# Restricted Estimating Equations

## 4.5 Empirical Analysis: The Fisher Relation

### 4.5.2 Restricted Estimating Equations

For model-simulated data we consider two alternative versions of the ‘correctly specified’ estimating equation (4.11). Firstly, we impose the restriction β5=0 so

that the forward interest rate term is omitted. The estimating equation is therefore:

̅ ̅ ̅ ̅ ̅

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and secondly the restriction β3=β₄=β₅=0 is imposed so that terms in the growth

rate of productive time, velocity growth and the forward interest rate term are omitted, as follows:

̅ ̅ ̅

(4.14) This second restriction produces an estimating equation which reflects the ‘long- run Taylor Condition’ presented as equation (4.7) above. As noted at that point, second moment terms, which reflect the extent to which nominal bonds act as suitable ‘insurance’ against periods of low consumption growth and which feature in Canzoneri et al.'s (2007) empirical specification, are absent from equation (4.7) due to the log-linearization procedure used to derive the Taylor Condition (4.1). Crowder and Hoffman (1996) also drop second moment terms from their analysis of the Fisher relation on the basis that previous empirical studies have found such terms to be only marginally significant while Arnwine and Yigit (2008, p.194) argue that the omission of such terms is justifiable for “a low inflation risk economy”.32

Table 4.8 shows the consequences of imposing these two sets of restrictions upon the correctly specified estimating equation.33 The first restriction (β5=0) yields a

lower mean inflation coefficient for both the high frequency component and for the medium-term cycle relative to the estimates reported above for the ‘correctly specified’ empirical model. The mean estimate for the medium-term cycle now takes a value of 1.025 compared to 2.068 previously (Table 4.4) and these coefficients are estimated no less precisely under this restriction (1000 statistically significant estimates). This estimate is similar to the GMM estimate of 0.963 presented on the right hand side of Table 3.5 in Chapter-III for the same

32 Crowder and Wohar (1999) also omit these second moment terms and Crowder and Hoffman

(1996, p.106) state that: “The evidence given by Shome, Smith, and Pinkerton (1988) implies that the effects of the risk premium on the “Fisher effect” estimates are inconsequential.” However, they subsequently caution that: “rejections of the implications of the textbook Fisher relation may suggest that these factors are indeed important.” (ibid).

33 Once again, the instrument set is appropriately adapted to reflect each modification to the

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restricted estimating equation under a 2-15 band pass filter. The high-frequency component (adjusted) mean estimate of 0.370 would now be inadmissible in the context of the Benk et al. (2010) model, for which may not fall below one, whereas the corresponding estimate stood at 2.221 in Table 4.4, well in excess of unity. Not surprisingly given that the estimating equation is now misspecified, the estimated inflation coefficient no longer resembles the theoretical prediction. The forward interest rate term turns out to be a crucial element of the full estimating equation (4.11), as was established in Chapter-III. Finally, the estimated inflation coefficient for the medium-frequency component (β₁=0.949) is now similar to the medium-term cycle estimate but, as with Table 4.4, one would need to treat this estimate with caution because of the high R-squared, low D-W statistic combination which continues to characterise the results obtained from this frequency band.

[Table 4.8 here]

Table 4.8 also shows that an (adjusted) mean inflation coefficient of 0.916 is obtained from the medium-term cycle under the second restricted estimating equation (β3=β4=β5=0), as is consistent with theoretical prediction of Canzoneri et

al.’s (2007, p.1866) Euler equation and Arnwine and Yigit’s (2008, eq.6) “augmented Fisher relation”. On the other hand, the mean consumption growth coefficient, which in principle now provides a direct estimate of the coefficient of relative risk aversion, is substantially smaller (0.128, adjusted mean) than the calibrated value (θ=1). However, it should be noted that the validity of the instrument set now appears to deteriorate and the estimated residual series show symptoms of autocorrelation according to both the D-W and Q statistics. The latter finding might indicate that the ‘long-run Taylor Condition’ (4.7) is better thought of as a long-run equilibrium condition for model-simulated data. This suggestion shall be explored using alternative econometric techniques later on in this section.

Considering the first of the two restricted specifications (β5=0) against U.S. time

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one for the pre- and post-Volcker subsamples in a similar manner to Clarida et al.’s (2000) findings for a forward-looking interest rate rule (Table 4.9). The ‘pre- Volcker’ estimate of 0.89 for the inflation coefficient is comparable to Clarida et al.’s estimate of 0.83, while the ‘CGG post-Volcker’ estimate of 1.97 compares to their estimate of 2.15 for this same period (Clarida et al., 2000, Table II).34 It is

also apparent from Table 4.9 that omitting the forward interest rate term leads to coefficients on consumption growth which are not statistically significant but that the coefficients on the ‘productive time’ term remain significant at the 1% level for all estimation periods.

[Table 4.9 here]

Turing to the second restricted specification for the ‘long-run Taylor Condition’, this estimating equation produces a higher pre-Volcker estimate for the inflation coefficient compared to the first restriction (Table 4.10).35 Consequently, although

the coefficient on inflation does increase when moving from pre- to post-Volcker subsamples in keeping with Clarida et al.’s (2000) estimates, the pre-Volcker point estimate no longer falls decisively below unity. The pre-Volcker point estimate for the coefficient on inflation is 1.048 and the post-Volcker point estimates all exceed two. Over the full sample period, the point estimate for the coefficient on inflation takes a value of unity in keeping with the ‘long-run Taylor Condition’. However, the estimated residual series is potentially autocorrelated over the full sample because the Q-statistic produces a P-value of 0.052 and so only marginally fails to reject at the 5% level of significance, even with two lagged dependent variable terms added to the specification. This is similar to the

34 Diagnostic tests suggest that two lagged dependent variable terms are required for the ‘pre-

Volcker’ subsample but that one lagged dependent variable term is sufficient for all other estimation periods. ‘Pre-Volcker’: a zero lag specification produces a D-W statistic of 0.532 and a Pr(Q) statistic of 0.000; adding one lagged dependent variable produces Pr(Q)=0.045. ‘CGG-post’: D-W=0.573 and Pr(Q)=0.000 for a zero lag specification; ‘Taylor-post’: D-W=0.546, Pr(Q)=0.000; ‘extended-post’: D-W=0.272, Pr(Q)=0.000; full sample: D-W=0.140, Pr(Q)=0.000.

35 Two lagged dependent variable terms are now required for the ‘pre-Volcker’ and ‘full sample’

periods and one lagged dependent variable is required for each of the post-Volcker subsamples. ‘Pre-Volcker’: a zero lag specification produces a D-W statistic of 0.405 and a Pr(Q) of 0.000; adding one lagged dependent variable gives Pr(Q)=0.001; ‘CGG-post’: D-W=0.575 and Pr(Q)=0.000 for a zero lag specification; ‘Taylor-post’: D-W=0.588, Pr(Q)=0.000; ‘extended-post’: D-W=0.316, Pr(Q)=0.000; ‘full sample’: D-W=0.156 and Pr(Q)=0.000 with zero lags and Pr(Q)=0.004 with a single lag.

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autocorrelation reported for the same estimating equation for model-simulated data in Table 4.8. To deal with such concerns we next consider an alternative econometric framework which casts (4.7) as a long-run equilibrium condition.

[Table 4.10 here]

Outline

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