RDP 8608: Exchange Rate Regimes and the Volatility, of Financial Prices: The Australian Case 2. The VAR Methodology
July 1986
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In general we will be concerned with an (nxl) vector of n endogenous variables Y_{t} containing domestic and foreign financial price variables. We assume that Y_{t} is generated by the m^{th} order vector-autoregression,
where D_{t} is a (nxl) vector representing the deterministic component of Y_{t} (generally a polynomial in time), B_{t} are (nxn) matrices and ε_{t} is a (nxl) 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. Given the choice of variables 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.^{[7]}
Tests which are commonly applied to VARs are tests for Granger-causality which test whether a variable, say Y_{1t} is useful in forecasting another variable, say Y_{2t}. The variable 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−1},
where M_{j} is a (nxn) 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). If one can assume some 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}) = ϕ where ϕ is a diagonal (nxn) matrix. For example, if we have a VAR with a foreign variable and a domestic variable and assume that the domestic variable does not contemporaneously cause the foreign variable, then the foreign variable will be ordered above the domestic variable in Y_{t} and G will be of the form,
where ρ is the estimated coefficient in the regression equation,
ε_{1t} is the innovation in the foreign variable, ε_{2t} the innovation in the domestic variable and u_{2t} the orthogonalised innovation in the domestic 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.
For the purposes of this paper, however, the second device of innovation accounting will be used. This 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.
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.
Footnote
On the basis of tests for within, and across, equation serial correlation and tests for the significance of B_{m}, from the zero matrix. 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. [7]