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Estimating a New Zealand NAIRU - WP 04/10

4  Model specification and results

This section outlines the models and data used to estimate the NAIRU and examines the results of these models. The basic models used are similar to the reduced-form models used in Richardson et al (2000) with the key difference being the estimation of the signal-to-noise ratio using Stock and Watson’s (Stock and Watson 1998) procedure. Once the signal-to-noise ratio is estimated it is then used in the reduced-form models to estimate the NAIRU. To evaluate the NAIRU estimates of these models against each other and the HP filter alternative, we then compare the fit of ordinary least squares (OLS) equations attempting to explain unexpected inflation using the various NAIRU estimates.

4.1  Data and model specification

The model used in this paper is:



where Δ is the first difference operator,πt is the inflation rate,πet is expected inflation, ( is the unemployment gap and represents short-term supply shocks. The basic model is similar to that used by the Bank of England and OECD, with the key exception being the inclusion of survey based measure of inflation expectations.

Equation (3) represents a Phillips curve relationship and models unexpected inflation as a function of one lag, the deviation of the unemployment rate () from the NAIRU (), and proxies for short-term supply shocks. Equation (4) specifies the time-series process generating the unobservable NAIRU, which is assumed to follow a random walk process.

Three measures of inflation are considered in this paper and they are the Consumer Price Index excluding interest costs (CPIX), CPI excluding government charges (CPIG), and the consumption deflator (COND).[5] Inflation is measured by annualised quarterly growth rates of the price indexes. The data used in the paper covers the period 1988:3 to 2003:4 for both CPIX and COND. For CPIG, the data used covers 1989:1 to 2003:4. The inflation expectation variable used is “expected annual CPI one year from now”, obtained from the Reserve Bank New Zealand survey of expectations.

As changes in Goods and Services Tax (GST) and government housing policy occurred during the sample period, we first removed the impact of the two policy changes on the dependent variable by regressing on two dummy variables. The first dummy variable is a GST dummy variable which is used to capture the increase in GST in 1989 Q3. The second one is to capture the change in government housing policy in 2001 Q1.[6]

Short-term supply shocks are defined as those that would be expected to revert to zero over one to two years, as we are attempting to estimate the medium term NAIRU concept. As it is impossible to control for all short-term shocks in reality our measure is a mix of the short and medium-term NAIRU definitions. The main proxies for short-term supply shocks used in this study are GDP imports deflator (NMPD) and petrol prices (PTGAS). Both variables enter Equation (3) as contemporaneous annualised quarterly growth rates. Another supply shock variable is the rate of change in local authority rates (LAR).

To avoid simultaneity issues, the unemployment gap should enter as lagged values in Equation (3). In this study, two model specifications are considered. In the first specification (Model 1), we allow only a contemporaneous unemployment gaps. In the alternative specification (Model 2), the first lag of the unemployment gap is included in the model. The decision on how many lags of is included is based on the significance of the last lag included and the significance of the autocorrelated errors.

Selection of the signal-to-noise ratio

One of the key issues in the estimation of an unobserved-components model is to choose the ratio of the variances of the error terms in the two equations, the signal-to-noise ratio. The ratio measures the volatility or variance of the NAIRU relative to the variance of changes in inflation and determines how the NAIRU can move around over time. A very high signal-to-noise ratio implies a NAIRU that moves a lot and helps to explain almost all of the variance in inflation. A very low signal-to-noise ratio implies a NAIRU that is very constant over time, explaining less of the variation in inflation. Therefore, the estimation results are very sensitive to the choice of the signal-to-noise ratio. Following Stock and Watson’s (1998) procedure, we obtain the median-unbiased estimates of the signal-to-noise ratio.

The first step of the procedure is to rewrite Equation (3) as follows:


where c is a constant.

In the second step, we compute the exponential Wald statistic of Andrews and Ploberger (1994) for an intercept shift at unknown date in Equation (5). We then use Stock and Watson’s result to obtain the medium-unbiased estimate of λ. The estimate of the standard deviation of is (for further detail see Stock and Watson (1998)):

Table 1 – The results of the Stock and Watson’s median-unbiased estimate of l
 Model 1Model 2
Exponential Wald Statistic1.680.743.452.111.174.47
Median-unbiased Estimate of λ6.83.510.

Table 1 above reports the values of the exponential Wald statistic testing for an intercept shift in Equation (5). Using the median-unbiased estimate of λ, we impose the signal-to-noise ratio when estimating the remaining model parameters. The results from the final-stage of estimation are presented in Table 2.


  • [5]Note that you could also use a measure of wage inflation. We have not done this because we did not feel that neither the Statistics New Zealand Quarterly Employment Survey or Labour Cost Index are appropriate and because there is not a long enough time series of the unadjusted Labour Cost Index at this time.
  • [6]There was also a major change in government housing policy in 1994. However, the coefficient on this dummy variable was not statistically significant.
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