Evaluating McCallum's rule when monetary policy matters

TOM STARK DEAN CROUSHORE Federal Reserve Bank of Philadelphia Philadelphia, Pennsylvania

Evaluating McCallum's Rule When Monetary Policy Matters* This paper provides new evidence on the usefulness of McCallum's proposed rule for monetary policy. The rule targets nominal GDP using the monetary base as the instrument. We analyze the rule using three very different economic models to see if the rule works well in different environments. Our results suggest that while the rule leads to lower inflation than there has been over the last 30 years, instability problems suggest that the rule should be modified to feed back on the growth rate of nominal GDP rather than the level.

1. Introduction

The standard approach used by economists to discuss monetary policy is to write down a model of the economy, demonstrate what optimal monetary policy is in the model, and show how different the optimal policy is from the policy that was actually in place. Since each economist engaged in such a task typically comes up with a unique model, we have a large number of views of optimal monetary policy, all of which are model-specific. After decades of following this strategy, it's time for us to admit that it isn't working. Fundamentally, macroeconomists disagree about the forces that drive the economy, and they are unlikely to ever discover the "true" model. Thus, they are unlikely to find "'the" optimal rule on which to base monetary policy. Just as with research on finding a cure for the common cold, the expected payoff to continued research on discovering the right model has become very small. This lack of a generally accepted model creates a key tension within the Federal Reserve since neither policymakers nor economic staff at the *We thank Bennett McCallum, Ray Fair, Todd Clark, Adrian Fleissig, Alfred Guender, Joe Mason, and two anonymous referees for comments on an earlier draft. We also thank Herb Taylor for suggesting some modifications to his original model mad Jordi Gali for helping us with some estimation issues. Thanks also to participants at the 1995 ASSA meetings, the 1996 SEA meetings and to the staff at the Federal Reserve Bank of Philadelphia for their suggestions. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of Philadelphia or of the Federal Reserve System.

Journal of Macroeconomics, Summer 1998, Vol. 20, No. 3, pp. 451-485 Copyright © 1998 by Louisiana State University Press 0164-0704/98/$1.50

451

Tom Stark and Dean Croushore

Fed share a common theory of macroeconomies. Thus, there are different views of what monetary policy ought to do at any point in time given the current state of the economy. As a result, policy outcomes depend on consensus-building, for it is seldom the ease that Keynesian, monetarist, new Keynesian, and new classical models point in the same direction. MeCallum (1987, 1988, 1990, 1993, 1995) suggests a new approach. He has proposed a specific rule for the Federal Reserve to follow in setting monetary policy, which he developed outside the confines of a single model, and which he suggests would work well in many different economic models. The rule would have the Fed target the level of nominal GDP using the monetary base as its instrmnent. Using a variety of empirical models, MeCallum has shown that, had the rule been in place instead of the actual discretionary monetary policy, nominal GDP growth would have been substantially lower over the past 30 years, implying a lower average rate of inflation. These results have been verified in additional models by Judd and Motley (1991, 1992, 1993). ~ If MeCallum's proposed rule works well in all of these types of models, polieymakers or economists of any theoretical predisposition could subscribe to the rule. This paper addresses four aspects of McCallum's proposed rule: (1) it expands the set of models used to evaluate MeCallmn's rule; (2) it shows how impulse responses can be used to evaluate the stabilization properties of MeCallum's rule in our models; (3) it provides a more detailed accounting of how the rule affects the endogenous variables in our models; and (4) it shows how targeting the growth rate of nominal GDP may be superior to targeting the level of nominal GDP. Following MeCallum and Judd-Motley, we also use our models to conduct stochastic simulations, with McCallum's rule determining the monetary-policy response to disturbances. One problem with the studies of MeCallum and Judd-Motley is that they draw upon economic models designed solely for the purpose of evaluating the rule; in particular, in their structural models they usually estimate equations relating the monetary base directly to nominal GDP. The main idea of this article is to expand the set of economic models on which MeCallum's rule has been tested. In particular, we examine economic models developed for purposes other than testing MeCallum's rule, in which the relationship between the monetary base and nominal GDP is less direct. If the rule performs well in these models, such evidence will be more convincing than finding that the rule works well in models designed specifically to test it. 1However,the results have been challenged by Hess, Small, and Brayton (1992). Also, a number of variations on McCanum's rule have been tested, including the use of the federal funds rate instead of the monetarybase as the policyinstrument. 452

Evaluating McCaUum's Rule The second innovation of our paper is to use impulse responses to analyze the problem of model instability. McCallum, Judd-Motley, and others working in this literature have found eases where either a target variable (like nominal GDP) or the instrument variable (the monetary base) suffer from explosive oscillations, As we show below, it isn't always clear how to identify this instability. We propose looking at impulse responses. Third, we evaluate the rule over a larger number of dimensions than previous researchers have done. MeCallum has argued that economists do not yet have an adequate understanding of the breakdown of nominal GDP into prices and real GDP. Thus, he proposes analyzing his rule by simulating its effect on the target variable, nominal GDP. While we are sympathetic to MeCallum's view, we also believe that polieymakers have a particular interest in the behavior of prices and real output. Thus, in our evaluation, we report how the rule affects inflation and real output (using a battery of summary statistics), as well as how it affects MeCallum's preferred variable, nominal GDP. Finally, we find that MeCallum's rule is highly effective in controlling the long-run rate of growth in nominal GDP and the price level. However, the tendency of the original rule to yield explosive oscillations in some models and large (but stable) short-run fluctuations in real GDP in other models leads us to prefer a modified version of the rule. Thus, our findings suggest that targeting the growth of nominal GDP is preferable to targeting the level of nominal GDP. The models we use are all models in which monetary policy has real effects. We don't examine real-business-cycle models or other types of flexible-price models because the real effects of monetary policy in those models tend to be nonexistent or very small. As such, MeCallum's rule no doubt works quite well in maintaining a low rate of inflation with no significant consequences for real output. So the question we're really seeking to answer in this paper is, in the class of models in which monetary policy potentially has real effects, how well does MeCallum's rule work? The Lueas critique can be directed powerfully to this research. Following previous literature, we conduct eounterfaetual policy experiments while leaving behavioral equations unchanged, which is precisely what Lueas warned about. There is no defense against this critique except to build models that can deal fully with behavioral changes in response to policy changes. None exist yet, though much research is under way on new classical models with money. So we must treat our results as preliminary until economists develop more complete Inacroeeonomie models in which money matters. In the next section we describe MeCallum's rule and the three models we use to examine it. The following section introduces the stochastic simulation technique we use to evaluate the rule and shows the results for each 453

Tom Stark and Dean Croushore model. Then we discuss the issue of stability and how varying the nature of feedback in the rule may be necessary.

2. McCallum's Rule and Our Models MeCallum's rule is specified by the equation ABt = (AID* + AY*) - AV-Bt + ~,(Xt*1

-

-

Xt_t) ,

(1)

where all variables are in logarithms, B t is the monetary base at time t, P is the price level, Y is real GDP, AV'B is an estimate of the long-run growth rate of monetary base velocity (equal to nominal GDP divided by the monetary base), ~, is a parameter that we call the monetary response factor, X is nominal GDP, and an asterisk (*) denotes either the targeted value of a variable or a desired rate of growth. The growth rate of the monetary base is determined by the desired rate of inflation (AP*) plus the rate of growth of potential real GDP (AY*), the growth rate of monetary-base velocity (a 16-quarter moving average), which helps to prevent the price level from drifting in response to a permanent shock to money demand, and the deviation of nominal GDP from target. Following McCallum, we define the target path for nominal GDP recursively by X* = X*-I + (AY* + at'*), with X~ = X0.z Our philosophy rests on choosing models that reflect a variety of beliefs about how the economy works. Thus, we choose to examine a Keynesian model, a reduced-form model, and a structural VAR model. If McCallum's rule works well in all three models, then policymakers should be interested in the rule; if it works only in certain types of models, then only policymakers who believe in the theory underlying such a model are likely to be interested in the rule. As stated earlier, an interesting feature of this paper is that we borrow these models from the existing literature, rather than drawing up new models. In general, we've kept the models as close to their original form as we could, though we've had to adapt them slightly to enable them to handle McCallum's rule. A Keynesian Model Our first model, taken from Friedman (1977), is Keynesian in spirit. The model is grounded in the IS/LM framework and is augmented with aggregate supply, money supply, and term structure relations. Aggregate ~rhis requires us to specify a path for potential real GDP. There are many alternatives one could choose from; we choose to use the potential real GDP series developed at the Federal Reserve Board for use in the P* model (see Hallman, Porter, and Small 1991).

454

Evaluating McCaUum's Rule spending depends (apart from dynamics) on long-term interest rates. On the LM side, money demand depends on short-term interest rates and on output; money supply depends on short-term rates and on the monetary base. Inflation is given by a Phillips-curve relationship, and long- and short-term rates are related through a term structure equation. The appendix provides a detailed description of all models and parameter estimates. A Reduced-Form Model Our second model, taken from Taylor (1992), is a bit more monetarist in spirit. The model imposes a (classical) long-run equilibrium in which output grows along trend, M2 velocity is constant, long-term real rates are constant, and there is a constant spread between long- and short-term rates. The price level is, in the long-run, proportional to the money stock. Temporary deviations from long-run equilibrium are allowed in a system of shortrun equations for output, prices, M2 velocity, and interest rates. A Structural VAR Model Our third model is a structural VAIl, taken from Gali (1992). Tile model imposes a sufficient number of restrictions to just-identify an aggregate supply relation and, on the demand side, IS, money demand, and money supply relations. Aggregate supply is identified by imposing the requirement that demand shocks have no long-run effect on the level of output, and the demand side of the model is identified by imposing restrictions on contemporaneous interactions.

3. Simulation Results

This section reports the results of simulating our three models over the period 1963 to 1993 under the supposition that McCallum's rule had determined monetary policy. The simulation proceeds by drawing random shocks from a normal distribution and attaching them to each equation in a model for each date. The shocks are mean zero and their variance is the same as the variance of the residuals in the estimated model. The models are solved to yield simulated values of all the endogenous variables over the sample period. In Taylor's reduced-form model, we allow for covariance among the shocks. Rather than looking at just one simulation, which could have come from a particularly bad set of shocks, we simulate each model 500 times and examine the median and 95% confidence intervals for the simulations. The Fed's desired rate of inflation (AP*) is assumed to be 0% and, thus, the desired nominal GDP path (X*) is assumed to grow at a rate equal to the rate of growth of potential real GDP. As a result, we expect the rule's 455

Tom Stark and Dean Croushore biggest impact to be a lower average simulated inflation rate than the actual average inflation rate. The rule should drive inflation to zero. 3 From the perspeetive of a policymaker, we are interested in two key questions. First, how suseeptible are the models to dynamic instability? Second, when stable, how variable are nominal GDP and its components, prices and real GDP? The key to determining this is the size of the monetary response factor (;~). Given that our goal is to see if the rule works well in a wide variety of models, we need the same monetary response factor to work in all different models. This would support MeCallum's claims about the value of the rule.

Keynesian Model Results As originally specified by Friedman (1977) and as shown in the appendix, the model treats the monetary base as exogenous. For us to examine what would happen if McCallum's rule had been in effect, we introduce Equation (1), MeCallum's rule, into the model, making the monetary base endogenous. In implementing the rule, we first choose a monetary response factor 0~) of 0.25. This value worked well in McCallum's research. We begin each eounterfactual simulation in 1963:ii and carry it out to 1993:ii. The results for nominal GDP, real GDP, and the price level are shown in Figure 1.4 For each variable, the solid lines show the median value of the 500 simulations at each date and the upper and lower bounds defining the 95% range of the simulations. The long-dash line shows the target path for nominal GDP in the top figure which, in most cases, cannot be distinguished from the median--and the level of potential real GDP in the middle figure. A short-dash line shows the historical value of each variable. Several results are readily apparent from the figure. First, mirroring the findings of MeCallum and Judd-Motley, nominal GDP is kept quite close to target (upper panel); furthermore, the 95% range of the simulations is below the historical path of nominal GDP. Second, the median path of real GDP is permanently below its potential level throughout the simulation (middle panel). This reflects, in part, the lack of a natural-rate mechanism in the model and initial conditions that call for an initial period of eontraetionary monetary policy to reduce inflation from its 1963 rate" of roughly 1.5% to its desired rate of 0%. Finally, inflation is lower in the simulations than it was historically, so the rule is effective in controlling inflation (lower panel). 3Strictly speaking, this is true only in models in which monetary policy has no permanent effect on real output. Both the reduced-form and structural VAR models have such a property, while the Keynesian model does not. 4Our Keynesian model simulations are conducted by first setting to zero the exogenous variables in the model.

456

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457

Tom Stark and Dean Croushore TABLE 1. Summary Statistics, Keynesian Model, 2 = 0.25 (Annualized Percentage Points) McCallum's Rule Level Target M

L

H

Constant Base Growth Rule M

Mean Growth of: NGDP 2.8 2.4 3.1 3.5 RGDP 2.6 1.2 4.2 3.0 P 0.2 - 1 . 4 1.5 0.5 Standard Deviation of Growth of: NGDP 4.7 3.8 5.9 4.3 RGDP 4.6 3.8 5.7 4.0 P 1.9 1.5 2.6 2.0 Root-Mean-Square-Deviation: NGDP 4.8 2.9 7.8 19.3 RGDP 13.1 5.6 33.8 11.0

McCallum's Rule Growth Target

L

H

M

L

H

1.2 1.8 -1.8

5.8 4.3 2.7

2.7 2.5 0.2

2.2 1.0 -1.3

3.2 4.0 1.8

3.6 3.5 1.5

5.2 4.7 2.9

4.3 4.2 2.0

3.6 3.5 1.5

5.1 5.0 2.8

5.3 4.2

55.3 27.5

6.6 13.4

3.7 5.0

11.2 37.9

NOTES: Each entry is expressed in annualized percentage points. M indicates the median value over 500 simulations; L and H indicate the ranges (Low and High) into which 95% of the simulated values fall.

To get a different perspective on the results, Table 1 presents some useful summary statistics. It shows, for the simulations with McCallum's rule (the column labeled "Level Target"), the average values of nominal GDP growth (NGDP), real GDP growth (RGDP), and inflation (P), plus the standard deviations of quarter-over-quarter log first-differences of nominal GDP, real GDP, and the price level. To get a summary empirical estimate of the degree to which the economy hits its target, we measure the rootmean-square deviation (RMSD) of nominal GDP from its target; to get a summary measure of the economy's divergence from its full-employment level, we calculate the RMSD of real GDP from its potential level. For each measure, we show the median value (denoted by M in the table), as well as the lower bound (L) and upper bound (H) representing the range into which 95% of the values fall. A natural question to ask about McCallum's rule is: How does McCallum's rule compare to other rules that might be worth considering? In Table 1, we compare it to a natural yardstick: a rule that sets the monetary base growth rate at a constant level equal to the growth rate of potential GDP. This is a rule without feedback it does not allow the monetary authority to respond to economic conditions, as does McCallum's rule. The 458

Evaluating McCallum's Rule constant-base-growth simulations, reported in Table 1, highlight the crucial role of proportional feedback in MeCallum's rule: the 95% ranges for both nominal GDP growth and inflation are noticeably wider than was the ease with MeCallum's rule. Not surprisingly, the RMSD of nominal GDP is much higher. Comparing standard deviations across rules, the constant-basegrowth simulation indieates that nominal and real GDP growth are a bit less variable on a quarter-over-quarter basis. Taken together, our Keynesianmodel simulations suggest that MeCallum's rule with feedback on the level of nominal GDP affords a high degree of control over the long-run growth of nominal variables (prices, nominal GDP) but that this control may come at the expense of increased quarter-over-quarter variability in real GDP. 5 Our constant-base-growth metric holds MeCallum's rule to a very high--and perhaps unreasonable--standard. MeCallum (1987) notes that two important features of any policy rule are that the rule be operational and that the rule not assume the absence of technological ehange in the financial services industry. While MeCallum's rule satisfies both of these requirements, our constant-base-growth rule does not. In particular, our simulations assume unrealistically that the Fed would have known in 1963 the exact rate of monetary-base growth required to yield zero inflation on average over the following 30 years. Thus, to even the playing field and cheek the robustness of our results, we redefined the constant-base-growth rule by adding one annualized percentage point of base growth to the old rule and conducted a new stochastic simulation. We think of this experiment as one in which the Fed in 1963 underestimates future average velocity growth and, based on that estimate, overestimates the rate of growth in the monetary base consistent with a zero-inflation objective. We find that, compared with our original constant-base-growth specification, the new specification yields higher average simulated growth in nominal variables, a rise in the RMSD of nominal GDP (from 19.3% to 23.8%), but unchanged quarter-overquarter variability. Thus, we conclude that McCallum's rule does even better relative to the constant-base-growth rule than Table 1 suggests. A similar result obtains in our other models. A final important feature of MeCallum's rule is that it does not generate economic instability in this model. Research on rules for monetary policy often finds that following rules leads to unstable eeonomic performance. The instability could come from one of two places: it could be inherent in the model, or it could be induced by the policy rule. To demonstrate that this model has no such problem, we examine impulse responses 5Judd-Mofley (1991) also find an increase in short-run real GDP variability. Note that our findings on the behavior of real GDP are likely to be particularl) susceptible to the Lucas critique; McCallum (1990) contains an insightful discussion of this point.

459

Tom Stark and Dean Croushore to see how the economy responds to a shock. These are plotted in Figure 9., which traces the response over time of the Keynesian model's endogenous variables to two different shocks: a shock to the IS equation and a shock to the AS equation. We do this for two different cases: one in which we treat the monetary base as exogenous (which corresponds to our constant-basegrowth simulation), and another in which we impose MeCallum's rule. If the model itself is causing instability, both impulse response functions should show accelerating fluctuations over time. But if MeCallum's rule is inducing fluctuations, then the model treating the base as exogenous should not display accelerating fluctuations. As Figure 2 indicates, when the monetary base is taken to be exogenous (solid line), the impulse responses eventually settle down--an indication that the model is stable. The impulse responses also settle down when MeCallum's rule is in place (dashed line), which suggests that McCallum's rule does not induce instability in the model. Overall, then, MeCallum's rule with ~ = 0.25 works well in keeping inflation low and maintaining economic stability in the Keynesian model. Reduced-Form Model Results The 500 simulations of this model with )~ = 0.25 show that McCallum's rule works quite well in stabilizing nominal and real GDP as well as the price level (Figure 3). 6 The median simulation path for real GDP is quite close to its potential level. The price level is also stabilized quite well. Inflation is close to zero as a result of using McCallum's rule. Table 2 illustrates these results more clearly. A comparison of RMSDs obtained by simulating the model under McCallum's rule and under a constant-base-growth rule suggests that McCallum's rule is on average better able to hit the nominal GDP target and that real GDP is closer, on average, to its potential level. As was the case in the Keynesian model, holding nominal GDP closer to target comes at the cost of a higher standard deviation of quarter-over-quarter changes in both real and nominal GDP. McCallum's rule keeps the average rate of inflation close to its desired rate of zero, with a much tighter 95% range and with a lower standard deviation than does the constant-base-growth rule. The impulse responses (Figure 4) in this model show clearly how the rule may induce short-run fluctuations in the economy in response to an output shock or to an inflation shock. When the monetary base is exogenous (solid line), the response of variables over time to the shock displays a smooth return to equilibrium in the ease of an output shock, or damped oscillations in the ease of an inflation shock. In contrast, with McCallum's rule and )~ 6For simulation purposes, we set to zero the exogenous variables in the model.

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462

Evaluating McCaUum's Rule TABLE 2. Summary Statistics, Reduced-Form Model, 2 = 0.25 (Annualized Percentage Points) McCallum's Rule Level Target M

L

H

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L

H

McCallum's Rule Growth Target M

L

H

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2.2 2.2 0.8

3.0 3.5 0.2

5.1 4.8 2.0

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6.2 6.3 2.2

5.3 5.2 1.6

7.7 8.2 3.2

8.6 55.2 6.3 20.9

7.3 9.3

4.5 5.8

10.7 13.8

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= 0.25 (dashed line), the shocks lead to sharper short-run movements in real and nominal GDP, though there's no sign of instability.

Structural VAR Model Results Simulations of the structural VAR model with )~ = 0.25 (Figure 5) are similar to those from the Keynesian model. As in the Keynesian model, the 95% ranges for real GDP and the price level widen slightly over time. This widening of the ranges occurs for nominal GDP as well, unlike the Keynesian model. In the structural VAR, unlike in the other models, McCallum's rule introduces instability, as demonstrated by the impulse responses (Figure 6). Shocks to the IS relationship or the AS relationship induce increasing oscillations in real GDP, prices, and nominal GDP (dashed line). This is a serious problem for McCallum's rule, because it suggests that the rule isn't robust across reasonable models of the economy. Importantly, the model is not inherently unstable, as indicated by the constant-base-growth impulse responses (solid line); instead, the instability comes from the rule's interaction with the model.

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8

-0.0036

~ ~ _ / ~'----~---7"--'~ ~

[

so

/

3B. L off Nom//zTIC D P

---I

--I

]/~...~JlO'2g - -I/.~ \ \

iiExoc

II°-~

-~-

2B. Z offPrice L eve/

2e

- -

-0.00 l 8 1

o.oooo

o.oo18

0.0038

-0.007S

-O.OOSO

-0.002S

o.oooo

O.O02S EXOG

-o,oo2s

0.0000 1

o.oo2s

o.oo olI -21 0.007S "EXOG

Figure 6. Structural VAR Model Impulse Responses

~

"

"" /

./

/

(Sohd Lines: Exogenous Monetary Base Growth Dash Lines: McCallum's Rule, X = 0.25, Level Target)

"

~

A S Shock IB Z ,oqReal CDP --'

/

%

72

72

\

\

-..-

Z2

// ..\

~

/

/

'

% ./

/

94

/ .~

94

/

94'

\

Evaluating McCaUum's Rule 4. Stability and Modifications of McCallum's Rule In all three models, MeCallum's rule is useful in reducing inflation, on average. The average level of real output is about at its potential level, while the priee level is much lower in the simulations than it was historically. However, in the structural VAR model, MeCallum's rule introduces instability. Is there a way to modify the rule to alleviate the instability problem? There are two possibilities: (1) we could ehange the monetary response factor (~); (2) we could modify the form of the rule. A. W. Phillips (1957), working with eontinuous-time systems, found that proportional-feedback policy rules can often be destabilizing and suggested modifying such rules to contain feedback on the time derivative of the target variable. Below, we implement this suggestion by incorporating feedbaek on the log first differenee of nominal GDP. McCallum (1993) finds that such a modification reduces the variability of the monetary base and also allows for larger values of~. How do such variations work in our set of models? First, we examine variations in the monetary response faetor (~). Then, we look at what happens when we target nominal GDP growth instead of the level of nominal GDP. We begin by conducting some additional simulations with different values of the monetary response factor 0~). Doing a crude search across values of ;~, we find that the structural VAR model is dynamically stable under MeCallum's rule with a L between 0.40 and 0.80. Such a large value of ~, implies a large policy response to any deviation of nominal GDP from its target, so that nominal GDP hits its target very elosely (Figure 7). Both real GDP and the price level are stabilized quite well, and the range of the simulations is quite narrow. The impulse responses in Figure 8, using )~ = 0.80, indicate that the endogenous variables continue to exhibit relatively higher short-run variability in response to shocks compared with the constant-base-growth rule and that the model with MeCallum's rule is dynamically stable. Thus, our results indicate that we need a higher value for the monetary response factor than McCallum's base value of)~ = 0.25 to make the structural VAR model stable. Raising )~ creates no problems in the Keynesian model. However, as we increase ~ to alleviate the problem of instability in the structural VAR model, we find instability in the reduced-form model. In fact, even a )~ as low as 0.35 introduces instability into the reduced-form model, as illustrated by the impulse responses in Figure 9. Thus, our investigation of modifying the size of the monetary response factor has left us with the result that there is no value of K for which all three models are stable. For that reason, we now examine McCallum's (1993) 467

1. Slmuloted,HIstorlcolo/TdTorffe£L offNomlnol CDP

9.0 8.5

I

8.0

J J

7,5

I

7,0 8.5 8,0 1984"

1970

1978

1982

1988

2. Slmuloted ~ndHlsLorlcolL ~ Reol GDP

9.S0 9.25 9.00 8.75 8.50 8.25 8.00 7.75

7.50 t 984

1970

1976

1982

1988

3, SlmulatedandHletorlcol Log PrIce Level

O.S

0.0 I J

-O.S

I

-1.0

m

-1,5 -2.0

1984

1970

1978

1982

1988

Figure 7. Structural VAR Model Simulations, ~. = 0.80, Level Target (Solid Lines: 95% range and median Short Dash Lines: historical values Long Dash Lines: target values)

468

II Ex°c

.IIExo~"

t

8

-0.032 "

0.000 1 -0.018 ]

/

--

s

0.032 ,IIEXOG 0.018 -110"80

-0.032

-0.016

0.000

0.018 JlO.80

0.032

8

°'°°°°ll

o.oo, -I

•

0 0024

IS Shock

28

50

28

50

20

-----~--I

5o

3A. L offNominol C D P

':1

2A. L og Price L eve/

--,

--I

/,11. L ogReolCDP

-o.oozs

-0.0050

6

s

0.0025 £XOG 0.0000 0.80

S

Figure 8. Structural VAR Model Impulse Responses

94

84

94

5o

28

I

50

28

50

38. LOg Nem/nolC D P

----

2B. L offPriceL eve/

28

AS Shock IB. L offReel C O P

(Solid Lines: Exogenous Monetary Base Growth Dash Lines: McCallum's Rule, % = 0.80, Level Target)

z2

72

72

-0.0025

0.0000

0,0025

0.0050 080

0.0075 EXOG

72

;;2

Z2

94

04

94

o~

0.01~ 0.010 0.005 0.000 -0.005 -0.010

0.015 0.010 0.005 0.000 -0.005 -0.010

-0.0036

-0.0018

0.0000

0.0018

0.0038 , '"

G

so

28

SO

3/I. L offNora/no~ C,L2P

2s

"t . j L . ~ . L ~ _ . / _ _ ~ " "

S

2/1.L offPrice L eve/

,"G ,,' 28',l ,j, SO,,,

~

"l

,:

10 -:31~'

,~ IExdq I t

--II

.'~1 ' ~

,;

-- --I

94

;' ' ~

94

I

II

-- --I

-0.012 [

0.000

0.012

0.024 [

0.036

0.012 1

0.02'1 1

0.036 [

6

6

.

-

%k /

•

Figure 9. Reduced-Form Model Impulse Responses

I"'

EXOG 10.3~

~xo~.

10.3~;

-

•- 0 . 0 1 0

-O.OIS

-

-O.OOS

0.000

O.OOS I

28

.

'/

. SO

28

50

3B.LoffNom/nolGDP

72

72

EXOG 0.3S

EXOC 0.35

lO.3S

EXOG

94

94

I I

---~-H

--If

/ ~ " ~ - L ~ I It "/~ ~ I ~ . ~."-'!.--..:-.-t.--~'1 ~" ~'" ' / ~1

2R. L off Price L eve/

.

Inflation Shock

lB. L z~jRa~I GDP

(Solid Lines: Exogenous Monetary Base Growth Dash Lines: McCallum's Rule, ~ = 0.35, Level Target)

72

l'~',

z2

72

:9"1 , I

,--~, m ,-7-T-7, I

:l

^

tt

'.

't

~

,

- I

~L_3 ! '

"~ t

OutputSt~ck

/A. L oqReo/CDP

Evaluating McCallum's Rule other suggested strategy: a rule that targets the growth rate, rather than the level, of nominal GDP. This represents a modification of Equation (1) to the following: z~,

= ( A e * + AY*) -

AVBt + ~ ( ~ * - 1 - aX,_I).

(9)

McCallum has found that this variation of the rule seems to work well. It has the potential, however, of allowing drift in the level of nominal GDP. In all three of our models, this variation of McCallum's rule works well with the monetary response factor set at 9~ = 0.25. All the models are stable, as shown by the impulse responses in Figure 10 (Keynesian model), Figure 11 (reduced-form model), and Figure 12 (structural VAR model). In Tables 1, 2, and 3, summary statistics for the stochastic simulations are shown in the columns labeled "Growth Target." In the Keynesian model, Table 1 shows that the switch to the growth target from the level target leaves mean growth rates about the same, reduces the standard deviations of real and nominal GDP growth but raises the standard deviation of inflation, and raises the RMSD of both nominal and real GDP. Since we still measure the target as the level of nominal GDP and real GDP, it isn't surprising that the RMSDs rise when we modify the rule to feed back on the growth rate of nominal GDP rather than its level. In the reduced-form model, Table 2 shows results similar to that of the Keynesian model, with the same qualitative differences between the results with the level target and the growth target. In the structural VAR model, Table 3 shows that the growth-rate rule is better than the level rule in all aspects: the mean growth rates have a narrower range, the standard deviations are much lower, and the RMSDs are also lower. The improved all-around performance is attributable to the fact that following the growth-rate rule eliminates the instability problem found with the level rule. Thus, our results provide evidence in favor of McCallum's rule in growth-rate form. The results confirm the intuition of Phillips (1957) that feedback on various forms (levels, as well as growth rates) of a state variable can help avoid instability problems in a policy rule.

5. Alternative Benchmarks As noted previously, our empirical strategy of using a constantmonetary-base-growth benchmark is not perfect because it is not obvious what would have been the optimal base growth rate. Other imperfect options exist. For example, one could compare our simulated average growth and variability measures with their historical counterparts. However, this option 471

EXOG

0.028

27

.0.028

.0,014

0.000

0.028

f

8

-- --

48

"~s

2z

~s

3A. L offNominal C D P

2z

----I

2A. / off Price L eve/

oo1,ll'oX°,o --I

.0 028 "

-0.014t

0.000 "

0.0141

0.25C

e

0.~5 C

EXOG

-o.ots

-0.008

0.000

0.008

0.016

IS Shock

I A. Zor:jReol C.DP

90

90

.0.050

.0.025

0,000

0.025

0.000 /

°°2s I J

e

6

Inflation Sl~ock

48

2z

48

3B. Loff A/omir~/CDP

27

2B. L offPrice L eve/

/8. L OffReol COD

Figure 10. KeynesianModelImpulseResponses (SolidLines:ExogenousMonetaryBaseGrowth Dash Lines:McCallum'sRule,X = 0.25, GrowthTarget)

ss

ss

ss

.0.02 -0.04

0.00

0.04 0.02

88

89

ss

EXOG 0.25C

0.25G EXOG

Exo

0.25 ~

So

So

9o

I

----II

'1

63

0.000 -0.005 -0.010

0,010 O,OOS

0.015

-~010 t

1

6

6

e

Ii

1

oloto,

0015

/ oo,o 1

-0.0036

o0.0018

0.0000

0.0018

0.0036

Output S~ck

SO

So

2B

60

3A. Loff/VominolCDP

2e

2,~. Z offPrice L eve/

28

/ A. £ off Raa/ dDP

94

J -0.012 I

0,000

0.012

0.024

0.036

-0.012

0.000

0.012

0.024

0,036

-O.OOS1

0.006 1 0.000 /

e

6

6

Figure 11. Reduced-Form Model Impulse Responses

,i

s4

94

----

Ii 28

so

.

SO

2B

SO

38. Zoff/Vom/nt7/COP

28

26. L offPr/ce Zevel

(Solid Lines: Exogenous Monetary Base Growth Dash Lines: McCallum's Rule, 9~ = 0.25, Growth Target)

72

EXOG

0.2SC

i2

EXOG 0.2S C

;'2

,xoo

10_2613

InflationSMock 18. Z offReal C D P

Z2

EXOG 0.26 C

72

O.~SC

72

ExoG lo.2sc

94

94

. - .I 94

.

26

So

_

~

....

.

.

94

i

0.0025

z2

.

94

-0.002S

.

!i!i

.

S

.

O.Ol6 11o.25o

0 032

2e

_-71

So

3A. Log Nam/na/COP

°0.0036

S

EXOC 0.25 G

0.25 C

Figure 12. Structural VAR Model Impulse Responses

64

6

[l

ExoG

10.25

IIEx°~"

AS SMock

I

28

--I

50

so

26

_---.~1

so

3B. LoffNomi/'~IGDP

2s

--~I

2R. L offPrice L 9P'el

--

/ B. L offR ~ / COP

(Solid Lines: Exogenous Monetary Base Growth Dash Lines: McCallum's Rule, ~, = 0.25, Growth Target)

72

-0.0018

0.0000

0.0018

0.0036

-0.0075

So

.

0.0000

0.0025

-0.032

2s

---:I

-0,0050

EXOG

0.~5 C

E×oc

z2

-0.016

0.000

0.016

0,032

2A. L offPrice L eve/

-0.0024

"" ~ ~ 0,0000

-"

0.0050

0.0075

-0.0025

e

IS SMock

IA. L O q R ~ l CLTP

-0.0012

0.0000

0.0024

z2

72

72

S4

94

94

Evaluating McCaUum's Rule Summary Statistics, Structural VAR Model, 2 = 0.25 (Annualized Percentage Points) TABLE 3.

McCallum's Rule Level Target M

L

H

Mean Growth of: NGDP 2.7 1.7 3.9 RGDP 2.8 1.8 3.9 P -0.1 -1.6 1.1 Standard Deviation of Growth of: NGDP 7.6 4.5 15.1 RGDP 5.9 3.9 10.4 P 6.5 3.0 14.0 Root-Mean-Square-Deviation: NGDP 8.9 3.4 18.9 RGDP 10.9 5.6 22.7

Constant Base Growth Rule M

L

2.9 2.8 0.1

1.1 2.0 -1.9

4.6 3.7 3.6 15.9 8.8

H

McCallum's Rule Growth Target M

L

H

4.5 3.5 2.0

2.7 2.9 -0.1

2.1 2.0 -1.2

3.3 3.6 0.8

3.8 3.2 2.4

6.2 4.2 5.8

4.9 4.0 4.1

3.9 3.4 2.3

7.0 4.9 7.3

5.8 3.4

36.3 21.1

6.6 9.0

3.4 4.1

12.3 19.9

NOTES: Each entry is expressedin annualized percentage points. M indicates the median value over 500 simulations; L and H indicate the ranges (Lowand High) into which 95% of the simulated values fall.

fails to control for the ability of the models to replicate the actual first and second moments of the data. v Thus, since the models are designed only to capture general aspects of the U.S. economy, we argue strongly against this strategy. Another option is to simtdate our models by feeding in the actual historical values of the monetary base. Indeed, both McCallum and Judd/ Motley have used such a strategy. While this was useful in McCalhim's simulations--in which estimated residuals are used as shock estimates in conducting a single simulation--we argue that this approach is less useful in the context of our multiple stochastic simulation framework because it does not allow base growth to respond to different shock realizations that occur in each of our 500 simulations. In this section, we describe briefly the results of an additional simulation that we believe provides a reasonable alternative to our constant-basegrowth benchmark, s Two of our models (Taylor's 1992 reduced-form model and Gali's 1992 structural VAR) include a Federal Reserve reaction function. We use these estimated reaction functions in conducting another set of 7We thank Ray Fair for makingthis point in his commentson an earlier versionof this paper. SSpacelimitations preclude any more than a brief description of these results. More detailed results are availableupon request. We thank an anonymousreferee for suggestingthis section. 475

Tom Stark and Dean Croushore

benchmark simulations (that is, we simulate these two models, allowing monetary policy to be given by the estimated reaction function, rather than by the constant base growth rule or by MeCallum's rule). In both models, the 95% ranges for average simulated inflation and nominal GDP growth are much wider when monetary policy is given by the estimated reaction function than was the case under the constant-basegrowth rule. In the reduced-form model, the ranges are - 1.1% to 6.0% for inflation (compared to -2.3% to 2.7% for the constant-base-growth rule) and 1.8% to 8.8% for nominal GDP growth (compared to 0.6% to 5.4% for the constant-base-growth rule). The comparison is even more pronounced in the structural VAR. We conclude that MeCallum's rule provides strong control over the growth of nominal variables relative to both of our benchmarks. Concerning our short-run variability measure (the standard deviation of quarter-over-quarter log first differences), we find that simulation with the estimated reaction function in the structural VAR produces standard deviations about the same as those reported in Table 3 for the constantbase-growth rule and for the growth-rate form of MeCallum's rule. However, in the reduced-form model, standard deviations obtained by simulating with the estimated reaction function are much lower than those obtained under any of the simulations reported in Table 2. Thus, in the reduced-form model, MeCallum's rule gains control over nominal variables at the expense of greater variability in real variables.

6. Summary and Conclusions We began our investigation concerned about the robustness of McCallum's rule for the growth in the monetary base in three established macroeconomic models that have not been used to test the rule. We found some problems of dynamic instability with McCallum's original formulation of the rule, which targets the level of nominal GDP. In addition, our stochastic simulations indicated a tendency toward higher short-run variability in nominal and real GDP compared with the variability produced in a baseline constant-base-growth simulation. When modified to feed back on the growth rate of nominal GDP instead of the level, as suggested in McCallum (1993), the rule appears to work well both in terms of dynamic stability and in our short-run variability measure--across our set of very different models. Our analysis of the models' impulse responses was shown to be useful in demonstrating when dynamic instability is an issue and in describing how McCallum's rule affects the models' responses to several types of shocks; we find that all the models are stable with MeCallum's rule in growth-rate form. Just as with all other policy changes, the Lucas critique is an important limitation to our simulation results. We don't have a good idea of how peo476

Evaluating McCallum's Rule pie's behavior would change if the Fed were to implement McCallum's rule. As macroeeonomists develop new theories of behavior, we may be better able to simulate the effects of using MeCallum's rule. McCallum's rule is a potentially useful rule for setting monetary policy. Had the rule been followed over the past 30 years, inflation would have been much lower than it actually was. Given the concern over inflation that arose in the 1970s and early 1980s, MeCallum's rule could serve as a useful device to guide monetary policy today. Received: June 1996 Final version. May 1997

References Fair, Ray C. "The Estimation of Simultaneous Equation Models with Lagged Endogenous Variables and First Order Serially Correlated Errors." Econometrica 38 (May 1970): 507-16. . Specification, Estimation, and Analysis of Macroeconometric Models. Cambridge, MA: Harvard University Press, 1984. Friedman, Benjamin M. "The Inefficiency of Short-Run Monetary Targets for Monetary Policy." Brookings Papers on Economic Activity 2 (1977): 293-335. Gall, Jordi. "How Well Does the IS-LM Model Fit Postwar U.S. Data?" Quarterly Journal of Economics 107 (May 1992): 709-38. Hallman, Jeffrey j., Richard D. Porter, and David H. Small. "Is the Price Level Tied to the M2 Monetary Aggregate in the Long Run?" American Economic Review 81 (September 1991): 841-58. Hess, Gregory D., David H. Small, and Flint Brayton. "Nominal Income Targeting with the Monetary Base as Instrument: An Evaluation of MeCallum's Rule." Manuscript. Federal Reserve Board of Governors, June 1992. Judd, John P., and Brian Motley. "Nominal Feedback Rules for Monetary Policy." Federal Reserve Bank of San Francisco Economic Review (Summer 1991): 3-17. --. "Controlling Inflation with an Interest Rate Instrument." Federal Reserve Bank of San Francisco Economic Review no. 3 (1992): 3-22. . "Using a Nominal GDP Rule to Guide Discretionary Monetary Policy." Federal Reserve Bank of San Francisco Economic Review no. 3 (1993): 3-11. Lanrent, Robert D. "An Interest Rate-Based Indicator of Monetary Policy." Federal Reserve Bank of Chicago Economic Perspectives 12 (January/ February 1988): 3-14. 477

Tom Stark and Dean Croushore McCallum, Bennett T. "The Case for Rules in the Conduct of Monetary Policy: A Concrete Example." Federal Reserve Bank of Richmond Economic Review (September/October 1987): 10-18. --. "Robustness Properties of a Rule for Monetary Policy." CarnegieRochester Conference Series on Public Policy 29 (Autumn 1988): 173204. --. "Could a Monetary Base Rule Have Prevented the Great Depression?" Journal of Monetary Economics 26 (August 1990): 3-26. --. "Specification and Analysis of a Monetary Policy Rule for Japan." Bank of Japan Monetary and Economic Studies 11 (November 1993): 145. --. "Monetary Policy Rules and Financial Stability." In Financial Stability in a Changing Environment, edited by K. Sawamoto, Z. Nakajima and H. Taguchi, New York, NY: St. Martin's Press, 1995. Phillips, A. W. "Stabilization Policy and the Time-Forms of Lagged Responses." Economic Journal 67 (June 1957): 265-77. Stark, Tom, and Dean Croushore. "Evaluating McCallum's Rule When Monetary Policy Matters." Federal Reserve Bank of Philadelphia Working Paper 96-3, January 1996. Taylor, Herb. "PSTAR + : A Small Macro Model for Policymakers." Federal Reserve Bank of Philadelphia Working Paper 92-26, December 1992.

Appendix This appendix presents parameter estimates for the models that we use to evaluate McCalhim's rule. Variable definitions follow. See Stark and Croushore (1996) for an extended discussion of the models.

A Keynesian Model Ben Friedman (1977) developed a complete Keynesian model of the economy. The following equations describe our version of the model, which we've estimated on quarterly U.S. data from 1959:i to 1993:ii (the sample was determined by data availability). The equations are estimated using twostage least squares, just as Friedman used in his original model. There is a correction for serial correlation, following the method of Fair (1970, 1984).9 The results of the estimation are IS.

A~'t =

1.35 + 0.542AYt_l + 0 . 0 5 2 7 A E t -

(3.33) j5 = - 0.282

(2.28) '

(7.07) /~2 = 0.08

(1.26)

1.98ARBAAt-

(1.85)

O.0338APIMt-1. (1.07)

(A1)

9As instruments, we use the set of instruments required by Fair, as well as other predetermined variables in the system.

478

Evaluating McCallum's Rule AS:

APt = 0.275APt_ 1 + 0.267APt_ 2 + 0.291APt_ 3 + 0.094APt_ 4 (3.24) (3.10) (3.41) (1.14) + O.0546APIM,_, + 0.0850[(L_1 - Y*-I) - (Yt-2 - ¥*-2)].

(4.18)

(2.50)

(~)

/]2 = 0.72. MD:

AR1912t =

0.783ARM2t_l + 0.143AYt(8.92)

= -0.275

(1.26)

Pt2 = 0.41,

2.19ARTBO3t.

(4.63)

RM2t = M2t - Pt

(2.51) ' MS:

AM2 t = - 0 . 2 1 1 A M 2 t _ 1 +

(2.16) 15 =

TS:

0.817 (13.3) '

/~2 = 0.39

' (A3)

1.17ABt- 0.261ARTBO3t. (7.37) (0.71) ' (A4)

RBAAt = 0.215 + 0.867RBAAt_1 + 0.264RTBO3t- O.097RTBO3t_I. (1.73) (40.8) (4.64) (1.80) ' = 0.336

/]2 = 0.99

(3.61) '

(A5)

In all the regressions here and later in the appendix, the terms in parentheses are the absolute values of the t-statistics, and p is the first-order autocorrelation coefficient. Further, all variables, except interest rates, are in logarithms, and all growth rates are expressed in annualized terms. A Reduced-Form Model

Herb Taylor (1992) developed a restricted reduced-form model, which he called PSTAR +, for use in aiding monetary policy decisions. The model is based on two main hypotheses: (1) a theory developed by Laurent (1988) that the spread between the federal funds interest rate and long-term interest rates is closely related to subsequent output growth; and (2) the P* (pronounced P-star) model developed by Hallman, Porter, and Small (1991). The P* model predicts future inflation using the monetarist theory that, in the long run, the price level is proportional to the M2 measure of the money supply. The model consists of a system of long-term equations that describe the steady state and a system of short-term equations that describe variation around the steady state. The following equations describe the long-term relationships in the model: 479

T o m Stark and Dean Croushore

Yt = Yt* + z~;

(A6)

V2~ = V2* + z y ;

(A7)

Pt = P* + z f ;

(AS)

RTt~IO t = 2.337 + APt -

1.132D65 - 0.734D74 + 5.261Dso

- 1.501D9o + ztaa ,

(A9)

R F F t = -0.692 + R T B I O t + ztas .

(A10)

Equations (A6) to (A8) suggest that the logs of real GDP (Y), M 2 velocity (V2), and the price level (P) all deviate from their equilibrium levels by the amount of a random shock (denoted by z ~', z v, or ze). The term Y* is the log of potential output, the same variable used in the Keynesian model described in the previous section. The model asserts that long-run M2 velocity is constant, so we take V2* to be the log of the mean value of velocity over the period from 1959:i to 1990:iv; V2* = 0.495.1° Finally, P* itself is just the log of the price level consistent with constant velocity, output at its potential level, and the current money supply:

P*=-M2t

+ V2* - Y * .

(All)

Equation (A9) suggests that the interest rate on 10-year Treasury bonds (BTBIO) is equal to a constant plus the rate of inflation (AP) plus dummy terms representing breaks in the real interest rate series in 1965:iii, 1974:i, 1980:ii, and 1990:ii. The inclusion of the inflation rate means the equation is consistent with a constant long-run equilibrium real interest rate. Following Taylor, the dummy variables correspond to breaks in the potential real GDP series. Finally, Equation (A10) suggests that the term structure has a fixed slope in the long run between the federal funds rate (RFF) and the 10-year T-bond interest rate. The short run is modeled by the following equations, which we estimate with seemingly unrelated techniques: 1°WeeshmateV2* over the period 1959:ito 1990:ivto get the samevaluefor velomtyas in Hallman, Porter, and Small(1991). The resultsaren't affectedat all by this choice,as opposed to estimatingvelocityoverthe entire sampleperiod from 1959:ito 1993:ii.

480

Evaluating McCallum's Rule

A~t . Ay* .

_ 16.54zL . .1 (2.28)

40.54zV 1 (5.19)

Rs2 + 0.706ARTBIOt_ 1 0.768zt_ (5.80) (1.55)

+ O.198ARTBIOt_a

(0.57) /~o = 0.37.

(AI2)

= -33.54zV_1 + 1.326ARFF t + 0.737ARFF~_I + 2.125ARTBlOt_ 1 (3.35) (6.87) (3.21) (3.62) + 0.954ARTBIOt_20.141(AYt_ I -AY*_ 1) + 0.770DUMTIME t (2.20) (2.28) (5.75) ' /~2 = 0.41. =-10.34zte_l (3.96)

-

(A13) +

0.192z~e~_l- 0 . 5 4 4 A z P t _ l (2.32) (6.00)

0.0363AV2~_2 (1.47)

+

0.198A2Pt_2 (2.62)

O.O0591APOIL~_2

(2.67)

'

/12 = 0.39.

(A14)

ART1)IOt = -0.0480zt~_~ +

(2.27)

0.0684ztRS~ +

(3.27)

O.218ARFF t

(7.23) (A15)

t]2 = 0.43. A R P F t = 14.18ztY ~ -

(4.54)

0.185ztBL1-

0.~,35ZtB_S1 -- 0.120~2P/_1

(3.35)

(4.25)

(2.36)

+ O.O0428APOILt_I + 0.634ARTBIOt_ 1 -

(2.66)

(4.08)

0.614ARTBIOt_2

(4.00)

~2 = 0.34.

(A16)

In simulating the reduced-form model, we must make one modification to the model. Since we plan to use McCallum's rule in the model instead of the federal funds rate reaction function in the original model (A16), we need to add a relationship between the federal funds rate, the monetary base, and the M2 measure of the m o n e y supply. We do this with a m o n e y multiplier specification: AM2f/12t =

0.583AMM2t_ 1 +

(7.10) /i 2 = 0.58,

0.252AMM2t_ 2 -

(2.98)

0.578ARFF t

(3.06) (A17)

481

Tom Stark and Dean Croushore

where MM2 is the M2 money multiplier, RFF is the federal funds rate, and all variables are expressed in annualized terms. Equation (A17) is the most parsimonious of several alternative specifications that we estimated. A Structural V A R Model

Our third model of the economy, based on Jordi Gali's (1992) study, is a structural vector autoregressive model intended to reflect a dynamic macroeconomic theory. Both demand shocks and supply shocks affect output in the short run, but only supply shocks have a permanent effect. The four variables of interest in the model are the log of real GDP (Y), the interest rate on three-month Treasury bills (RTB03), the CPI inflation rate (AP), and the log of real M1 money balances (RM1). There are four structural shocks in the model, representing shocks to the main equations of a Keynesian model: an aggregate supply shock (dis), a money supply shock (eMS), a money demand shock (eMD), and a spending shock (ds). The unrestricted VAR representation of the model is: [I - B(L)]Zt = vt,

(A18)

where Z is the (4 × 1) vector of variables [AY, ARTB03, RTB03 - AP, ARM1]', I is a (4 X 4) identity matrix, B(L) is a (4 x 4) matrix polynomial in the lag operator with B(0) = 0, and v is a (4 x 1) vector of reduced-form disturbances for which Evtv [ = Z.. The reduced-form disturbances are related to the structural disturbances (e) by the equation vt = Set,

(A19)

where e is the (4 x 1) vector [dis, e Ms, e MD, eIS] ', Eete; = Ze, and S is the (4 x 4) matrix of contemporaneous structural coefficients. Using Equation (A19) in Equation (A18) and multiplying through by S-1 gives the structural vector autoregressive (SVAR) relationship A*(L)Zt = et,

(A20)

where A*(L) =- S - I [ I - B(L)]. To identify the model given by (A20), we impose the same restrictions as those imposed by Gall, as described in more detail in Stark and Croushore (1996). We follow Gall in using his method and in using beginning-of-quarter data to estimate the model. Estimation is difficult because the identifying restrictions are represented by a set of nonlinear equations that must be

482

Evaluating McCallum's Rule

solved simultaneously. Using repeated iterations of a simplex routine yields the following solution for the S matrix: -2.35

/

0.00

0.20

¢=/-0.70

0.00

-1.97]

0.58 0.36

-0.27 /

0.44 1.41 o.70/

1_-0.52

-2.92

2.36

(A21)

-0.30_1

These results can be used to generate the contemporaneous relationships in the structural version of the model: 11 AYt = 2.48 + 3.06ARTBO3t

-

1.44(RTB03

AP) t

-

+ 0 . 3 9 A R M 1 t + etaS/s n ,

(A22)

ARTBO3t = - 0 . 0 3 - 0.06AYt - 0 . 3 8 ( R T B 0 3 - AP) t

(A23)

+ 0.38ARM1 t + eMS/s 22 ' (RTB03

-

AP) t

=

-1.388

+

0.20AY

t -

3.15ARTBO3t

-

0.78ARMlt

(A24)

+ eMD/S 33 , A R M 1 t = 0.07 -

1.54AYt - 7.80ARTBO3t

+ 3.68(RTB03 -

AP)t + eItS/s 44 ,

(A25)

where S ii is the ith element on the main diagonal of the matrix S-1. To incorporate McCalhim's rule into the structural VAR model, we again add a money-multiplier equation. This provides a relationship between the M1 measure of the money supply, which is the monetary variable in the structural VAR model, and the monetary base, which enters McCalhim's rule. As in the reduced-form model, we allow the money multiplier to depend on interest rates--in this case the three-month T-bill rate. The equation for the money multiplier is: 11Standard errors of the estimates are not presented here, as is also true in Galfs paper, because doing so would require far more computer power than we have available. In principle, such standard errors could be calculated using Monte Carlo methods, but the nature of the simultaneous nonlinear equations needed to estimate the model makes even these methods quite time- and computer memory-intensive.

483

Tom Stark and Dean Croushore AMY/I1 t =

O.022AMMlt_ 1 +

(0.25) -

O.172AMMI,_ 2 +

(1.97)

0.754ARTBO3t_ 1 -

(2.64)

O.187AMMIt_ 3

(2.13)

0.479ARTB03t_2-

(1.65)

R2 = 0.11.

0.594ARTBO3t_ 3

(2.14) (A26)

Variable Definitions

D65, Dv4, Dso, D9o = DUMTIME

=

E

=

MM1

=

MM2

=

M1

=

M2 p

= =

PIM POIL

=

=

Log of file St. Louis Federal Reserve Bank's monetary base. Dummy variables set equal to 1 for all dates on or after 1965:iii, 1974:i, 1980://, and 1990:ii, respectively, and to zero before those dates. Dummy time-trend variable initialized at i in 1991:i. Log of real cyclically adjusted Federal government expenditures (deflated with P). Log of the M1 money multiplier, eonstructed as M1 - B. Log of the M2 money multiplier, constructed as M2 - B. Log of the nominal M1 money stock. Log of the nominal M2 money stock. Log of the implicit priee deflator for GDP (in the structural VAR model, log of the consumer price index). Log of the implicit price deflator for imports. Log of the real price per barrel of imported petroleum and products (deflated with P). Interest rate on Moody's corporate Baa bonds. Interest rate on Federal funds. Log of real M1 money balances (deflated with i°). Log of real M2 money balances (deflated with 10). Interest rate on 3-month Treasury bills. Interest rate on 10-year Treasury bonds. Log of monetary base velocity, constructed as P

RBAA RFF

= =

RM1

=

RM2

=

RTB03 RTB10

= =

VB

=

V2

= Log of M2 velocity, eonstrueted as P + Y M2.

+Y-B.

484

Evaluating McCallum's Rule X X* Y y*

= = = =

Log of nominal GDP, constructed as P + Y. Target log nominal GDP, as defined in text. Log of real GDP. Log of potential real GDP.

All data are obtained from standard government sources, with the exception of the nominal price of oil and potential real GDP (obtained from the Federal Reserve Board). In the Keynesian and reduced-form models, data available monthly are averaged over the quarter (before conversion to natural logs). In the structural VAR model, quarterly observations for the M1 money stock, the consumer price index, the 3-month Treasury-bill rate, and the M1 money multiplier correspond to the first month of the quarter. All data, with the exception of interest rates, are seasonally adjusted. The first difference operator, A, when applied to a variable in log form, is expressed in annualized percentage points.

485