In the following, we will focus on the results in GGL for the Euro area. Our replication of their estimates is given in (7.12), using the same set of instruments: five lags of inflation, and two lags of the wage share, detrended output, and wage inflation.
Apt = 0.681Apt+i + 0.281Apt_ 1 + 0.019wst + 0.063 (7.12)
(0.073) (0.072) (0.027) (0.069)
GMM, T = 107 (1971(3) to 1998(1))
Xj(8) = 8.01[0.43],
where Xj(') is Hansen’s (1982) J-test of overidentifying restrictions. The role of the wage share (as a proxy for real marginal costs) is a definable trait of the NPCM, yet the empirical relevance of wst is not apparent in (7.12): it is statistically insignificant. Note also that the sum of the coefficients of the two inflation terms is 0.96. Taken together, the insignificance of wst and the near unit-root, imply that (7.12) is almost indistinguishable from a pure time-series model, a dVAR. On the other hand, the formal significance of the forward term, and the insignificance of the J-statistic corroborate the NPCM. The merits of the J-statistic are discussed in Section 7.5: in the rest of this section we conduct a sensitivity analysis with regards to GMM estimation methodology.
The results in (7.12) were obtained by a GMM procedure which computes the weighting matrix once. When instead we iterate over both coefficients and weighting matrix, with fixed bandwidth, we obtain
Apt = 0.731Apt+1 + 0.340Apt_ 1 - 0.042wst - 0.102 (7.13)
(0.052) (0.069) (0.029) (0.070)
GMM, T = 107 (1971(3) to 1998(1))
Xj(8) = 7.34[0.50].
As before, there is clear indication of a unit root (the sum of the two inflation coefficients is now slightly above one). The wage share coefficient is wrongly signed, but it is still insignificantly different from zero, though.
Next, we investigate the robustness with regard to the choice of instruments. We use an alternative output-gap measure (emugapt), which is a simple transformation of the one defined in Fagan et al. (2001) as real output relative to potential output, measured by a constant-return-to-scale Cobb-Douglas production function with neutral technical progress. We also omit the two lags
of wage growth. Apart from yet another sign-change in the ws coefficient, the results respond little to these changes in the set of instruments:
Apt = 0.60Apt+1 + 0.35Apt_ і + 0.03 wst + 0.08 (7.14)
(0.06) (0.06) (0.03) (0.06)
GMM, T = 107 (1972(4) to 1997(4))
Xj(6) = 6.74[0.35].
Finally, we investigate the robustness with respect to estimation method. Since the NPCM is a linear model, the only real advantage of choosing GMM as opposed to 2SLS as estimation method is the potential necessity to correct for autocorrelated residuals. Autocorrelation is in line with the rational expectations hypothesis, implied by replacing EtApt+1 with Apt+1 in estimation—see Blake (1991) and Appendix A.2—but it may also be a symptom of mis-specification, as discussed in Nymoen (2002). As shown below, the estimates are robust with respect to estimation method, even though the standard errors are doubled, since the model suffers from severe autocorrelation:
Apt = 0.66Apt+1 + 0.28Apt_ 1 + 0.07 wst + 0.10 (7.15)
(0.14) (0.12) (0.09) (0.12)
2SLS, T = 104 (1972(2) to 1998(1))
The p-value of the Sargan specification test, Xj, a|, is 0.06, and indicates that (7.15) could be mis-specified, since some of the instruments could be potential regressors. The Fire| is the F-statistic from the first stage regression of Apt+1 against the instrument set and indicates no ‘weak instruments’ problem, although it is only strictly valid in the case of one endogenous regressor—see Stock et al. (2002).
We conclude from the range of estimates that the significance of the wage share is fragile and that its formal statistical significance depends on the exact implementation of the estimation method used. The coefficient of the forward variable on the other hand is pervasive and will be a focal point of the following analysis. Residual autocorrelation is another robust feature, as also noted by GGL. But more work is needed before we can judge whether autocorrelation really corroborates the theory, which is GGL’s view, or whether it is a sign of econometric mis-specification.