Working Paper Deriving the Taylor principle when the central bank supplies money


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1 eonstor Der OpenAessPublikationsserver der ZBW LeibnizInformationszentrum Wirtshaft The Open Aess Publiation Server of the ZBW Leibniz Information Centre for Eonomis Davies, Ceri; Gillman, Max; Kejak, Mihal Working Paper Deriving the Taylor priniple when the entral bank supplies money Cardiff Eonomis Working Papers, No. E2012/20 Provided in Cooperation with: Cardiff Business Shool, Cardiff University Suggested Citation: Davies, Ceri; Gillman, Max; Kejak, Mihal (2012) : Deriving the Taylor priniple when the entral bank supplies money, Cardiff Eonomis Working Papers, No. E2012/20 This Version is available at: StandardNutzungsbedingungen: Die Dokumente auf EonStor dürfen zu eigenen wissenshaftlihen Zweken und zum Privatgebrauh gespeihert und kopiert werden. Sie dürfen die Dokumente niht für öffentlihe oder kommerzielle Zweke vervielfältigen, öffentlih ausstellen, öffentlih zugänglih mahen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter OpenContentLizenzen (insbesondere CCLizenzen) zur Verfügung gestellt haben sollten, gelten abweihend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrehte. Terms of use: Douments in EonStor may be saved and opied for your personal and sholarly purposes. You are not to opy douments for publi or ommerial purposes, to exhibit the douments publily, to make them publily available on the internet, or to distribute or otherwise use the douments in publi. If the douments have been made available under an Open Content Liene (espeially Creative Commons Lienes), you may exerise further usage rights as speified in the indiated liene. zbw LeibnizInformationszentrum Wirtshaft Leibniz Information Centre for Eonomis
2 Cardiff Eonomis Working Papers Working Paper No. E2012/20 Deriving the Taylor Priniple when the Central Bank Supplies Money Ceri Davies, Max Gillman and Mihal Kejak August 2012 Cardiff Business Shool Aberonway Building Colum Drive Cardiff CF10 3EU United Kingdom t: +44 (0) f: +44 (0) business.ardiff.a.uk This paper an be downloaded from eonpapers.repe.org/repe:df:wpaper:2012/20 This working paper is produed for disussion purpose only. These working papers are expeted to be published in due ourse, in revised form, and should not be quoted or ited without the author s written permission. Cardiff Eonomis Working Papers are available online from: eonpapers.repe.org/paper/dfwpaper/ and business.ardiff.a.uk/researh/aademisetions/eonomis/workingpapers Enquiries:
3 Deriving the Taylor Priniple when the Central Bank Supplies Money Ceri Davies Cardiff Business Shool Mihal Kejak CERGEEI July 23, 2012 Max Gillman Cardiff Business Shool Abstrat The paper presents a humanapitalbased endogenous growth, ashinadvane eonomy with endogenous veloity where exhange redit is produed in a deentralized banking setor, and money is supplied stohastially by the entral bank. From this it derives an exat funtional form for a general equilibrium Taylor rule. The inflation oeffi ient is always greater than one when the veloity of money exeeds one; veloity growth enters the equilibrium ondition as a separate variable. The paper then suessfully estimates the magnitude of the oeffi ient on inflation from 1000 samples of Monte Carlo simulated data. This shows that it would be spurious to onlude that the entral bank has a reation funtion with a strong response to inflation in a Taylor priniple sense, sine it is only meeting fisal needs through the inflation tax. The paper also estimates several deliberately misspeified models to show how an inflation oeffi ient of less than one an result from model misspeifiation. An inflation oeffi ient greater than one holds theoretially along the balaned growth path equilibrium, making it a sharply robust priniple based on the eonomy s underlying strutural parameters. JEL Classifiation: E13, E31, E43, E52 Keywords: Taylor rule, veloity, forwardlooking, misspeifiation bias We thank Hao Hong and Vo Phuong Mai Le for researh assistane; Samuel Reynard, Warren Weber, Paul Whelan and James Cloyne for disussion; presentations at the CDMA onferene, Louvain, the Monetary Poliy Conferene in Birmingham, the University of London, Birkbek, the FEBS onferene, London, and the Bank of England. 0
4 1 Introdution Interest rate rules are widely onsidered as monetary poliy reation funtions that represent how the entral bank adjusts a shortterm nominal interest rate in response to the state of the eonomy. The magnitude of the reation funtion oeffi ients are interpreted to reflet a poliymaker s attitude towards variation in key maroeonomi variables suh as inflation and the output gap. It has been suggested that poliymakers ought to adhere to the Taylor priniple, whereby inflation above target is met by a morethanproportional inrease in the shortterm nominal interest rate and hene an inrease in the real interest rate. Suh an interest rate rule forms one of the three ore equations of the prominent New Keynesian modelling framework, suh as in Woodford, 2003, Clarida et al., 1999; and Clarida et al. (2000). One wellknown finding omes from the latter paper whih onludes that the Taylor priniple holds for a VolkerGreenspan sample of U.S. data but that it is violated for a prevolker sample during whih the Fed was deemed to be aommodating in its reation funtion. In ontrast, a historial strand of literature going bak to Poole (1970), and updated for example by Alvarez et al. (2001) and Chowdhury and Shabert (2008), onsiders interest rate rules and money supply rules as two ways of implementing the same monetary poliy. This paper perhaps most losely follows Alvarez et al. (2001) by deriving the equilibrium nominal interest rate in a rule form within a general equilibrium eonomy in whih the entral bank onduts poliy by stohastially supplying money. Instead of an exogenous fration of agents being able to use bonds as in Alvarez et al., here the onsumer purhases goods with an endogenous fration of banksupplied intratemporal redit that avoids the inflation tax on exhange. This ashinadvane monetary eonomy is also extended to inlude endogenous growth, along with endogenous veloity, as in Benk et al. (2010). The resulting equilibrium nominal interest rate ondition nests the standard Taylor rule within a more general forwardlooking setting that endogenously inludes traditional monetary elements, suh as the (exogenous) veloity in Alvarez et al., and the money demand in MCallum and Nelson (1999). The endogenous growth aspet implies that the target terms of the equilibrium Taylor ondition, suh as the inflation rate target or the potential output level, are the balaned growth path (BGP ) equilibrium values of the related variables. In addition, the oeffi ients of the Taylor ondition are a funtion of the model s utility and tehnology parameters along with the BGP money supply growth rate. This in essene fulfills Luas s (1976) goal of postulating poliy rules with oeffi ients that depend expliitly upon the eonomy s underlying utility and tehnology oeffi ients plus a key poliy hoie, in this ase the BGP rate of money supply growth. Aestheti as suh a formulation of the Taylor ondition may be, Luas s researh agenda provides a solid result here: a theoretial derivation of the Taylor priniple where the oeffi ient on the inflation term always exeeds or equals one. The priniple holds for any given nonfriedman (1969) optimum BGP money supply growth rate, it equals one only at the Friedman optimum, and never falls below one. Similarly, the 1
5 inflation oeffi ient always exeeds one when the endogenous veloity exeeds one sine the ashinadvane veloity rises above one for any nonfriedman optimal rate of money supply growth. In general, the inflation oeffi ient rises with the BGP veloity level. Another entral result is that the expeted veloity growth rate itself enters the Taylor ondition as an additional term, in ontrast to standard Taylor rules. Omitting this term an ause misspeifiation bias in estimated Taylor rules within the eonomy. Having derived the Taylor ondition, the paper then estimates it by applying three onventional estimation proedures to one thousand samples of artifiial data simulated from the baseline model, where the simulated data is passed through three standard filters prior to estimation. The results verify the theoretial form of the Taylor ondition along several key dimensions. In partiular, the oeffi ient on inflation is greater than one and lose to its theoretial magnitude for all three estimation tehniques and for all three data filters. Satisfying the Taylor priniple in this fashion, robustness tests explore the impat of estimating two alternative Taylor onditions. This involves two ad ho, deliberately misspeified equations relative to the true theoretial expression: the first hanges just one of the variables in the Taylor ondition while the seond posits a standard Taylor rule that involves multiple misspeifiation errors. Using the same artifiial data, estimating the two misspeified models results in the oeffi  ient on inflation falling below one, ausing the Taylor priniple to fail. In the ontext of atual data, this result would typially be interpreted as the entral bank being passive or weak towards inflation. Here, the paper shows that suh an interpretation ould be spurious in that it ould our simply beause of a misspeified estimating equation. 1 The estimated Taylor rule emerges even though the entral bank is merely satisfying fisal needs through the inflation tax. This implies the entral point of the paper: it would be spurious within this eonomy to assoiate the Taylor ondition with a reation funtion for the nominal interest rate sine in the model the entral bank just stohastially prints money. Seond, failure of the soalled Taylor priniple in numerous published empirial studies may be a result of model misspeifiation rather than behavioral hanges by the entral bank per se. Indeed, our urrent preliminary extension of this work, not presented here, shows that estimation with atual US data of Taylor rules whih inlude the unonventional terms implied by the theory of this paper  partiularly veloity growth  an reverse the result that the oeffi ient on inflation falls below unity during periods of maroeonomi instability. 2 Related work is vast but inludes Taylor (1999), who alludes to the possibility that an interest rate rule an be derived from the quantity theory of money. Sørensen and WhittaJaobsen (2005, pp ) present suh a derivation under the assumption of onstant money growth whereby the oeffi ients of 1 Estimation of simulated data is onduted by Fève and Auray (2002), for a standard CIA model, and Salyer and Van Gaasbek (2007), for a limited partiipation model. We are indebted to Warren Weber for the suggestion to follow suh an approah here. 2 Clarida et al. s (2000) prevolker sample, for example, orresponds to a period of high and variable inflation. 2
6 the rule relate to elastiities of money demand rather than the preferenes of poliymakers. Fève and Auray (2002) and Shabert (2003) onsider the link between money supply rules and interest rate rules in standard ashinadvane models with veloity fixed at unity. Alternatively, the paper ould be viewed in light of Canzoneri et al. (2007) in that it shows how the puzzle of estimating the Euler equation for the nominal interest rate an be solved by ombining that equilibrium ondition with the stohasti asset priing kernel to derive an expression for a Taylor ondition that an be suessfully estimated. Setion 2 desribes the eonomy, as in Benk et al. (2008, 2010). Setion 3 derives the model s Taylor ondition and Setion 4 provides the baseline alibration. Setion 5 desribes the eonometri methodology whih is applied to modelsimulated data and presents the orresponding estimation results. Setion 6 derives speial theoretial ases of the more general (Setion 2) model to show how alternative Taylor onditions an be derived. Setion 7 presents a disussion and Setion 8 onludes. 2 Stohasti Endogenous Growth with Banking The representative agent eonomy is as in Benk et al (2008, 2010) but with a deentralized banking setor that produes redit as in Gillman and Kejak (2011). By ombining the business yle with endogenous growth, stationary inflation lowers the output growth rate as supported empirially in Gillman et al. (2004) and Fountas et al. (2006), for example. Further, money supply shoks an ause inflation at low frequenies, as in Haug and Dewald (2011) and as supported by Sargent and Surio (2008, 2011), whih an lead to output growth effets if the shoks are persistent and repeated. This allows shoks over the business yle to ause hanges in growth rates and in stationary ratios. The shoks to the goods setor produtivity and the money supply growth rate are standard, while the third shok to redit setor produtivity exists by virtue of the model s endogenous money veloity via the prodution funtion used extensively in the finanial intermediation miroeonomis literature starting with Clark (1984). The shoks our at the beginning of the period, are observed by the onsumer before the deision making proess ommenes, and follow a vetor firstorder autoregressive proess. For goods setor produtivity, z t, the money supply growth rate, u t, and bank setor produtivity, v t : Z t = Φ Z Z t 1 + ε Zt, (1) where the shoks are Z t = [z t u t v t ], the autoorrelation matrix is Φ Z = diag {ϕ z, ϕ u, ϕ v } and ϕ z, ϕ u, ϕ v (0, 1) are autoorrelation parameters, and the shok innovations are ε Zt = [ɛ zt ɛ ut ɛ vt ] N (0, Σ). The general struture of the seondorder moments is assumed to be given by the varianeovariane matrix Σ. These shoks affet the eonomy as desribed below, and as alibrated in Benk et al. (2010). 3
7 2.1 Consumer Problem A representative onsumer has expeted lifetime utility from onsumption of goods, t, and leisure, x t ; with β (0, 1), ψ > 0 and θ > 0, this is given by: U = E 0 t=0 β ( tx ψ t ) 1 θ. (2) 1 θ Output of goods, y t, and inreases in human apital, are produed with physial apital and effetive labor eah in CobbDouglas fashion; the bank setor produes exhange redit using labor and deposits as inputs. Let s Gt and s Ht denote the frations of physial apital that the agent uses in the goods prodution (G) and human apital investment (H), whereby: s Gt + s Ht = 1. (3) The agent alloates a time endowment of one between leisure, x t, labor in goods prodution, l Gt, time spent investing in the stok of human apital, l Ht, and time spent working in the bank setor (F subsripts for Finane), denoted by l F t : l Gt + l Ht + l F t + x t = 1. (4) Output of goods an be onverted into physial apital, k t, without ost and so is divided between onsumption goods and investment, denoted by i t, net of apital depreiation. Thus, the apital stok used for prodution in the next period is given by: k t+1 = (1 δ k )k t + i t = (1 δ k )k t + y t t. (5) The human apital investment is produed using apital s Ht k t and effetive labor l Ht h t, with A H > 0 and η [0, 1], suh that the human apital flow onstraint is h t+1 = (1 δ h )h t + A H (s Ht k t ) 1 η (l Ht h t ) η. (6) With w t and r t denoting the real wage and real interest rate, the onsumer reeives nominal inome of wages and rents, P t w t (l Gt + l F t ) h t and P t r t (s Gt + s Qt ) k t, a nominal transfer from the government, T t, and dividends from the bank. The onsumer buys shares in the bank by making deposits of inome at the bank. Eah dollar deposited buys one share at a fixed prie of one, and the onsumer reeives the residual profit of the bank as dividend inome in proportion to the number of shares (deposits) owned. Denoting the real quantity of deposits by d t, and the dividend per unit of deposits as R F t, the onsumer reeives a nominal dividend inome of P t R F t d t. The onsumer also pays to the bank a fee for redit servies, whereby one unit of redit servie is required for eah unit of redit that the bank supplies the onsumer for use in buying goods. With P F t denoting the nominal prie of eah unit of redit, and q t the real quantity of redit that the onsumer an use in exhange, the onsumer pays P F t q t in redit fees. 4
8 With other expenditures on goods, of P t t, and physial apital investment, P t k t+1 P t (1 δ k )k t, and on investment in ash for purhases, of M t+1 M t, and in nominal bonds B t+1 B t (1 + R t ), where R t is the net nominal interest rate, the onsumer s budget onstraint is: P t w t (l Gt + l F t ) h t + P t r t s Gt k t + P t R F t d t + T t (7) P F t q t + P t t + P t k t+1 P t (1 δ k )k t + M t+1 M t +B t+1 B t (1 + R t ). The onsumer an purhase the goods by using either money M t or redit servies. With the lump sum transfer of ash T t oming from the government at the beginning of the period, and with money and redit equally usable to buy goods, the onsumer s exhange tehnology is: M t + T t + P t q t P t t. (8) Sine all ash omes out of deposits at the bank, and redit purhases are paid off at the end of the period out of the same deposits, the total deposits are equal to onsumption. This gives the onstraint that: d t = t. (9) Given k 0, h 0, and the evolution of M t (t 0) as given by the exogenous monetary poliy in equation (17) below, the onsumer maximizes utility subjet to the budget, exhange and deposit onstraints (7)(9). 2.2 Banking Firm Problem The bank produes redit that is available for exhange at the point of purhase. The bank determines the amount of suh redit by maximizing its dividend profit subjet to the labor and deposit osts of produing the redit. The prodution of redit uses a onstant returns to sale tehnology with effetive labor and deposited funds as inputs. In partiular, with A F > 0 and γ (0, 1): q t = A F e vt (l F t h t ) γ d 1 γ t, (10) where A F e vt is the stohasti fator produtivity. Subjet to the prodution funtion in equation (10), the bank maximizes profit Π F t with respet to the labor l F t and deposits d t : Equilibrium implies that: Π F t = P F t q t P t w t l F t h t P t R F t d t. (11) ( PF t P t ) ( ) γ 1 lf γa F e vt t h t = w t ; (12) d t 5
9 ( PF t P t ) ( ) γ lf (1 γ) A F e vt t h t = R F t. (13) d t ( PF t These indiate that the marginal ost of redit, P t ), is equal to the marginal w fator prie divided by the marginal fator produt, or t ) γ 1, and γa F e v t ( lf t h t d t that the zero profit dividend ( yield ) paid on ( deposits ) is equal to the fration of the marginal ost given by PF t P t (1 γ) qt d t. 2.3 Goods Produer Problem The firm maximizes profit given by y t w t l Gt h t r t s Gt k t, subjet to a standard CobbDouglas prodution funtion in effetive labor and apital: y t = A G e zt (s Gt k t ) 1 α (l Gt h t ) α. (14) The first order onditions for the firm s problem yield the standard expressions for the wage rate and the rental rate of apital: ( ) 1 α sgt k w t = αa G e zt t, (15) l Gt h t ( ) α sgt k r t = (1 α)a G e zt t. (16) l Gt h t 2.4 Government Money Supply It is assumed that the government poliy inludes sequenes of nominal transfers as given by T t = Θ t M t = (Θ + e ut 1)M t, Θ t = [M t M t 1 ]/M t 1, (17) where Θ t is the growth rate of money and Θ is the stationary gross growth rate of money. 2.5 Definition of Competitive Equilibrium The representative agent s optimization problem an be written reursively as: V (s) = max {u(, x) + βev,x,l G,l H,l F,s G,s H,q,d,k,h,M (s )} (18) subjet to the onditions (3) to (9), where the state of the eonomy is denoted by s = (k, h, M, B; z, u, v) and a prime ( ) indiates the nextperiod values. A ompetitive equilibrium onsists of a set of poliy funtions (s), x(s), l G (s), l H (s), l F (s), s G (s), s H (s), q(s), d(s), k (s), h (s), M (s), B (s) priing funtions P (s), w(s), r(s), R F (s), P F (s) and a value funtion V (s), suh that: (i) the onsumer maximize utility, given the priing funtions and the poliy funtions, so that V (s) solves the funtional equation (18); 6
10 (ii) the goods produer maximizes profit similarly, with the resulting funtions for w and r being given by equations (15) and (16); (iii) the bank firm maximizes profit similarly in equation (11) subjet to the tehnology of equation (10) (iv) the goods, money and redit markets lear, in equations (7) and (14), and in (8), (17), and (10). 3 General Equilibrium Taylor Condition The Taylor ondition is now derived as an equilibrium ondition of the Benk et al. (2010) model desribed in the previous setion. Beginning from the firstorder onditions of the model, we obtain: { θ t+1 1 = βe xψ(1 θ) t+1 t θ t x ψ(1 θ) t R t R t+1 R t+1 π t+1 }, (19) where R and π are gross rates of nominal interest and inflation, respetively; R t is 1 plus a weighted average osts of exhange, with weights of m on the money ost R t 1 and 1 m on the average redit ost of γ (R t 1) : R t = 1 + m ( t (R t 1) + γ 1 m ) t (R t 1). t t m t t is the onsumption normalized money demand, i.e. the inverse of the onsumption veloity of money. In effet, equation (19) augments a standard onsumption Euler equation with the growth rate of this average ost of exhange. If all transations are onduted using money (m t / t = 1) then equation (19) reverts bak to the familiar onsumption Euler equation whih would feature as an equilibrium ondition of a standard CIA model without a money alternative. 3 For any variable z t, define ẑ t ln z t ln z, where the absene of a time subsript denotes a BGP stationary value, and define ĝ z,t+1 ln z t+1 ln z t, whih approximates the growth rate at time t + 1 for suffi iently small z t. Consider a loglinear approximation of (19) evaluated around the BGP : 0 = E t { θĝ,t+1 ψ (1 θ) ĝ x,t+1 + ĝ R,t+1 R t+1 + π t+1 }. Rearranging this in terms of R t gives the Taylor ondition expressed in logdeviations from the BGP equilibrium: R t = E t {Ω π t+1 + Ωθĝ,t+1 Ωψ (1 θ) ĝ x,t+1 (20) (1 γ) ( ) 1 m [ m ]} + R [ 1 (1 γ) ( )] 1 m (R 1) 1 m ĝ m,t+1 R t+1, 3 The nominal interest rate and inflation both enter equation (19) with a one period lead. This is onsistent with Carlstrom and Fuerst s (2001) "ash in advane timing" whih ontrasts with their "ash when I m done timing", where both the nominal interest rate and inflation enter the Euler equation ontemporaneously. Carlstrom and Fuerst (2001) rejet the latter for CIA models. 7
11 where Ω 1 + (1 γ)(1 m ) R[1 (1 γ)(1 m )] 1. The Taylor ondition (20) an now be expressed in net rates and absolute deviations from the BGP equilibrium, as demonstrated by the following proposition. Proposition 1 An equilibrium ondition of the eonomy is in the form of the Taylor Rule (Orphanides, 2008) that sets deviations of the shortterm nominal interest rate from some baseline path in proportion to deviations of target variables from their targets: ( R t R = ΩE t (π t+1 π) + ΩθE t g,t+1 g ) Ωψ (1 θ) E t g x,t+1 (21) (1 γ) ( ) 1 m [ m + R [ 1 (1 γ) ( ( )] 1 m (R 1) 1 m E t g m,t+1 E t Rt+1 R )]. where Ω 1, and for a given w, then Ω Ω R > 0 and A F > 0, and the target values are equal to the balaned growth path equilibrium values. 4 Proof. Sine the BGP solution for normalized money demand is: 0 m = 1 A F ( (R 1) γaf w ) γ 1 γ 1, then Ω 1 + (1 γ)(1 m ) Ω 1 and, given w, R[1 (1 γ)(1 m )] R 0 and Ω A F 0. For a linear prodution funtion of goods, w is the onstant marginal produt of labor, but more generally w is endogenous and will hange; however this hange in w is quantitatively small ompared to hanges in R and A F, so that the derivatives above almost always hold true. Note that for a unitary onsumption veloity of money, the latter two veloity growth and forward interest terms drop out of the equation (21) The term in π in equation (21) an be ompared to the inflation target that features in many interest rate rules (e.g. Taylor, 1993; Clarida et al., 2000). This is usually set as an exogenous onstant in a onventional rule but represents the BGP rate of inflation in the Taylor ondition. 5 The term in onsumption growth is similar, but not idential, to the first differene of the output gap that features in the soalled speed limit rule (Walsh, 2003). Alternatively, the term in the growth rate of leisure time an be ompared to the unemployment rate whih sometimes features in onventional interest rate rules in plae of the output gap. 6 Equation (21) also ontains two terms whih are not usually found in standard monetary poliy reation funtions. First, there is a term in the growth 4 This is the the Brookings projet form of the Taylor rule as desribed in Orphanides (2008). 5 Although see Ireland (2007) for an example of a onventional interest rate rule with a timevarying inflation target. 6 For example, Mankiw (2001) inludes the unemployment rate in an interest rate rule and Rudebush (2009) inludes the unemployment gap. 8
12 rate of the real (onsumption normalized) demand for money. Conventional interest rate rules are usually onsidered in the ontext of models whih omit monetary relationships and thus money demand does not feature diretly in the model, let alone the poliy rule. 7 Seondly, the Taylor ondition ontains a term in the expeted future nominal interest rate. This ontrasts with the lagged nominal interest term whih is often used to apture interest rate smoothing in a onventional rule (e.g. Clarida et al., 2000). In general, the oeffi ient on inflation in (21) exeeds unity (Ω > 1). This repliates the Taylor priniple whereby the nominal interest rate responds more than oneforone to (expeted future) inflation deviations from target. However, the inflation oeffi ient in the Taylor ondition is a funtion of the BGP nominal interest rate (R), the onsumption normalized demand for real money balanes (m/) and the effi ieny with whih the banking setor transforms units of deposits into units of the redit servie, as refleted by the magnitude of (1 γ). Furthermore, higher produtivity in the banking setor (A F ) auses a higher veloity and implies a larger inflation oeffi ient in the Taylor ondition. The magnitude of Ω learly does not reflet a poliymaker reation to inflation in the onventional, reation funtion sense. 8 Equation (21) an alternatively be rewritten in terms of the onsumption veloity of money, V t t m t, and the produtive time, or employment, growth rate (l l G + l H + l F = 1 x). Using the fat that x t = 1 x l x t : ( R t R = ΩE t (π t+1 π) + ΩθE t g,t+1 g ) l + Ωψ (1 θ) 1 l E tg l,t+1 ( Ω V E t g V,t+1 (Ω 1) E t Rt+1 R ). (22) Where overbarred terms again denote net rates and ( (R 1) (1 γ) m ) Ω V R γ + (1 γ) m. Proposition 2 For the Taylor ondition of equation (22), it is always true that 0 Ω V 1 Ω. 7 Shifts in the demand for money are perfetly aommodated by adjustments to the money supply in order to maintain the ruleimplied nominal interest rate. This, it is laimed, renders the evolution of the money supply an operational detail whih need not be modelled diretly (e.g. Woodford, 2008). 8 Unlike Sørensen and WhittaJaobsen s (2005, pp ) quantity theory based equilibrium ondition, the inflation oeffi ient in (21) exeeds unity for any (admissible) interest elastiity of money demand. In their expression, the inflation oeffi ient falls below unity if the interest (semi) elastiity of money demand exeeds one in absolute value. In the Benk et al. (2010) model, the oeffi ient on inflation would exeed unity even in this ase but the entral bank would not wish to inrease the money supply growth rate to this extent beause seigniorage revenues would begin to reede as the elastiity inreases beyond this point. 9
13 Proof. (1 γ) ( ) 1 m [ ] γ Ω 1 + R[1 (1 γ) ( m ) 1 m 1; ] = 1 A 1 1 γ (R 1) γ 1 γ F 1; w ( ) ( 1 (1 γ) 1 m ) m (R 1) (1 γ) 0; 0 Ω V R 1 (1 γ) ( ) 1 m 1; 0 Ω V 1 Ω. Note that at the Friedman (1969) optimum of (gross) R = 1, then m = 1, ω = 0, and the veloity oeffi ient is Ω V = 0. The veloity growth term only matters when the nominal interest rate and inflation differ from the Friedman ((1969) optimum and ) flutuate. In turn, this has impliations for Ω = 1 + (1 γ)(1 m ), sine when R = 1, then (1 γ) ( ) 1 m = 0, and Ω = 1. R[1 (1 γ)(1 m )] For m below one (i.e. veloity above one), whih is true for most pratial experiene, the model s equivalent of the Taylor priniple, Ω > 1, holds. Corollary 3 Given w, then Ω ΩV R 0, R 0, Ω A F 0, ΩV A F 0. Proof. This omes diretly from the definitions of parameters above. A higher target R an be aomplished only by a higher BGP money supply growth rate. This would in turn make the inflation oeffi ient Ω larger, and so also the onsumption growth oeffi ient (Ωθ), and the forward interest rate and veloity oeffi ients would beome more negative. A higher redit produtivity fator A F, and so a higher veloity, auses a higher inflation oeffi ient and a more negative response to the forwardlooking interest term but a less negative oeffi ient on the veloity growth term. Note that with exogenous growth, the above Taylor ondition would appear to look idential. However, under exogenous growth the targeted inflation rate and growth rate of the eonomy are unrelated and exogenously speified. Under endogenous growth, the targets are instead the endogenously determined BGP values: for inflation, the growth rate, and the nominal interest rate. And all of these are determined in part by the long run stationary money supply growth rate Θ, whih is exogenously given. In turn, this Θ translates diretly into a long run inflation target aepted by the entral bank, suh as two perent. So the model assumes only this target of a long run money supply growth, or alternatively, the long run inflation rate target. 3.1 Misspeified Taylor Condition with Output Growth It is not surprising to find that the growth rate of onsumption appears in equation (22) rather than the output growth rate given that the derivation of the Taylor ondition begins from the onsumption Euler equation (19). However, the Taylor ondition an be rewritten to inlude an output growth term and thus orrespond more losely to standard Taylor rule speifiations, in partiular the 10
14 speed limit rule onsidered by Walsh (2003). To derive this alternative rule, onsider that the equation, y t = t + i t, implies that ŷ t = y ĉt + i y ît, with î t = k i [ kt (1 δ) k ] t 1. The growth rate of investment an be understood as the aeleration of the growth of apital gross of depreiation. The Taylor ondition rewrites as [ y R t R = ΩE t (π t+1 π) + Ωθ E ( t gy,t+1 g ) i E ( t gi,t+1 g )] (23) +Ωψ (1 θ) l 1 l E tg l,t+1 Ω V E t g V,t+1 (Ω 1) E t ( Rt+1 R ). A term in investment growth does not appear in standard, exogenously speified Taylor rules but plays a role as part of what is interpreted as growth in the output gap in this Taylor ondition with output growth. Equation (23) forms the basis for the two misspeified estimating equations onsidered in Setion 5. The first misspeified estimating equation simply replaes the onsumption growth term in equation (22) with an output growth term as follows: R t R = ΩE t (π t+1 π) + Ωθ [ E t ( gy,t+1 g )] (24) +Ωψ (1 θ) l 1 l E tg l,t+1 Ω V E t g V,t+1 (Ω 1) E t ( Rt+1 R ). As the omparison between equation (23) and equation (24) shows, suh an estimating equation erroneously overlooks the weighting on the output growth rate ( y ) and omits the term in the investment growth rate. Replaing onsumption growth with output growth without an additional term in investment therefore misrepresents the struture of the underlying, Benk et al. (2010) model and as suh equation (24) is misspeified. Note that with no physial apital in the eonomy, suh a Taylor ondition as above would be the orret equilibrium ondition of the eonomy. 3.2 Misspeified Standard Taylor Rule The seond misspeified model erroneously imposes yet more restritions on equation (23). Imposing the same restritions used to arrive at equation (24) but also dropping the terms in produtive time and veloity gives: R t R = ΩE t (π t+1 π) + Ωθ [ E t ( gy,t+1 g )] (25) (Ω 1) E t ( Rt+1 R ). This an be interpreted as a onventional interest rate rule with a forwardlooking interest rate smoothing term; the additional restrition that Ω = 1 would repliate a standard interest rate rule without interest rate smoothing. One again, equation (25) does not aurately represent an equilibrium ondition of the Benk et al. (2010) eonomy and is therefore misspeified. The 11
15 Preferenes θ 1 Relative risk aversion parameter ψ 1.84 Leisure weight β 0.96 Disount fator Goods Prodution α 0.64 Labor share in goods prodution δ k Depreiation rate of goods setor A G 1 Goods produtivity parameter Human Capital Prodution ε 0.83 Labor share in human apital prodution δ h Depreiation rate of human apital setor A H 0.21 Human apital produtivity parameter Banking Setor γ 0.11 Labor share in redit prodution A F 1.1 Banking produtivity parameter Government σ 0.05 Money growth rate Table 1: Parameters first two terms would indeed be the orret equilibrium Taylor ondition if the eonomy had neither physial apital or the ability to use exhange redit to avoid the inflation tax. Then Ω = 1, there is no veloity or forward interest rate term, and the output growth term would enter as above. 4 Calibration We follow Benk et al. (2010) in using postwar U.S. data to alibrate the model (Table 1) and alulate a series of target values onsistent with this alibration (Table 2); please see Benk et al. for the shok proess alibration. Subjet to this alibration, we derive a set of theoretial preditions for the oeffi ients of the Taylor ondition (22). These values will subsequently be ompared to the oeffi ients estimated from artifiial data simulated from the model. Consider first the inflation oeffi ient (Ω). Aording to the alibration and target values presented in tables 1 and 2, its theoretial value is (1 γ) ( ) 1 m Ω = 1+ R [ 1 (1 γ) ( (1 0.11) (1 0.38) )] 1 m = (1 (1 0.11) (1 0.38)) = And for R = 1, only ash is used so that m = 1 and Ω reverts to its lower bound of one. This also happens with zero redit produtivity (A F = 0), in whih ase only ash is used in exhange. 12
16 g Avg. annual output growth rate π Avg. annual inflation rate R Nominal interest rate l G Labor used in goods setor l H 0.20 Labor used in human apital setor l F Labor used in banking setor i/y Investmentoutput ratio in goods setor m/ 0.38 Share of money transations x 0.55 Leisure time l 1 x 0.45 Produtive time Table 2: Target Values The remaining oeffi ients, exept for veloity, are simple funtions of the inflation oeffi ient. The onsumption growth oeffi ient is Ωθ, whih with θ = 1 for logutility should simply take the same magnitude as the oeffi ient on inflation (θω = 2.125). The oeffi ient on the produtive time growth rate is 0 = Ω (1 θ) ψ l 1 l beause of log utility. However with leisure preferene alibrated at 1.84, and produtive time (1 x l) along the BGP equal to 0.45, the estimated value of the produtive time oeffi ient an be interpreted as implying a ertain θ as fatored by Ωψ l 1 l = (2.125) (1.84) = Given the magnitude of the inflation oeffi ient, the oeffi ient on the forward interest term is simply (Ω 1) = 1.125; the veloity oeffi ient Ω V is : (R 1) R ( ) (1 γ) m [ ( )] 1 (1 γ) 1 m ( ( ) = (1 0.11) 0.38 (1 (1 0.11) (1 0.38)) At the Friedman (1969) optimum (R = 1), Ω V = 0. In this ase the omission of the term in veloity growth in the estimation exerises that follow would be innouous but this is not true in general. 5 Artifiial Data Estimation The strutural model whih underpins the general equilibrium Taylor ondition forms the basis for the data generating proess of the simulated data. In partiular, the model introdued in Setion 2 is simulated using the alibration provided in Setion 4 in order to generate 1000 alternative joint histories for eah of the variables in equation (22), where eah history is 100 periods in length. To do so, 100 random sequenes for the shok vetor innovations are generated. Control funtions of the loglinearized model are then used to ompute sequenes for eah variable. Eah observation within a given history may be thought of as an annual period given the frequeny onsidered by the Benk et al. (2010) model. The data set an therefore be viewed as omprising of 1000, 100year, samples of artifiial data. ). 13
17 5.1 Estimation Methodology This setion presents the results of estimating a orretly speified estimating equation based upon the true theoretial relationship (22) against artifiial data generated from the Benk et al. (2010) model. 9 In a similar manner, two alternative estimating equations are evaluated using this same data set. Sine these alternative estimating equations differ from the expression based upon the true theoretial relationship, they neessarily onstitute misspeified empirial models. 10 Prior estimation, the simulated data is filtered alternatively by 1) a Hodrik Presott (HP) filter with a smoothing parameter seleted in aordane with Ravn and Uhlig (2002); 2) a Christiano and Fitzgerald (2003) band pass filter whih uses a 38 window that is standard for the business yle frequeny; and 3) a Christiano and Fitzgerald (2003) band pass filter whih uses a 215 window in order to retain lower frequeny trends in the data as well, in partiular as in Comin and Gertler s (2006) mediumterm yle. 11 A priori, the 215 band pass filter may be regarded as the most relevant to the underlying theoretial model beause shoks in the model an ause low frequeny events during the business yle suh as a hange in the permanent inome level without it returning to its previous level. 12 The first estimation tehnique onsidered is OLS, as used by Taylor (1999) in the ontext of a ontemporaneous interest rate rule. However, beause expeted future variables may be orrelated with the error term we seek a suitable set of instruments to proxy for these terms. 13 Two instrumental variables (IV) tehniques are onsidered and eah differs by the instrument set employed. The first is a two stage least squares (2SLS) estimator under whih the first lags of inflation, onsumption growth, produtive time growth and veloity growth and the seond lag of the nominal interest rate (sine the first lag is the dependent variable) are used as instruments. Adding a onstant term to the instrument set yields a 2SLS estimator with no overidentifying restritions. In using lagged 9 The exerise onduted here is similar to those onduted by Fève and Auray (2002), for a standard CIA model, and Salyer and Van Gaasbek (2007), for a limited partiipation model. 10 We aknowledge that in a full information maximum likelihood estimation that uses all of the equilibrium onditions of the eonomy we may be able to reover almost exatly the theoretial oeffi ients of the Taylor ondition; we leave that exerise as an important part of future researh that enompasses the entire alternative model; and then we ould also ompare it to the standard three equation entral bank poliy model. 11 Their mediumterm yle is defined using a band pass filter for quarterly data; the general priniple is to retain elements of the data that the HP and 38 filters would ordinarily onsign to the trend. 12 The filtering proedure takes aount of the Siklos and Wohar (2005) ritique of empirial Taylor rule studies whih do not address the nonstationarity of the data. Standard ADF and KPSS tests (results not reported) suggest that the filtered data onsidered here is stationary. Aordingly, the filters do not implement a detrending proedure. 13 Empirial studies usually deal with expeted future terms either by replaing them with realised future values and appealing to rational expetations for the resulting onditional foreast errors (e.g. Clarida et al., 1998, 2000) or by using private setor or entral bank foreasts as empirial proxies (e.g. Orphanides, 2001; Siklos and Wohar, 2005). 14
18 terms as instruments we exploit the fat that suh terms are predetermined and thus not suseptible to the simultaneity problem whih motivates the use of IV tehniques. The 2SLS proedure applies a NeweyWest adjustment for heteroskedastiity and autoorrelation (HAC) to the oeffi ient ovariane matrix. The seond IV proedure is a generalized method of moments (GMM) estimator under whih three additional lags of inflation, onsumption growth, produtive time growth and veloity growth and two further lags of the nominal interest rate are added to the instrument set. 14 Expanding the instrument set in this manner redues the sample size available for eah of the 1000 estimation runs but enables the validity of the instrument set to be assessed using the Hansen Jtest. The GMM estimator employed iterates on the weighting matrix in two steps and applies a HAC adjustment to the weighting matrix using a Bartlett kernel with a NeweyWest fixed bandwidth. 15 A similar HAC adjustment is also applied to the ovariane weighting matrix. Results are presented in three sets of tables, one set for eah estimating equation, and are further subdivided aording to the statistial filter applied to the data. Alongside the estimates obtained from an unrestrited estimating equation, eah table also reports estimates obtained from a restrited estimating equation whih arbitrarily omits the forward interest rate term (β 5 = 0). This arbitrary restrition demonstrates the importane of the dynami term in equation (22). Eah table of results presents mean oeffi ient estimates along with the standard error of these estimates (as opposed to the mean standard error). The figures in square brakets report the number of oeffi ients estimated to be statistially different from zero at the 5% level of signifiane and this term is used as an indiation of the preision of the estimates rather than the mean standard error. An adjusted mean figure is also reported for eah oeffi ient; this is obtained by setting non statistially signifiant oeffi ient estimates to zero when alulating the averages. The tables also report mean Rsquare and mean adjusted Rsquare statistis along with mean Pvalues for the Fstatisti for overall signifiane (these annot be omputed for the GMM estimator) and mean Pvalues for the Hansen Jstatisti whih tests the va 14 Carare and Thaidze (2005, p.15) note that the fourlagsasinstruments speifiation is the standard approah in the interest rate rule literature (e.g. Orphanides, 2001). However, we preserve the sample size of eah estimation run rather than add an additional (fifth) lag of the expeted future nominal interest rate to the GMM instrument set. The 2SLS and GMM estimators yield idential estimates if the instrument set for the latter is restrited to math that of the former. Note that although the GMM proedure in general orrets for autoorrelation and heteroskedastiity for atual data estimation, in estimating with simulated data we use lags as instruments for predetermined variables that are free from simultaneity bias. The instruments may be good beause the data is serially orrelated but no further lags are needed for the estimating equation itself. For atual data, Clarida et al. (QJE, 2000, p.153) use a GMM estimator "with an optimal weighting matrix that aounts for possible serial orrelation in [the error term]" but they also add two lags of the dependent variable to their estimating equation on the basis that this "seemed to be suffi ient to eliminate any serial orrelation in the error term." (p.157), implying that the GMM orretion was insuffi ient. 15 Jondeau et al. (2004, p.227) state that: "To our knowledge, all estimations of the forwardlooking reation funtion based on GMM have so far relied on the twostep estimator." They proeed to onsider more sophistiated GMM estimators but nevertheless identify advantages to the "simple approah" (p.238) adopted in the literature. 15
19 lidity of the instrument set (these an only be alulated in the presene of overidentifying restritions), and mean DurbinWatson (DW) statistis whih test for autoorrelation. The number of estimation runs for whih the null hypothesis of the Fstatisti is rejeted and the number for whih the Jstatisti is not rejeted are reported alongside the relevant Pvalues and the number of estimation runs for whih the DW test statisti exeeds its upper ritial value is reported alongside this test statisti General Taylor Condition Tables 35 present estimates obtained from the following orretly speified estimating equation: R t = β 0 +β 1 E t π t+1 +β 2 E t g,t+1 +β 3 E t g l,t+1 +β 4 E t g V,t+1 +β 5 E t R t+1 +ε t. (26) Expeted future variables on the right hand side are obtained diretly from the model simulation proedure and are instrumented for as desribed above. The key result is that Tables 35 onsistently report an inflation oeffi ient whih exeeds unity for the empirial model whih most aurately reflets the underlying theoretial model. This result is found to be robust to the statistial filter applied to the data and to the estimator employed, subjet to the estimator providing a preise set of estimates. The forward interest rate term is also found to be important in terms of generating a oeffi ient on inflation onsistent with the underlying, Benk et al. (2010), model. Arbitrarily omitting this dynami term yields muh smaller estimates of the inflation oeffi ient to the extent that most of the estimates now fall below the eonomially signifiant threshold of unity. In terms of the general features of the results obtained from the unrestrited speifiation, the OLS and GMM proedures tend to generate a greater number of statistially signifiant estimates than the 2SLS estimator. Fousing on Table 5, the 2SLS estimator provides a statistially signifiant estimate for the inflation oeffi ient for only 580 of the 1000 simulated histories in Table 5 while the OLS and GMM estimators both return 1000 signifiant estimates. The onstant term, for whih very few statistially signifiant estimates are generated, stands as an exeption but this finding is onsistent with equation (22). The OLS and GMM proedures generate reasonably large Rsquare and adjusted Rsquare statistis, whereas negative Rsquare statistis are obtained from the simple 2SLS estimator, possibly symptomati of an inadequate instrument set. Expanding the instrument set in order to implement the GMM proedure leads to 1000 rejetions of the Jtest for instrument validity aross all three filters. One might also be wary of the low number of deisive DW statisti rejetions produed by the OLS proedure, although the mean DW statisti remains reasonably large in eah ase; 1.56 for the 215 filter, for example. The results for 16 The DW ount inludes only deisive rejetions, i.e. it exludes test statistis whih lie in the inonlusive region of the test s ritial values. 16
20 the 38 band pass filter in Table 4 are unusual in the sense that all three estimation proedures produe a very low number of DW statisti rejetions. For the other two filters, this undesirable result is speifi to the OLS estimator. Consider in partiular the results presented in Table 5 for the 215 band pass filter, i.e. the filter whih retains more of the low frequeny omponents of the simulated data. The mean oeffi ient on inflation is estimated to be using the OLS estimator and using the GMM estimator. 17 These estimates ompare favorably to the theoretial value of Ω = The right hand side of Table 5 shows that the mean of the estimated inflation oeffi ients falls below unity when the the forward interest rate term is arbitrarily omitted from the estimating equation; a preise mean estimate of is obtained from the OLS proedure and a similarly preise mean estimate of is obtained from the GMM proedure. Similar OLS and GMM estimates are obtained for the oeffi ient on inflation under the two alternative band pass filters in Tables 4 and 5, both in terms of the mean oeffi ient estimates for the unrestrited speifiation and in terms of the deline in magnitude indued when β 5 = 0 is arbitrarily imposed upon the estimating equation. Compared to the estimated inflation oeffi ients, the estimated oeffi ients on onsumption growth and produtive time growth diverge more from their theoretial preditions for the unrestrited estimating equation. Under log utility (θ = 1), the former should take the same magnitude as the oeffi ient on inflation and the latter should take a value of zero. The mean estimates of both of these theoretial parameters an be used to bakout an estimate of the oeffi ient of relative risk aversion (θ). Firstly, using the mean GMM estimate for the oeffi ient on onsumption growth of (Table 5) and the orresponding estimate of Ω, an implied estimate of θ an be alulated as β 2 β = = 0.131, whih is smaller than the baseline alibration of θ = 1. Alternatively, the relationship β 3 = β 1 ψ (1 θ) l/(1 l), whih is obtained from equation (22) with Ω replaed by its estimate β 1, an also be used to obtain an implied value of θ. Using the estimates presented in Table 5, the implied estimate of θ is 1.103, whih is muh loser to the alibrated value of θ = 1. Table 5 also reports that both the OLS and GMM proedures generate 1000 statistially signifiant estimates for the oeffi ient on veloity growth under the unrestrited estimating equation and that the mean of the estimated oeffi ients is orretly signed for both estimators. The mean of the point estimates are reported as and for OLS and GMM respetively; these are smaller than the theoretial predition of Similar estimates are obtained under the HP and 38 filters. Finally, Table 5 reports mean estimates of (OLS) and (GMM) for the forward interest rate oeffi ient ompared to a theoretial value of The mean estimates are therefore orretly signed but, again, smaller than the theoretial predition. 17 The disussion fouses on the OLS and GMM estimators beause they produe more preise estimates and also beause the OLS estimator tends to rejet the null hypothesis of the Fstatisti more frequently than the 2SLS estimator (1000 vs. 907 rejetions in Table 5, for example). The OLS regressions are possibly affl ited by autoorrelation, however, as disussed above. 17
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