General Taylor Condition

The left hand side of Tables 3.3-3.5 present estimates obtained from using the estimation procedures described above to evaluate the following ‘correctly specified’ estimating equation:

̅ ̅ ̅ ̅ ̅ ̅

(3.31) against HP, 3-8 band pass and 2-15 band pass filtered data respectively. Expected future values are obtained from the model simulation procedure and are used directly under the OLS estimator or are instrumented for under the IV techniques, as described above.

[Table 3.3 here] [Table 3.4 here]

15 _{The D-W count excludes cases for which the test statistic falls in the inconclusive region of the}

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[Table 3.5 here]

The key result in Tables 3.3-3.5 is that the mean inflation coefficient consistently exceeds unity for the estimating equation (3.31) which accurately reflects the underlying theoretical relationship (3.27); furthermore, the magnitude of these estimates is broadly consistent with the theoretical prediction derived in Section 3.4 ( =2.125), subject to the caveat that the coefficients are estimated precisely. This result is found to be robust to both the statistical filter applied to the data and to the econometric procedure employed to generate the estimates. Generally speaking, the OLS and GMM estimators produce a greater number of statistically significant coefficient estimates than the 2SLS estimator. In Table 3.5 for the 2-15 filter, for example, the 2SLS estimator provides a statistically significant estimate for the inflation coefficient for only 580 of the 1000 simulated samples whereas the OLS and GMM estimators both return 1000 statistically significant estimates. The OLS and GMM procedures generate reasonably large R-squared and adjusted R-squared statistics (greater than 0.77 in all cases), whereas negative R-squared statistics are repeatedly obtained for the 2SLS estimator. The OLS estimator also rejects the null hypothesis of the F-statistic more frequently than the 2SLS estimator (1000 vs. 907 rejections in Table 3.5, for example). However, one might be wary of the low number of D-W null hypothesis non-rejections produced by the OLS estimator. The results for the 3-8 band pass filter (Table 3.4) are unusual in the sense that all three estimation procedures produce a high number of D-W test rejections; for the other two filters this undesirable result is confined to the OLS estimator. On the whole, one might therefore be inclined to favour the GMM estimates out of the three alternatives. Furthermore, the instrument set used for the GMM estimator shows no signs of being invalid since 1000 non-rejections of the null hypothesis of the J-statistic for instrument validity are obtained across all three filters. Under the GMM procedure, the mean inflation coefficients are estimated to be 2.299, 2.423 and 2.306 in Tables 3.3, 3.4 and 3.5 respectively; these estimates compare favourably to the predicted value of 2.125.

The forward interest rate term is found to be an important element of the estimating equation (3.31) in terms of generating an inflation coefficient which is

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consistent with the underlying Benk et al. (2010) model. The right hand side of Tables 3.3, 3.4 and 3.5 show that the mean estimate of the inflation coefficient falls below unity for the OLS and GMM estimators under the β5=0 restriction.

Precise mean estimates of 0.614 and 0.963 (adjusted mean for 999 statistically significant coefficients) are obtained from the OLS and GMM procedures under the 2-15 filter, for example. Similar estimates are obtained for the inflation coefficient under the two alternative filters in Tables 3.3 and 3.4, both in terms of the mean coefficient estimates for the unrestricted specification and in terms of the decline in magnitude induced by the arbitrary restriction, although the GMM estimate for the inflation coefficient tends to be smaller under the HP and 3-8 band pass filters at 0.614 (adjusted mean for 925 statistically significant estimates) and 0.679 (for 974 estimates).

In contrast to the inflation coefficients, the estimated coefficients for consumption growth and productive time growth diverge from their theoretical predictions for the unrestricted estimating equation (3.31). Under log utility (θ=1), the former should take the same magnitude as the coefficient on inflation and the latter should take a value of zero. The coefficient estimates obtained can be used to ‘back-out’ an estimate of the coefficient of relative risk aversion (θ). Firstly, using the mean GMM estimate for the coefficient on consumption growth of 0.302 (Table 3.5) and the corresponding estimate of 2.306 for , an implied estimate of θ can be calculated as β2/β1=0.131, which is substantially smaller than the

baseline calibration of θ=1. Alternatively, the relationship β₃=β₁ψ(1–θ)l/(1-l), which is obtained from equation (3.27) with replaced by its estimate β1, can

also be used to obtain an implied estimate of θ. Using the estimates presented in Table 3.5, the implied estimate would be θ=1.103, which is much more in keeping with the calibrated value. Table 3.5 also reports that both the OLS and GMM procedures generate 1000 statistically significant estimates for the coefficient on velocity growth under the unrestricted estimating equation and that the mean estimate is correctly signed for both estimators. The mean coefficient estimates are reported as -0.196 and -0.269 for OLS and GMM estimators respectively; these estimates are somewhat smaller than the theoretical prediction of -0.065. Similar mean coefficient estimates are obtained from the HP and 3-8 filters. Finally, Table

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3.5 reports mean estimates of -1.761 and -1.729 (OLS and GMM respectively) for the forward interest rate coefficient compared to a theoretical prediction of - 1.125. The mean estimates are therefore correctly signed but, again, somewhat smaller than the theoretical prediction.

For a standard interest rate rule, the magnitude of the coefficient on inflation is deemed to reflect the strength of monetary policymakers’ dislike of inflation deviations from target and a magnitude above one, in particular, is deemed to satisfy the Taylor principle. However, this interpretation is not applicable to the Taylor Condition. The result that the estimated coefficient on inflation exceeds unity in (3.31) is a consequence of a money growth rule, not an interest rate rule. Similarly, the break-down of the Taylor principle under the ‘restricted’ estimating equation (β5=0) cannot be interpreted as a softening of policymakers’ attitude

towards inflation; this result simply emanates from model misspecification.

Unlike the estimates obtained from filtered data, each of the three econometric procedures described above generates an estimated inflation coefficient which falls below unity when equation (3.31) is evaluated against the simulated data in unfiltered form. Specifically, the OLS procedure generates a mean coefficient estimate of 0.837 (1000 statistically significant coefficient estimates), the 2SLS procedure generates a mean coefficient estimate of 0.101 (adjusted mean for 107 statistically significant coefficient estimates) and the GMM procedure produces a mean coefficient estimate of 0.478 (adjusted mean for 804 statistically significant coefficient estimates).16 _{Under the restriction on the forward interest rate term}

considered previously (β5=0), all three procedures generate a mean inflation

coefficient of less than one in a manner consistent with a conventional interest rate rule which violates the Taylor principle (OLS: 0.702 for 1000 statistically significant coefficient estimates; 2SLS: 0.729, adjusted mean for 905 statistically significant coefficient estimates; GMM: 0.782 for 1000 statistically significant coefficient estimates). Again, this Taylor principle interpretation does not apply in the present context. One cannot draw inferences about the conduct of monetary policy from these estimates because they are generated simply as a product of a

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misspecified estimating equation. Monetary policy continues to be characterised by the money growth rule described in Section 3.2.4.

We interpret the low estimated inflation coefficients obtained from unfiltered data as a reflection of the fact that all frequency components of the simulated data are considered jointly if a filter is not applied. On the other hand, the ‘high frequency filters’ used in this chapter isolate the ‘short-run relationship’ between the nominal interest rate and inflation, a relationship which is typically interpreted as an ‘interest rate rule’ when recovered from actual time series data. In Chapter-IV we adjust the calibration of the band pass filter in order to retain lower frequency trends in the simulated data. To preview the results obtained there, these extended frequency ranges allow us to identify a long-run Fisher relation in the simulated data. However, as Table 4.7 of Chapter-IV will show, if the frequency range considered is extended beyond a certain point (approximately 50 periods or ‘years’) then the mean estimated inflation coefficient falls below one, as is the case when the simulated data is evaluated in unfiltered form. As one would expect, the effect of the filters used in this chapter is to isolate the high frequency relationship between the nominal interest rate and inflation and the nominal interest rate and the other terms which feature in the Taylor Condition (3.27).