Estimating the uncertainty of the Phillips curve NAIRU

This section describes three approaches for estimation of a ‘confidence region’ of a (time independent) Phillips curve NAIRU. As noted by Staiger et al. (1997) the reason for the absence of confidence intervals in most NAIRU calculations has to do with the fact that the NAIRU (e. g. in (4.4)) is a non-linear func­tion of the regression coefficients. Nevertheless, three approaches can be used to construct confidence intervals for the NAIRU: the Wald, Fieller, and likeli­hood ratio statistics.[23] The Fieller and likelihood ratio forms appear preferable because of their finite sample properties.

The first and most intuitive approach is based on the associated standard error and t ratio for the estimated coefficients, and thus corresponds to a Wald statistic; see Wald (1943) and Silvey (1975, pp. 115-18). This method may be characterised as follows. A wage Phillips curve is estimated in the form of (4.1) in Section 4.2. In the case of full pass-through of productivity gains on wages, and no ‘money illusion’, the Phillips curve NAIRU uphl1 is (3w0/(3w1, and its estimated value pu is I3w0//3w1, where a circumflex denotes estimated values.

As already noted, (4.1) is conveniently rewritten as:

Awt - Aat - Aqt = - f3w1(ut - uphl1) + £wU (4.13)

where uphl1 may be estimated directly by (say) non-linear least squares. The result is numerically equivalent to the ratio (3w0/ (3w1 derived from the linear estimates (/3w0,/?w1) in (4.1). In either case, a standard error for fiphl1 can be computed, from which confidence intervals are directly obtained.

More generally, a confidence interval includes the unconstrained/most likely estimate of uphl1, which is /3w0/(3w1, and some region around that value. Heuristically, the confidence interval contains each value of the ratio that does not violate the hypothesis

Hw : = uphl1 (4.14)


too strongly in the data. More formally, let FW(uphl1) be the Wald - based F-statistic for testing Hw, and let Pr(-) be the probability of its argument. Then, a confidence interval of (1 — a)% is [up^uhlgh] defined by Pr(Fw(uphl1)) < 1 - a for uphl1 Є, uhhgh].

If wi, the elasticity of the rate of unemployment in the Phillips curve, is precisely estimated, the Wald approach is usually quite satisfactory. Small sample sizes clearly endanger estimation precision, but ‘how small is small’ depends on the amount of information ‘per observation’ and the effective sample size. However, if f3w1 is imprecisely estimated (i. e. not very significant statistic­ally), this approach can be highly misleading. Specifically, the Wald approach ignores how /3w0/f3w1 behaves for values of /3w1 relatively close to zero, where ‘relatively’ reflects the uncertainty in the estimate of f3w 1. For European Phillips curves, the w1 estimates are typically insignificant statistically, so this concern is germane to calculating Phillips curve natural rates for Europe. In essence, the problem arises because jlu is a non-linear function of estimators (/3w0, /?w1) that are (approximately) jointly normally distributed; see Gregory and Veall (1985) for details.

The second approach avoids this problem by transforming the non-linear hypothesis (4.14) into a linear one, namely:

Hf: pwo - /?w1uphl1 =0. (4.15)

This approach is due to Fieller (1954), so the hypothesis in (4.15) and cor­responding F-statistic are denoted HF and Ff(uphl1). Because the hypothesis (4.15) is linear in the parameters w0 and w1, tests of this hypothesis are typically well-behaved, even if /3w1 is close to zero. Determination of confidence intervals is exactly as for the Wald approach, except that the F-statistic is constructed for f3w0 - ew1 uphl1. See Kendall and Stuart (1973, pp. 130-2) for a summary.

The third approach uses the likelihood ratio (LR) statistic (see Silvey 1975, pp. 108-12), to calculate the confidence interval for the hypothesis Hw.

That is, (4.13) is estimated both unrestrictedly and under the restriction HW, corresponding likelihoods (or residual sums of squares for single equations) are obtained, and the confidence interval is constructed from values of wphl1 for which the LR statistic is less than a given critical value.

Three final comments are in order. First, if the original model is linear in its parameters, as in (4.1), then Fieller’s solution is numerically equivalent to the LR one, giving the former a generic justification. Second, if the estimated Phillips curve does not display dynamic homogeneity, (3w0/(3w is only a com­ponent of the NAIRU estimate that would be consistent with the underlying theory, cf. the general expression (4.10). This complicates the computation of the NAIRU further, in that one should take into account the covariance of terms like (3w0/(3w, and ((3'w3 + (3'w4 — 1)/(3'w4. However, unless the departure from homogeneity is numerically large, [wfoh1, Mh! gh] may be representative of the degree of uncertainty that is associated with the estimated Phillips curve natu­ral rate. Third, identical statistical problems crop up in other areas of applied macroeconomics too, for example, in the form of a ‘Monetary conditions index’; see Eika et al. (1996).

Section 4.6 contains an application of the Wald and Fieller/LR methods to the Phillips curve NAIRU of the Norwegian economy.

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