**4.5 Empirical Analysis: The Fisher Relation**

**4.5.5 Fitting the ECM to Model-Simulated Data**

The ECM is now applied to the 1000 samples of artificial data simulated from the Benk et al. (2010) model. The econometric framework is analogous to that of the previous section except for the fact that the long-run equilibrium condition is now specified to be:

̅ ̅ ̅

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in a manner consistent with the ‘long-run Taylor Condition’, equation (4.7) above but where the simulated data is now considered in unfiltered form so as not to remove any trends in the series.

The application of the ECM to simulated data might be questioned on the basis
that Table 4.3 suggests that the majority of the simulated series are stationary.
Firstly, out of the seven variables tested in Table 4.3, the nominal interest rate and
inflation series are the ‘most likely’ to be non-stationary because they produce the
smallest number of ADF rejections and KPSS non-rejections. Nevertheless,
treating the simulated series as stationary, as we did in Section 4.4, still seems to
be reasonable based upon the absolute number of rejections and non-rejections
*obtained. Secondly, the key requirement for the ECM is that the error term (ςt* in

equation 4.18) is stationary. If any of the variables in the long-run equilibrium
(4.18) are non-stationary then cointegration is required for this to be true but if
*all of the series in (4.18) are stationary then ςt* is also guaranteed to be stationary.

It is not necessarily the case that ECMs can only be used with non-stationary
data.46_{ We therefore proceed to apply the ECM on the understanding that the }

simulated data is in all likelihood stationary but that this econometric framework can still be legitimately applied and might indeed be useful for ameliorating the autocorrelation reported in Table 4.8 (right-hand side).

As with the actual data, we must first investigate whether the two ‘non-stationary’ series are cointegrated. Table 4.16 repeats for simulated data the exercise conducted in Table 4.12 for actual data. The results indicate that the residual series derived from a simple regression of the nominal interest rate on inflation are stationary, as are the residuals obtained from this regression with dependent and independent variables reversed. The null hypothesis for the ADF test – that the residual series in question is integrated of order one – is rejected for 993 out

46_{ De Boef and Keele (2008) have argued for wider use of error correction methods in their field, }

political science, where the data are often persistent but unlikely to be integrated. They state that: “ECMs suffer from benign neglect in stationary time series applications in political science. Intimately connected with and applied almost exclusively to cointegrated time series, it seems analysts have concluded that ECMs are only suited to estimating statistical relationships between two integrated time series. In fact, ECMs may be used with stationary data to great advantage.” (p.189).

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of the 1000 samples while the null hypothesis for the KPSS test – that the residual series in question is stationary – is not rejected for 848 out of 1000 samples. Of course, this result would also be expected if we were to regress one stationary series on another stationary series. Either way, we may proceed to implement the ECM (4.17) for the long-run relationship (4.18). We also note the difference between the coefficient estimate derived from this cointegrating relationship and the coefficient obtained from the estimates derived from the levels regression in the right-hand side of Table 4.8. The consumption growth term would appear to be important for obtaining the Fisher relation result for the simulated data, as Arnwine and Yigit (2008) argue, given that the regression without ̅ produces a smaller mean coefficient estimate.

*[Table 4.16 here] *

The final step is to estimate the ECM for each of the 1000 simulated samples. As
for the actual data above, we consider a GMM procedure along with a simple OLS
estimator.47_{ In order to choose an appropriate lag length we again search across }

*27 different specifications by varying τ1, τ2 and τ3* between 0 and 2. The residuals

obtained from the OLS procedure exhibit a high degree of serial correlation for all
prospective specifications but Table 4.17 presents the ‘preferred specification’
based on the AIC. On the other hand, the GMM estimator successfully ameliorates
the autocorrelation problem and the resulting point estimate for the coefficient on
*inflation, ρ1* in (4.18), is 1.033. This estimate is consistent with a one-to-one Fisher

relation. However, the estimate for the coefficient on consumption growth is lower than the equivalent estimated presented in Table 4.15; 0.447 compared to (approximately) one previously. Other features of the GMM estimation appear to be satisfactory – the R-squared is reasonably high (0.806) and the instrument set satisfies the J-test for 933 out of 1000 samples.

*[Table 4.17 here] *

47_{ As was the case for actual data, the instrument set for the GMM estimator consists of a constant, }

two lagged first differences of the nominal interest rate, four lagged first differences of inflation, four lagged first differences of consumption growth and one lagged level of the nominal interest rate, inflation and consumption growth.

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**4.6 Conclusion **

It has become increasingly common to interpret the short-run relationship between the nominal interest rate and inflation as an interest rate rule. This interpretation has the advantage of conforming to the way in which prominent central banks around the world communicate their monetary policy decisions to the public (Mehrling, 2006). However, Lucas (2003) has questioned whether such rules offer any new insight given that the Fisher relation already explains the link between the nominal interest rate and (expected future) inflation. In recent empirical work, Islam and Ali (2012) have drawn a connection between well- known empirical results in the distinct Taylor rule and Fisher relation literatures. In this chapter, we use the structural model of Benk et al. (2010), outlined in detail in Chapter-III, to provide a unified theoretical framework for the relationship between these two literatures.

In essence, the standard consumption Euler equation, which we have shown in Chapter-I to be central to the ‘equivalence’ between interest rate rules and money supply rules in a flexible price environment, is augmented by a term which reflects the endogenous money-credit choice modelled by Benk et al. (2010). From this, we derive the interest rate rule implicit in the model, which we label as the Taylor Condition. As shown in Chapter-III, in general the ‘Taylor principle’ holds in the short-run in that the coefficient on inflation in the Taylor Condition exceeds one and reverts to its minimum value of one at the Friedman (1969) optimum, at which the net nominal interest rate is zero and money balances carry no opportunity cost. A coefficient of one on inflation is also obtained for the ‘long- run Taylor Condition’ introduced in this chapter and, as such, this long-run form corresponds to Arnwine and Yigit’s (2008) ‘augmented Fisher relation’ which links the nominal interest rate to inflation and consumption growth.

For model-simulated data, we have extended the results of the previous chapter to apply to statistical filters which retain some of the lower frequency fluctuations that the standard ‘business cycle’ filters applied previously would have discarded. As described in Section 4.4, we extend the band pass filter in order to construct a ‘medium-term cycle’, similar in the spirit to Comin and Gertler (2006), although

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initially using a narrower filter window than the one they initially specified. The Taylor principle result holds for extended filter windows of 2-20 and 2-25 years but the one-for-one relationship between the nominal interest rate and (expected future) inflation emerges as the window extended to conform to their original, 2- 50 year, medium-term cycle. The Fisher relation is also found to hold for simulated data when the estimating equation is restricted to conform to the ‘long- run Taylor Condition’.

Support for the theoretical restrictions implied by the underlying structural model has also been provided for U.S. time series data. One particularly striking result is that the variation in the coefficient on inflation reported by Clarida et al. (2000), and which is central to the New Keynesian account of the historical record, is not replicated for the Taylor Condition implicit in the Benk et al. (2010) model. Instead, the Taylor principle holds for both ‘pre-Volcker’ and ‘post- Volcker’ subsamples of our full, 1960-2011, period of study, as long as the correct empirical model is specified. In addition, we replicate the well-known result in the Fisher relation literature that there exists a one-for-one relationship between the nominal interest rate and inflation over the full 51 year period. To do so, however, we needed to resort to error correction methods to deal with the non-stationary nature of the nominal interest rate and inflation series over this lengthy time period and also applied a tax adjustment to the nominal interest rate series. Our unified approach to the Taylor and Fisher relations has involved a careful consideration of monetary variables. Nelson (2008a) argues that the New Keynesian model cannot offer a coherent account of the long-run without reference to monetary variables. Similarly, we have shown that a variable velocity of money is crucial to the account of the short-run offered by our framework. More broadly, our alternative framework casts further doubt upon the notion that estimated interest rate rule can be used to reveal the underlying behaviour of policymakers.

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**C**

**HAPTER**

**-V:****G**

**ENERAL**

**C**

**ONCLUSION**

**5.1 Contribution **

In this thesis we have provided a structural framework to account for empirical results consistent with both the Taylor principle and the Fisher relation. In so doing, we have questioned the conventional ‘reaction function’ interpretation of the relationship between the nominal interest rate and inflation, motivated by theoretical, empirical and practical reservations as to whether monetary policy analysis can be conducted without reference to monetary variables.