معادلات اویلر و نرخ های بهره بازار پول: یک چالش برای مدل های سیاست های پولی

Standard macroeconomic models equate the money market rate targeted by the central bank with the interest rate implied by a consumption Euler equation. We use U.S. data to calculate the interest rates implied by Euler equations derived from a number of specifications of household preferences. Correlations between these Euler equation rates and the Federal Funds rate are generally negative. Regression results and impulse response functions imply that the spreads between the Euler equation rates and the Federal Funds rate are systematically linked to the stance of monetary policy. Our findings pose a fundamental challenge for models that equate the two.

The consumption Euler equation of a representative household is a fundamental building block of many macroeconomic models, including the new neoclassical synthesis (NNS) models that are now a standard framework for the analysis of monetary policy.1 NNS models typically equate the money market rate targeted by the central bank with the interest rate in the Euler equation; thus the Euler equation provides a direct link between monetary policy and consumption demand. In this paper, we use U.S. data to calculate the interest rate implied by the Euler equation, and we compare this Euler equation rate with a money market rate. We find the behavior of the money market rate differs significantly from the implied Euler equation rate. This poses a fundamental challenge for models that equate the two rates. The fact that the two interest rate series do not coincide—and that the spread between the Euler equation rate and the money market rate is generally positive—comes as no surprise; these anomalies have been well documented in the literature on the “equity premium puzzle” and the “risk free rate puzzle”.2 And the failure of consumption Euler equation models should come as no surprise; there is a sizable literature that tries to fit Euler equations, and generally finds that the data on returns and aggregate consumption are not consistent with the model.3 If the spread between the two rates were simply a constant, or a constant plus a little statistical noise, then the problem might not be thought to be very serious. The purpose of this paper is to document something more fundamental—and more problematic—in the relationship between the Euler equation rate and observed money market rates. In Section 2, we compute the implied Euler equation rates for a number of specifications of preferences and find that they are strongly negatively correlated with money market rates.4 This suggests that something, or some things, are systematically moving the two rates in opposite directions. One possible explanation is apparent in the figures we present in Section 2. During the Volcker tightening in the early 1980s, the Euler equation rates fell while the money market rates rose, and during the Greenspan easing in the early 2000s, just the opposite occurred. These easily identified episodes suggest that the spread may be systematically linked to the stance of monetary policy. In Section 3, we document the statistical link between the interest rate spread and the stance of monetary policy. We do this in two ways. First, we regress the spread on standard measures of the stance of monetary policy, and then we generate impulse response functions for monetary policy shocks. The regressions imply that a monetary tightening decreases the spread, and the impulse response functions imply that a monetary tightening increases the money market rate and decreases the Euler equation rate. The intuition for why the spread is systematically linked to monetary policy is clearest if the representative consumer has additively separable CRRA utility and consumption is lognormally distributed. In this case, the consumption Euler equation implies that the real interest rate is proportional to the expected growth of real consumption. The empirical literature shows that a monetary tightening has a small effect on consumption in the first quarter following the tightening. In the following few quarters, the consumption falls more rapidly so that expected consumption growth declines. A decline in expected consumption growth will reduce the real interest rate implied by the Euler equation. The empirical literature shows that money market rates respond in the opposite direction. Changing the form of preferences can, in principle, address this problem. Adding habit persistence is an attractive alternative because doing so has proven useful in several other contexts. Consumption growth continues, however, to play a key role in the Euler equations obtained from alternative preferences. As a result this problem also plagues models with more general preferences, and the same intuition appears to apply. Both of our results—the negative correlation between the Euler equation rate and money market rate, and the sensitivity of the spread to monetary policy—pose a major challenge for models of monetary policy that equate the Euler equation rate and the rate targeted by the central bank. In Section 4, we summarize our results, we relate our work to some of the more recent literature on macroeconomic modeling, and we discuss ways in which NNS models might be modified to meet the challenge documented here.

Interest rates implied by combining the dynamics of consumption and inflation observed in U.S. data with Euler equations derived from several specifications for preferences exhibit behavior that differs strikingly from that of money market interest rates. We first showed that the rates implied by consumption Euler equations and the money market rate are not highly correlated. Instead, the correlation between the two rates is strongly negative, except for preferences that imply the Euler equation rate is extremely volatile, which virtually eliminates any correlation at all. This result raises a problem for standard macroeconomic models, which equate the Euler equation rate with the money market rate. Next we showed that the difference between the implied consumption Euler equation rates and the Federal Funds rate is systematically related to monetary policy. This second result raises a problem that is especially severe for models, such as NNS models, that examine the effects of monetary policy. A large empirical literature shows that monetary policy has a liquidity effect—that is, an unexpected monetary tightening raises nominal and real money market rates. The same literature finds that a monetary tightening reduces consumption and its rates of growth for several quarters. In this paper we showed that it is difficult to reconcile these two facts with models that equate Euler equation rates with money market rates. The problem arises because a decline in expected consumption growth appears to be associated with a decline in real interest rates in all of the Euler equations we considered; adding habit formation to consumer preferences does not seem to change this basic result. Neither do the recursive preferences used by Tallarini (2000) and Epstein and Zin, 1989 and Epstein and Zin, 1991. It is of course possible that some other preferences (with or without habit) could resolve the puzzle, but doing so would require that the impact of expected consumption growth be reversed. Adding habit formation to consumer preferences has yielded a significant payoff in recent macroeconomic modeling. Fuhrer (2000) and Christiano et al. (2005) find that habit persistence is instrumental in allowing their models to generate macroeconomic responses to monetary policy shocks that are consistent with the responses found in unrestricted VARs. Boldrin et al. (2002) find that habit persistence allows their two sector model to generate a mean riskless rate and equity premium consistent with those observed in the data; it also improves their model's ability to reproduce key aspects of business cycles. And Tallarini (2000) finds that including modified Epstein–Zin preferences allows his real business cycle model to resolve the equity premium and riskless rate puzzles without adversely affecting the moments of the model's aggregate quantities. In light of this success, our results may come as a bit of a surprise. Of course, we are looking at a correlation that has not been considered previously—the correlation between a model's Euler equation rate and an observed money market rate. But still, it is worth thinking about how our results might, or might not, be consistent with the recent literature on macroeconomic modeling with habit formation. First, there is no necessary conflict between Christiano et al.'s (2005) results and our finding of excessive volatility in the Euler equation rate implied by their specification of preferences. Indeed, the volatility in that rate is no surprise: Boldrin et al. (2001) note that the very same preferences yield excessive volatility, but they question whether this fact will ultimately prove to be a problem for models with habit, citing the work of Abel (1999) and Campbell and Cochrane (1999). Our results suggest that the problem is fundamental: alternative specifications of preferences can eliminate the excessive volatility, but they yield an Euler equation rate that is strongly negatively correlated with the money market rate. Second, our finding that the spread between the Euler equation rate and the money market rate responds systematically to the stance of monetary policy might at first appear to be inconsistent with the success reported by Christiano et al. (2005). However, we think there is an explanation for the difference in our results, and it is rather subtle. The difference lies in the fact that our calculations use the impulse response for consumption from the unrestricted VAR. In contrast, Christiano, Eichenbaum, and Evans estimate the parameters of their model so as to match as closely as possible all of the impulse responses to a monetary policy shock. In doing so they trade off deviations in the impulse responses of both consumption and interest rates from those found in the unrestricted VAR. The model's interest rate response can be brought closer into alignment by allowing the consumption response to differ from that in the unrestricted VAR. And given the sensitivity of interest rates to the path of consumption that is characteristic of models with habit, relatively small differences in the path of consumption can have large effects on the Euler equation rate. Consider, for example, the impulse response function of the real Euler equation rate implied by the preferences used by Edge (2002) and Christiano et al. (2005). As we note above, the impact effect of a positive Federal Funds rate shock on the Euler equation rate is significantly negative. In fact, not a single one of the 1,000 bootstrap replications produce a positive initial response. Nonetheless, we can produce a positive initial Euler equation response—indeed we could produce one that is identical to the change in the real Federal Funds rate—by judiciously choosing consumption responses well within our estimated confidence bands. In this paper, we have tried to resolve the problem posed by the failure of implied Euler equation rates to match the behavior of money market rates by changing consumer preferences. As an alternative, Daniel and Marshall, 1997 and Daniel and Marshall, 1998 note several possible frictions may account for the failure of consumption-based models to produce either an equity premium or a riskless money market rate that correspond to those observed in the data. They argue that if market frictions of some kind are behind these problems then the failures ought to be more prevalent in quarterly data than in annual or biennial data. They then show that models with habit persistence do a considerably better job of matching the mean and variance of both the equity premium and the riskless rate at one- and two-year horizons. We have also considered one- and two-year horizons and found that the negative correlation problem is reduced, but not completely eliminated. If market frictions are important, as these results suggest, then modeling these frictions is essential for monetary policy models, which necessarily focus on relatively high (quarterly) frequency fluctuations in interest rates and macroeconomic aggregates. Limited participation models and models attributing liquidity services to money market assets are two alternatives within the representative agent paradigm.22 Both model a wedge between the CCAPM rate and the money market rate. Lucas (1990), Fuerst (1992), and Christiano and Eichenbaum, 1992, Christiano and Eichenbaum, 1995 and Christiano et al., 1997 assume that households do not adjust their money holdings immediately following a monetary policy shock. Instead, the impact of a monetary shock falls on financial intermediaries, which, in turn, adjust their lending to firms. As a result, money market interest rates are no longer given by a consumption Euler equation. Bensal and Coleman (1996), Canzoneri and Diba (2005), and Canzoneri et al. (2006) introduce a spread by allowing bonds to provide transactions services. In Canzoneri and Diba's model, an expansionary open market operation increases the ratio of money to bonds; this in turn lowers the money market interest rate by changing the marginal transactions services of money and bonds. In practice, it remains to be seen if this prediction is empirically significant.