3  Estimation issues

Estimation of the NAIRU is imprecise as the NAIRU cannot be directly observed and it appears to change over time. This means that estimates of the NAIRU tend to have wide error bands. For example, Staiger, Stock and Watson (1996) found a typical 95% confidence interval to be a gap of 2.6%. They point out three explanations for the large degree of uncertainty in estimates of the NAIRU. First, there is uncertainty arising from not knowing the correct parameters of the model (as with any econometric estimation). Secondly, the NAIRU is potentially stochastic (random). Thirdly, there is uncertainty over model specification, there are a number of potential models of the NAIRU and we do not know which one is “true”. For example, the NAIRU can be estimated as a constant, a slowly changing function of time, an unobserved random walk, as a function of other labour market variables etc, but all measures retain a large degree of uncertainty (Staiger et al 1996).

An additional area of imprecision is that most models require an estimate of inflationary expectations. One way to get around this is to assume that expectations are adaptive and equal to the previous period’s inflation so . Other options are to use a survey measure of inflation expectations (which is used in this paper), or some function of the mean of inflation adjusted by the previous period’s inflation.

3.1  Theoretical models

There are a several techniques used to estimate the NAIRU. In this section, we discuss commonly used techniques and explain why this study uses the reduced form technique over structural or purely statistical models. Section 3.1.4 goes into more detail on the Kalman filtering technique used in this study. Section 3.2 outlines examples of studies which used the reduced form technique to estimate the NAIRU.

3.1.1  Structural models

Structural models compute the NAIRU as the equilibrium of a model of aggregate wage and price setting behaviours (Fabiani and Mestre 2000, Richardson et al 2000). The equilibrium level of unemployment is derived as the set of values for which inflation is stable. Structural models usually assume full adjustment of firms and workers to all shocks, so the derived equilibrium measure of unemployment corresponds more closely to the long-run equilibrium rate of unemployment described in section 2.1.

The key advantage of structural models is that they provide more information on the determinants of the NAIRU, as they are based on a theoretical framework that explains how macroeconomic shocks and policy instruments affect the long-term equilibrium rate of unemployment. However, structural models are not generally considered to be a preferred estimation method as they are very complex to estimate; are likely to be inaccurate as they rely on assumptions about the underlying behaviour of economic agents for which there is no general consensus; they face a number of econometric and measurement issues; and cannot generally be calculated in a timely manner (for a more detailed discussion see Richardson et al (2000)).

3.1.2  Pure statistical models

Pure statistical models split the actual unemployment rate into trend and a cyclical component and ignore all other information, including the relationship between the unemployment rate and inflation. Pure statistical models differ from the structural models discussed as they are only intended to be predictive, whereas the structural models attempt to help explain the relationship between inflation and unemployment. They rely on the assumption that equilibrating forces are strong enough to bring the unemployment rate back to trend relatively quickly so that on average the actual unemployment rate should fluctuate around the NAIRU.

Pure statistical models have the advantage of being timely and relatively simple to calculate, an example included in the results section of this paper uses the Hodrick Prescott (HP) filter to estimate trend unemployment as a weighted moving average of actual unemployment. The difficulty with these models include a lack of consensus about how the estimated trend is modelled in terms of its variance and relationship with the cyclical component; as filters tend to behave as a moving average they respond very slowly to apparent changes in unemployment; and most importantly, all information other than unemployment is ignored, including the relationship between inflation and unemployment.

3.1.3  The reduced form approach

Reduced form approach has several advantages over a pure statistical model including being directly related to the theoretical definition of the NAIRU as it is based on the expectation-augmented Phillips curve, and also because it allows us to control for a range of factors wider than the inflation/unemployment relationship. In addition, the simplicity of the approach means that it is consistent with a range of models and is therefore likely to be more robust than the structural approach which is based on a single model.

These advantages make the reduced form approach the most popular technique in recent studies. However, the reduced form approach also has a number of disadvantages. It requires some form of estimation of inflation expectations. The approach is atheoretical which has the advantage of not relying on assumptions about behaviour but the disadvantage is that it leaves the interaction between unemployment and inflation indeterminate so may not provide a measure of underlying NAIRU. The filters lack precision for end of sample estimates. The final disadvantage of the reduced form approach is that the results may be sensitive to arbitrary choices in the model. Note that if the unemployment gap and all shock variables prove to be insignificant then the reduced form model becomes a purely statistical model. The key parameter that is usually chosen in an arbitrary manner is the signal-to-noise ratio, which in this study we have instead estimated using Stock and Watson’s (1998) procedure.

3.1.4  Kalman (1960) filter

The previous section discussed how using filtering techniques on reduced form equations are currently the generally preferred NAIRU estimation method as it is based on the expectations augmented Phillips curve but also allows us to control for more factors than just the inflation/unemployment relationship. This paper uses the Kalman filtering technique, which is outlined in this section. The Kalman filter is the most commonly used reduced form filtering technique for estimating the NAIRU due to its simplicity of estimation (Greenslade, Pierse and Saleheen 2003, Richardson et al 2000).

The filtering process uses the rule that certeris paribus stable inflation (when inflation is equal to inflation expectations) implies an unemployment rate that is at the NAIRU but rising (falling) inflation is suggestive of an unemployment rate that is below (above) the NAIRU. However, controlling for temporary shocks such as oil prices may allow the short-term NAIRU to deviate from the medium-term NAIRU concept referred to in Section 2.1.

For example, a general specification of this framework would be:

(1)

(2)

The first equation is a Phillips curve and it models unexpected inflation as a function of: shocks , controlling for the appropriate temporary supply shocks allows us to calculate a NAIRU that is compatible with non-increasing inflation in the absence of temporary supply shocks; the unemployment gap , the NAIRU is time varying and its movement is modelled by Equation (2).

Inflation expectations are not model endogenous, which are picked up by the lagged inflation or a survey measure. The random exogenous events or ‘noise’ which are iid, normally distributed with a mean zero and variances and, and are uncorrelated with each other.

The unemployment gap can be thought of as the demand component of the equation, shocks generally being related to supply, and lagged inflation and/or inflation expectations picking up an inertia effect.

Equation (2) specifies how the NAIRU can change over time. The ratio of the variance of the two error terms is the signal-to-noise ratio. This measures volatility in the NAIRU in relation to volatility in inflation, so determines the smoothness of the estimated series. 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. One of the key assumptions required for estimation is setting the signal-to-noise ratio. Past studies have tended to arbitrarily select the signal-to-noise ratio, but this paper uses Stock and Watson’s (1998) procedure.

If you can assume normally distributed errors, then the Kalman filter can compute a log-likelihood function that allows estimation of the parameters using the maximum likelihood method. For more detail on the Kalman filter see Harvey (1990).

Note that the Hodrick-Prescott multivariate filter (HPMV) is also sometimes used but the Kalman filter is generally preferred as it is more flexible (Richardson et al 2000). The HPMV filter requires a third equation specifying the gap between unemployment and the NAIRU, this means that the NAIRU estimates tend to follow actual unemployment more closely and without adjustments estimates tend to be drawn towards end point values (Richardson et al 2000).