2.2  Structure and identification issues

VAR models are equivalent to a system of reduced form equations relating each endogenous variable to lagged endogenous (predetermined) and exogenous variables. Since the model contains 13 variables, there are several challenges in recovering robust estimates of the parameters in the structural form equations from the estimated parameters in the reduced form equations.

The approach used is to assume the economy is described by a structural form equation, ignoring constant terms, given by

(1)    

where is an order matrix polynomial in the lag operator , such that . is a non-singular matrix normalised to have ones on the diagonal and summarises the contemporaneous relationships between the variables in the model contained in the vector . is a vector of variables and is a vector of mean zero serially uncorrelated structural disturbances. The variance of is denoted by . is a diagonal matrix where diagonal elements are the variances of structural disturbances; therefore the structural disturbances are assumed to be mutually uncorrelated.

Associated with this structural model is the reduced form VAR which is estimated

(2)    

where is a matrix polynomial in the lag operator ; is a vector of serially uncorrelated reduced form disturbances; and . The relationships between the components of equations (1) and (2) are as follows

(3)    

and

(4)    

Recovering the structural parameters of the VAR model specified by equation (1) from the estimated reduced form coefficients requires that the model is either exactly identified or over-identified. Exact identification requires the same number of free parameters in and as there are independent parameters in the covariance matrix () from the reduced form model.

Using equations (3) and (4), the parameters in the structural form equation and those in the reduced form equation are related by

(5)    

and

(6)    

Maximum likelihood estimates of and can be obtained only through sample estimates of . The right-hand side of equation (6) has free parameters to be estimated. Since contains , we need at least restrictions. By normalising diagonal elements of to ones, we need at least restrictions on to achieve identification. This is a minimum requirement. In the structural VAR approach, can be any structure as long as it has sufficient restrictions.

There are several ways of specifying the restrictions to achieve identification of the structural parameters. One procedure for determining appropriate restrictions to identify a structural VAR is to use the restrictions that are implied from a fully specified macroeconomic model. The structural VAR models estimated by Blanchard and Watson (1986), who used theory to incorporate short run restrictions, Shapiro and Watson (1988) and Blanchard and Quah (1989), who used theory to justify the inclusion of long-run restrictions, and Gali (1992), who used theory to justify both short-run and long-run restrictions, are examples of this approach. Garratt, Lee, Pesaran and Shin (1998), Huh (1999) and Buckle, Kim and Tam (2001) also followed this approach.

The alternative and more common approach is to choose the set of variables and identification restrictions that are broadly consistent with the preferred theory and prior empirical research. The metric used to evaluate the appropriateness of the variables and restrictions is whether the behaviour of the dynamic responses of the model is consistent with the preferred theoretical view of the expected response. Recent attempts to identify monetary policy effects in small open economies by Kim and Roubini (2000) and Brischetto and Voss (1999) are some of the many examples of this second approach.

This alternative approach has been described by Leeper, Sims and Zha (1996) as an informal approach to applying more formal prior beliefs to econometric modelling. They argue that the approach is in principle no different from other specification methods used in modelling, so long as the modeller does not fail to disclose the methods used to select the model. As Brischetto and Voss point out however, there are still several concerns about the identification restrictions that have been applied to structural VAR models in this manner. These include the robustness of the conclusions to alternative reasonable identification restrictions (see Faust, 1998 and Joiner, 2002), and the difficulty of clearly interpreting what aspects of the model arise from restrictions imposed on the model and what arise from the data. However, these concerns can arise in most multi-equation models and are not restricted to structural VAR models.

A popular and straightforward method is to orthogonalize reduced form errors by Choleski decomposition as originally applied by Sims (1980). However, this approach to identification requires the assumption that the system of equations follows a recursive structure, that is, a Wold-causal chain.

In some circumstances, Choleski decomposition may coincide with the prior theoretical view of the appropriate model structure. This procedure can therefore be viewed as a special case of a more general approach. However, there are many circumstances where restrictions resulting from Choleski decomposition will be unreasonable. It would not be appropriate for example if there were contemporaneous interaction between variables. In those circumstances, if monetary policy for example was implemented according to an explicit policy rule, such as a Taylor Rule, the Choleski decomposition would not enable private sector responses, such as the responses of GNE and GDP, to shocks to foreign variables and to monetary policy in a small open economy to be differentiated.

A more general method for imposing restrictions was suggested by Blanchard and Watson (1986), Bernanke (1986) and Sims (1986), while still giving restrictions on only contemporaneous structural parameters. This method permits non-recursive structures and the specification of restrictions based on prior theoretical and empirical information about private sector behaviour and policy reaction functions.

This more general approach has subsequently been extended to the small open economy by Cushman and Zha (1997) and Dungey and Pagan (2000) in their structural VAR models of Canada and Australia respectively. Their approach is to impose two blocks of structural equations. One block represents the international economy. The other block of structural equations represents the domestic economy. Dependent variables in the domestic economy block are completely absent from equations in the international block. This follows naturally from the small open economy assumption.

There are several potential advantages to be gained from this block exogeneity approach for the small open economy. One advantage claimed by Cushman and Zha (1997), Kim and Roubini (2000) and Brischetto and Voss (1999) is it helps identify the monetary policy reaction function for a small open economy. This approach enables monetary policy to react contemporaneously to a variety of domestic and international variables whose data are likely to be immediately available to the monetary authority. Kim and Roubini (2000), for example, included contemporaneous exchange rate and world price of oil (as a proxy for expected future inflation) variables in the domestic monetary policy reaction function. Cushman and Zha (1997) and Brischetto and Voss (1999) also included the contemporaneous US Federal Funds rate in their specification of the domestic monetary policy reaction function.

The block exogeneity approach also has the advantage of enabling a larger set of international variables to be included in the model, while reducing the number of parameters needed to estimate the domestic block. Cushman and Zha (1997) included four international variables in their model of the Canadian economy: United States (US) industrial production, US consumer prices, US Federal Funds rate and world total commodity export prices. Dungey and Pagan (2000) included real US GDP, real US interest rates, the Australian terms of trade, and the Dow Jones Index deflated by the US consumer prices index. They also treated Australia’s real exports as exogenous to the domestic economy, making a total of 5 variables in their international block.

The lessons from the recent developments in open economy VAR modelling is the inclusion of more foreign variables is likely to be important for correct specification, for better identification of contemporaneous interactions, and for a richer set of lagged responses.