**2.2 Implied Nominal Interest Rate and Discrepancy Series**

**2.2.4 Implicit Rule Discrepancies**

As documented in Figure 2.3, the discrepancy series produced by the ‘interest rate rule’ implicit in the unit-velocity CIA model shares similar features to the

15_{ The first round of quantitative easing (announced in December 2008) included the purchase of }

$1.25 trillion in mortgage-backed securities and $300 billion in long-term treasury bonds and the second round (announced in November 2010) involved the purchase of another $600 billion in long-dated Treasury bonds.

16_{ Chung et al. (2011) express policymakers’ attempts to influence long-term interest rates in }

terms of the short term rate, even though the latter is ‘stuck’ at zero. To do so they regress quarterly changes in the 10-year Treasury yield on quarterly changes in the federal funds rate for the period 1987-2007 and find that a 100 basis point reduction in the short term rate is equivalent to a 25 basis point reduction in long term yields. In this way they translate ‘unconventional’ monetary policy back into ‘conventional’ form. Similarly, Rudebusch (2010, Figure 3) provides an estimate as to what extent the discrepancy between rule-implied and actual nominal interest rate should be adjusted to take account of the attempts to ease monetary policy by ‘unconventional’ means. He argues that: “If the Fed’s purchases reduced long rates by ½ to ¾ of a percentage point, the resulting stimulus would be very roughly equal to a 1½ to 3 percentage point cut in the funds rate”.

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discrepancy series generated by a conventional interest rate rule. Given the
fundamental difference between the Taylor Rule (2.1) and the Implicit Rule (2.2),
it would plainly be inappropriate to interpret the discrepancy series obtained
from the latter as indicative of ‘policy mistakes’ in the same manner as Taylor’s
(1999) ‘historical approach’. We therefore interpret the discrepancy series
derived from (2.2) with reference to Lucas (2003), who poses the question as to
*whether the data supports a “stable real rate plus inflation premium, à la Irving *
*Fisher?” or a “Central Bank policy response to inflation?” His answer to this *
*question is, “100% Fisher” and he concludes that: “[the] figure contains no *
*information on central bank interest rate policy rules” (2003, slide-10). While the *
Implicit Rule (2.2) does feature a one-for-one relationship between the nominal
interest rate and inflation in a manner consistent with the Fisher relation, it also
features a term in consumption growth. This is consistent with Arnwine and
Yigit’s (2008) “augmented Fisher relation”. They insist that empirical applications
of the Fisher relation should make allowance for changes in consumption (or
*output if y=c) to account for pro-cyclical fluctuations in the real interest rate and *
that the expected one-for-one relation should only be expected to hold if such
provision is made.

Figure 2.13 adapts the figure presented by Lucas (2003, slide 9) to apply to the
Implicit Rule (2.2) by considering the rate of inflation plus one half times the
growth rate along the horizontal axis in accordance with the ‘original’ calibration
above. We also adjust the gap between the 45-degree line and the dashed line to
*be 2 percentage points (ρ=2) rather than 2.5 percentage points (which was Lucas’ *
assumed “stable real rate”) and distinguish between the various monetary policy
regimes defined above.17

*[Figure 2.13 here] *

17_{ It is not clear which precise series Lucas (2003) plots or what frequency and time period he }

considers but for the purposes of Figure 2.13 we interpret Lucas’ “Short-term Interest Rate” to be the effective federal funds rate and his “Inflation Rate (CPI)” to be the year-on-year percentage change in the CPI index, consistent with the quantitative analysis presented above.

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Figure 2.13 contains three regions, each of which can be defined by the following
*inequalities: ̅>ρ+( ̅+θ ̅) for plots to the northwest of the solid line; ̅< ̅+θ ̅ for *
*plots to the southeast of the dashed line; and ρ+( ̅+θ ̅)> ̅>( ̅+θ ̅) for plots lying *
between the solid and dashed lines. Historical periods characterised by relatively
small discrepancies from the parallel lines could be characterised as periods for
which the modified Fisher relation holds. In this way, we apply a (augmented)
Fisher relation interpretation to the data which does not depend upon ‘Taylor
*rule logic’. Treating the rate of time preference (ρ) as analogous to the equilibrium *
real interest rate, Figure 2.13 shows the tendency for low (<2%) and often
negative ex-post real interest rates during the pre-Volcker era and relatively high
(>2%) real interest rates during the Volcker disinflation. Observations relating to
the Greenspan-Bernanke era usually fall within or reasonably close to the 2
percentage point region thus indicating a low and stable real interest rate of
interest in the 0-2% range. Negative ex-post real interest rates are observed for
the crisis period because with the nominal interest rate close to zero, only a
modest rate of inflation is required to generate a negative ex-post real rate.

Alternatively, the Implicit Rule (2.2) can be interpreted in the context of a
consumption (output) Euler equation. As shown in Chapter-I, the Implicit Rule
above is essentially a re-arranged Euler equation. Canzoneri et al. (2007)
document the inability of conventional intertemporal Euler equations to track
*observed ex post real interest rates.*18_{ Figure 2.14 plots the equivalent to their }

“Fig. 1” in terms of the real interest rate, ̅ – ̅ , but uses the simpler expression (2.2) which does not include uncertainty.

*[Figure 2.14 here] *

Canzoneri et al. (2007) suggest that this poor empirical fit stems partly from the fact that the real interest rate derived from Euler equations such as (2.2) cannot capture the ‘liquidity effect’ associated with monetary policy adjustments. They point to two historical episodes in particular to illustrate their point (Fig. 1, p.1867): Firstly, during the late 1970s when the Euler-equation-implied real rate

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fell at a time when policymakers actually ‘tightened’ monetary policy by engineering increases in the short-term real interest rate (‘the Volcker tightening’, as they describe it); and secondly, in c.2001 when the model-implied real rate increased although monetary policy was actually ‘loosened’ (‘the Greenspan easing’). These two episodes are also observed in Figure 2.14 – the former episode generates large negative discrepancies between the observed and model- implied real interest rate while the latter generates large positive discrepancies (as shown by the highlighted regions).

Reynard and Schabert (2009) also consider the link (or lack thereof) between
short-term interest rates and Euler-equation-implied rates. They apply a similar
conditional log-normality assumption but express their figure (p.6) in terms of
the nominal rather than the real rate of interest. Therefore, their ‘Standard Euler
*interest rate’ is analogous to the series labelled as ‘IR (a=b) original’ and ‘IR (a=b) *
revised’ in Figure 2.2 above. They show that the spread between the model-
implied nominal rate and the observed nominal rate often varies inversely. Figure
2.15 shows that this result also emerges from the simplified expression (2.2).19_{ }

*[Figure 2.15 here] *

Reynard and Schabert (2009) proceed to construct a model which abstracts from the assumption that the nominal interest rate used to implement monetary policy and the nominal interest rate which enters the Euler equation coincide.