RDP 9009: An Empirical Model of Australian Interest Rates, Exchange Rates and Monetary Policy 3. The VAR Method

A vector autoregression (VAR) is a dynamic system of reduced form equations. VARs were popularized by Sims (1980), as a reaction to what he saw as the “incredible” restrictions necessary to identify structural models. The intended purpose of VARs was to provide a framework to study the historical dynamics of an economy, without imposing any prior structure on the problem at hand. The two principal tools for analysing these dynamics are impulse responses and variance decompositions. These are based on the moving average representation of the VAR.

Consider the following vector autoregressive representation^[5]

where y_t is a stationary stochastic process and L is the lag operator. Under suitable regularity conditions (1) can be written as a vector moving average representation

where the coefficients of the matrix a(L) are functions of the estimated autoregressive parameters b(L). a(L) at lag 0 is the identity matrix. u_t is the forecast error (innovation) of the autoregression given information at t – 1.

The impulse response is the dynamic effect on the system of a particular shock. For example, given a shock to the federal funds rate we can then trace, over time, the effects on the other variables in the system. The variance decomposition of the k-step ahead forecast is the proportion of the total forecast variance of one component of y_t+k associated with shocks to the moving average representation of another variable.

While a VAR is a set of reduced form equations, the early view that it is a mere atheoretical representation of the data is now known to be incorrect. Since, in general, the innovations u_t are correlated with each other, the effects on the system of a particular shock are difficult to interpret. The innovations therefore need to be orthogonalized; however, this orthogonolization imposes structure on the model. The model is then no longer atheoretical, and so the impulse responses stemming from the model are conditional on the structure that has been imposed.^[6]

Typically, orthogonalization is achieved via the Choleski decomposition, which (implicitly) places a recursive structure on the model. An n variable model has n! possible recursive structures. Valid use of the Choleski procedure therefore obligates us to make an explicit judgement about which of these structures is most appropriate. We choose the following – the variables are placed in the order: federal funds rate, U.S. 10 year bond rate, exchange rate, cash rate, Australian 10 year bond rate.

In other words, we take it to be (approximately) the case that the federal funds rate and the U.S. 10 year bond rate affect Australian interest rates and the $A/$US exchange rate contemporaneously, but not vice versa. This seems to us to be a reasonable assumption. The ordering of the variables within each country bloc is more problematic, and there is no fully satisfactory way to resolve this difficulty.^[7] In any case, with high frequency data, the issue of which variables are contemporaneously exogenous to each other is probably of only minor importance.

We use daily data with the exchange rate measured as an average of buy and sell rates at 4 p.m., Eastern Australian time. Domestic interest rates are recorded at 11 a.m., the federal funds rate and U.S. bond rate are recorded at the close of trading in the previous day in the United States. Deleted from the sample were days when there was no trading in Australia. The data come from the Reserve Bank of Australia's database, and are available from the authors on request.

Footnotes

This exposition is based on Runkle (1987). [5]

Cooley and Leroy (1985) provide an extensive discussion of this and related issues. [6]

This dilemma has led to the development of structural VARs e.g. Blanchard (1989) which resolve the issue by specifying explicit structural models in innovations of variables, and avoiding the Choleski method altogether. However, Keating (1990) argues that the exclusion restrictions used to identify structural VARs can yield inconsistent parameter estimates. [7]