A New Interpretation of Money Growth Targeting and the Monetarist Experiment

Transcription

1 A New Interpretation of Money Growth Targeting and the Monetarist Experiment John W. Keating Andrew Lee Smith January 3, 06 Abstract Does Friedman s k-percent money growth rule guarantee a unique equilibrium outcome? We show analytically the answer to this question is sensitive to the method of aggregation. Focusing on broad measures of money, we find that fixing the growth rate of the true monetary aggregate will generate a unique rational expectations equilibrium. The true monetary aggregate is parametric. We show, however, that this determinacy result extends to a growth rule based on the non-parametric Divisia monetary aggregate. Interestingly, Friedman s proposal to fix the growth rate of the broad simple-sum monetary aggregate results in indeterminacy stemming from that aggregate s inaccuracy in tracking the true monetary aggregate. Determinacy regions of interest rate rules that permit a reaction to the growth rate of monetary aggregates are also examined. For these rules a novel Taylor principle is shown to hold provided an appropriate monetary aggregate is used. All theoretical results are presented in the framework of the canonical New-Keynesian model. Our findings provide a new interpretation of U.S. experience with monetary targets and the conclusions that were drawn. Volcker has repeatedly claimed the Fed followed a policy of "practical Monetarism" based on simple sum monetary aggregates. We find that the Fed s emphasis on these measures induced indeterminacy causing key macroeconomic variables to become more volatilie. We conclude Friedman was correct to claim this period fails as a test of Monetarism. But our explanation is not that the Fed didn t target money but rather that the central bank targeted a poor measure of it. Keywords: Friedman s k-percent Rule, Determinacy, Monetary Aggregates, Taylor Rules JEL codes: C43, E3, E40, E44, E5, E5,E58, E60 Preliminary work on this project was presented at the Missouri Economics Conference at the University of Missouri, (sponsored by the Federal Reserve Bank of St. Louis), the European Economics Association Meetings at the University of Gothenburg. We would like to thank participants for their helpful comments and Joe Haslag in particular for an insightful conversation. Keating thanks the Federal Reserve Bank of Kansas City for inviting him to be a visiting scholar which helped to complete work on this paper. The views expressed herein are solely those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Kansas City or the Federal Reserve System. Department of Economics, The University of Kansas. jkeating@ku.edu Research Department, Federal Reserve Bank of Kansas City. andrew.smith@kc.frb.org

2 Introduction The Federal Reserve first established targets for monetary aggregates in 970 and initially did not make them public. In 975 the Fed began setting target ranges for money growth and making them publically known on a timely basis. However, US experience with monetary targets was short-lived. An M target was last set in 986 and in July of 993 Chairman Greenspan testified before Congress that "M has been downgraded as a reliable indicator of financial conditions in the economy, and no single variable has yet been identified to take its place." Friedman (996) asserts that with this Congressional testimony "the Federal Reserve publically acknowledged that it had downgraded even its broad money growth targets a change that most observers of U.S. monetary policy had noticed long before." The argument about whether a central bank should implement policy by means of a monetary aggregate or an interest rate has seemingly been settled ever since for most economists. In the 996 Robbins lectures Blinder (999) (p.8) went even further by pronouncing "the death of Monetarism" based on this downgraded status of money along with various empirical findings that illustrated weak and/or unstable relationships between simple sum monetary aggregates and nominal GDP. 3 The closest the Fed ever came to Monetarist policies was during the period from October 979 to October 98 when it implemented a non-borrowed reserves operating procedure 4. The stated purpose of this new policy was to bring inflation down by means of improved control of money growth. Volcker (978) outlines the rationale for what he called "practical Monetarism." Hence, this period is typically referred to as the Monetarist Experiment. The Fed clearly was doing something radically different, according to the anomalous behavior of interest rates. Figure reports time-varying standard deviations for the 3-Month Treasury Bill Rate based on a 3-month centered window. This volatility measure is reported for a span of years that encompasses the period the Fed claimed concern for money growth rates. 5 Shaded areas indicate NBER recessions. Interest rate volatility increased dramatically at the start of the Monetarist Experiment, remained exceptionally high throughout, and fell almost as dramatically as that period ended. As is well known average inflation came down quite rapidly, consistent with the claimed purpose of Fed policy. However, output experienced a double-dip recession that was the most severe postwar downturn until the Great Recession. And quite unexpectedly the money growth rates did not fall dramatically, in contrast to the substantial declines in inflation and output growth. This unusual behavior caused some economists to conclude that the period illustrated failures of Monetarist policies and flaws in designing policy around money growth targets. Even some Monetarists seemingly aquiesced to a reduced role for money in monetary policy. 6 However, those targets were eventually made public. See Table in Kozicki and Tinsley (009), for example. Statement to Congress, Federal Reserve Bulletin 79 (September), p Blinder (999) quotes Gerry Bouey, a former governor at the Bank of Canada, as saying "we didn t abandon the monetary aggregates, they abandoned us." 4 According to Bernanke (006) 5 We stop the volatility figures in 999. As noted by Bernanke (006) the Fed decided to "discontinue setting target ranges for M and other aggregates after the statutory requirement for reporting such ranges lapsed in McCallum (008) McCallum (008) concludes: "It is only in its emphasis on monetary aggregates that

3 Volatility of 3-Month Treasury Bills internal targets established 3-month centered window target ranges made public Monetarist Experiment end of M targeting Fed downgrades monetary aggregates Figure : shaded areas are NBER-dated recession periods. There is not universal agreement that this period serves as a reliable test of Monetarism. Friedman (983) claims Fed rhetoric was Monetarist, but its behavior was not. He argues a Monetarist policy would have stabilized the economy, however, he notes that inflation, output, and money growth volatilities each increased substantially during this so-called Monetarist Experiment. Figures, 3, and 4, respectively, report volatilities for monthly measures of those series. Once again, each graph covers the period in which the Fed exressed some concern for money, is based on 3-month centered windows, and contains shaded areas that denote NBER recessions. Notice the spike in volatility for the monthly Chicago Fed National Activity Index which coincides with the begining of the Monetarist Experiment. While this is close to the start of a recession, this is the only recession in this sample period for which output volatility rises dramatically near the beginning of a downturn. Inflation volatility exhibits a similar spike around the beginning of the Monetarist Experiment, and this increase is much larger than occurs for any other recession during the sample period. But Friedman points out that the strongest cast against policy being in any way Monetarist comes from the dramatic rise in money growth volatility. M growth volatility rises impresssively at the the start of the Monetarist Experiement and is nearly always higher during that period than for any other years the Fed established M money growth targets. On average the volatility is about twice as large in the period. While Friedman and others have alleged the monetarism is not being widely espoused and practiced today. The other main precepts of Monetarism such as long-run monetary neutrality, short-run monetary non-neutrality, and the distinction between real and nominal interest rates are key features in most of today s macro models." 3

4 .5 Volatility of Chicago Fed National Activity Index 3-month centered window.00 internal targets established target ranges made public Monetarist Experiment end of M targeting Fed downgrades monetary aggregates Figure : shaded areas are NBER-dated recession periods. Fed was merely paying lip service to Monetarism, 7 Volcker has continuously maintained the policy at that time was based on Monetarist principles. 8 This paper reexamines monetary targeting and the Monetarist Experiement in the context of modern macroeconomic theory. Friedman (960) - in his classic (960) work, A Program for Monetary Stability - argued the economy can be stabilized by stabilizing the growth rate of a money at k-percent, and this policy proposal has received considerable attention by monetary theorist. 9 Surprisingly though, the question of whether this rule could be implemented with a broad monetary aggregate in a manner which delivers a unique rational expectations equilibrium remains unanswered. If such a rule fails in this capacity, fixing the growth rate of a monetary aggregate may induce instability by allowing for sunspot equilibria. This paper provides an answer to this question within the preferred framework of modern monetary policy analysis, a New-Keynesian model. 7 Friedman (98) states: " In most countries that I know about, lip service, not actual adherence, has been paid to Monetarist policies. Essentially every major country, and many a minor one, proclaims monetary growth targets annually and pronounces its determination to stick to them. However, any relation between the targets and actual monetary growth is purely coincidental. The United States is a particularly egregious case" 8 See the interview in Mehrling (00), for example. 9 For example, Ireland (996) has shown that a policy of holding the growth rate of money constant can be optimal from a welfare perspective when prices are set in advance. Additionally, Carlstrom and Fuerst (995, 003) explore the welfare and determinacy properties of money growth rules in flexible price models. 4

5 6 internal targets established Volatility of CPI Inflation target ranges made public 3-month centered window Monetarist Experiment end of M targeting Fed downgrades monetary aggregates Figure 3: shaded areas are NBER-dated recession periods. We show analytically that Friedman s k-percent rule can deliver a unique rational expectations equilibrium when the true monetary aggregate is used. Since that aggregate depends on deep structural parameters, such a rule is not simple in the sense defined by (Gali, 008). A simple rule which could actually be used by central banks, 0 would be to fix the growth rate of the Divisia monetary aggregate. We show that such a rule inherits the determinacy properties of the k-percent rule based on using the true monetary aggregate. Another well-known non-parametric aggregate is the simple-sum monetary aggregate which was the method used to compute the M measure Friedman advocated. Interestingly, if this aggregate is used in place of the true monetary aggregate,we find Friedman s k-percent rule is likely to result in indeterminacy stemming from the simple-sum s error in tracking the true aggregate. We conclude that Friedman was correct in his judgment that fixing the growth rate of a broad monetary aggregate may stabilize the economy. However, we show in the context of a New-Keynesian model with multiple types of assets which provide monetary services that aggregation by simple-sum is in general a flawed approach. Most similar to this part of our theoretical analysis is Evans and Honkapohja (003). They analyze determinacy properties of k-percent rules in a New-Keynesian model when money is modeled as a single monetary asset. In this case, money in the model can be interpreted as base money since it earns zero interest. They show numerically that fixing the growth rate of this measure of money yields a unique rational expectations equilibrium 0 Arguably, one reason Friedman advocated a k-percent rule was its simplicity. 5

6 0 Volatility of Simple Sum M Growth internal targets established target ranges made public 3-month centered window Monetarist Experiment end of M targeting Fed downgrades monetary aggregates Figure 4: shaded areas are NBER-dated recession periods. under a broad range of values. However Friedman s k-percent rule calls for the central bank to fix the growth rate of a high-level monetary aggregate, not base money. In order to analyze determinacy properties of broad monetary aggregates, we require a model like the one developed by Belongia and Ireland (04) which provides a role for both currency and deposits as competing sources of monetary services. We use this as our framework to analyze the determinacy properties of monetary policy rules with broad money. In addition to providing analytic representations of the determinacy regions of various monetary aggregate growth rules, we also analyze interest rate rules reacting to inflation and the growth rate of monetary aggregates. Once again we find that "measurement matters (Belongia, 996) from a determinacy standpoint. Interest rate rules reacting to the growth rate of either the true monetary aggregate or the Divisia monetary aggregate satisfy a novel Taylor principle for monetary aggregates. The simple-sum monetary aggregate s error in tracking the true aggregate prevents an interest rate rule reacting to simple-sum from having a similar determinacy region. Instead, such rules have very small determinacy regions. Following this introductory section we present our model which is a standard New- Keynesian framework that allows for two different types of assets to provide monetary services. Section 3 examines determinacy properties of k-percent rules and compares outcomes obtained with different monetary aggregates. The fourth section of the paper investigates interest rate rules that allow for a reaction to monetary aggregates. Again we address determinacy properties associated with alternative monetary aggregates. Section 5 performs a 6

7 numerical analysis with more complicated monetary policy rules that allow policy to react to more information making it impossible to derive analytical results. Section 6 concludes the paper. A New-Keynesian Model We consider a canonical version of the linearized New-Keynesian (NK) model. Following Woodford s (003) notation we assume all exogenous disturbances are bounded in amplitude by ξ. All variables with a " " over them denote non-stochastic steady-state values. Variables with a "~" over them denote log-deviations from the non-stochastic steady state. As usual, this model consists of 3 non-policy equations when money is included in the model. ỹ e t = E t {ỹ e t+} [ r t E t { π t+ } r e t ] () π t = κỹ e t + βe t { π t+ } () m t p t = ỹ e t ũ t + υ t + z t (3) Equation () is the dynamic IS curve, equation () is the New-Keynesian Phillips Curve (NKPC) and equation (3) is the demand for the monetary aggregate. In the above equations ỹ e t = y t z t denotes the efficient output gap and r e t = E t { z t+ } E t {ã t+ } denotes the efficient interest rate. The variables z t, ã t, and υ t represent exogenous technology, preference and money-demand disturbances respectively. Each stochastic disturbance follows a stationary AR() process and is driven by orthogonal white noise shocks. This assumption is only made for convenience as all the results presented hold for general stationary processes. We depart from the textbook analysis of money. Following Belongia and Ireland (04) we model broad money so that (3) is the demand for the aggregate service flow from currency and interest bearing deposits. For this reason, (3) includes the user-cost of the monetary aggregate u t which in general depends on a vector of interest rates. Modeling money as a broader measure turns out to play a key role in our determinacy results. In addition to examining the effects of monetary aggregate growth rules we will also investigate interest rate rules which may react to the growth rate of a monetary aggregate. A constant elasticity of substitution (CES) function: m t = [ν ω n ω ω t ] ω + ( ν) ω ω ω d ω t (4) is assumed for the true monetary aggregate where n t and d t are non interest bearing currency and interest bearing deposits respectively. Meanwhile, ω > 0 is the elasticity of substitution between currency and deposits and 0 < ν < governs the steady-state shares of currency and deposits. The CES function is homogeneous of degree one in terms of n t and d t, an attractive property since that means changing both n t and d t by a certain percentage will change the monetary aggregate by exactly that same percentage. Of course, simple-sum also has that property. And when ω, m t = n t + d t. Hence, the simple-sum monetary aggregate is the appropriate monetary aggregate when currency and deposits are perfect substitutes. More generally, when ω <, currency and deposits are not perfect one-for-one substitutes, and then a simple-sum aggregate will not equal the true aggregate. 7

8 The aggregate dual user-cost of money is correspondingly defined by ( ) ( rt ω rt r u t = ν d ) ω t + ( ν) r t r t ω. (5) where r t is the gross nominal interest rate on loans, r d t is the deposit rate and u n t = (r t ) r t (6) is the user cost of currency. Following the structural model of Belongia and Ireland (04) with profit maximizing banks, the user cost of deposits can be expressed as u d t = r t r d t r t = (r t )τ t + x t r t (7) in which τ t and x t are exogenous financial disturbances that follow stationary stochastic processes with orthogonal white noise shocks. Specifically, τ t represents reserves demand disturbances with a mean τ and x t represents deposit cost disturbances with a mean x. To remain empirically relevant, we will assume throughout the paper that the stochastic processes are bounded so that at all times r t > r d t > ensuring that deposits are cheaper than currency. A log-linear approximation to (5) is given by ũ t = η r r t + η τ τ t + η x x t + O( ξ ) (8) where all the coefficients are positive. Substituting (8) into (3) yields a five variable dynamic model with three equations (, and 9). m t p t = ỹ e t η r r t η τ τ t η x x t + υ t + z t (9) The dimension of the model can be reduced by one variable after defining real balances as lt m t p t. As is standard, we close the model with a specification of monetary policy. 3 Friedman k-percent Rules In this section we consider the performance of various money growth rules. Initially we close the model with a constant money growth rule for the true monetary aggregate: m t = 0. (0) Such k-percent rules have been examined in New-Keynesian models by Evans and Honkapohja (003) and Gali (008) where they were shown to deliver a unique REE. However, these studies assume the existence of a single monetary asset rather than a variety of such assets. The multi-asset case is more relevant given the variety of monetary assets that pay different rates in the modern economy. Since previous work leaves unsettled the question of whether the determinacy properties extend to monetary measures that aggregate different types of monetary assets, it is unclear whether proposals to fix the growth rate of a broad aggregate, such as M growth as recommended by Friedman (9??), will lead to a determinate outcome. 8

9 This section shows three main results. First, the k-percent rule is always determinate when applied to the true monetary aggregate, regardless of the magnitude of the interest semi-elasticity. Second, this result of general determinacy extends to rules when the true aggregate is replaced by the Divisia monetary aggregate. Finally, we show that this strong result does not extend to the simple-sum monetary aggregate. Instead, a constant simplesum growth rule is indeterminate unless the interest semi-elasticity is sufficiently large. Consider the dynamic system consisting of (), (), (9) and (0). Proposition. For any 0 < β <, for any κ > 0 and for any η r 0 if the central bank follows the policy rule m t = 0 then there exists a unique REE. The proof of this proposition is in the appendix, and unless stated otherwise, all proofs for remaining propositions, lemmas, and corollaries are found in the appendix. Proposition shows that a constant monetary aggregate growth rule always delivers a unique REE if based on the true monetary aggregate. This result extends the findings of Evans and Honkapohja (003) and Gali (008) to monetary aggregates. Proposition is of little use to central banks which do not know the underlying parameters of the monetary aggregate. To circumvent this problem, suppose the central bank fixes the growth rate of the Divisia monetary aggregate, developed initially by Barnett (980), instead of fixing the growth rate of the true monetary aggregate. Definition. The growth rate of the Divisia monetary aggregate is given by: ( s ln(gt divisia n ) = t + s n ) ( ) ( t nt s d ln + t + s d ) ( ) t dt ln, () n t where s n t and s d t are the expenditure shares of currency and interest bearing deposits respectively defined by: s n (r t )n t t = and s d (r t )n t + (r t rt d t = s n t. () )d t The Divisia monetary aggregate is the expenditure share-weighed growth rate of currency and deposits. In contrast to the true aggregate, the Divisia aggregate is non-parametric and depends only on observable information. An important advantage of Divisia for policymakers is it doesn t require parameter estimation, in contrast to the true monetary aggregate. The following lemma shows that for the general CES specification of m t, the Divisia monetary aggregate tracks the growth rate of the true monetary aggregate to first order accuracy without error. Lemma. For any 0 < ν < and for any ω > 0, the difference between the Divisia monetary aggregate and the true monetary is given by ln(g divisia t ) ln(m t ) = O( ξ ). A central bank could estimate the parameters in equation 4, the monetary aggregator function, but that would induce parameter uncertainty from sampling error. Moreover, this approach leaves a central bank exposed to questions and criticisms with regards to how they estimate the parameters. Divisia avoids these concerns. 9 d t

10 The accuracy properties of the Divisia monetary aggregate are well known in Index number theory. Most notably, Divisia (96) showed in continuous time, the Divisia aggregate tracks any linearly homogenous function without error. Moreover, Diewert (976) classified the discrete time Divisia monetary aggregate defined above as superlative - meaning that it can track the growth rate of any twice differentiable linearly homogenous function up to second order accuracy without error. This lemma shows that for the CES function in particular, a linear approximation to the Divisia monetary aggregate tracks the true monetary aggregate up to first order accuracy without error. This result will prove useful in analyzing local determinacy which requires only a linear approximation to the non-linear model. Formally, suppose the central bank follows the rule g divisia t = 0 (3) in place of (0). The dynamic system consists of (), (), (9) and (3), and we obtain the following result. Corollary. For any 0 < β <, for any κ > 0 and for any η r 0 if the central bank follows the policy rule g t divisia = 0 then there exists a unique REE. The proof follows immediately from combining Lemma and Proposition. Corollary shows the central bank may replace the true aggregate with the Divisia monetary aggregate without any change in determinacy. This result obtains because determinacy is a local condition and Lemma shows that locally, the Divisia monetary aggregate exactly tracks the growth rate of the true monetary aggregate. Therefore, determinacy properties for growth rules based on the true monetary aggregate are inherited by rules which instead use the Divisia monetary aggregate. However, the same can not be said for the more common simple-sum monetary aggregate. Definition. The growth rate of the simple-sum monetary aggregate is defined by ( ) ln(g simple sum nt + d t t ) = ln. (4) n t + d t The simple-sum aggregate treats currency and interest bearing deposits as one for one perfect substitutes. Standard microeconomic theory dictates in such a case only the cheapest monetary asset would have a positive demand in equilibrium. Since this is not the case in this model, nor reality, it is not surprising the simple-sum aggregate will locally track the true aggregate with error. The following lemma defines and quantifies this error. Lemma. For any 0 < ν <, for any ω > 0 and for any ( τ, x) satisfying ( r ) > ( r ) τ + x so that ū n > ū d > 0, the difference between the growth rate of the simple-sum monetary aggregate and the growth rate of the true monetary aggregate is given by ln(g simple sum t ) ln(m t ) = ψ r (ω) r t ψ x (ω) x t ψ τ (ω) τ t + O( ξ ) (5) where ( ψ r (ω), ψ x (ω), ψ τ (ω) ) R 3 ++ with lim ω ψ r (ω) = ψ x (ω) = ψ τ (ω) = 0. 0

11 Intuitively, any change in the relative prices of currency and deposits results in a substitution between these assets from the household. Since the simple-sum monetary aggregate treats these assets as perfect one for one substitutes it is not able to internalize these relative changes in the service flow from the monetary aggregate. Only in the limiting case of perfect substitutes does this error disappear. From a determinacy standpoint, simple-sum s error in tracking the true monetary aggregate arising from the exogenous financial shocks is irrelevant. However, the error that depends on the endogenous interest rate - r t - will influence the determinacy region of a rule utilizing the simple-sum monetary aggregate. In particular, the following corollary summarizes determinacy properties when instead of (0) the policy rule is: g simple sum t = 0, (6) with the dynamic system now consisting of (), (), (9) and (6). Corollary. For any 0 < β <, for any κ > 0 if the central bank follows the policy rule g simple sum t = 0 then there exists a unique REE if and only if ψ r (ω) < η r +. Importantly, when using the simple-sum monetary aggregate in place of either the true monetary aggregate or the Divisia monetary aggregate a constant monetary aggregate growth rule may not be determinate. Indeterminacy will occur when the error coefficient (ψ r (ω)) is large relative to the interest semi-elasticity of money demand (η r ). This key money demand parameter varies widelyin the empirical literature. For example Ball (00) estimates η r =.05 while Ireland (009) estimates η r =.9. Given the substantial range of estimates for η r, central banks may be understandably averse to implementing Friedman s k-percent rule based on a simple-sum measure. Furthermore, Section 6 of this paper presents estimates of ψ r (ω) which combined with even the largest plausible estimates of η r suggest that the determinacy condition in Corollary is unlikely to be satisfied. In contrast, a k-percent rule with Divisia yields no indeterminacy concerns whatsoever. 4 Inflation Targeting with Money Growth In reality, the Volcker Fed was attempting to bring down the rate of inflation by lowering the growth rate of money. Unfortunately, analytical results in the New Keynesian model for rules which specify the use of a money growth target to achieve an inflation target are out of reach. However, the intuition for how indeterminacy can arise in such rules can be gleaned from an endowment economy with no nominal rigidities. In this Fisherian model of the price level, the non-policy block of the model can be described by the Fisher equation, r t = ( ρ e )ẽ t + E t { π t+ }, (7) where ẽ t = ρ e ẽ t + ε e t is the exogenous endowment which follows an autoregressive process. We assume sufficient seperability between the consumer s preferences for consumption goods and monetary services so that the Fisherian model can be augmented with a money demand equation, m t p t = ẽ t ũ t. (8)

12 Linearizing the user-cost of money ũ t along the same lines as in equation (7) yields a typical money demand equation which depends on output and the nominal interest rate (along with various financial sector disturbances), m t p t = ẽ t η r r t η τ τ t η x x t + υ t. (9) Equations (7) and (9), along with a description of monetary policy close this simple model of the price level. 4. Indeterminacy when using Money to target Inflation One of the earliest monetary theories posited that a central bank can achieve an inflation target by adjusting the growth rate of money. The so called, Quantity theory of money posits a one for one long-run relationship between money growth and inflation. Despite its theoretical appeal, quantity theory type relationships have only been exploited in practice in the U.S. during the Monetarist Experiment. In this section, we discuss how implementing a money-growth instrument regime can lead to multiple equilibria, depending on the the monetary aggregate that is targeted. Suppose the central bank implements a money growth rule to target the rate of inflation, m t = φ π π t (0) where φ π > 0 under the plausible mechanism whereby the central bank slows the growth rate of money to bring down inflation. The following proposition shows this policy rule always anchors expectations on a unique equilibrium. Proposition. For any 0 < β < and for any η r 0 if the central bank follows the policy rule m t = φ π π t with φ π 0 then there exists a unique REE. This result builds on Proposition to the extent that φ π > 0 implies that (0) is not a constant money growth rule but instead a money instrument rule which seeks to adjust the growth rate of M t to achieve an inflation target. Interestingly, there is no lower or upper bound on the aggressiveness with which policy makers have to react to deviations of inflation from their target. Intuitively, the lack of a lower bound for these rules stems from the fact that even a constant money growth rule requires policy makers to actively adjust the amount of outside money to achieve the desired growth rate of the broad aggregate. However, the broad aggregate the central bank chooses to target is important. The following corollaries to Proposition show the Divisia aggregate has the same determinacy property as the inflation targeting rule which uses the household s monetary aggregate. Meanwhile, the simple-sum monetary aggregate is subject to a lower bound on the inflation response which stems entirely from its error in tracking m t. Corollary 3. For any 0 < β < and for any η r 0 if the central bank follows the policy rule gt Divisia = φ π π t with φ π 0 then there exists a unique REE. The proof to Corollary 3 follows immediately from Lemma and Proposition. To reiterate, since determinacy is a local property of a rational expectations model and since the Divisia monetary aggregate tracks m t locally without error, the determinacy conditions are

13 interchangeable between the instruments m t and g t Divisia. Using similar reasoning, policy rules which use the simple-sum aggregate will not typically share the same determinacy regions as policy rules which use the household s, or the Divisia, monetary aggregate. The following corollary quantifies the difference between these determinacy regions. Corollary 4. For any 0 < β < and for any η r 0 if the central bank follows the policy rule g simple sum t = φ π π t with φ π 0 then there exists a unique REE if and only if φ π > (ψ r (ω) η r ). 4. Testing for Indeterminacy In this section, we use the determinacy conditions from our theoretical model to test the likelihood with which the conditions for determinacy were satisfied during the Monetarist Experiment. To do so, we follow a similar strategy to Clarida, Gali, and Gertler (000) and Coibion and Gorodnichenko (0) by developing the conditions for determinacy from within a theoretical model and then estimating the model s structural equations to get a sense if the estimated parameter values place the economy inside the determinacy region. In what follows we assume the Federal Reserve was following a rule of the form g simple sum t = φ π π t from October of 979 to October 98. We however, will let the data speak as to the value of φ π. Corollary 4 shows that knowledge of φ π itself is not enough to determine if the Monetarist Experiment subjected the economy to sunspots. We must also elicit measures of ψ r (ω) and η r. Table : Econometric Estimation of a Monetary Policy Reaction Function g simple sum = β 0 + β π t + ε mp t 979:0-98:0 ˆβ0 ˆβ No Instruments OLS Estimates Standard-Error Instrument Set A GMM Estimates Standard-Error Instrument Set B GMM Estimates Standard-Error The standard errors are computed as in Newey and West (987) using T 4 lags. Constant Volatility Estimates Some text here. Some text here. 3

14 Table : Econometric Estimation of the Simple-Sum Error Term ln(n t + d t ) ln(m t ) = β 0 + β r t + β x t + β 3 τ t + ε t 979:0-98:0 ˆβ0 ˆβ ˆβ ˆβ3 Measuring r t with the 3-Month T-Bill Rate No Instruments OLS Estimates Standard-Error Instrument Set A GMM Estimates Standard-Error Instrument Set B GMM Estimates Standard-Error Measuring r t with the C & I Loan Rate No Instruments OLS Estimates Standard-Error Instrument Set A GMM Estimates Standard-Error Instrument Set B GMM Estimates Standard-Error The standard errors are computed as in Newey and West (987) using T 4 lags. 4.3 Targeting Inflation with Money Although many studies have focused on the likelihood that monetary policy induced instability into the economy over this period of time using DSGE models, most assume the central bank s behavior can still be described by an interest rate rule, possibly with breaks around this period (Clarida, Gali, and Gertler, 000; Lubik and Schorfheide, 004; Coibion and Gorodnichenko, 0). However, these works implicitly acknowledge the difficulty in analyzing interest rate rules from by estimating interest rate rules from 98 onwards. Coibion and Gorodnichenko (0) explicitly acknowledge the problem of using an interest rate to describe monetary policy during the Monetarist Experiment. In their baseline estimation, they drop the period from from their sample and acknowledge that during this time, the Federal Reserve officially abandoned interest rate targeting in favor 4

15 of targeting monetary aggregates. All of these aforementioned studies find the probability of indeterminacy to be high over the period while assuming an interest rate instrument regime. In what follows, we seek to use the above insights from the theoretical model to shed new light on this period of time. In particular, we provide econometric evidence which, when placed within our theoretical model, is consistent with the hypothesis economic instability was induced by monetary policy over this time but not because of a passive reaction to inflation in an interest rate rule. Instead, we find that multiple equilibria emerged at that time from a measurement error in the monetary aggregate the Federal reserve was targeting. We view our interpretation of this period as more appealing than previous interpretations for three reasons. First, there is substantial narrative evidence from FOMC transcripts the Federal Reserve was targeting money growth, not interest rates from Second, our interpretation of this period explains why the Federal Reserve had great difficulty achieving their money growth targets during the Monetarist Experiment. Finally, our interpretation of this time period bridges the empirical work of Sims and Zha (006) who find statistical evidence that favors including money in the description of monetary policy during this time period with the previous work on indeterminacy in DSGE models. Our results are entirely complimentary to the previous literature (Clarida, Gali, and Gertler, 000; Lubik and Schorfheide, 004; Coibion and Gorodnichenko, 0). Our work is consistent with the finding the economy was indeterminate before 979 due to passive monetary policy high trend inflation and determinate after 98 due to more active monetary policy and lower trend inflation. We simply fill the gap from 979 to98 to understand how the Monetarist Experiment was an indeterminate regime despite falling inflation rates and a very aggressive anti-inflation stance by Paul Volcker. One shortcoming of our analysis is that we don t fully model the operating procedure used to target money growth. In particular (sources here) argue the Federal Reserve was using non-borrowed reserves to achieve their desired money growth rates which was a means to the end goal of lower inflation. Without a more intricately modeled financial sector we resort to the short-cut common in DSGE models which is to assume direct control of intermediate targets (such as monetary aggregates and interest rates) and place them on the left hand side of policy reaction functions and place final targets (such as inflation and output) on the right hand side. More fully modeling the monetary transmission mechanism from the instruments controlled by the Federal Reserve (such as reserves and government bonds) to final targets the FOMC seeks to achieve (such as inflation and output) as in Ireland (0) is a promising endeavor; however, our work represents an important step for understanding the behavior of the economy during the Monetarist Experiment. 5 Interest Rate Rules with Money In this section we consider interest rate feedback rules which react to lagged interest rates and the growth rate of a monetary aggregate. When Friedman proposed the k-percent rule, he also advocated changes in bank regulation which would make the rule easily implemented. For example, Friedman (960) argued for increasing the reserve requirements to 00% to make the monetary aggregates more easily controllable. Friedman (960) went on to say that: 5

16 Table 3: Econometric Estimation of a Monetary Policy Reaction Function a g simple sum = β 0 + β π t + ε mp t 979:0-98:0 ˆβ0 ˆβ Instrument Set A OLS Estimates Standard-Error Instrument Set B OLS Estimates Standard-Error a The standard errors are computed as in Newey and West (987) using T 4 lags. This [constant money growth] is not, under our present System, an easy thing to do. It involves a great many technical difficulties and there will be some deviations from it. If the other changes I suggested were made in the System, it would make the task easier; but even without those changes, it could be done under the present System. Under a fractional reserve banking system a pragmatic way of targeting the growth rate of money is with an interest rate reaction function whereby the central bank adjusts the interest rate as needed to correct deviations of money growth from the desired k-percent target. More recently a number of studies have included nominal money growth in interest rate rules such as Canova and Menz (0), Sims and Zha (006) and Fahr, Motto, Rostagno, Smets, and Tristani (03). Fewer papers have included Divisia monetary aggregates in interest rate feedback rules. One that has is Belongia and Ireland (0) which estimates structural VAR s with such rules. Keating and Smith (03) find that interest rate rules that react to Divisia may be optimal from a normative perspective if financial shocks are present and the natural rate of interest is unobservable. This result is consistent with the assertion of McCallum and Nelson (0) and Andres, Lopez-Salido, and Nelson (009) that nominal money growth provides valuable real-time information regarding the natural rate of interest. Despite this interest in rules that allow the policy rate to react to money, there are no results on the determinacy properties of such rules to guide policy makers. With this in mind, assume the central bank sets the policy rate according to r t = φ r r t + φ m m t. () Using () to close the dynamic model that still contains (), () and (9), we have the following result. Proposition 3. For any 0 < β <, for any κ > 0 and for any η r 0, if the central bank follows the policy rule r t = φ r r t + φ m m t with φ m 0 and φ r 0 then there exists a unique REE if and only if φ m + φ r >. 6

17 Intuitively, this result extends the Taylor Principle to monetary aggregates. Expectations will remain well-anchored if the central bank reacts more than one for one to nominal aggregate-money growth, in the long-run. Proposition provides a sufficient condition for determinacy that is independent of the magnitude of the interest semi-elasticity of money demand which is important given that the estimates of this parameter vary significantly in the empirical literature. Despite the neutrality of the result with regard to η r, this rule requires that a central bank measure the unobservable true monetary aggregate. To circumvent this issue suppose instead the central bank replaces the true monetary aggregate with the Divisia monetary aggregate r t = φ r r t + φ d g divisia t. () The policy rule in () could more easily be implemented by a central bank compared to () as the interest rate is responding to observable information and does not entail the estimation of any deep structural parameters. Furthermore, the following corollary to Proposition shows that a central bank can replace m with g t divisia without changing the determinacy condition. Corollary 5. For any 0 < β <, for any κ > 0 and for any η r 0 if the central bank follows the policy rule r t = φ r r t + φ d g t divisia with φ d 0 and φ r 0 then there exists a unique REE if and only if φ d + φ r >. The proof of this corollary follows immediately by combining Lemma with Proposition. This result is significant in terms of actually implementing interest rate rules reacting to money growth. Most notably, central banks can use the non-parametric Divisia aggregate in interest rate feedback rules like () and guarantee determinacy by setting φ d >. The same can not be said, however, for the well-known simple-sum monetary aggregate. In particular, the following corollary summarizes determinacy properties when the policy rule in (3) is used instead of () or () r t = φ r r t + φ ss g simple sum. (3) Corollary 6. For any 0 < β <, for any κ > 0 and for any η r 0 if the central bank follows the policy rule r t = φ r r t + φ ss g simple sum with φ ss 0 and φ r 0 then there exists a unique REE if and only if φ r + φ ss φ ss [ψ r (ω) η r ] > φ ss [ψ r (ω) η r ]. The proof is derived from Lemma and Proposition. This result highlights how once again simple-sum and Divisia monetary aggregates diverge in terms of determinacy properties. Namely, the determinacy region for interest rate rules reacting to the simple-sum monetary aggregate depend on simple-sum s error in tracking the true monetary aggregate. Intuitively, if ψ r (ω) is relatively large compared to the interest semi-elasticity of money demand then an upper and lower bound is placed on φ ss for determinacy. Meanwhile, if ψ r (ω) is relatively small compared to the interest semi-elasticity of money demand then only a lower bound is placed on φ ss for determinacy. For example, in the extreme case where the simple-sum and Divisia aggregates coincide (i.e. ω ), Corollary 3 shows that φ ss > 7

18 would be sufficient for determinacy. However, in the more general case, determinacy under the interest rate rule reacting to simple-sum may have a relatively small determinacy region as summarized below for the case in which ψ r (ω) η r >. The following numerical section of the paper provides empirical justification for this inequality. Table 4: Determinacy Regimes Under r t = φ r r t + φ ss g simple sum t Regime a Determinacy Conditions b Passive φ r + φ ss < φ ss > ψ r(ω) η r φ ss > φ ψ r(ω) η r r φ ss > + φ (ψ r(ω) η r) (ψ r(ω) η r) r Active I φ r + φ ss > φ ss < ψ r(ω) η r φ ss < φ ψ r(ω) η r r Active II φ r + φ ss > φ ss > ψ r(ω) η r φ ss < φ ψ r(ω) η r r φ ss < + φ (ψ r(ω) η r) (ψ r(ω) η r) r a We use the terms "Passive" and "Active" similar to Leeper (99) to describe a monetary regime where φ r + φ ss < or φ r + φ ss > respectively. b All of these conditions are sufficient for the existence of a unique REE under the conditions stated in Corollary 4 and the additional assumption that ψ r (ω) η r > The analysis in the following section shows this to be the empirically relevant case for the U.S. economy. To get an idea of how likely determinacy is under an interest rate rule reacting to the simple-sum aggregate Figure plots the determinacy region under the numerical calibration presented below. There are regions of determinacy. Using Leeper s (99) terminology, one region is in the passive regime while the other is in the active regime. What stands out the most is how unlikely the condition presented in Corollary 4 is satisfied. Under this interest rate rule with simple-sum money determinacy occurs within such a narrow band that it is much more the exception than the norm. 6 Numerical Analysis Potentially more realistic policy rules may include not only a response to the growth rate of a monetary aggregate but also a response to inflation. Unfortunately this slightly more complicated rule places analytic results out of reach so we must resort to numerical analysis of determinacy. Many of the parameters in the model are standard so we take their values from the previous literature. We vary the remaining parameters over a reasonable range to understand how determinacy may, or may not, be achieved for the US economy. More precisely, we set β =.99 and κ =.3. The two key non-policy parameters relevant for 8

19 5 Determinacy Region Under r t = ρ r r t + φ ss µ Simple Sum t φ ss ρ r Figure 5: The shaded areas is determinate while the white area is indeterminate. The determinacy region above is graphed for η r =.9 with the other parameters fixed at the values presented in Table 6. determinacy of the simple-sum rules are η r - the interest semi-elasticity of money demand and ψ r (ω) - simple-sum s endogenous error in tracking the true monetary aggregate. There is a large literature estimating the interest semi-elasticity of money demand. Ball (00) estimates η r =.05 and Ireland (009) estimates η r =.9. These important papers provide a reasonable range of parameter values: η r [0.05,.9]. To gain some insight regarding the size of simple-sum s error term in tracking the true monetary aggregate, we interpret ln(n t + d t ) ln(m t ) = β 0 + β r t + β x t + β 3 τ t + ε t (4) as a linear econometric relationship with the error in this equation assumed to be a stationary stochastic process. The linear functional relationship is all that is necessary to address determinacy issues. The estimate of β will provide a calibrated value for ψ r (ω). We estimate (4) using monthly data from 967:0 to 0:09 from the St. Louis Fed s FRED database. We use data on simple-sum M and the M MSI (Divisia) series to form the dependent variable. As for the independent variables we first construct a reserves ratio series - τ t - using the St. Louis Adjusted Reserves series and the non-currency components of M. We 9

20 then use this and the commercial and industrial loan rate, the M own rate and the structural equation in (7) to back out a time series for x t. In this simple model, r t is simultaneously equal to the loan rate, the benchmark rate and the policy rate. Hence, interpreting r t empirically is not straightforward. For this reason, we estimate (4) twice where first we use the 3-Month Treasury Bill secondary market series for r t and then we examine the robustness of our estimates by using the commercial and industrial loan rate for r t. Table 5 reveals the point-estimates are surprisingly similar. Table 5: Econometric Estimation of simple-sum Error Term a ln(n t + d t ) ln(m t ) = β 0 + β r t + β x t + β 3 τ t + ε t 967:0-0:09 ˆβ0 ˆβ ˆβ ˆβ3 Measuring r t with the 3-Month T-Bill Rate OLS Estimates Standard-Error Measuring r t with the C & I Loan Rate OLS Estimates Standard-Error :0-007:09 ˆβ0 ˆβ ˆβ ˆβ3 Measuring r t with the 3-Month T-Bill Rate OLS Estimates Standard-Error Measuring r t with the C & I Loan Rate OLS Estimates Standard-Error a The standard errors are computed as in Newey and West (987) using T 4 lags. The results in Table 5 support the theoretical findings in Lemma as the estimated coefficients all have the predicted sign. Moreover, the key parameter for the determinacy of the constant simple-sum growth rule is estimated to be significantly larger than the interest semi-elasticity estimates of Ball (00) and Ireland (009) over both the full-sample and the sub-sample. To give further credence to the above estimation, the value of ˆβ over the 984:0-007:09 period is nearly identical to the value of ψ r (ω) that is implied by the calibration put forth in Belongia and Ireland (04). For our numerical analysis we use the smaller estimate obtained from this sample period. The full sample estimates include heterogeneous monetary regimes and the financial crisis. For this reason, we also estimate equation (4) over a sub-sample which roughly captures the Great Moderation and stops just before the crisis. The Fed s policy over that sub-period can be characterized by a predictable interest rate rule with a low inflation target. 0