3 Estimation

3.1  Method

While some models are parameterised using estimated coefficients, others are calibrated to fit certain aspects of the data. With a model this small, incomplete specification is unavoidable - there are features of recent economic history that cannot be explained within the very narrow modelling framework considered here. But this does not mean it cannot be used for quantitative assessment. It simply implies that accepting the estimation results without some sensitivity to information that is available outside the small model would likely lead to bias.

Likewise, simply choosing the model parameters by applying judgement or with reference to theory would ignore the useful information contained within the data. Bayesian estimation serves as a bridge between calibration and estimation - the selection of priors allows for the incorporation of additional information available to the modeller, while the process of maximum likelihood estimation extracts some value from the data. In practice, the priors serve to guide the maximum likelihood estimate by placing more weight on certain areas of the parameter space. And the chosen prior variance acts to determine the weighting between the prior and the unconstrained maximum likelihood estimate contained within the posterior estimate.

My choice of prior distributions for the model parameters is informed by other studies relating to the New Zealand economy including, Lubik & Schorfheide (2007), Szeto (2013), Parkyn & Vehbi (2013) and Karagedikli et al (2013). I also refer to the simulation properties of the New Zealand Treasury Model and Reserve Bank of New Zealand forecasting models - see Ryan & Szeto (2009), Szeto et al (2003) and Benes et al (2009).

With regard to the Bayesian estimation process, I use data from the final quarter of 1993 to the third quarter of 2012 - avoiding New Zealand's disinflationary period but making use of data over the recent recession. The likelihood function is estimated using the Kalman filter and the Metropolis-Hastings algorithm is used to generate draws from the posterior distribution. 100,000 draws are run with the first 25,000 discarded as burn in. Table 1 presents both the choice of priors and the posterior estimates of the model parameters.[21]

Table 1 - Priors and posterior estimation results

Param.	Eq.	Description	Prior mean	Posterior mean	Posterior LCI1a	Posterior UCI1b	Prior S.E. 1c	Dist. 1d
αy	IS	Output gap persistence	0.80	0.87	0.83	0.91	0.03	Norm
αr	IS	Interest rate elasticity of demand	0.25	0.09	0.06	0.13	0.04	Norm
αq	IS	Exchange rate elasticity of demand	0.025	0.022	0.018	0.026	0.0025	Norm
αNR	IS	Degree of non-Ricardian behaviour	0.65	0.44	0.30	0.57	0.08	Norm
	IS	Demand shock	0.50	0.67	0.58	0.75	0.10	Inv G.
ςƒ	IS	Fiscal policy persistence	0.60	0.83	0.77	0.89	0.20	Norm
	IS	Fiscal policy shock	0.14	0.13	0.12	0.15	0.05	Inv G.
βπ	PC	Inflation persistence	0.20	0.22	0.14	0.29	0.05	Norm
βy	PC	Inflation sensitivity to output gap	0.25	0.26	0.15	0.29	0.10	Norm
βq	PC	Exchange rate sensitivity of inflation	0.025	0.025	0.021	0.029	0.0025	Norm
	PC	Inflation shock	1.40	1.40	1.23	1.55	0.10	Inv G.
δi	TR	Interest rate smoothing parameter	0.75	0.79	0.76	0.82	0.02	Norm
δy	TR	Interest rate sensitivity to output gap	0.50	0.57	0.28	0.85	0.20	Norm
δπ	TR	Interest rate sensitivity to inflation	1.50	1.48	0.84	2.09	0.40	Norm
	TR	Interest rate shock	0.60	0.70	0.63	0.77	0.05	Inv G.
δcs=τcs	TR/IE	Spread equality with base rate	1.00	0.98	0.83	1.14	0.10	Norm
ψq	RUIP	PPP convergence parameter	0.020	0.022	0.014	0.030	0.005	Norm
ψε	RUIP	Exchange rate shock persistence	0.85	0.85	0.86	0.81	0.90	Norm
	RUIP	Exchange rate shock	0.70	0.63	0.43	0.80	0.20	Inv G.
θcs	CS	Spread persistence	0.80	0.79	0.71	0.87	0.10	Norm
	CS	Credit spread shock	0.45	0.43	0.38	0.47	0.05	Inv G.
ςif	IF	Foreign interest rate persistence	0.90	0.90	0.85	0.94	0.10	Norm
	IF	Foreign interest rate shock	0.40	0.36	0.32	0.40	0.05	Inv G.
ςπ	PF	Foreign inflation persistence	0.15	0.12	0.04	0.19	0.05	Norm
	PF	Foreign price shock	2.00	2.29	1.99	2.58	0.50	Inv G.

^1a LCI refers to the lower 95 per cent confidence interval value

^1b UCI refers to the upper 95 per cent confidence interval value

^1c S.E refers to the prior standard error of the distribution

^1d Dist. refers to parameter distributions - prior distributions are assumed normal for model parameters and inverse gamma for shocks

3.2  Model evaluation

To evaluate the fit of the model I first estimate a two-lag SVAR model, which allows the data to speak with the minimum number of identifying restrictions applied. The data series included are the inflation deviation from target, the output gap, the real interest rate gap and the exchange rate gap, which is also the Cholesky ordering of the variables. Fiscal policy enters the SVAR exogenously since it does not theoretically depend on any other variable in the model.

This time, I use data from the final quarter of 1993 to the first quarter of 2008 to exclude the influences of the earlier disinflation associated with shifting to a new monetary policy framework and the global financial crisis.This model represents a baseline comparator from a time when the data were well-behaved and the SVAR results appear sensible Akaike information criteria results support the inclusion of two lags and unit root analysis suggests the model has a stable equilibrium.

I then compare the impulse responses from the SVAR with those of the estimated reduced-form model. I find that, broadly speaking, the impulse responses are consistent with one another. In particular, the humped responses of inflation to the output gap and the output gap to interest rates receive good empirical support. The dynamics of the exchange rate and the associated influences on output are less well-supported. Overall, the SVAR impulse responses generally support the dynamics of the reduced-form model - Figures A.1-A.16 in the Annex illustrate this in full.

There is good reason to suspect that the dynamics of the model associated with the exchange rate are muddied by commodity prices. International evidence such as that presented in IMF (2012) suggests that domestic output is positively correlated with commodity prices for exporters of primary goods, such as New Zealand. Because the exchange rate is also correlated with commodity prices the exchange rate is positively correlated with output - which is not consistent with theory.

The introduction of commodity prices - which are so persistent as to be indistinguishable, in practice, from a random walk - would affect the stability of the VAR and reduced-form modelling results. Instead, my priors over the exchange rate - drawn from evidence from other, larger models of the New Zealand economy, that include the effect of commodity prices - are imposed more strictly than priors relating to other parameters of the reduced-form model.

To provide an alternative model against which to compare the impulse responses, I estimate a sign-restricted VAR (SRVAR) in a similar way to Jaaskala & Jennings (2010). The associated impulse responses are presented alongside the responses of the other models in the Annex (Figures 14-17). The restrictions are presented in Table 4, within the Annex. The restrictions serve to eliminate both the exchange rate and price puzzles associated with the VAR estimates and leaves the magnitude of many of the impulse responses broadly consistent with those from the reduced-form model.[22] However, the approach also introduces an excess sensitivity of the exchange rate to endogenous factors, probably reflecting misspecification and the omission of relevant variables in the estimation process.

Finally, I am also interested to see whether the cross-equation restrictions of the reduced-form model have a significant bearing on its dynamics. To investigate whether this is the case I run a stochastic simulation of the model and record the data that is generated. I then estimate another SVAR using that simulation data and compare the impulse responses with those of the estimated model. I find that the impulse responses are broadly consistent with one another, suggesting the findings presented later are not overly dependent on the cross equation restrictions of the reduced-form model. Again, Figures A.1 - A.16 in the Annex present these impulse responses.