a (n×n) matrix of moving average coefficients. The response of the ith variable to a unit innovation in the kth variable j periods earlier is given by the ikth element of Mj. In general, however, there is likely to be some contemporaneous correlation among innovations, which is not taken into account in equation (2). By making an assumption about the contemporaneous causal ordering of the variables in Yt (such that contemporaneous causality is one way, i.e., recursive) one can obtain orthogonalised innovations ut where ut = Gεt, so that E(ut ut) = ϕ where ϕ is a diagonal (n×n) matrix. In this paper we always assume that the variable representing the business cycle does not contemporaneously cause the leading index. Thus if we order Yt such that the leading index is the first variable then G will in general be of the form

where ρ is the estimated coefficient in the regression equation

ε1t is the innovation in the leading index, ε2t the innovation in the business cycle variable and u2t the orthogonalised innovation in the business cycle variable (in the sense that it is orthogonal to u1t = ε1t).

In terms of orthogonalised innovations, ut, the moving average representation is

where the ikth element of Aj gives the response of variable i to an orthogonalised unit impulse in variable k, j periods earlier. litterman (1979), however, notes that unit innovations may be difficult to interpret, especially when the standard errors of the innovations are very small. For this reason we calculate a scaled version of equation (3) which gives the response of the system to innovations of one standard error in size. The impulse response functions obtained from this scaled version provide information regarding the length of time it takes for shocks in the leading index to show up in the activity variable. Hence, they provide some idea of the lead time between a movement in the leading index and the associated subsequent movement in activity.

The second device of innovation accounting relates to the decomposition of the k-step ahead forecast variance of each variable in the VAR, into percentages contributed by the innovations in each variable. A variable whose own innovations account for all or most of its own forecast variance would be said to be exogenous (in the Sims sense) to the system. Thus, if the leading indexes are useful in forecasting business cycle variables, then the innovations in the leading index should account for a (subjectively) large percentage of the k-step ahead forecast variance of business cycle variables.

The k-step ahead forecast variance may best be seen by considering the k-step ahead forecast error induced by forecasting Yt linearly from its own past

(in terms of orthogonalised innovations) where Et(Yt+k) is the linear least squares forecast of Yt+k given all information at time t. The k-step ahead forecast variance is

Because of the extensive orthogonality conditions built into the model, the k-step ahead forecast variance of each variable will be a weighted sum of the variances of the innovations to each variable. Thus we can obtain the percentage contribution of each variable's innovations to the variance of any other variable. Again, if the leading indexes are useful for forecasting activity variables at horizon k, their innovations will have a large contribution to the k-step ahead forecasting variance of these activity variables.


On the basis of tests for within, and across, equation serial correlation. The inverse autocorrelation function (i.e., the autocorrelation function of the dual model) is used to test for non-stationarity of the residuals. (See, for example, Priestley (1981). All of the empirical work is done using the macro facilities of version 5 of SAS. [4]