2  A structural VAR model of the business cycle

2.1  Basic features

The SVAR model we apply is the open economy structural VAR model developed by Buckle, Kim, Kirkham, McLellan and Sharma (2002). Compared to previous New Zealand VAR models, it includes an expanded set of international, financial and domestic variables with the object of capturing and identifying shocks emanating from key geographical and industrial characteristics of the New Zealand economy.

The model contains 13 variables and each variable is explained by a structural equation that has an error term associated with it. The error term for each equation is interpreted as representing a particular innovation or shock. These shocks are labelled according to the structural equation from which they derive. For example, the error term derived from the equation for domestic interest rates, which is specified to capture the central bank’s reaction function in order to identify monetary policy, is given the name ‘domestic interest rate shock’.

The model focuses on the identification of shocks that lead to temporary deviations of economic activity from its long-run growth path. In other words, in line with the structural VAR models of Sims (1980), Bernanke and Blinder (1992) and Dungey and Pagan (2000), departures from trend are viewed as transient. To make this approach operational, all variables in the SVAR model are detrended using the Hodrick-Prescott filter (Hodrick and Prescott, 1997) with the exception of the climate variable, which was detrended by removing the long-run average for each quarter. This is consistent with understanding the dynamic impact of temporary shocks around a long-run growth path. Shocks may be persistent, but they are transient.

2.2  Structure and identification issues

Structural VAR models are derived from 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)      B(L) y_t = u_t

where B(L) is a P^th order matrix polynomial in the lag operator L, such that B(L) = B₀ - B₁L - B₂L² - ...- B_pL^p.  B_o 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 y_t . y_t is a n vector of variables and u_t is a n vector of mean-zero serially uncorrelated structural disturbances. The variance of u_t is denoted by Λ, a diagonal matrix where elements are the variances of structural disturbances; therefore the structural disturbances are assumed to be mutually uncorrelated. Structural disturbances cannot be observed and must be inferred from the reduced form shocks.

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

(2)     A(L) y_t = ε_t

where A(L) is a matrix polynomial in the lag operator L; ε_t is a n vector of serially uncorrelated reduced form disturbances; and var( ε_t) = Σ. The relationship 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 parameters in B₀and Λ as there are independent parameters in the covariance matrix (Σ) 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 B₀ and Λ can be obtained only through sample estimates of Σ. The right-hand side of equation (6) has n(n+1) free parameters to be estimated. Since Σ contains n[(n+1)/2], we need at least n[(n+1)/2] restrictions. By normalising each diagonal element of B₀ to 1, we need at least n[(n+1)/2] restrictions on B₀ to achieve identification. This is a minimum requirement. In the structural VAR approach, B₀ 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 is to use the restrictions implied by a fully specified macroeconomic model. An alternative procedure is to choose the set of variables and identification restrictions that are broadly consistent with the preferred theory and prior empirical research. This approach has been described by Leeper, Sims and Zha (1996) as an informal approach to applying more formal prior beliefs to econometric modelling and it is the approach applied in this paper.

The particular method we use to impose identifying restrictions is similar to that suggested by Cushman and Zha (1997) and Dungey and Pagan (2000) in their structural VAR models of Canada and Australia respectively.[3] This is a more flexible method than the Choleski decomposition procedure originally suggested by Sims (1980) while still giving restrictions on only contemporaneous structural parameters. It permits non-recursive structures and the specification of restrictions based on prior theoretical and empirical information about private sector behaviour and policy reaction functions.

The small open economy extension developed by Cushman and Zha (1997) and Dungey and Pagan (2000) is to impose two blocks of structural equations. One block represents the international economy. The other block of structural equations represents the domestic economy. Variables appearing 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 that 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.[4] Another advantage of this block exogeneity approach is that it enables a larger set of international variables to be included in the model, while reducing the number of parameters needed to estimate the domestic block.[5] The lessons from these recent developments in open economy VAR modelling is that 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.