2 The rationale for including the IGBC

This section discusses the pitfalls of excluding the level of the debt variable in a standard VAR framework. Following the exposition in FG, we show why a VAR that excludes the level of the debt is likely to be misspecified and might eventually imply explosive paths for the debt-to-GDP ratio.

Consider that the reduced-form fiscal VAR model with k lags is described by the system:

             (1)

where Y_t = is a five-dimensional vector that includes government spending, taxes, output, inflation and interest rates respectively, C_i is a coefficient matrix of size 5 x 5 and u_t is the vector of the reduced form residuals representing unexpected movements in the components of Y_t.

Excluding the level of the debt-to-GDP ratio, d_t, in (1) would imply that this variable is instead contained in u_t along with other exogenous shocks. However, this is problematic since the level of debt and the variables in Y_t such as government spending, taxes and interest rates are inherently tied via the government's budget constraint. For example, in cases when the rate of growth of the economy is not equal to the average cost of financing the debt, a feedback from the level of debt to fiscal variables is inevitable. Furthermore, interest rates may be affected by changes in the debt dynamics via changes to the risk premium.

The resulting correlation between the error terms and the dependent variables constitutes a violation of a basic assumption of OLS estimation; namely that the regressors and error terms should be uncorrelated. This, in turn, will result in biased estimates of the C_i coefficients.

Including the level of the debt ratio in (1) alone, on the other hand, is not sufficient and the evolution of the debt dynamics (d_t) in relation to the variables in Y_t should also be included as an identity:

           (2)

where i_t is the nominal interest rate, Δp_t is inflation, Δy_t is real GDP growth and G_t and T_t are the logs of government expenditure (excluding interest payments) and government revenues (net of transfers) respectively. Equation 2 shows that the evolution of debt-to- GDP ratio depends on two sets of factors. The first one represents the previous debt level multiplied by the ratio of the real interest rate and the inverse of the growth rate . The second part is the primary deficit as a ratio of GDP. The exponentials are used as these variables are expressed in logarithms. The implication of the debt identity is that when real interest rates are higher than the growth rate, a primary surplus is needed to keep the debt to GDP ratio constant (see Blanchard et al., 1990 for details). The structural form of the system to be estimated is therefore:

           (3)

where matrix A describes the contemporaneous relationships among the variables, the non-zero off-diagonal elements of Β allow some shocks to affect more than one endogenous variable in the system and ρ denotes the number of lags used in the SVAR. The reduced form representation can then be obtained by multiplying both sides of (3) by A^-1:

             (4)

Where

The presence of d_t-i will amplify the dynamic effect of fiscal shocks and the impulse response calculated from (1) and (2) as the system will diverge from those calculated when such feedback is omitted. The degree of this divergence, on the other hand, will be dependent on the strength of the feedback from debt to macroeconomic variables. FG finds that this feedback plays an important role in the case of US. We find that this feedback is relatively less important for New Zealand given its relatively low debt-to-GDP ratio. Another implication of excluding the debt ratio in (1) is that simulated values for fiscal variables such as government spending and tax revenues from such a system might imply incredible paths for the debt-to-GDP ratio. As an example, we conduct an empirical exercise using New Zealand data, similar to the one reported in FG for the US case. Initially, we estimate the five-variable VAR defined in (1) for the period 1986:1-2010:4. Then, we simulate data for each variable for 80 quarters and calculate the implied debt-to-GDP ratio using (2). The results are presented in Figure 1. It can be seen that the VAR without the debt feedback produces an explosive path for the debt-to-GDP ratio. In such cases, it is likely that the impulse responses calculated from the system will not be reliable (ie, calculated along implausible paths for the debt ratio). On the other hand, imposing the feedback and linking the variables that constitute the IGBC by the identity (2) creates a relatively more stable debt-to-GDP profile. It is important to note that the explosive behaviour is heavily dependent on the corresponding values of the fiscal variables at the starting point of the simulation.

Figure 1: Actual and simulated debt-to-GDP ratios (with and without feedbacks)

Source: Authors' calculations