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3 Data and Parameterisation

The model is estimated using quarterly data from 1994q1 to 2012q3. Benes et al (2010) used the year-on-year rate of the core measure of the Consumer Price Index (CPI) in their paper. In this paper, the inflation used in this paper is the annualised quarterly percentage change of the 10% trimmed mean of the Consumer Price Index (CPI).

The reason for using quarterly percentage changes in CPI is to avoid the problem of overlapping data[2] with the year-on-year rate of inflation. As the quarterly percentage change in the core measure of CPI is very volatile for New Zealand, we use the 10% trimmed mean in our baseline model.

As the rate of Goods and Services Tax increased from 12.5% to 15% in the December quarter 2010, we removed the impact of the policy change on the inflation by lowering quarterly inflation by 1.8 percentage points for that quarter. The inflation expectations variable is obtained from expected annual CPI two years from now in the Reserve Bank New Zealand (RBNZ) survey of expectations. The data sources of the other variables can be found in the following table:

Table 3 - Data and sources
Variable	Source
Y_t	Real Production GDP (sa)
π_t	10% trimmed mean CPI inflation (sa)
	Expected annual CPI 2 years from now from the RBNZ Survey of Expectations
U_t	Unemployment rate from the Household Labour Force Survey (sa)
C_t	NZIER's Quarterly Survey of Business Opinion (QSBO) - economy-wide capacity utilisation (sa)

The model is estimated using regularised maximum likelihood (Ljung, 1999), which requires users to specify priors for each parameter. The advantage of using regularised maximum likelihood is that if a model contains parameters that lack identification, the penalty function in the regularised maximum likelihood helps to shape up the likelihood so that its estimate is pushed towards the initial value. It is equivalent to a Bayesian methodology[3].

As part of the estimation process, you need to specify the lower bound, the upper bound and the penalty expression of each parameter. It is important to stress that like any Bayesian estimation, the prior density for each parameter plays a significant role in determining the estimates and confidence intervals for potential output and other latent variables. For example, if one believes that potential output growth is less volatile, one way to achieve this is to reduce the size of the priors on the standard deviations of the shocks of the potential output process, and .

The following table presents prior distributions and estimated posterior distributions for the model.

Table 4 - Maximum regularised likelihood
Parameter	Mode	Dispersion	Mode	Dispersion
	Prior		Posterior
α	0.900	0.158	0.898	0.022
β	0.400	0.316	0.321	0.045
Ω	0.500	0.316	0.458	0.053
Ρ1	1.200	0.158	1.207	0.022
Ρ2	-0.400	0.158	-0.390	0.022
Ρ3	5.000	3.162	4.982	0.442
κ₁	0.100	0.632	0.061	0.076
κ₂	1.500	1.581	0.676	0.124
Ø₁	0.800	0.158	0.767	0.022
Ø₂	0.300	0.158	0.228	0.023
τ	0.100	0.158	0.042	0.018
δ	0.500	0.158	0.501	0.023
ω	3.000	1.581	2.928	0.247
λ	2.000	3.162	2.013	0.440
γ	0.600	1.897	0.400	1.014
η	0.600	1.897	0.936	0.037
σ_ε^γ	0.700	0.316	0.995	0.047
	1.000	3.162	0.280	0.030
	0.080	0.158	0.155	0.021
	0.050	0.158	0.105	0.021
	0.400	0.316	0.710	0.036
	0.250	0.158	0.426	0.022
	0.075	0.032	0.131	0.004
	0.500	0.316	0.989	0.044
	0.300	0.316	0.161	0.011
	0.040	0.032	0.084	0.004
	0.067	0.032	0.122	0.005

Notes

[2]The problem of using overlapping data could introduce a moving average term in the error term (see Hansen and Hodrik, 1980).
[3]See Appendix 2 for technical details.

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3 Data and Parameterisation

Notes