Constructing and estimating a realistic optimizing model of monetary policy

Journal of Monetary Economics 45 (2000) 329}359

Constructing and estimating a realistic optimizing model of monetary policyq Jinill Kim* Department of Economics, University of Virginia, Charlottesville, VA 22903, USA Received 17 January 1997; received in revised form 25 February 1999; accepted 11 March 1999

Abstract A dynamic stochastic general-equilibrium (DSGE) model with real and nominal, both price and wage, rigidities succeeds in capturing some key nominal features of U.S. business cycles. Additive technology shocks, as well as multiplicative shocks, are introduced and shown to be crucial. Monetary policy is speci"ed as an interest rate targeting rule following developments in the structural vector autoregression (VAR) literature. The interaction between real and nominal rigidities is essential to reproduce the liquidity e!ect of monetary policy. The model is estimated by maximum likelihood on U.S. data, and its "t is comparable to that of an unrestricted "rst-order VAR. Besides producing reasonable impulse responses and second moments, this model replicates a feature of U.S. business cycles, never captured by previous research with DSGE models, that an increase

q

This is a revised chapter of my dissertation at Yale University. Special thanks to Christopher Sims for his guidance and support. Thanks also to William Brainard, Giancarlo Corsetti, Jordi Gali, Frederico Galizia, Mark Gertler, Robert King, Nobuhiro Kiyotaki, Robert Shiller, T.N. Srinivasan, an anonymous referee, and the co-editor, David Backus, for valuable comments. I also thank seminar participants at the Board of Governors of the Federal Reserve System, Federal Reserve Banks of New York and Richmond, the Bank of Canada, the University of Cambridge, Universitat Pompeu Fabra, SUNY at Stony Brook, Washington University in St. Louis, the University of Pennsylvania's Wharton School, Korea}America Economic Association summer conference, Indiana University, the University of Virginia, and Princeton University. Any remaining errors are my own. The "rst version of this paper circulated under the title &Monetary Policy in a Stochastic Equilibrium Model with Real and Nominal Rigidities' in the Fall 1995. * Tel.: #1-804-924-7581; fax: #1-804-982-2904. E-mail address: [email protected] (J. Kim) 0304-3932/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 3 2 ( 9 9 ) 0 0 0 5 4 - 9

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in interest rates predicts a decrease in output two to six quarters in the future. ( 2000 Published by Elsevier Science B.V. All rights reserved. JEL classixcation: E50; E32; C51 Keywords: Monetary policy; Optimizing model; Additive technology shocks

1. Introduction The comovement of monetary and real aggregates and the inverse relation between the movements of money growth and nominal interest rates are two prominent nominal features of business cycles in the United States and many other countries.1 In this paper, we will try to explain these features through the two channels of monetary policy } the output e!ect and the liquidity e!ect. The output e!ect, de"ned here as the positive response of aggregate output to expansionary monetary policy, has been a key question for economists who have searched for a monetary explanation of the business cycle. The liquidity e!ect, de"ned as the decrease in interest rates in response to monetary expansion, has been an important issue in empirical macroeconomics.2 Stimulated by Kydland and Prescott (1982) and Long and Plosser (1983), dynamic stochastic general-equilibrium (DSGE) models have become a useful tool for macroeconomic analysis, especially for business cycle analysis. Previous work using a #exible-price competitive DSGE models have provided a reasonable description of the data on real variables. One stream of recent work incorporates outside money in a #exible-price competitive DSGE model. Money is introduced in a cash-in-advance economy by Cooley and Hansen (1989) to study the e!ects of in#ation. Sims (1994) introduces money through a transaction-cost framework. Using a simple money-in-the-utility-function model without nominal rigidities, Benassy (1995) shows analytically that the dynamics of the real variables are exactly the same as those in a model without money. Such models do not provide a good description of the money}output correlation and cannot reproduce reasonable impulse responses to the shocks in monetary policy, because of the following generic implication. If money growth

1 Even if the "rst feature is universally accepted, the presence of the second feature is somewhat controversial. It depends on the choice of monetary aggregates and trending mechanisms. See Chari et al. (1995) as an example. Cooley and Hansen (1995) summarize additional stylized facts of the nominal features. 2 This de"nition of the liquidity e!ect as a causal relation is, of course, not universal. For example, Ohanian and Stockman (1995) use the term to refer to the statistical correlation.

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displays a positive persistence, then shocks to the growth rate of money drive output down and nominal interest rate up through an anticipated in#ation e!ect. To generate the output e!ect, nominal rigidities are introduced into DSGE models.3 There are two alternative ways of introducing nominal rigidities. The "rst is to replace an equilibrium equation matching demand and supply with an equation describing the determination of price and/or wage. The staggered contract theory of Fischer (1977) is usually adopted. Cho (1993), Cooley and Hansen (1995), Benassy (1995), and Cho and Cooley (1995) study the implications of nominal wage contracts for the transmission of monetary shocks, and Yun (1994) explains the comovement of in#ation and output with a staggered multi-period price setting model. Leeper and Sims (1994) also experiment with both price and wage rigidities by postulating equations describing price and wage movements. Another way of introducing nominal rigidities is via adjustment costs. For an economic agent to have control over the price and/or wage, some form of imperfect competition is needed. Following Blanchard and Kiyotaki (1987), the monopolistic competition framework has been widely used. Hairault and Portier (1993) show that monetary shocks are necessary to reproduce some stylized facts of business cycles, and Rotemberg (1996) presents a model that is consistent with a variety of facts concerning the correlation of output, prices and hours of work. All the above models of nominal rigidities do indeed generate the output e!ect of monetary policy. However, as shown in Kimball (1995), the presence of nominal rigidities by themselves does not produce the liquidity e!ect. To generate the liquidity e!ect, I assume real rigidities in the form of adjustment costs for capital. This conjecture is found in King (1991) and implemented in Novales (1992), Dow (1995), and King and Watson (1996). This paper constructs and estimates a DSGE model of monetary policy. It tries to build a realistic and usable model, in the spirit of Leeper and Sims (1994). The model features the two e!ects of monetary policy.4 It also captures many interesting U.S. business cycle facts. Section 2 presents the model and the solution. Households maximize their utility and "rms maximize their pro"t.

3 Besides introducing nominal rigidities, another way to produce the output e!ect is to treat monetary shocks as a source of confusion that makes it di$cult for agents to separate relative price changes from aggregate price changes, as in the &Lucas island model'. Cooley and Hansen (1995) "nd that monetary shocks do not appear to play a quantitatively important role in driving business cycles in a model based on this theory. 4 Christiano and Eichenbaum (1992a) propose a DSGE model with an alternative transmission mechanism of monetary policy. The model generates both the output and liquidity e!ects due to participation constraints in the "nancial market. Dotsey and Ireland (1995) provide a skeptical view of the model. Beaudry and Devereux (1995) also construct a model which features the two e!ects by assuming predetermined prices as a way of resolving indeterminacy.

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Government action is characterized by monetary and "scal policies. There are two real and two nominal shocks a!ecting the economy. Section 3 illustrates the model's qualitative and quantitative properties. Because the time-series and policy implications critically depend on the choice of the parameters, maximum likelihood estimates are computed. The impulse responses evaluated at the estimated parameters feature the two e!ects of monetary policy: the output and the liquidity e!ects. The "t of the model and its variance decompositions are comparable to those from vector autoregression (VAR) models. Cyclical implications are considered by comparing the second moments of the model with those of the data. Finally, one of the challenges in King and Watson (1996) is satisfactorily met. According to the estimated model, an increase in interest rates in the current period predicts a decrease in real economic activity two to six quarters in the future. None of their three models captures this feature of the U.S. business cycle. Section 4 concludes.

2. The model We develop a DSGE model consisting of a government and three types of private agents } an aggregator, I households indexed by i, and J "rms indexed by j. The aggregator serves two functions in this economy. In the labor market, it transforms heterogeneous labor into a &composite labor' usable for "rms' production. In the goods market, it collects heterogeneous goods to make a &composite good' which households can consume and invest. Demand for an input by the aggregator depends on the relative price of the input. Each household has monopoly power over its own type of labor, facing the demand by the aggregator. In both capital and goods markets, households are a price taker. It also accumulates capital and rents it to "rms. Each "rm has monopoly power over its own product, facing the demand by the aggregator. It acts competitively in factor markets. The government derives revenue from issuing money and debt and expends its revenue through transfers and interest payments on outstanding debt. The assumptions of monopoly power of households ("rms) in the labor (goods) market allow nominal rigidities to arise in the form of adjustment costs for wages (prices). Following the literature on menu costs, we assume that it is costly for "rms to adjust their output price. Similarly, wage adjustment costs on the part of households are assumed to capture wage rigidities. Meanwhile, adjustment costs for capital give rise to real rigidities. The model has four shocks. On the households' side, a preference shock is identi"ed as a shock in money demand. The production function of the "rms involves two shocks } a multiplicative shock and an additive shock. Finally, there is a shock in monetary policy. The information structure is such that variables dated t are always known at time t.

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2.1. The aggregator The aggregator purchases di!erentiated inputs which are described by an N-dimensional vector, (H , H ,2, H ), and transforms them into H units of the 1 2 N composite output. The functional form follows Dixit and Stiglitz (1977):

A

B

h@(h~1) N H" + H(h~1)@h . (1) n n/1 This is a constant returns to scale (CRS) and constant elasticity of substitution (CES) production function which has been used in the literature on monopolistic competition. The parameter h is the elasticity of substitution between the di!erent inputs (possibly di!erent for goods and labor). To guarantee the existence of an equilibrium, h is restricted to be greater than unity. The supplier of each di!erentiated input sets a price for it; the collection of these prices describes a price vector conformable with the vector of inputs purchased. The pro"t of the aggregator is5 N P"PH! + P H , (2) n n n/1 where P is the price index and P is the price of the nth input. The "rst-order n condition with respect to H reduces to a constant-elasticity inverse demand n function:6

A B

H ~1@h P "P n . n H

(3)

In the following, same symbols are used for every agent to avoid the need for separate equations specifying market clearing conditions of the equilibrium. 2.2. Households Households are identical, except for the heterogeneity of labor. Having monopoly power over one unit of its own labor, each household enters at time t with a predetermined capital stock, money holdings and bond holdings. The

5 The price index is de"ned as P"(+N P1~h)1@(1~h). n/1 n 6 In the case of labor market where the elasticity of substitution is h , (3) is interpreted as follows: L = "= (¸ /¸ )~1@hL , where = is the real wage of household i, = is the real wage index, ¸ is the it t it t it t it labor supply of household i, and ¸ is the amount of the composite labor supplied by the aggregator, t all at time t. Likewise, in the goods market where the elasticity of substitution is h , (3) is written as Y follows: P "P (> /> )~1@hY , where P is the output price of "rm j, P is the price index, > is the jt t jt t jt t jt output supply of "rm j, and > is the amount of the composite good supplied by the aggregator, all at t time t.

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household receives rental income, wage income, a lump-sum transfer, and a constant share of pro"ts. It also pays the costs of adjusting its capital and wage. Preferences are given by the utility function ; which represents the expectai0 tion of the discounted sum of instantaneous utilities, conditional on the information at time zero:

C C

A

BD

= M ; "E + bt; C , it , ¸ i0 0 it P it t t/0 = ((CH)a(1!¸ )1~a)1~p1 it "E + bt it , 0 1!p 1 t/0 where

A

A B

D

(4)

B

M (p2 ~1)@p2 p2 @(p2 ~1) it CH" C(p2 ~1)@p2 #b . (5) it it t P t The discount factor b is between 0 and 1. Households prefer the present to Consumption, C , and real balances, M /P , interact through a CES function. it it t The CES parameter, p (50), decides the elasticity of money demand. Instan2 taneous utility is a constant relative risk aversion (CRRA) transformation of a Cobb}Douglas function of the CES bundle, CH, and the amount of leisure, it (1!¸ ). The CRRA coe$cient, p , is assumed to be positive. We adopt it 1 a money-in-the-utility-function framework. However, this model can be converted into a transaction-cost framework without altering any implications, where the variable representing consumption is not C but CH. it it The only stochastic element of households' preferences is in b . The variable t decides the importance of consumption relative to real balances and it follows that log(b )"o log(b )#(1!o )log(b)#e , (6) t b t~1 b bt where the e 's are i.i.d. random variables distributed N(0,p2). Since b decides the b t bt level of money demand, the innovations in b are identi"ed as money demand t shocks; as a matter of fact, the other shocks a!ect money demand only indirectly. Another element of money demand is p , which determines the elasticity of 2 money demand. Money demand shocks could be introduced via p , but it is 2 more reasonable to introduce a randomness in the level, rather than in the elasticity. Utility maximization is subject to several constraints. Each household accumulates capital and rents it to "rms. The accumulation technology is given by the following equation: K "I #(1!d)K , (7) it i,t~1 i,t~1 where K is the amount of capital stock and I is the amount of investit it ment, both at time t. The existing capital depreciates at a constant rate of d.

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Adjustment costs for capital have been used to provide a rigorous foundation for the q-theory of investment. In this paper, adjustment costs are internal to the households and are given by

A B

2 / I ACK" K it I , (8) it it 2 K it where / is the adjustment cost scale parameter for capital. This functional K form, satisfying the local assumptions in Abel and Blanchard (1983), produces strictly positive steady-state adjustment costs, 5% of investment in the estimation result. An alternative formulation would make installation costs a function of net investment as in (/ /2)(I /K !d)2I . This formulation produces zero K it it it steady-state adjustment costs and therefore simpler dynamics as shown in Kim (1998). Quadratic costs are justi"ed on the grounds that it is easier to absorb new capacity into the "rm at a slow rate. Adjustment costs for capital play an important role in determining nominal and real interest rates. More investment makes installed capital more valuable due to adjustment costs. However, the price of capital is expected to return to the steady state in the future. This expected decrease in the price of capital moves the real interest rate down. This mechanism is crucial in generating the liquidity e!ect. Expansionary monetary policy increases investment and thus decreases the real interest rate. If this o!sets the anticipated in#ation e!ect, the nominal interest rate goes down: the liquidity e!ect. Wage rigidities are introduced through the cost of adjusting nominal wages, assumed to be quadratic and zero at the steady state. The real total adjustment cost for household i is given by

A

B

2 / P= t it !k = , ACW" W (9) it it 2 P = t~1 i,t~1 where / is the adjustment cost scale parameter for the wage and k is the W steady-state growth rate of money. The multiplicative term, = , makes the cost it grow at the growth rate of the real wage index. This real cost, ACW, enters the it budget constraint of the household in a similar way to adjustment costs for capital. Wage rigidities are a source of the output e!ect of monetary policy. Facing the increased labor demand induced by expansionary monetary policy, households would increase wages proportionately if it were not for adjustment costs. However, households increase wages less than proportionately due to adjustment costs. Their supply of labor increases and so does output. The budget constraint of household i is given by

A

A BB

2 / I M B C #I 1# K it # it # it #ACW it it it 2 K P P it t t M r B J "= ¸ #Z K #¹ # i,t~1 # t~1 i,t~1 # + s P , it it t it it ij jt P P t t j/1

(10)

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where B is government debt, Z is the rental rate, ¹ is the government it t it lump-sum transfer payment, r is the gross nominal interest rate, and s is its t ij constant share of real pro"ts P of "rm j. Government debt earns the nominal jt interest rate and money is not an interest bearing asset. A No-Ponzi-Game condition is imposed on households' borrowing; it requires debt not to increase asymptotically faster than the interest rate. This prevents households from borrowing to the level that makes the marginal utility of consumption zero at every period. Another constraint for the households is the labor demand (via the aggregator) by "rms. From the "rst-order conditions, a standard money demand equation is derived.

A B

P C 1@p2 t it . (11) M it The elasticity of real money balances with respect to the net interest rate is approximately (!p ). Also, to a "rst-order approximation, the following rela2 tion holds between the nominal interest rate and the rental rate:7 r~1"1!b t t

A

B

(1!d)E [1#3/ (I /K )2]#E [Z #/ (I /K )3] 2 K t`1 t`1 t t t`1 K t`1 t`1 [1#3/ (I /K )2] 2 K t t P t`1 . ]E (12) t P t It is clear from this relation that an expected decrease in the investment price implies a lower nominal interest rate and that real rigidities are important in generating the liquidity e!ect. r" t

C D

2.3. Firms Firms are identical, except for the heterogeneity of outputs. Each good is produced by one "rm only, and this "rm takes all other output prices as given. In the market for its inputs, capital and labor, these "rms behave competitively. A "rm's decision regarding inputs results in a certain amount of output and this amount, in turn, determines the output price from the demand curve of the aggregator. The "rm also pays the cost of adjusting its output price.

7 This expression is obtained by invoking certainty equivalence, i.e. E[f (>)]"f (E[>]), which is justi"ed by the assumption that the variances are very small. This could be achieved by manipulating deterministic "rst-order conditions and restoring the expectational operators. From this expression, the real interest rate is the "rst term on the right-hand side of the equation.

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Firm j produces > units of net output under a common increasing-returnsjt to-scale technology with both multiplicative and additive technology shocks. The production function is (13) > "A (Ka (gt¸ )1~a)c#U Ct!FCt, jt t jt t jt with the restrictions that 0(c(a~1, g51, and C51.8 K is the capital stock jt and ¸ is the quantity of composite labor. A and U , common to all "rms, jt t t follow the stochastic processes:

A B

log

A B

A A t "o log t~1 #e , A At A A

(14)

(15) U "oU U #eU , t t t~1 where A('0) is the steady-state value, and e and eU are both i.i.d. variables t At distributed N(0,p2 ) and N(0,p2U ), respectively. Since both of these two shocks A appear in the production function, e and eU are assumed to be correlated with t At the correlation coe$cient o U . All other pairs of errors in the model are assumed A to be uncorrelated. A is the multiplicative productivity shock, which is conventionally used in the t DSGE literature. In this paper, another type of productivity shock is also introduced. It is an &additive' productivity shock, U . Although the additive t shock is not generally used in the literature emphasizing the comovement features among real variables, its introduction is very important in capturing the relation between real and nominal variables over business cycles, independent of the multiplicative shock. Since the multiplicative shock a!ects the slope as well as the level of the production function, it a!ects the interest rate via changing the marginal product of capital. The additive shock does not change the marginal product directly, although it a!ects the level of output directly. Such a di!erence turns out to be important in explaining the dynamics between real and nominal variables, notably output and the interest rate. In a general-equilibrium context, a model with additive technology shocks can be interpreted as a model with shocks in goods demand and a conventional example of goods demand shocks is government spending.9 Positive shocks

8 C is the growth rate of nonstationary real variables. To guarantee the existence of a balanced growth path, C"g(1~a)c@(1~ac). This functional form produces a steady-state path with a geometric trend, and the data need not be pre"ltered for the estimation. By incorporating a trending mechanism inside the model as above, the model integrates both growth and business cycles. See Cooley and Prescott (1995) on this point. 9 A model with aggregate demand shocks is Henderson and Kim (1999) and examples of government spending shocks are Aiyagari et al. (1992), Christiano and Eichenbaum (1992b), and Rotemberg and Woodford (1995, 1998).

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to additive productivity is observationally equivalent to negative shocks in government spending. The government spending shock has the same absolute value as the additive shock. The transfer in the new model with government spending would be the addition of the transfer in the original model and the additive shock. Output would exclude the part of the additive shock. Both of these two equivalent interpretations are used in this paper, depending on which is more convenient in the particular context. Regarding the identi"cation of the shocks, this paper is in line with Rotemberg and Woodford (1998) in not trying to identify by using the data on government spending in the national income account. Estimation results for the coe$cient of serial correlation are strikingly similar. Our estimate for oU is 0.911 and theirs is 0.92. The parameter F('0) represents a &"xed cost' component, which makes it possible for the "rms to earn zero pro"t. In each period, the amount FCt is used up for administrative purposes just to keep production going and this is independent of how much output is produced. This &"xed cost' implies increasing returns to scale. An additional source of increasing returns to scale is embodied in c, whenever c'1. Kim (1997) shows analytically that the two types of returns to scale have di!erent implications. Each "rm sells its output to the aggregator in a monopolistically competitive market. Without any other assumptions, the "rm's problem is essentially a static one. We now introduce price rigidities through price adjustment costs. As with wage rigidities, the real adjustment cost for "rm j is given by

A

B

/ P k 2 jt ! ACP " P > , jt jt 2 P C j,t~1

(16)

where / is the adjustment cost scale parameter for price and k/C is the P steady-state gross in#ation rate. Like wage rigidities, price rigidities are a source of the output e!ect of monetary policy. Once price adjustment costs are introduced, the problem of "rm j becomes dynamic. The "rm maximizes its value which is the expectation of the discounted sum of its pro"t #ows, conditional on the information at time zero: < "E j0 0

C

D

= + oPP , t t jt t/0

(17)

where P P "P > !P Z K !P = ¸ !P ACP . jt t jt jt jt t t jt t t jt t

(18)

The maximization is subject to all the constraints including the output demand function. The "rm discount factor is given by a stochastic process, Mo N. In the t equilibrium, it represents a pricing kernel for contingent claims.

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2.4. The government The budget constraint of the government is

A

B A

B

M !M B !r B t t~1 # t t~1 t~1 "¹ , t P P t t

(19)

where I I I M"+ M , B"+ B and ¹ " + ¹ . t it t it t it i/1 i/1 i/1 Among the "ve variables in the constraint, (M , P , B , r , ¹ ), three restrictions t t t t t are imposed by the budget constraint and the optimizing behavior of households and "rms. The government has two degrees of freedom and its behavior is described by monetary policy and "scal policy. Outside the DSGE framework, some researchers use structural VAR models to isolate the e!ects of monetary policy. Traditionally, the stance of monetary policy was measured by the growth rate of a monetary aggregate. This identi"cation created a &liquidity puzzle', which means that monetary expansion is associated with rising rather than falling interest rates. This problem with money-based measures of monetary policy has led Bernanke and Blinder (1992) and Sims (1992) to identify monetary policy with innovations in interest rates. This identi"cation scheme, while successful in producing reasonable results in some dimensions, produces other theoretical and empirical problems. For example, an expansionary monetary policy is associated with a strong and persistent drop in the price level. Recently, Gordon and Leeper (1994), Strongin (1995) and Sims and Zha (1995) have identi"ed monetary policy in such a way that the stock of money depends on the innovations in money demand as well as in monetary policy. My speci"cation of monetary policy hinges on this type of identi"cation, by including both a monetary aggregate and the nominal interest rate in the policy rule. In a model without nominal rigidities, Coleman (1996) shows that such endogeneity of money supply is very important in capturing the observed relation between nominal and real variables, except for the money}output correlation. He suggests that money endogeneity be accompanied by nominal rigidities to explain the relationship between money and output. This paper con"rms the role of money endogeneity and also shows that combining it with nominal rigidities improves the performance of the model signi"cantly. The equation for monetary policy can be interpreted either as one deciding money supply or as one deciding the interest rate. Interpreted as determining money supply, the money growth rate #uctuates around a convex combination of last period's money growth rate and the steady-state money growth rate. Other elements of monetary policy concern the role of the interest rate and

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a shock in monetary policy. It enters the policy equation as a function of this period's interest rate, two previous periods' interest rates, and its steady state. The resulting speci"cation constrains the government when setting its two policy variables as follows:

A B

log

A B C AB

M M t "o log t~1 #(1!o )log k#(1#l)log k M M t M M t~1 t~2 r r r #l log t !(o #o )log t~1 #o o log t~2 r R r R r r r

A B

A BD

,

(20)

log k "o log k #e , (21) t k t~1 kt where r is the steady-state value of r and the e 's are i.i.d. variables distributed t kt N(0,p2). k Two parameters, o and o , determine how much the monetary authority r R smooths the level and the di!erence of the interest rate. The parameter l('0) embodies the sensitivity of the authority to interest rate movements. The traditional money-based measure of monetary policy corresponds to the speci"cation where l is set to zero. Most DSGE research on money follows this speci"cation. Empirical results show that l is signi"cantly di!erent from zero. The point estimate is 0.576 and its standard error is 0.266. Even if our speci"cation is richer than that of the traditional exogenous money supply, it is still limited in considering the feedback from the real side. In the model, the feedback is only through the interest rate. It would be more realistic to include the rate of in#ation and measures of real activities, income or consumption, in the monetary policy reaction function. This limitation turns out to be important in the analysis of the variance decompositions. It is widely recognized that both monetary and "scal policies in#uence the price level and, in models with nominal rigidities, real variables. For example, Sims (1994) shows analytically which combinations of monetary and "scal policies produce a unique equilibrium, not only locally but also globally, in a more simple model. In our model, "scal policy is speci"ed as follows: B ¹ "¹Ct!q t~1 , (22) t P t where ¹ and q are constant coe$cients. The conditions for a unique equilibrium are checked locally and numerically. 2.5. The equilibrium Completion of the system requires a relation between the stochastic discount factor of the households and that of the "rms. It is conventionally assumed that

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every agent in the economy has access to a complete contingent asset market. Then there is a unique market discount factor, which implies the following equation at all states:

A BA B

o P b; t`1 t`1 " Ci,t`1 . (23) ; it o P C t t There might potentially be equilibria in which identical agents behave di!erently due to di!erent initial conditions. However, in view of the symmetry of the environment, it is both reasonable and practical to scrutinize a symmetric equilibrium and to normalize both I and J to 1. Following Hornstein (1993), we assume that there are no pro"ts at the steady state. This assumption is equivalent to determining the "xed cost as follows:

C A

BD

1 F" 1!c 1! A(kM a¸M 1~a)c, (24) h Y where the lower case letter and an upper bar represent the transformed stationary variable and the steady-state value. Since money is neutral at the steady state, m6 and p6 cannot be determined independently. However, money is not super-neutral at the steady state due to the in#ation tax channel. Since the equilibrium cannot be solved for analytically, I log-linearize the system around the steady state. Now the model can be cast in the form P x( "P x( #P e #P (x( !E [x( ]), (25) 0 t 1 t~1 2 t 3 t t~1 t where x( is the percentage deviation of x from its steady state.10 Note that the t t coe$cient matrices, (P , P , P , P ), are nonlinear functions of the deep 0 1 2 3 parameters. The system is solved following the QZ decomposition method by Sims (1995).11 The solution, if there is a unique equilibrium, takes the following form: x( "W x( #W e , (26) t 1 t~1 2 t where there is no expectational term. The solution is a restricted VAR in the sense that the coe$cient matrices, (W ,W ), are functions of the deep parameters. 1 2 Note that the solution is equivalent to the following relation among the nontransformed variables: log(X )"[(I!W )log x6 #W log c ]#[(I!W )log c ]t t 1 1 X 1 X #W log(X )#W e , 1 t~1 2 t 10 If the steady state is zero as for pro"ts, the hatted variable is the variable itself. 11 The QZ decomposition method, analytically more general and numerically more stable than the method by Blanchard and Kahn (1980), is implemented by a modi"ed version of the MATLAB program gensys.m written by Christopher Sims. The program reads (P , P , P , P ) from (25) 0 1 2 3 as inputs and writes (W ,W ) in (26) as outputs. 1 2

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where the log of a vector is the vector containing logs of the components. This relation is a "rst-order VAR with a constant and a linear time trend. The "t of the DSGE model is compared with an unrestricted VAR of a similar form.

3. E4ects of monetary policy The structural VAR literature on the identi"cation of monetary policy conventionally uses four variables. They are the interest rate, money stock, the price level and output. Restricting the analysis to the above four variables preserves comparability of our model with other results in the literature.12 Since the error structure of the model comprises four shocks, using more than four variables almost always makes the covariance matrix of the data singular and so the likelihood becomes degenerate. For the model to be well de"ned, the parameters are restricted within the region speci"ed in Appendix A. The requirement that the model has a unique solution forces additional restrictions on the parameter space which cannot be expressed in an analytic form. For any parameter vector which produces no equilibrium or multiple equilibria, the likelihood value is set to a very small number so as not to a!ect the maximum of the likelihood function. Therefore, the likelihood function is discontinuous on the boundary.13 In this section, we start with impulse responses of various versions of the model to understand the role of the real and nominal rigidities. The bene"ts and limits of the rigidities are discussed. Then we assess the "t of the model from various perspectives, including impulse responses at the estimated parameter values. Since the model includes four structural shocks, various decompositions show how important each shock is in explaining the innovation of the variables. Finally, cyclical implications are presented with an emphasis on the relation between the nominal interest rate and output. 3.1. Impulse responses Before analyzing the quantitative implications of the estimated model, it is interesting to study the impulse responses for several restricted versions. These impulse responses are drawn with respect to a 1% expansionary temporary shock in monetary policy. The role of rigidities in producing the business-cycle

12 We do not pay attention to other variables such as consumption, investment and pro"ts. See Christiano et al. (1996) for the behavior of other variables in a model with nominal rigidities. 13 Such discontinuities are problematic for usual optimization routines based on a gradient method. The routine used in this paper, implemented by an MATLAB program csminwel.m written by Christopher Sims, is robust to discontinuities even if it is based on a gradient method.

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Table 1 Restrictions for rigidities

Prototype DSGE Monopolistic Comp. Real rigidities Wage rigidities Price rigidities All rigidities

(h ,h ,c) L Y

/ K

/ W

/ P

(R,R,1) Free (R,R,1) Free Free Free

0 0 Free 0 0 Free

0 0 0 Free 0 Free

0 0 0 0 Free Free

properties of monetary policy becomes clearer in this exercise. The parameters are set to arbitrary but plausible values and then restricted as follows. Six speci"cations of the model are considered, each summarized by a row in Table 1. The "rst row describes a prototype DSGE model without monopolistic competition or any rigidities. The second corresponds to a model featuring monopolistic competition but no rigidities. The third introduces real rigidities alone, without monopolistic competition or nominal rigidities. Then, monopolistic competition and wage rigidities are considered in the fourth model, without real rigidities. Analogously, the "fth speci"cation has monopolistic competition and price rigidities, without real rigidities. Finally, the full model with all parameters free has all the rigidities. Each column of Fig. 1 corresponds to a row in Table 1. The "rst three columns illustrate how the economy responds to a monetary shock when there are no nominal rigidities. In these cases, expansionary monetary policy decreases output and increases the nominal interest rate, which is the reverse of the output e!ect and the liquidity e!ect } the two e!ects of monetary policy we are trying to capture. In the "rst column, called RBC in the "gure, an increase in the interest rate and a decrease in output are quantitatively very small. This results from the fact that anticipated in#ation, through a substitution e!ect, moves the interest rate up and output down by a small amount.14 The responses of the second column (M.C.) are similar to those of the prototype DSGE model, except for the di!erent magnitude of the output movement. The responses in the Real R. column are almost the same as the prototype DSGE model, even with real rigidities.15 The 14 The "rst three columns have scale factors in the response of the interest rate and output. Recall the result of Benassy (1995) that, when prices and wages are #exible, the movements of real variables are not a!ected by nominal shocks at all. There is no room for the substitution e!ect since the utility function is separable among consumption, real balances, and labor. 15 This result is robust to the size of / , since the e!ect of real rigidities through the price of K investment is too small compared with the anticipated in#ation e!ect. This is not true if there are nominal rigidities.

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Fig. 1. Rigidities and impluse response (the responses are with respect to a 1% expansionary shock in monetory policy).

only di!erence is that the magnitude of the output movement is a little larger than that in the prototype DSGE model. Thus, real rigidities by themselves do not seem to signi"cantly impact the behavior of the nominal variables. However, as we shall see, when real rigidities are combined with nominal rigidities, the behavior of the model is drastically di!erent. The next three columns consider how the economy responds to a monetary shock when there are nominal rigidities. Nominal rigidities produce the output e!ect, but the liquidity e!ect is likely to appear only when real rigidities are added to nominal rigidities. In the fourth column, called Wage R., wage rigidities are introduced together with monopolistic competition. The impulse responses display the output e!ect of monetary policy. After an expansionary monetary shock, the households adjust wages only gradually due to the presence of adjustment costs. Therefore, they partially increase hours of work and this, in turn, increases output. Also note that prices do not adjust instantaneously. However, the liquidity e!ect does not appear. In the "fth column (Price R.), price rigidities replace wage rigidities. This model also exhibits the output e!ect of monetary policy, through the

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behavior of the "rms. The "rms will not adjust their price fully and would rather produce more output. The movement of prices is slower than in the model with wage rigidities. The liquidity e!ect does not appear, either. A common feature of columns 4 and 5 is the signi"cantly positive responses of the interest rate.16 This is due to a dramatic increase in investment, not matched by an equal increase in savings. Investment increases since higher output demand in subsequent periods is expected, which then increases the rental rate.17 Savings do not increase as much, despite the fact that there is a temporary increase in income. Furthermore, since prices adjust only gradually, expectations of future in#ation also push the nominal interest rate up. The last column (All R.) corresponds to the full model with all the rigidities. Combining real and nominal rigidities, we "nally capture the liquidity e!ect of monetary policy. The output e!ect is also quite sizable. The instantaneous increase of output is much smaller than in the models with only nominal rigidities. 3.2. Assessing the xt We have seen how the log-linearized "rst-order conditions produce the solution, (26), which is a restricted "rst-order VAR of all the variables in the model. The evaluation of the likelihood is nonstandard as follows. Since only the four variables are observed in the data set, the model implies an in"niteorder VAR for the observed variables.18 Therefore, the structural residuals cannot be expressed as an analytic function of the data. Moreover, the dimension of the variance}covariance matrix of the full data is so large that the matrix cannot be stored on a personal computer. Therefore, we need a recursive way of computing the likelihood which avoids storing the matrix. In this paper, the likelihood is calculated by the Kalman "ltering method.19 Quarterly U.S. data from 1959:I to 1995:I are extracted from the Citibase. Appendix A contains the data descriptions and the parameter estimates. Since only one out of the four

16 This feature is, of course, not insensitive to the parameter values. Important parameters are the serial correlation coe$cients of monetary policy, o and o . However, assigning plausible values to M k these parameters does not produce the liquidity e!ect. Note also that the positive response of the interest rate in Price R. induces the overshooting behavior of the money stock. 17 This increase of the real interest rate is not a generic result of a sticky price model. In a cash-in-advance model of Ohanian and Stockman (1995), monetary expansion decreases the real interest rate. 18 See Anderson et al. (1995) for details. 19 An alternative method based on approximating an in"nite-order VAR with a "nite-order VAR can be used instead. Kim (1995) explains both methods and compares them from a computational perspective.

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data variables is a real variable, the parameters which are mainly related with real variables are loosely estimated. Since the VAR structure allows us to interpret the likelihood as a measure of the &normalized mean squared error of forecasts', it is natural to compare the "t of the DSGE model with that of a comparable unrestricted VAR "tted to the data variables.20 Previous DSGE models, Leeper and Sims (1994) for example, do not "t the data as well as unrestricted VAR models. There are 30 free parameters in our DSGE model, and its likelihood value at the best "t is 2775. A "rst-order reduced-form VAR is estimated with a constant term and a time trend. Its maximum likelihood value is 2790 and it has 34 free parameters. According to the likelihood criterion, the DSGE model forecasts only 0.9% worse than the VAR. The maximum likelihood value of a VAR without a time trend is the same as that of the DSGE model. The VAR has 30 free parameters. Moreover, if the likelihood is de"ned conditional on !10th period (10 periods before the sample), the DSGE model performs better than the two VARs.21 Even though standard t-tests and model selection criteria reject a "rst-order VAR in favor of higher-order models, most DSGE models "t much worse than even "rst-order VARs and "tting the data as well as a "rst-order VAR is a contribution to the DSGE literature. The time series charts in Fig. 2 compare the residuals of the DSGE model with the innovations from the VAR with a time trend. Overall, the DSGE model predicts no better than the VAR, but it does nearly as well and is superior over certain time ranges. If we compare the performance over each decade by averaging the residuals, the DSGE model does worse in the 1960s, better in the 1970s, about as good in the 1980s, and again worse in the 1990s. The chart does not contain the residuals for the "rst three periods corresponding to the "rst year of the data. The DSGE model does especially worse than the VAR in those early unreported periods, which is a generic problem when "tting a DSGE model. This is because a DSGE model implies an in"nite order VAR for the data variables. Therefore, it is di$cult to explain the movements with

20 Standard classical tests such as a likelihood ratio test may not be applied, since the two models are not nested. 21 See the following table for the exact likelihood values. Meanwhile, model selection criteria which penalize the number of parameters produce similar results. Three criteria are used: the Akaike information criterion, the Hannan}Quinn criterion, and the Schwarz criterion. Observation conditioned on

DSGE

VAR with a trend

VAR without a trend

the 1st observation the !10th observation

2775 2786

2790 2756

2775 2778

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Fig. 2. Prediction errors of the DSGE and the VAR (reported are the residuals in the percentage terms). Table 2 Forecast standard deviations Standard deviation

r

M

P

>

The DSGE model The VAR with a trend The VAR without a trend

0.00256 0.00247 0.00259

0.00673 0.00740 0.00752

0.00422 0.00417 0.00426

0.00797 0.00686 0.00693

only a few prior observations. In other words, since our data vector does not include all the variables of the DSGE model, the "rst observation does not capture either short-run or long-run information completely. Note also that the likelihood is de"ned conditional on the "rst observation. In a "rst-order VAR where all variables are available in the data, the "rst observation captures all information precisely. For convenience, Table 2 reports the Cholesky decomposition of the covariance matrices, which enables comparisons across the models. The ordering of the variables for Cholesky decomposition is (r, M, P, >). The reported elements are interpreted as the standard deviation of the one-step-ahead

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Table 3 Data means and the steady state

U.S. data means Model steady state

r

M

P

>

1.0166 1.0259

2.869 2.444

24.367 26.752

19.610 16.231

forecast errors. The DSGE model produces a better "t for money stock than the VAR with a time trend. Compared with the VAR without a time trend, the DSGE model improves on the interest rate, money stock and the price level. The DSGE model captures the movement of the nominal variables pretty well. Overall, we conclude that our DSGE model explains the movements of the nominal variables better than those of the real variables. Another way to assess the "t is to compare the steady state of the model with the means of the data. The estimated parameters imply a steady state that matches the means of the detrended data in some aspects, as reported in Table 3. The steady state of the interest rate is di!erent from the mean by less than 1%. The steady states of money and output are lower than their means since much of their movements are explained by persistent shocks. This is consistent with the fact that the steady state of the price level is higher than the mean by 9%. For the model's implications to be credible, the estimates must produce sensible dynamic responses to the exogenous shocks. Calculation of impulse responses and variance decompositions should involve the decision on how to treat the covariation in any two innovations. For our model, there is only one covariation between the multiplicative technology shock, e t , and the additive A technology shock, eUt . Since the additive shock is equivalent to a government spending shock, the multiplicative shock is ordered before the additive shock. This implies that government spending is a!ected by the multiplicative technology shock, but not vice versa. The orthogonalization of the additive shock in Appendix A calculates how much of the government spending is a response to the multiplicative shock. Fig. 3 reports the impulse responses evaluated at the estimated parameters. All responses are with respect to a temporary shock of one standard deviation.22 The "rst column corresponds to the responses to a positive monetary policy shock. Both the liquidity and output e!ects are evident in the impulse responses.

22 Error bands are not drawn here. Due to parameter restrictions, the posterior is not normal but truncated normal, which makes the calculation of probability regions complicated. This also applies to the variance decompositions and the second moments.

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Fig. 3. Impulse responses at the estimated parameters (the responses are with respect to onestandard-deviation shocks).

The nominal rigidities are a source of slow adjustment of prices. The second column shows that a positive money demand shock moves the interest rate up and output down. In the third column, a positive multiplicative technology shock decreases the interest rate. This is mainly because the multiplicative shock is positively correlated with, and also ordered before, the additive technology shock. Since additive technology shocks are a novel feature of the paper, we pay particular attention to the impulse responses with respect to those shocks. Since a positive additive shock is equivalent to a decrease in government spending, this results in a decrease in the interest rate. This decrease and the wealth e!ect makes consumers less willing to work. Since capital is predetermined, a decrease in labor amounts to a decrease in output. This decrease in labor is so large that it o!sets the direct increase in output due to the additive shock. This o!setting is equivalent to the Keynesian-multiplier e!ects of government spending. Aiyagari et al. (1992) show that the multiplier is larger than unity when the shocks are highly serially correlated. The intuition is that the expected low labor would incur a decrease in the future capital stock, hence a decrease in investment. Wage rigidities also help magnify this multiplier e!ects by making labor respond more.

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Table 4 Interest rate decompositions Interest rate

1

2

3

4

5

10

20

Monetary policy Money demand Multiplicative technology Additive technology

0.02 0.20 0.04 0.74

0.03 0.17 0.07 0.73

0.04 0.15 0.10 0.71

0.04 0.13 0.12 0.70

0.04 0.12 0.15 0.70

0.03 0.08 0.24 0.64

0.03 0.06 0.32 0.59

3.3. Variance decompositions Once estimated, the model can be used to evaluate the underlying exogenous sources of #uctuations over the sample period. The fraction of the variance attributable to each shock is readily computed from the solution, (26), evaluated at the estimated parameters. In the following four tables, the variances of the interest rate, money stock, the price level, and output are decomposed into fractions that are explained by the shocks in monetary policy, money demand, multiplicative technology, and additive technology. The percentages of each variable's forecast error variance due to the four shocks are reported for several forecast horizons. The report is both in the short and medium runs } from 1 to 5 quarters, 10 quarters, and 20 quarters. The "rst line of Table 4 can be read as follows. 2% of the variance of the interest rate is explained by the monetary policy shock when the forecast horizon is 1 quarter and 3% is explained by the shock when the forecast horizon is 2 quarters. Note that the four numbers in each column add up to 100%. Table 4 gives the results for interest rate decompositions. The additive shock accounts for more than 70% of interest rate movement in the short run. As the forecast horizon lengthens, its importance decreases but is still more than 50%. This is partly because the additive technology shock strongly a!ects expected de#ation. An intuitive explanation of this is that a positive additive shock, equivalent to a decrease in government spending, in#uences the interest rate directly. This importance of additive technology shocks explains the failure of Ireland (1997) in explaining the behavior of the interest rate, as discussed in Leeper (1997). The model of Ireland (1997) does not have an additive technology shock or its equivalent. The second most important shock depends on the forecast horizon. It is the shock in money demand in the short run, and the multiplicative technology shock in the medium run. The shock in monetary policy does not seem to be important for the movement of the interest rate. Including a feedback mechanism of in#ation and the real activities into the monetary policy reaction function, as in Henderson and Kim (1999), would make monetary policy more important. With the new reaction function, the

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Table 5 Money decompositions Money

1

2

3

4

5

10

20

Monetary policy Money demand Multiplicative technology Additive technology

0.95 0.01 0.00 0.04

0.98 0.00 0.00 0.02

0.99 0.00 0.00 0.01

0.99 0.00 0.00 0.01

0.99 0.00 0.00 0.00

0.99 0.00 0.00 0.00

0.99 0.00 0.00 0.00

Price

1

2

3

4

5

10

20

Monetary policy Money demand Multiplicative technology Additive technology

0.17 0.12 0.66 0.05

0.17 0.12 0.65 0.05

0.17 0.12 0.65 0.05

0.17 0.12 0.65 0.05

0.18 0.12 0.65 0.05

0.19 0.12 0.65 0.05

0.20 0.12 0.64 0.05

Table 6 Price decompositions

multiplicative and the additive technology shocks would account for less of interest rate movements since part of the old propagation is endogenized by the feedback of the real activities. This conjecture is con"rmed by calibrating the parameter representing the sensitivity of the feedback. Table 5 shows that the movement of the money stock is almost entirely due to the shock in monetary policy. Even if money is signi"cantly endogenous from a statistical point of view, the endogeneity is not very sizable from an economic point of view. Other shocks play only a small role. The variance decomposition of money in our model is quite di!erent from the results in the structural VAR literature, where a shock in money demand plays an important role. Here, the money demand shock does not contribute at all, even if its variance is large relative to the variance of the monetary policy shock. Including the feedback mechanism into the monetary policy reaction function would also change the importance of other shocks in the movement of the money stock. The most important factor for price movements is the multiplicative technology shock. It explains steadily over 60% of the movements as shown in Table 6. The shock in monetary policy explains 15}20% and the shock in money demand explains 12% of price movements. The additive shock has a small e!ect, 5%, on price movements. Here again, the new speci"cation of monetary policy including the feedback through in#ation and the real activities would increase the importance of the monetary policy shock. Table 7 reports that the additive technology shock explains 40}55% of output #uctuations in the short run. This re#ects the fact that some movements of

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Table 7 Output decompositions Output

1

2

3

4

5

10

20

Monetary policy Money demand Multiplicative technology Additive technology

0.21 0.20 0.03 0.55

0.20 0.19 0.11 0.50

0.19 0.17 0.20 0.44

0.17 0.15 0.31 0.38

0.15 0.13 0.41 0.32

0.07 0.07 0.72 0.14

0.03 0.03 0.89 0.05

output do not accompany the changes of the marginal products, e.g. the interest rate. The shocks in government spending, i.e. additive technology, are more important than those in multiplicative technology. If the multiplicative and additive shocks are combined and called the real shocks, these explain less than 70% of the #uctuations, if the forecast horizon is less than 1 year. However, in the medium run, the multiplicative shock accounts for 70}90% of output #uctuations. In the short run, the shock in monetary policy and the shock in money demand are also important in understanding output #uctuations. Each explains 15}20% of output movements in the short run, but much less in the medium run. This variance decomposition of output is similar to the results of the structural VAR literature. 3.4. Cyclical implications It is standard in the DSGE literature to focus on the summary statistics describing the relative volatility of the series and their correlation with output. The formal maximum likelihood estimation procedure obviously does not place all the weight on this small set of statistics, so it is worthwhile to see what the parameter estimates imply for these commonly studied statistics. Therefore, the standard deviations and the contemporaneous correlations of the model variables are computed analytically and compared with those of the data. The tth period model forecasts are conditioned on the sample up to period t!1, and the parameters are set to the full-sample maximum likelihood estimates. For consistency with the linear trend of our model, both the model and data series are logged and di!erenced before calculation, so these statistics are in percentage terms.23 Table 8 shows that the standard deviation of output predicted by the model exactly matches that of the data. For the other three variables, the standard

23 Di!erent "ltering methods, including a Hodrick}Prescott "lter, produce similar results.

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Table 8 Contemporaneous second moments Model forecast (data) Interest rate Money

Interest rate

Money

0.0034 (0.0026) (!0.1276)

!0.3918

0.3753

0.0198

0.0866

0.1524

0.0099 (0.0066) (!0.2455)

!0.4229

Price

(0.0954)

0.0097 (0.0086) (0.0231)

Output

(0.3488)

(0.1360)

Price

Output

0.0089 (0.0089)

deviations of the model forecasts are larger than those from the data. The absolute values of the correlations of the model forecasts are also higher, except for the interest-rate}output pair. One possible reason for this is that the variables are highly persistent even after logarithmic di!erencing and that the estimation may be "tting this feature at the cost of "tting higher frequencies. Overall, the model matches all the signs of the second moments. Considering that the estimation procedure is designed to "t all aspects of the data, it is not obvious ex ante how well the estimates are able to match the small set of statistics. King and Watson (1996) analyze the nominal features of business cycles with three di!erent DSGE models. While the models have diverse successes and failures, none can account for the fact that the nominal and real interest rates are &inverted leading indicators' of real economic activity. That is, none of their three models captures the U.S. business cycle fact that a high nominal or real interest rate in the current quarter predicts a decrease in real economic activity two to six quarters in the future. To see if our model improves on that aspect, serial correlation coe$cients between the interest rate and output are computed. Table 9 shows that our model explains the feature that the nominal interest rate is an inverted leading indicator. The serial correlation coe$cients from the model reproduce the pattern of the data that an increase in the interest rate in the current period predicts a decrease in output two to six quarters in the future. The ability of the model to do this is mainly due to the introduction of the additive technology shock, which induces an instantaneous decrease in the interest rate and a future increase in output as shown in the fourth column of Fig. 3. A recent paper by Rotemberg and Woodford (1998) also features the property of the inverted leading indicator and it also introduces additive technology shocks in the form of government spending shocks.

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Table 9 Serial correlations between interest rate and output Corr(* log r , * log > ) t t`d d (quarters) Data Model

1 0.072 0.042

2 !0.297 !0.177

3 !0.188 !0.173

4 !0.209 !0.193

5 !0.198 !0.153

6 !0.177 !0.109

It is remarkable to see how crucial a role the additive technology shock plays in the analysis. For example, it is the most important shock in the variance decomposition of the interest rate at all horizons and accounts for about 50% of short-run output movements. Moreover, it is this shock which enables the model to produce the feature that the interest rate is a negative leading indicator. Note also that we have sometimes referred to it as a negative shock to government spending. However, as an empirical matter, it behaves in a quite di!erent way than a shock to government spending. Very di!erent results are obtained if we estimate an AR(1) process for government spending. Government spending is far smoother than the additive technology shock.24 Therefore, it is a topic of further research to "gure out what the additive technology shock really represents.

4. Conclusion Previous work using a #exible-price, competitive DSGE model has provided reasonable descriptions of the data on real variables. However, such work has not captured the nominal features of business cycles adequately. Typically, expansionary monetary policy produces neither a positive response of aggregate output nor a negative response of interest rates. This paper aims at "lling this gap with a DSGE model extended to allow for real and nominal rigidities. The introduction of additive technology shocks and the endogeneity of money are important in analyzing monetary business cycles. In order to select the

24 We use the government purchases of goods and services (ggeq in the Citibase) as the data for quarterly government spending. The variance of an innovation to government spending is more than 200 times smaller than the estimated variance of an innovation to the additive technology shock, and the unconditional variance of government spending is about 15 times smaller than that of the additive technology shock.

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magnitude of the e!ects of monetary policy in a data-dependent manner, the model is estimated using maximum likelihood on U.S. data. The estimated model exhibits reasonable impulse responses and its forecasts produce second moments similar to those of the data. As a by-product, it also reproduces the fact that an increase in interest rates in the current period predicts a decrease in real economic activity two to six quarters in the future, a feature of U.S. business cycles which has never been captured by previous research using DSGE models. It would be interesting to estimate the model for a sub-period and see how well the estimated model explains the out-of-sample data. This is particularly interesting since monetary policy regimes are said to have changed several times, e.g. the October 1979 Volker disin#ation. Fig. 2 shows that the model's implications for money and the price level fall apart after 1990. It would be more helpful to randomize the policy regimes. For further research, adding more structure into the model and using more data for estimation are likely to produce a better-behaving estimated model. Since this paper deals with only four variables as the data and three out of the four are nominal variables, it would be particularly interesting to add more real variables in the estimation and to see how well an augmented model could explain the additional variables. This exercise would evaluate additive technology shocks in explaining real variables. Introduction of additive technology shocks, which is a key factor in monetary business cycles as shown in this paper, could help or hurt the behavior of real variables.

Appendix A A.1. Parameter restrictions (Table 10) Table 10 Parameters

Region

a, b, d, a c, C (or g), k, h , h L Y A, / , / , / , b, m6 , l K W P o , o , oU , o U , o , o , o , o b A A k M r R p2, p2 , p2U , p2, p , p b A k 1 2

(0, 1) (1,R) (0,R) (!1,1) (0,R)

A.2. The data The following time series are extracted from the Citibase database (Table 11).

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Table 11 Series!

Title

fy! fm2 gd gnpq p16

Interest rate: federal funds (e!ective) (% per annum, NSA) Money stock: M2 (Bil. $) Implicit price de#ator: gross national product Gross national product (Bil. 1987$) (T.1.6) Population: total civilian noninstitutional (Thous., NSA)

!The variables of the models are de"ned as follows: r "1#(fyff), t 400 M "1000(fm2), t p16 P "gd, t > "1000(gnpq). t p16

A.3. Orthogonalization of the additive shock The model includes the additive shocks which are equivalent to the negative government spending shocks. Since the shocks are correlated with the multiplicative productivity shocks, we need to orthogonalize the two shocks. Consistent with the impulse-responses assumption that the multiplicative shocks are ordered before the additive shocks, the additive shocks are orthogonalized as follows:

A

B

p U p U eUt " A e t # eU ! A e t . t p2 A p2 A A A

(27)

Note that the two terms on the right-hand side are uncorrelated. Interpreting the additive shocks as government spending shocks, the intuition of orthogonalization is straightforward. The "rst part, ((p U /p2 )e t ), A A A is a response to the multiplicative technology shocks. The positive estimated covariance indicates the countercyclical component of government spending. This part is attributed to the multiplicative shocks when the impulse responses and the variance decompositions are calculated. The other part, (eU !(p U /p2 )e t ), is an innovation independent of the multiplicative shocks. t A A A The weights of each part in terms of contributing to the variance are o2 U and A (1!o2 U ).25 A

25 The estimation results produce o2 U "0.4303. A

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A.4. Estimation results Parameters are described in the same order as they are introduced in the model (Table 12). The standard errors are from the noninformative, #at prior over the whole real line regardless of the parameter restrictions.26 Therefore, standard t-tests should not be applied to test for the contribution of the parameter. Table 12 Parameter h L h Y b a p 1 p 2 b o b p2 b d / K / W a c C o A p2 A A oU p2U o U A / P o M l o r o R o k p2 k

Estimate 12.37 3.203 0.999 0.672 14.22 0.112 6]10~21 0.990 0.008 0.019 312 0.153 0.225 1.223 1.002 0.981 0.0003 7.01 0.911 18.328 0.656 0.806 !0.158 0.576 0.999 0.980 0.487 0.00002

Standard error 13.38 1.211 0.011 0.658 8.33 0.060 1]10~19 0.018 0.008 0.055 4 0.297 0.260 0.856 0.002 0.025 0.0002 4.84 0.029 7.331 0.298 1.815 0.085 0.266 3.777 3.582 0.006 0.00001

26 The Hessian of the log likelihood function is evaluated at the estimates and then inverted. The standard errors are read as the roots of the diagonal elements.

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References Abel, A.B., Blanchard, O.J., 1983. An intertemporal model of saving and investment. Econometrica 51 (3), 675}692. Aiyagari, S.R., Christiano, L.J., Eichenbaum, M., 1992. The output, employment, and interest rate e!ects of government consumption. Journal of Monetary Economics 30, 73}86. Anderson, E.W., Hansen, L.P., McGrattan, E.R., Sargent, T.J., 1995. On the mechanics of forming and estimating dynamic linear economies. Research Department Sta! Report 198, Federal Reserve Bank of Minneapolis; also in Amman, H.S., Kendrick, D.A., Rust, J. (Eds.), Handbook of Computational Economics, Vol. 1. Elsevier, Amsterdam, 1996. Beaudry, P., Devereux, M.B., 1995. Monopolistic competition, price setting and the e!ects of real and monetary shocks. Mimeo, University of British Columbia. Benassy, J., 1995. Money and wage contracts in an optimizing model of the business cycle. Journal of Monetary Economics 35, 303}315. Bernanke, B.S., Blinder, A.S., 1992. The federal funds rate and the channels of monetary transmission. American Economic Review 82, 901}921. Blanchard, O.J., Kahn, C.M., 1980. The solution of linear di!erence models under rational expectations. Econometrica 48, 1305}1311. Blanchard, O.J., Kiyotaki, N., 1987. Monopolistic competition and the e!ects of aggregate demand. American Economic Review 77 (4), 647}666. Chari, V.V., Christiano, L.J., Eichenbaum, M., 1995. Inside money, outside money, and short-term interest rates. Journal of Money, Credit and Banking 27 (4), 1354}1386. Cho, J., 1993. Money and the business cycle with one-period nominal contracts. Canadian Journal of Economics 26, 638}659. Cho, J., Cooley, T.F., 1995. The business cycle with nominal contracts. Economic Theory 6, 13}33. Christiano, L.J., Eichenbaum, M., 1992a. Liquidity e!ects and the monetary transmission mechanism. American Economic Review 82 (2), 346}353. Christiano, L.J., Eichenbaum, M., 1992b. Current real-business-cycle theories and aggregate labormarket #uctuations. American Economic Review 82 (3), 430}450. Christiano, L.J., Eichenbaum, M., Evans, C., 1996. Sticky price and limited participation models of money: a comparison. Mimeo, Northwestern University. Coleman, W.J., II, 1996. Money and output: a test of reverse causation. American Economic Review 86 (1), 90}111. Cooley, T.F., Hansen, G.D., 1989. The in#ation tax in a real business cycle model. American Economic Review 79 (4), 733}748. Cooley, T.F., Hansen, G.D., 1995. Money and business cycle.. In: Cooley, T.F. (Ed.), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ. Cooley, T.F., Prescott, E.C., 1995. Economic growth and business cycles. In: Cooley, T.F. (Ed.), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ. Dixit, A.K., Stiglitz, J., 1977. Monopolistic competition and optimum product diversity. American Economic Review 67, 297}308. Dotsey, M., Ireland, P., 1995. Liquidity e!ects and transactions technologies. Journal of Money, Credit and Banking 27 (4), 1441}1457. Dow Jr., J.P., 1995. The demand and liquidity e!ects of monetary shocks. Journal of Monetary Economics 36, 91}115. Fischer, S., 1977. Long-term contracts, rational expectations and the optimal money supply rule. Journal of Political Economy 85, 191}205. Gordon, D.B., Leeper, E.M., 1994. The dynamic impacts of monetary policy: an exercise in tentative identi"cation. Journal of Political Economy 102 (6), 1228}1247. Hairault, J., Portier, F., 1993. Money, new-Keynesian macroeconomics and the business cycle. European Economic Review 37, 1533}1568.

J. Kim / Journal of Monetary Economics 45 (2000) 329}359

359

Henderson, D., Kim, J., 1999. Exact utilities under alternative monetary rules in a simple macro model with optimizing agents. International Tax and Public Finance 6, 507}535. Hornstein, A., 1993. Monopolistic competition, increasing returns to scale and the importance of productivity shocks. Journal of Monetary Economics 31, 299}316. Ireland, P., 1997. A small, structural, quarterly model for monetary policy evaluation. Carnegie}Rochester Conference Series on Public Policy 47, 83}108. Kim, J., 1995. A note on likelihood calculation of a general stochastic linear di!erence equation. Mimeo. Kim, J., 1997. Three sources of increasing returns to scale. Mimeo. Kim, J., 1998. Adjustment costs of investment in general equilibrium: analytic results. Mimeo. Kimball, M.S., 1995. The quantitative analytics of the basic neomonetarist model. Journal of Money, Credit and Banking 27 (4), 1241}1277. King, R.G., 1991. Money and business cycles. Mimeo, University of Rochester. King, R.G., Watson, M.W., 1996. Money, prices, interest rates and the business cycle. Review of Economics and Statistics 78, 35}53. Kydland, F.E., Prescott, E.C., 1982. Time to build and aggregate #uctuations. Econometrica 50, 1345}1370. Leeper, E.M., 1997. A small, structural, quarterly model for monetary policy evaluation: a comment. Carnegie}Rochester Conference Series on Public Policy 47, 109}120. Leeper, E.M., Sims, C.A., 1994. Toward a modern macroeconomic model usable for policy analysis. In: Fischer, S., Rotemberg, J.J. (Eds.), NBER Macroeconomics Annual 1994. The MIT Press, Cambridge, MA. Long Jr., J.B., Plosser, C., 1983. Real business cycles. Journal of Political Economy 91, 39}69. Novales, A., 1992. Equilibrium interest-rate determination under adjustment costs. Journal of Economics Dynamics and Control 16 (1), 1}25. Ohanian, L.E., Stockman, A.C., 1995. Theoretical issues of liquidity e!ects. Federal Reserve Bank of St. Louis Review 77 (3), 3}25. Rotemberg, J.J., 1996. Prices, output and hours: an empirical analysis based on a sticky price model. Journal of Monetary Economics 37, 505}533. Rotemberg, J.J., Woodford, M., 1995. Dynamic general equilibrium models with imperfectly competitive product markets. In: Cooley, T.F. (Ed.), Frontiers of Business Cycle Research. Princeton University Press, Princeton, NJ. Rotemberg, J.J., Woodford, M., 1998. An optimization-based econometric framework for the evaluation of monetary policy: expanded version. Mimeo, Princeton University. Sims, C.A., 1992. Interpreting the macroeconomic time series facts: the e!ects of monetary policy. European Economic Review 36, 976}1000. Sims, C.A., 1994. A simple model for study of the determination of the price level and the interaction of monetary and "scal policy. Economic Theory 4, 381}399. Sims, C.A., 1995. Solving linear rational expectations models. Mimeo, Department of Economics, Yale University. Sims, C.A., Zha, T., 1995. Does monetary policy generate recessions?: Using less aggregated price data to identify monetary policy. Mimeo, Department of Economics, Yale University. Strongin, S., 1995. The identi"cation of monetary policy disturbances explaining the liquidity puzzle. Journal of Monetary Economics 35, 463}497. Yun, T., 1994. Nominal price rigidity, money supply endogeneity, and business cycles. Journal of Monetary Economics 37, 345}370.