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

May 1986

In general we will be concerned with a (n×1) vector of n endogenous variables y_{t} 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 Y_{t} is generated by the following m^{th} order vector-autoregression

where D_{t} is a (n×1) vector representing the deterministic component of Y_{t} (generally a polynomial in time), B_{j} are (n×n) matrices and ε_{t} 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 B_{j}. matrices. Rather, the B_{j}'s are uniquely determined under the orthogonality conditions E(ε_{t} ) = 0 and E(Y_{t-j}ε_{t}) = 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 Y_{t}, 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 ε_{t} 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 Y_{1t} is useful in forecasting another variable, say Y_{2t}. Y_{1t} is said to be useful in forecasting Y_{2t} if the inclusion of lags of Y_{1t}. in the equation for Y_{2t} significantly reduces the forecast variance. Thus it tests whether lags of Y_{1t}. contain any additional information on Y_{2t} which is not already contained in the lags of Y_{2t }itself.

The model presented in equation (1) is difficult to describe in terms of the B_{j} 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 Y_{t−j}

where M_{j} is a (n×n) matrix of moving average coefficients. The response of the i^{th} variable to a unit innovation in the k^{th} variable j periods earlier is given by the ik^{th} element of M_{j}. 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 Y_{t} (such that contemporaneous causality is one way, i.e., recursive) one can obtain orthogonalised innovations u_{t} where u_{t} = Gε_{t}, so that E(u_{t} u_{t}) = 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 Y_{t} 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

ε_{1t} is the innovation in the leading index, ε_{2t} the innovation in the business cycle variable and u_{2t} the orthogonalised innovation in the business cycle variable (in the sense that it is orthogonal to u_{1t} = ε_{1t}).

In terms of orthogonalised innovations, u_{t}, the moving average representation is

where the ik^{th} element of A_{j} 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 Y_{t} linearly from its own past

(in terms of orthogonalised innovations) where E_{t}(Y_{t+k}) is the linear least squares forecast of Y_{t+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.

## Footnote

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]