This section gives a step in this direction by specifying and estimating a Bayesian version
of our basic model (2.10), which avoids ad hoc permanent corrections of the response to
debt accumulation. A Bayesian updating scheme is a sensible approach in our small
sample setting, where low degrees of freedom and difficulties to apply asymptotic results
or to establish long run statistical properties of time series make questionable the
application of classical econometric methods.
Model (2.10) has a vector autoregressive form with government debt and output gap as
exogenous variables. This allows a straight application of the Bayesian Vector
Autoregressive (BVAR) methodology.11 BVAR models are well designed to handle the
trade-off between over and under parameterization in contexts of limited size samples,
providing an objective scheme for the updating of prior estimates.
The specification of prior beliefs is in fact the critical point once we move to a Bayesian
framework. With long samples, the prior effect on posterior estimates is negligible
because sample information ends up shaping the final results. However, with small
samples, the prior might be the determinant factor (arguably leading to ad hoc final
results when prior information is ad hoc). This would be the case if the specified prior is
so tight (very small variance) that final results are unaffected by sample observations. On
the other extreme, a too loose (large variance) prior could lead to very volatile posterior
estimates, largely affected by new sample observations, as it is the typical case in a
context of scarce degrees of freedom. Ideally we want a not too loose/not too tight prior.
A prior that deals with the degrees of freedom problem, but is at the same time sensitive
to new sample observations that call for its modification. This is one of the underlying
principles in BVAR models. A second principle is that the prior odds are tilted towards
own lags, which have a higher prior variance.
With these principles in mind our prior specification treats the parameter vector in (2.10),
β = (ρ, δ, γ, c), as a multivariate normal with independent components and the following
mean vector and covariance matrix:
cov( β) = diag [0.5σ ε 0.01(0.5<σ ε ) 0.01(0.5σ ε ) 40(0.5<σ ε )] (2.11)
10 )
where ρ and δ are the least square estimates in Table 2.6 and σ^ε is the least square
estimate of the error term variance in an autoregressive regression of the primary surplus
with one lag, a scaling factor helpful to control for the units of measurement of variables.
In line with BVAR models, the covariance in (2.11) gives more weight to the own lag
coefficient (50% of the error term variance) than to the coefficients of other variables (1%
of the own lag coefficient variance), and specifies a flat prior for the constant term. As for
the prior mean vector, a common practice in the BVAR literature is to specify a prior
mean equal to one for the own first lag and equal to zero for the rest of the coefficients in
the equation. The vector in (2.11) incorporates a slight deviation from this practice by
taking as useful prior information the estimates in Table 2.6 for the inertia component
and, our focus of attention, the reaction to debt accumulation.
For a description of the methodology see e.g. Doan et al. (1986) and Ballabriga (1997).
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