Deriving the Taylor Principle when the Central Bank Supplies Money Ceri Davies Cardi¤ Business School
Max Gillman Cardi¤ Business School
Michal Kejak CERGEEI March 5, 2013
Abstract The paper presents a humancapitalbased endogenous growth, cashinadvance economy with endogenous velocity where exchange credit is produced in a decentralized banking sector, and money is supplied stochastically by the central bank. From this it derives an exact functional form for a general equilibrium ‘Taylor rule’. The in‡ation coe¢ cient is always greater than one when the velocity of money exceeds one; velocity growth enters the equilibrium condition as a separate variable. The paper then successfully estimates the magnitude of the coe¢ cient on in‡ation from 1000 samples of Monte Carlo simulated data. This shows that it would be spurious to conclude that the central bank has a reaction function with a strong response to in‡ation in a ‘Taylor principle’sense, since it is only meeting …scal needs through the in‡ation tax. The paper also estimates several deliberately misspeci…ed models to show how an in‡ation coe¢ cient of less than one can result from model misspeci…cation. An in‡ation coe¢ cient greater than one holds theoretically along the balanced growth path equilibrium, making it a sharply robust principle based on the economy’s underlying structural parameters. JEL Classi…cation: E13, E31, E43, E52 Keywords: Taylor rule, velocity, forwardlooking, misspeci…cation bias
We thank Hao Hong and Vo Phuong Mai Le for research assistance; Samuel Reynard, Warren Weber, Paul Whelan and James Cloyne for discussion; presentations at the CDMA conference, Louvain, the Monetary Policy Conference in Birmingham, the University of London, Birkbeck, the FEBS conference, London, the Bank of England and SUNY Bu¤alo.
0
1
Introduction
What does it mean to say that the central bank should be following the Taylor (1993) rule? And what does it mean when argued that current policy is wrong if it does not follow the Taylor rule? If a central bank should adhere to a Taylor rule over the long run, then can the Taylor rule simply be added onto neoclassical models as an ad hoc feature or used as a central component of short run Phillips curve models with sticky prices in the New Keynesian mold? Further, if in the long run the central bank targets in‡ation at some rate such as zero or two percent, and if the Taylor rule should be followed in the long run, then does that not translate into simply saying that central banks should track the market’s ‡uctuating real interest rate in the long run using the Taylor rule? (See Fama’s, 2012, perspective on this). It is conventional to view interest rate rules as monetary policy ‘reaction functions’ that represent how the central bank adjusts a shortterm nominal interest rate in response to the state of the economy. The magnitude of the reaction function coe¢ cients are interpreted to re‡ect a policymaker’s preferences towards variation in key macroeconomic variables such as in‡ation and variously de…ned output gaps. It has been suggested that policymakers ought to adhere to the ‘Taylor principle’, whereby in‡ation above target is met by a morethanproportional increase in the shortterm nominal interest rate and hence an increase in the real interest rate. Such an interest rate rule forms one of the three core equations of the prominent New Keynesian modelling framework, such as in Woodford (2003), Clarida et al. (1999); and Clarida et al. (2000). One wellknown …nding from the latter paper concludes that the Taylor principle holds for a ‘VolckerGreenspan’sample of U.S. data but that it is violated for a ‘preVolcker’ sample during which the Fed was deemed to be accommodating in its reaction to in‡ation. Davig and Leeper (2007) provide a reduced form model view of how Taylor rules can hold in the long run including possible shifts between "active" and "passive" monetary regimes. FernándezVillaverde and RubioRamírez (2007) estimate a New Keynesian DSGE model with the results that the structural parameters shift and so are not structural in a deep sense. From a di¤erent angle, an historical strand of literature going back to Poole (1970), and updated by Alvarez et al. (2001) and Chowdhury and Schabert (2008), for example, considers interest rate rules and money supply rules as two ways of implementing the same monetary policy. This paper perhaps most closely follows Alvarez et al. (2001) by deriving the equilibrium nominal interest rate in ‘rule form’ within a general equilibrium economy in which the central bank conducts monetary policy by stochastically supplying money. Instead of an exogenous fraction of agents being able to use bonds as in Alvarez et al., here the consumer purchases goods with an endogenous fraction of banksupplied intratemporal credit that avoids the in‡ation tax on exchange. This cashinadvance monetary economy is also extended to include endogenous growth, along with endogenous velocity, as in Benk et al. (2010). The resulting equilibrium condition for the nominal interest rate ‘nests’the standard Taylor 1
rule within a more general forwardlooking setting that endogenously includes traditional monetary elements, such as the (exogenous) velocity in Alvarez et al., and the money demand in McCallum and Nelson (1999). The endogenous growth aspect implies that the ‘target’ terms of the equilibrium ‘Taylor condition’, such as the in‡ation target or the ‘potential’output level, are the balanced growth path (BGP ) equilibrium values of the relevant variables. In addition, the coe¢ cients of the Taylor condition are a function of the model’s utility and technology parameters along with the BGP money supply growth rate. This in essence ful…lls Lucas’s (1976) goal of postulating policy rules with coe¢ cients that depend explicitly upon the economy’s underlying utility and technology coe¢ cients plus a key policy choice, in this case the BGP rate of money supply growth. This di¤ers from Leeper and Zha’s (2003) novel approach to the Lucas critique in monetary policy in that they study whether the variance of the money supply growth may be changing, interpreting the Lucas critique as holding when the variance is unchanged. Our model assumes a single money supply variance for the entire period, and within this context shows how the average growth rate of money supply on the BGP is a structural policy parameter that the consumer understands as part of the equilibrium conditions used to determine their behavior. The Taylor principle results from the structural parameters within a policy function, or optimal control law, or policy rule, as Lucas and Sargent (1981) state it variously, given a certain distribution of policy control processes, in this case the money supply growth rate.1 The framing of the Taylor condition for a given constant variance distribution could be said to complement the Leeper and Zha (2003) Lucas critique focus on possible changes in the money supply rule variance distribution per se. It provides a solid theoretical result that could be said to exist within each regime, or within just a single regime if that holds over the entire period, as is modeled here: a structural derivation of the ‘Taylor principle’ whereby the coe¢ cient on the in‡ation term always exceeds one for any given nonFriedman (1969) optimum BGP money supply growth rate, equals one only at the Friedman optimum, and never falls below one. Equivalently, the in‡ation coe¢ cient always exceeds one when the endogenous velocity exceeds one since the cashinadvance velocity rises above one for any nonFriedman optimal rate of money supply growth. In general, the in‡ation coe¢ cient rises with the BGP velocity level. Another central result is that the expected velocity growth rate itself enters the Taylor condition as an additional term, in contrast to standard Taylor rules. Omitting this term can cause misspeci…cation bias in estimated Taylor rules within the economy. Having derived the Taylor condition, the paper then estimates it by applying three conventional estimation procedures to one thousand samples of arti…cial data simulated from the baseline model, where the simulated data is passed through three standard statistical …lters prior to estimation. The results verify 1 See FernándezVillaverde and RubioRamírez (2007) for a discussion of types of suspect structural parameters, to which our utility and technology parameters do not belong.
2
the theoretical form of the Taylor condition along several key dimensions. In particular, the coe¢ cient on in‡ation is greater than one and close to its theoretical magnitude for all three estimation techniques and for all three data …lters. Robustness tests explore the impact of estimating two alternative Taylor conditions. This involves the use of two ad hoc, deliberately misspeci…ed equations which di¤er from the ‘true’theoretical expression: the …rst changes just one of the variables in the Taylor condition while the second posits a standard Taylor rule which involves multiple misspeci…cation errors. Using the same arti…cial data, the two misspeci…ed models produce an estimated coe¢ cient on in‡ation which falls below one, a violation of the ‘Taylor principle’. In the context of actual data, this result would typically be interpreted as the central bank being ‘passive’, or ‘weak’, or ‘accommodative’towards in‡ation. Here, the paper shows that such an interpretation could be spurious as this result occurs simply due to misspeci…cation in the estimating equation.2 The estimated ‘Taylor rule’emerges even though the central bank is merely satisfying the government’s …scal needs through the in‡ation tax. This implies the central point of the paper: it would be spurious within this economy to associate the Taylor condition with a ‘reaction function’for the nominal interest rate since in the model the central bank just stochastically prints money. Also, failure of the socalled Taylor principle in numerous published empirical studies may be a result of model misspeci…cation rather than behavioral changes by the central bank per se. Indeed, our current preliminary extension of this work, not presented here, shows that estimation with actual U.S. data of Taylor rules which include the unconventional terms implied by the theory of this paper  particularly velocity growth  can reverse the result that the coe¢ cient on in‡ation falls below unity during periods of macroeconomic instability.3 Related work is vast but includes Taylor (1999), who alludes to the possibility that an interest rate rule can be derived from the quantity theory of money. Sørensen and WhittaJacobsen (2005, pp.502505) present such a derivation under the assumption of constant money growth whereby the coe¢ cients of the ‘rule’relate to elasticities of money demand rather than the preferences of policymakers. Fève and Auray (2002) and Schabert (2003) consider the link between money supply rules and interest rate rules in standard cashinadvance models with velocity …xed at unity. Alternatively, the paper could be viewed in the context of Canzoneri et al’s. (2007) account of the shortcomings of estimated Euler equations because it shows how the Euler equation in combination with the stochastic asset pricing kernel can be used to derive a Taylor condition which can be estimated successfully. Also related is the long history of literature on whether money demand can be described as a stable function. During the …nancial deregulation of the early 1980s in the US and countries like the UK, money demand in the US was deemed 2 Estimation of simulated data is conducted by Fève and Auray (2002), for a standard CIA model, and Salyer and Van Gaasbeck (2007), for a ‘limited participation’model. We are indebted to Warren Weber for the suggestion to follow such an approach here. 3 Clarida et al.’s (2000) ‘preVolcker’ sample, for example, corresponds to a period of high and variable in‡ation.
3
instable in that no cointegrating relation could be found for a standard money demand depending on interest rates and income (Friedman and Kuttner, 1992). Such …ndings appeared to have helped usher in the popularity of interest rate rules and the dropping of the use of money in monetary policy. Research continued to try to show that money demand was stable using changes in …nancial intermediation productivity, either by making exchange credit a substitute to money with a price that could be included in the money demand function (Gillman and Otto, 2007), by extending standard BaumolTobin models (Alvarez and Lippi, 2009), by de…ning the aggregates so that they included only the money elements, as in Barnett et al. (1984) and Lucas and Nicolini (2012), or through liquidity e¤ects (Kelly et al. 2011, Alvarez and Lippi, 2011). In comparison to this literature, we model stochastic …nancial productivity through exchange credit production, with the same stable money demand function of Gillman and Otto even when bank productivity falls as in the recent bank crisis. As a result, the Taylor condition of the paper also is stable, rather than one with di¤erent regimes. At the same time our model includes in‡ation tax targeting through a stationary long run money supply growth rate in the optimal tax sense of Lucas and Stokey (1983). Section 2 describes the economy, as in Benk et al. (2008, 2010). Section 3 derives the model’s ‘Taylor condition’ and Section 4 provides the baseline calibration. Section 5 describes the econometric methodology applied to modelsimulated data and presents the estimation results. Section 6 derives theoretical special cases of the more general (Section 2) model to show how alternative Taylor conditions can be derived. Section 7 presents a discussion and Section 8 concludes.
2
Stochastic Endogenous Growth with Banking
The representative agent economy is as in Benk et al (2008, 2010) but with a decentralized banking sector that produces credit as in Gillman and Kejak (2011). By combining the business cycle with endogenous growth, stationary in‡ation lowers the output growth rate as supported empirically in Gillman et al. (2004) and Fountas et al. (2006), for example. Further, money supply shocks can cause in‡ation at low frequencies, as in Haug and Dewald (2012) and as supported by Sargent and Surico (2008, 2011), which can lead to output growth e¤ects if the shocks are persistent and repeated. This allows shocks over the business cycle to cause changes in growth rates and in stationary ratios. The shocks to the goods sector productivity and the money supply growth rate are standard, while the third shock to credit sector productivity exists by virtue of the model’s endogenous money velocity. Exchange credit is produced via a functional form used extensively in the …nancial intermediation microeconomics literature starting with Clark (1984) and promulgated by Berger and Humphrey (1997) and Inklaar and Wang (2013), for example. The shocks occur at the beginning of the period, are observed by the consumer before the decision making process commences, and follow a vector …rst
4
order autoregressive process. For goods sector productivity, zt ; the money supply growth rate, ut ; and bank sector productivity, vt : Zt =
Z Zt 1
+ "Zt ;
(1)
0
where the shocks are Zt = [zt ut vt ] , the autocorrelation matrix is Z = diag f'z ; 'u ; 'v g and 'z ; 'u ; 'v 2 (0; 1) are autocorrelation parameters, and 0 the shock innovations are "Zt = [ zt ut vt ] N (0; ) : The general structure of the secondorder moments is assumed to be given by the variancecovariance matrix . These shocks a¤ect the economy as described below, and as calibrated in Benk et al. (2010).
2.1
Consumer Problem
A representative consumer has expected lifetime utility from consumption of goods, ct ; and leisure, xt ; with 2 (0; 1) ; > 0 and > 0; this is given by: U = E0
1 X (ct xt )1 1 t=0
:
(2)
Output of goods, yt , and increases in human capital, are produced with physical capital and e¤ective labor each in CobbDouglas fashion; the bank sector produces exchange credit using labor and deposits as inputs. Let sGt and sHt denote the fractions of physical capital that the agent uses in goods production (G) and human capital investment (H), whereby: sGt + sHt = 1:
(3)
The agent allocates a time endowment of one between leisure, xt ; labor in goods production, lGt , time spent investing in the stock of human capital, lHt , and time spent working in the bank sector (F subscripts for Finance), denoted by lF t : lGt + lHt + lF t + xt = 1: (4) Output of goods can be converted into physical capital, kt ; without cost and is thus divided between consumption goods and investment, denoted by it ; net of capital depreciation. The capital stock used for production in the next period is therefore given by: kt+1 = (1
k )kt
+ it = (1
k )kt
+ yt
ct :
(5)
The human capital investment is produced using capital sHt kt and e¤ective labor lHt ht ; with AH > 0 and 2 [0; 1] ; such that the human capital ‡ow constraint is: 1 ht+1 = (1 (lHt ht ) : (6) h )ht + AH (sHt kt ) With wt and rt denoting the real wage and real interest rate, the consumer receives nominal income of wages and rents, Pt wt (lGt + lF t ) ht and Pt rt sGt kt ; 5
a nominal transfer from the government, Tt ; and dividends from the bank. The consumer buys shares in the bank by making deposits of income at the bank. Each dollar deposited buys one share at a …xed price of one, and the consumer receives the residual pro…t of the bank as dividend income in proportion to the number of shares (deposits) owned. Denoting the real quantity of deposits by dt ; and the dividend per unit of deposits as RF t ; the consumer receives a nominal dividend income of Pt RF t dt : The consumer also pays to the bank a fee for credit services, whereby one unit of credit service is required for each unit of credit that the bank supplies the consumer for use in buying goods. With PF t denoting the nominal price of each unit of credit, and qt the real quantity of credit that the consumer can use in exchange, the consumer pays PF t qt in credit fees. With other expenditures on goods, of Pt ct ; and physical capital investment, Pt kt+1 Pt (1 Mt ; k )kt ; and on investment in cash for purchases, of Mt+1 and in nominal bonds Bt+1 Bt (1 + Rt ), where Rt is the net nominal interest rate, the consumer’s budget constraint is: Pt wt (lGt + lF t ) ht + Pt rt sGt kt + Pt RF t dt + Tt PF t qt + Pt ct + Pt kt+1 Pt (1 Mt k )kt + Mt+1 +Bt+1 Bt (1 + Rt ):
(7)
The consumer can purchase goods by using either money Mt or credit services. With the lump sum transfer of cash Tt coming from the government at the beginning of the period, and with money and credit equally usable to buy goods, the consumer’s exchange technology is: Mt + Tt + Pt qt
Pt ct :
(8)
Since all cash comes out of deposits at the bank and credit purchases are paid o¤ at the end of the period out of the same deposits, total deposits are equal to consumption. This gives the constraint that: dt = ct :
(9)
Given k0 , h0 ; and the evolution of Mt (t 0) as given by the exogenous monetary policy in equation (17) below, the consumer maximizes utility subject to the budget, exchange and deposit constraints (7)(9).
2.2
Banking Firm Problem
The bank produces credit that is available for exchange at the point of purchase. The bank determines the amount of such credit by maximizing its dividend pro…t subject to the labor and deposit costs of producing the credit. The production of credit uses a constant returns to scale technology with e¤ective labor and deposited funds as inputs. In particular, with AF > 0 and 2 (0; 1): 6
qt = AF evt (lF t ht ) d1t
;
(10)
where AF evt is the stochastic factor productivity. Subject to the production function in equation (10), the bank maximizes pro…t F t with respect to the labor lF t and deposits dt : Ft
= PF t qt
Pt wt lF t ht
Pt RF t dt :
(11)
Equilibrium implies that: PF t Pt PF t Pt
AF evt
(1
1
l F t ht dt
= wt ;
l F t ht dt
) AF evt
(12)
= RF t :
(13)
These indicate that the marginal cost of credit, PPFtt , is equal to the marginal wt factor price divided by the marginal factor product, or 1 ; and lF t ht v AF e
t
dt
that the zero pro…t dividend yield paid on deposits is equal to the fraction of the marginal cost given by PPFtt (1 ) dqtt :
2.3
Goods Producer Problem
The …rm maximizes pro…t given by yt wt lGt ht rt sGt kt ; subject to a standard CobbDouglas production function in e¤ective labor and capital: yt = AG ezt (sGt kt )1
(lGt ht ) :
(14)
The …rst order conditions for the …rm’s problem yield the standard expressions for the wage rate and the rental rate of capital:
rt = (1
2.4
1
sGt kt lGt ht
wt = AG ezt
)AG ezt
;
sGt kt lGt ht
(15)
:
(16)
Government Money Supply
It is assumed that government policy includes sequences of nominal transfers as given by: Tt =
t Mt
=(
+ eut
1)Mt ;
where t is the growth rate of money and rate of money. 7
t
= [Mt
Mt
1 ]=Mt 1 ;
(17)
is the stationary gross growth
2.5
De…nition of Competitive Equilibrium
The representative agent’s optimization problem can be written recursively as: V (s) =
max
c; x; lG; lH; lF ;sG; sH ;q;d;k0 ;h0 ;M 0
fu(c; x) + EV (s0 )g
(18)
subject to the conditions (3) to (9), where the state of the economy is denoted by s = (k; h; M; B; z; u; v) and a prime (’) indicates nextperiod values. A competitive equilibrium consists of a set of policy functions c(s), x(s), lG (s), lH (s), lF (s), sG (s), sH (s); q(s), d(s), k 0 (s), h0 (s), M 0 (s), B 0 (s) pricing functions P (s), w(s), r(s); RF (s); PF (s) and a value function V (s), such that: (i) the consumer maximizes utility, given the pricing functions and the policy functions, so that V (s) solves the functional equation (18); (ii) the goods producer maximizes pro…t similarly, with the resulting functions for w and r being given by equations (15) and (16); (iii) the bank …rm maximizes pro…t similarly in equation (11) subject to the technology of equation (10) (iv) the goods, money and credit markets clear, in equations (7) and (14), and in (8), (17), and (10).
3
General Equilibrium Taylor Condition
The ‘Taylor condition’is now derived as an equilibrium condition of the Benk et al. (2010) model described in the previous section. Beginning from the …rstorder conditions of the model, we obtain: ) ( (1 ) ~ ct+1 xt+1 Rt Rt+1 ; (19) 1 = Et (1 ) ~ Rt+1 t+1 c x t
t
where R and are gross rates of nominal interest and in‡ation, respectively. ~ t represents (one plus) a ‘weighted average cost of exchange’ as The term R follows: mt ~ t = 1 + mt (Rt 1) + R 1 (Rt 1): ct ct 1) where a weight of m c is attached to the opportunity cost of money (Rt and a weight of (1 m ) is attached to the average cost of credit, (R 1) ; t c t and m is the real consumption normalized demand for money (i.e. the inverse ct of the consumption velocity of money). In e¤ect, equation (19) augments a standard consumption Euler equation with (the growth rate of) the weighted average cost of exchange. If all goods purchases are conducted using money (mt =ct = 1) then equation (19) reverts back to the familiar consumption Euler equation which would constitute an equilibrium condition of a standard, unit velocity cashinadvance model without a money alternative. For any variable zt ; de…ne zbt ln zt ln z; where the absence of a time subscript denotes a BGP stationary value, and de…ne gbz;t+1 ln zt+1 ln zt ; which 8
approximates the growth rate at time t + 1 for su¢ ciently small zt . Consider a loglinear approximation of (19) evaluated around the BGP : n o bt+1 + bt+1 : 0 = Et gbc;t+1 (1 ) gbx;t+1 + gbR;t+1 R ~
bt gives the Taylor condition expressed in logRearranging this in terms of R deviations from the BGP equilibrium: bt R
= Et f bt+1 + +
where
1+
gbc;t+1
(1 ) 1 R 1 (1 ) 1 )(1
(1
m c
(1
m c
(R
m c
) gbx;t+1
1)
m c
m c
1
)
gb mc ;t+1
bt+1 R
)
(20)
;
1: The Taylor condition (20) can now be )] expressed in net rates (denoted by overbarred terms) and absolute deviations from the BGP equilibrium, as demonstrated by the following proposition. R[1 (1
)(1
m c
Proposition 1 An equilibrium condition of the economy takes the form of a Taylor Rule which sets deviations of the shortterm nominal interest rate from some baseline path in proportion to deviations of variables from their targets: Rt
R
=
Et ( +
)+
t+1
Et g c;t+1
(1 ) 1 R 1 (1 ) 1
m c
(R
m c
g 1)
(1 m c m c
1
) Et g x;t+1
Et g mc ;t+1
Et Rt+1
@ @ where 1; and for a given w; then @R > 0 and @A > 0; and the target F values are equal to the balanced growth path equilibrium values.4
Proof. Since the BGP solution for normalized money demand is: 0 (1
m =1 c )(1
m c
AF
(R
1) AF w
1
1;
)
@ @ 1 and, given w; @R 0: 0 and @A F )] For a linear production function of goods w is the constant marginal product of labor but more generally w is endogenous and will change; however this change in w is quantitatively small compared to changes in R and AF ; so that the derivatives above almost always hold true. Note that for a unitary consumption velocity of money, the velocity growth and forward interest terms drop out of equation (21) The term in equation (21) can be compared to the in‡ation target that features in many interest rate rules (e.g. Taylor, 1993; Clarida et al., 2000).
then
1+
R[1 (1
)(1
m c
4 This is the the Brookings project form of the Taylor rule as described in Orphanides (2008).
9
(21) R
:
This is usually set as an exogenous constant in a conventional rule but represents the BGP rate of in‡ation in the Taylor condition.5 The term in consumption growth is similar, but not identical to, the …rst di¤erence of the output gap that features in the socalled ‘speed limit’rule (Walsh, 2003). Alternatively, the term in the growth rate of leisure time can be compared to the unemployment rate which sometimes features in conventional interest rate rules in place of the output gap.6 Equation (21) also contains two terms which are not usually found in standard monetary policy reaction functions. Firstly, there is a term in the growth rate of the real (consumption normalized) demand for money. Conventional interest rate rules are usually considered in the context of models which omit monetary relationships and thus money demand does not feature directly in the model.7 Secondly, the Taylor condition contains a term in the expected future nominal interest rate. This contrasts with the lagged nominal interest term which is often used to capture ‘interest rate smoothing’in a conventional rule (e.g. Clarida et al., 2000). In general, the coe¢ cient on in‡ation in (21) exceeds unity ( > 1). This replicates the ‘Taylor principle’whereby the nominal interest rate responds more than oneforone to (expected future) in‡ation deviations from ‘target’. However, the in‡ation coe¢ cient in the Taylor condition does not re‡ect policymakers’preferences. Rather, it is a function of the BGP nominal interest rate (R), the consumption normalized demand for real money balances (m=c) and the e¢ ciency with which the banking sector transforms units of deposits into units of the credit service, as re‡ected by the magnitude of (1 ). Furthermore, higher productivity in the banking sector (AF ) causes a higher velocity and implies a larger in‡ation coe¢ cient in the Taylor condition. The magnitude of clearly does not re‡ect a response to in‡ation in the conventional ‘reaction function’sense.8 Equation (21) can alternatively be rewritten in terms of the consumption ct velocity of money, Vt mt , and the productive time, or ‘employment’, growth rate (l lG + lH + lF = 1 x). Using the fact that x bt = 1 x b lt : x
5 Although
see Ireland (2007) for an example of a conventional interest rate rule with a timevarying in‡ation target. 6 For example, Mankiw (2001) includes the unemployment rate in an interest rate rule and Rudebusch (2009) includes the ‘unemployment gap’. 7 Speci…cally, shifts in the demand for money are perfectly accommodated by adjustments to the money supply in order to maintain the ruleimplied nominal interest rate. This, it is claimed, renders the evolution of the money supply an operational detail which need not be modelled directly (e.g. Woodford, 2008). 8 Unlike Sørensen and WhittaJacobsen’s (2005, pp.502505) quantity theory based equilibrium condition, the in‡ation coe¢ cient in (21) exceeds unity for any (admissible) interest elasticity of money demand. In their expression, the in‡ation coe¢ cient falls below unity if the interest (semi) elasticity of money demand exceeds one in absolute value. In the Benk et al. (2010) model, the coe¢ cient on in‡ation would exceed unity even in this case but the central bank would not wish to increase the money supply growth rate to this extent because seigniorage revenues would begin to recede as the elasticity increases beyond this point.
10
Rt
R
=
Et ( V
)+
t+1
Et g V;t+1
Et g c;t+1
(
g +
1) Et Rt+1
(1
)
l 1
l
Et g l;t+1
R :
(22)
Where overbarred terms again denote net rates and: (R V
(1 )m c + (1 )m c
1) R
:
Proposition 2 For the Taylor condition of equation (22), it is always true that 0 1 : V Proof. 1+ 1 )
(1 ) 1 m c R[1 (1 ) 1
(1
) 1
0
V
m c 1
m c
]
0; ) 0
1;
m =1 c (R
V
1
AF1
(R
1)
1
1;
w
1) (1 R
m c
) 1
(1
) 1
:
At the Friedman (1969) optimum for the gross nominal interest rate (R = 1), = 1; ! = 0; and the velocity coe¢ cient ( V ) takes a value of zero. The velocity growth term only enters the Taylor condition when the nominal interest rate di¤ers from the Friedman (1969) optimum and ‡uctuates. In turn, (1 )(1 m ) ; since when R = 1; this has implications for = 1 + R 1 (1 ) 1 c m [ ( c )] (1 ) 1 m = 0; and = 1: For m c c below one (velocity above one), which is true for most practical experience, the model’s equivalent of the ‘Taylor principle’( > 1) holds. m c
Corollary 3 Given w; then
@ @R
0; @@RV
@ 0; @A F
0; @@AVF
0:
Proof. This comes directly from the de…nitions of parameters above. A higher target nominal interest rate can be accomplished only by a higher BGP money supply growth rate. This would in turn make the in‡ation and consumption growth coe¢ cients larger and the forward interest rate and velocity coe¢ cients would become more negative. A higher credit productivity factor AF ; and so a higher velocity, leads to a higher in‡ation coe¢ cient and a more negative response to the forwardlooking interest term but a less negative coe¢ cient on the velocity growth term. The Taylor condition above would look identical with exogenous growth. However, under exogenous growth the targeted in‡ation rate and growth rate of the economy are unrelated and exogenously speci…ed. Under endogenous 11
m c
!
1;
growth the targets are instead the endogenously determined BGP values for in‡ation, the growth rate, and the nominal interest rate and each of these are determined, in part, by the long run stationary money supply growth rate ; which is exogenously given. In turn, translates directly into a long run in‡ation target accepted by the central bank, such as the two percent target often incorporated into conventional interest rate rules (for example, Taylor, 1993). So the model assumes only a long run money supply growth target, or alternatively, a long run in‡ation rate target.
3.1
Misspeci…ed Taylor Condition with Output Growth
It is not surprising to …nd that the growth rate of consumption appears in equation (22) rather than the output growth rate given that the derivation of the Taylor condition begins from the consumption Euler equation (19). However, the Taylor condition can be rewritten to include an output growth term and hence correspond more closely to standard Taylor rule speci…cations, in particular the ‘speed limit’rule considered by Walsh (2003). To derive this alternative ct + yi bit ; where rule, consider that the identity yt = ct + it implies that ybt = yc b h i bit = k b kt (1 )b kt 1 : The growth rate of investment can be understood i
as the acceleration of the growth of capital gross of depreciation. The Taylor condition can be rewritten as:
Rt
R
=
Et ( +
)+
t+1
(1
)
l 1
l
y Et g y;t+1 c Et g l;t+1
V
g
Et g V;t+1
i Et g i;t+1 c (
g
1) Et Rt+1
(23) R :
A term in investment growth does not appear in standard Taylor rules but plays a role as part of what is interpreted as the output gap growth rate in this modi…ed Taylor condition. Equation (23) forms the basis for the two misspeci…ed estimating equations considered in Section 5: The …rst misspeci…ed estimating equation simply replaces the consumption growth term in equation (22) with an output growth term as follows: Rt
R
=
Et ( +
t+1
(1
)+ Et g y;t+1 g l ) Et g l;t+1 V Et g V;t+1 1 l
(24) (
1) Et Rt+1
R :
Comparing equation (23) and equation (24) shows that the latter erroneously overlooks the weighting on the output growth rate ( yc ) and omits the term in the investment growth rate. Replacing consumption growth with output growth without the additional term in investment therefore misrepresents the structure of the underlying Benk et al. (2010) model and as such equation (24) is misspeci…ed. Note that with no physical capital in the economy, equation (24) would be a valid equilibrium condition of the economy. 12
3.2
Misspeci…ed Standard Taylor Rule
The second misspeci…ed model erroneously imposes the same restrictions used to arrive at equation (24) but also drops the terms in productive time and velocity, giving: Rt
R
=
Et ( (
)+
t+1
1) Et Rt+1
Et g y;t+1
g
(25)
R :
This can be interpreted as a conventional interest rate rule with a forwardlooking ‘interest rate smoothing’ term; the additional restriction that = 1 would replicate a standard interest rate rule without interest rate smoothing. Once again, equation (25) does not accurately represent an equilibrium condition of the Benk et al. (2010) economy and is therefore misspeci…ed. Equation (25) with = 1 would be the correct equilibrium condition if the economy featured neither physical capital nor exchange credit.
4
Calibration
We follow Benk et al. (2010) in using postwar U.S. data to calibrate the model (Table 1) and calculate a series of ‘target values’consistent with this calibration (Table 2); see Benk et al. for the shock process calibration. Subject to this calibration, we derive a set of theoretical ‘predictions’ for the coe¢ cients of the Taylor condition (22). These values will subsequently be compared to the coe¢ cients estimated from arti…cial data simulated from the model. Consider …rst the in‡ation coe¢ cient ( ): According to the calibration and target values presented in tables 1 and 2, its theoretical value is: = 1+
(1 ) 1 m c R 1 (1 ) 1
m c
= 1+
(1 1:0944 (1
0:11) (1 0:38) = 2:125 (1 0:11) (1 0:38))
And for R = 1; only cash is used so that m reverts to its lower bound c = 1 and of 1: This also happens with zero credit productivity (AF = 0), in which case only cash is used in exchange. The remaining coe¢ cients, except for velocity, are simple functions of the in‡ation coe¢ cient. The consumption growth coe¢ cient is ; which with = 1 for logutility should simply take the same magnitude as the coe¢ cient on in‡ation ( = 2:125). The coe¢ cient on the productive time growth rate should take a value of zero with log utility. However with leisure preference calibrated at 1:84, and productive time (1 x l) equal to equal to 0:45 along the BGP; the estimated value of the productive time coe¢ cient can be l 0:45 interpreted as implying a certain factored by 1 l = (2:125) (1:84) 0:55 = 3:199. Given the magnitude of the in‡ation coe¢ cient, the coe¢ cient on the forward interest term is simply ( 1) = 1:125: The velocity coe¢ cient
13
Preferences 1 1.84 0.96
Relative risk aversion parameter Leisure weight Discount factor
Goods Production 0.64 0.031 k AG 1
Labor share in goods production Depreciation rate of goods sector Goods productivity parameter
Human Capital Production " 0.83 0.025 h AH 0.21
Labor share in human capital production Depreciation rate of human capital sector Human capital productivity parameter
Banking Sector AF
0.11 1.1
Labor share in credit production Banking productivity parameter
0.05
Money growth rate
Government Table 1: Parameters
g R lG lH lF i=y m=c x l 1
x
0.024 0.026 0.0944 0.248 0.20 0.0018 0.238 0.38 0.55 0.45
Avg. annual output growth rate Avg. annual in‡ation rate Nominal interest rate Labor used in goods sector Labor used in human capital sector Labor used in banking sector Investmentoutput ratio in goods sector Share of money transactions Leisure time Productive time Table 2: Target Values
14
(
V
) is
(R
0:065 using:
1) R
1
(1 (1
)m c ) 1
m c
!
(1:0944 1) 1:0944
=
(1
(1 0:11) 0:38 (1 0:11) (1 0:38))
At the Friedman (1969) optimum (R = 1); V = 0: In this case the omission of the term in velocity growth in the estimation exercises that follow would be innocuous but this is not true in general.
5
Arti…cial Data Estimation
The Benk et al. (2010) model presented in Section 2 is simulated using the calibration provided in Table 1 in order to generate 1000 alternative ‘joint histories’ for each of the variables in equation (22), where each history is 100 periods in length. To do so, 100 random sequences for the shock vector innovations are generated and control functions of the loglinearized model are used to compute sequences for each variable. Each observation within a given history may be thought of as an annual period given the frequency considered by the Benk et al. (2010) model. The data set used to estimate the coe¢ cients of the Taylor condition can therefore be viewed as comprising of 1000, ‘100year’, samples of arti…cial data.
5.1
Estimation Methodology
This section presents the results of estimating a ‘correctly speci…ed’ estimating equation based upon the true theoretical relationship (22) against arti…cial data generated from the Benk et al. (2010) model.9 In a similar manner, two alternative estimating equations are evaluated using the same data set. Since these alternative estimating equations di¤er from the expression based upon the true theoretical relationship, they necessarily constitute misspeci…ed empirical models.10 Prior to estimation, the simulated data is …ltered by either 1) a HodrickPrescott (HP) …lter with a smoothing parameter selected in accordance with Ravn and Uhlig (2002); 2) a 3 8 period ("year") Christiano and Fitzgerald (2003) band pass …lter for ‘business cycle frequencies’; or 3) a 2 15 year Christiano and Fitzgerald (2003) band pass …lter which retains more of the lower frequency trends in the data than the 3 8 year …lter, in the spirit of Comin and 9 The exercise conducted here is similar to those conducted by Fève and Auray (2002), for a standard CIA model, and Salyer and Van Gaasbeck (2007), for a ‘limited participation’ model. 1 0 We acknowledge that in a full information maximum likelihood estimation that uses all of the equilibrium conditions of the economy we may be able to recover the theoretical coe¢ cients of the Taylor condition almost exactly; we leave that exercise as an important part of future research that encompasses the entire alternative model; and then we could also compare it to the standard three equation central bank policy model.
15
:
Gertler’s (2006) ‘mediumterm cycle’.11 A priori, the 2 15 band pass …lter might be regarded as the ‘most relevant’ to the underlying theoretical model because shocks in the model can cause low frequency events during the business cycle, such as a change in the permanent income level without a reversion to its previous level.12 The …rst estimation technique considered is OLS, as used by Taylor (1999) in the context of a contemporaneous interest rate rule. However, if expected future variables are correlated with the error term then a suitable set of instruments are required to proxy for these forwardlooking terms.13 Two instrumental variables (IV) techniques are considered and each di¤ers by the instrument set employed. The …rst is a two stage least squares (2SLS) estimator under which the …rst lags of in‡ation, consumption growth, productive time growth and velocity growth and the second lag of the nominal interest rate are used as instruments. Adding a constant term to the instrument set provides a ‘just identi…ed’2SLS estimator. In using lagged variables as instruments we exploit the fact that such terms are predetermined and thus not susceptible to the simultaneity problem which motivates the use of IV techniques. The 2SLS procedure applies a NeweyWest adjustment for heteroskedasticity and autocorrelation (HAC) to the coe¢ cient covariance matrix. The second IV procedure is a generalized method of moments (GMM) estimator under which three additional lags of in‡ation, consumption growth, productive time growth and velocity growth and two further lags of the nominal interest rate are added to the instrument set.14 Expanding the instrument set in this manner reduces the sample size available for each of the 1000 simulated sample periods but the overidentifying restrictions can now be used to test the validity of the instrument set using the Hansen Jtest. The GMM estimator used iterates on the weighting matrix in two steps and applies a HAC 1 1 Comin and Gertler’s ‘mediumterm cycle’ is de…ned using a wider 2200 quarter …lter. However, the 215 …lter will still retain periodicities that the HP and 38 …lters consign to the ‘trend’. 1 2 In principle, the …ltering procedure takes account of the Siklos and Wohar (2005) critique of empirical Taylor rule studies which do not address the nonstationarity of the data. However, standard ADF and KPSS tests suggest that the simulated data is stationary prior to …ltering (results not reported). Accordingly, the …lters do not implement a detrending procedure. 1 3 Empirical studies usually deal with expected future terms either by replacing them with realised future values and appealing to rational expectations for the resulting conditional forecast errors (e.g. Clarida et al., 1998, 2000) or by using private sector or central bank forecasts as empirical proxies (e.g. Orphanides, 2001; Siklos and Wohar, 2005). 1 4 Carare and Tchaidze (2005, p.15) note that the fourlagsasinstruments speci…cation is the standard approach in the interest rate rule literature (e.g. Orphanides, 2001). Although the GMM procedure in general corrects for autocorrelation and heteroskedasticity, in estimating with simulated data we use lags as ‘valid’instruments for predetermined variables. These instruments might prove to be ‘relevant’because the data is serially correlated but no further lags are needed for the estimating equation itself. For actual data, Clarida et al. (QJE, 2000, p.153) use a GMM estimator "with an optimal weighting matrix that accounts for possible serial correlation 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 su¢ cient to eliminate any serial correlation in the error term." (p.157), implying that the GMM correction was insu¢ cient for this purpose.
16
adjustment to the weighting matrix using a Bartlett kernel with a NeweyWest …xed bandwidth.15 A similar HAC adjustment is also applied to the covariance weighting matrix. The results are presented in three sets of tables, one set for each estimating equation, and are further subdivided according to the statistical …lter applied to the simulated data. Alongside the estimates obtained from an ‘unrestricted’ estimating equation, each table also reports estimates derived from a ‘restricted’ estimating equation which arbitrarily omits the forward interest rate term ( 5 = 0): This arbitrary restriction demonstrates the importance of the dynamic term in equation (22). Each table of results reports mean coe¢ cient estimates along with the standard error of these estimates (as opposed to the mean standard error). The …gures in square brackets report the number of coe¢ cients estimated to be statistically di¤erent from zero at the 5% level of signi…cance and this count is used as an indication of the ‘precision’of the estimates. An ‘adjusted mean’ …gure is also reported for each coe¢ cient; this is obtained by setting non statistically signi…cant coe¢ cient estimates to zero when calculating the averages. The tables also report mean Rsquare and mean adjusted Rsquare statistics along with the mean Pvalue for the Fstatistic for overall signi…cance (these cannot be computed for the GMM estimator), the mean Pvalue for the Hansen Jstatistic which tests the validity of the instrument set (these can only be calculated in the presence of overidentifying restrictions), and the mean DurbinWatson (DW) statistic which tests for autocorrelation. The number of estimations for which the null hypothesis of the Jstatistic is not rejected  i.e. the instrument set is not found to be invalid  is reported alongside its mean Pvalue and the number of simulated series for which the DW statistic exceeds its upper critical value  i.e. the null hypothesis that the residuals are serially uncorrelated cannot be rejected  is reported alongside the mean DW statistic.16
5.2
General Taylor Condition
Tables 35 present estimates obtained from the following ‘correctly speci…ed’ estimating equation: Rt =
0 + 1 Et t+1 + 2 Et gc;t+1 + 3 Et gl;t+1 + 4 Et gV;t+1 + 5 Et Rt+1 +"t :
(26)
Expected future variables on the right hand side are obtained directly from the model simulation procedure and are instrumented for as described above. The key result is that Tables 35 consistently report an in‡ation coe¢ cient which exceeds unity for the estimating equation which accurately re‡ects the 1 5 Jondeau et al. (2004, p.227) state that: "To our knowledge, all estimations of the forwardlooking reaction function based on GMM have so far relied on the twostep estimator." They proceed to consider more sophisticated GMM estimators but nevertheless identify advantages to the "simple approach" (p.238) adopted in the literature. 1 6 The DW count excludes cases for which the test statistic falls in the inconclusive region of the test’s critical values.
17
underlying theoretical model. This result is found to be robust to the statistical …lter applied to the data and to the estimator employed, subject to the estimator providing a ‘precise’set of estimates. The forward interest rate term is also found to be important in terms of generating a coe¢ cient on in‡ation consistent with the underlying Benk et al. (2010) model. Arbitrarily omitting this dynamic term yields much smaller estimates for the in‡ation coe¢ cient to the extent that the mean estimate often falls below unity. In terms of the general features of the results obtained from the unrestricted speci…cation, the OLS and GMM procedures tend to generate a greater number of statistically signi…cant estimates than the 2SLS estimator. Focusing on Table 5 for the 2 15 …lter, the 2SLS estimator provides a statistically significant estimate for the in‡ation coe¢ cient for only 580 of the 1000 simulated histories while the OLS and GMM estimators both return 1000 statistically signi…cant estimates. The OLS and GMM procedures generate reasonably large Rsquare and adjusted Rsquare statistics, whereas negative Rsquare statistics are obtained from the simple 2SLS estimator. Expanding the instrument set in order to implement the GMM procedure leads to 1000 rejections of the Jtest for instrument validity across all three …lters. One might also be wary of the high number of DW null hypothesis rejections produced by the OLS estimator, although the mean DW statistic remains ‘reasonably large’in each case; 1:56 for the 2 15 …lter, for example. The results for the 3 8 band pass …lter (Table 4) are unusual in the sense that all three estimation procedures produce a high number of DW test rejections. For the other two …lters, this undesirable result is con…ned to the OLS estimator. Table 5 reports that the mean estimate for the in‡ation coe¢ cient is 2:179 using the OLS estimator and 2:306 using the GMM estimator.17 These estimates compare favorably to the theoretical prediction of = 2:125. The right hand side of Table 5 shows that the mean estimate of the in‡ation coe¢ cient falls below unity for the OLS and GMM estimators when the forward interest rate term is arbitrarily omitted from the estimating equation. A precise mean estimate of 0:614 is obtained from the OLS estimator and a similarly precise mean estimate of 0:964 is obtained from the GMM procedure. Similar OLS and GMM estimates are obtained for the in‡ation coe¢ cient under the two alternative …lters in Tables 3 and 4, both in terms of the mean coe¢ cient estimates for the unrestricted speci…cation and in terms of the decline in magnitude induced by the arbitrary restriction. In contrast to the estimated in‡ation coe¢ cients, the estimated coe¢ cients for consumption growth and productive time growth diverge from their theoretical predictions for the ‘unrestricted’estimating equation. Under log utility ( = 1), the former should take the same magnitude as the coe¢ cient on in‡ation and the latter should take a value of zero. The coe¢ cient estimates 1 7 The discussion focuses on the OLS and GMM estimators because they produce more ‘precise’ estimates and also because the OLS estimator tends to reject the null hypothesis of the Fstatistic more frequently than the 2SLS estimator (1000 vs. 907 rejections in Table 5, for example). The OLS regressions are possibly a icted by autocorrelation however, as discussed above, thus one might favor the GMM estimates.
18
can be used to ‘backout’an estimate of the coe¢ cient of relative risk aversion ( ): Firstly, using the mean GMM estimate for the coe¢ cient on consumption growth of 0:302 (Table 5) and the corresponding estimate of , an implied estimate of can be calculated as 2 = 0:302 2:306 = 0:131, which is substantially 1 smaller than the baseline calibration of = 1. Alternatively, the relationship ) l=(1 l), which is obtained from equation (22) with replaced 3 = 1 (1 by its estimate 1 , can also be used to obtain an implied estimate of . Using the estimates presented in Table 5, the implied estimate would be = 1:103; which is much closer to the calibrated value. Table 5 also reports that both the OLS and GMM procedures generate 1000 statistically signi…cant estimates for the coe¢ cient on velocity growth under the unrestricted estimating equation and that the mean estimate is correctly signed for both estimators. The mean coe¢ cient estimates are reported as 0:196 and 0:269 for OLS and GMM estimators respectively; these estimates are somewhat smaller than the theoretical prediction of 0:065. Similar estimates are obtained under the HP and 3 8 …lters. Finally, Table 5 reports mean estimates of 1:761 (OLS) and 1:729 (GMM) for the forward interest rate coe¢ cient compared to a theoretical prediction of 1:125. The mean estimates are therefore correctly signed but, again, smaller than the theoretical prediction. In a standard interest rate rule an in‡ation coe¢ cient in excess of unity is interpreted to re‡ect policymaker’s dislike of in‡ation deviations from target. However, this interpretation is not applicable to the Taylor condition. The result that the coe¢ cient on in‡ation exceeds unity is a consequence of a money growth rule not an interest rate rule. Similarly, the breakdown of the Taylor principle under the ‘restricted’ estimating equation ( 5 = 0) cannot be interpreted as a softening of policymakers’ attitude towards in‡ation; this result simply emanates from model misspeci…cation.
5.3
Taylor Condition with Output Growth
The same estimation procedure is now applied to an estimating equation which replaces consumption growth in equation (26) with output growth as follows: Rt =
0 + 1 Et t+1 + 2 Et gy;t+1 + 3 Et gl;t+1 + 4 Et gV;t+1 + 5 Et Rt+1 +"t :
(27)
The simulated data remains unchanged, therefore equation (27) represents a misspeci…ed version of the ‘correct’ estimating equation, which continues to be equation (26).18 In particular, equation (27) can be seen to correspond to the misspeci…ed Taylor condition, equation (24). The results are similar across the HP, the 3 8 band pass and the 2 15 band pass …lters but as the latter …lter gives the most statistically signi…cant coe¢ cient estimates only estimates from the 2 15 …lter are presented (Table 6). Comparing the general features of the results to those presented in Tables 1 8 The instrument sets used for the 2SLS and GMM estimators are modi…ed by replacing consumption growth with output growth but remains unchanged in terms of the number of lags included.
19
HP …ltered data, where HP
= 6:25
0
Standard error Adjusted mean
Et
t+1
Unrestricted
Assumed
5
=0
OLS
2SLS
GMM
OLS
2SLS
GMM
9.68E07 [0] 2.87E05
2.04E07 [0] 2.15E05
8.09E07 [17] 3.15E05
7.60E07 [0] 2.39E05
4.78E07 [0] 1.67E05
7.57E08 [8] 4.29E05


4.27E08


2.59E07
2.019 [1000]
2.309 [691]
2.299 [1000]
0.315 [830]
0.757 [397]
0.621 [925]
Standard error
0.248
1.488
0.268
0.126
0.856
0.265
Adjusted mean
2.019
1.800
2.299
0.293
0.475
0.614
0.251 [1000] 0.024
0.336 [959] 0.096
0.293 [1000] 0.020
0.172 [1000] 0.020
0.313 [989] 0.048
0.231 [1000] 0.025
0.251 0.243 [890]
0.324 0.536 [774]
0.293 0.374 [997]
0.172 0.281 [864]
0.311 0.530 [774]
0.231 0.427 [996]
Et gc;t+1 Standard error Adjusted mean
Et gl;t+1 Standard error
0.094
0.321
0.079
0.100
0.231
0.111
Adjusted mean
0.236 0.137 [990] 0.031
0.448 0.267 [800] 0.228
0.374 0.212 [1000] 0.033
0.265 0.098 [889] 0.036
0.453 0.317 [888] 0.109
0.427 0.190 [992] 0.056
0.137 1.819 [1000]
0.229 2.338 [646]
0.212 2.005 [1000]
0.093 N/A
0.293 N/A
0.190 N/A
Standard error
0.221
2.282
0.277
Adjusted mean
1.819
1.692
2.005
0.789 0.778 2.35E15 (1000) N/A
<0 <0 0.015 (974) N/A
0.796 0.785 N/A 0.258 {1000}
0.544 0.525 3.93E09 (1000) N/A
<0 <0 0.003 (992) 0.159 {482}
0.482 0.459 N/A 0.269 {1000}
1.474 <151> 99
2.243 <1000> 98
2.194 <970> 96
1.732 <419> 99
2.145 <999> 98
2.047 <882> 96
Et gV;t+1 Standard error Adjusted mean
Et Rt+1
M ean; Rsquare Adjusted Rsquare Pr(Fstatistic) Pr(Jstatistic) DurbinWatson Sample size (1000x) Notes: ‘Standard error’measures the variation in the coe¢ cient estimates. ‘Adjusted mean’assigns a value of zero to non statistically signi…cant estimates. Fstatistic: null hypothesis of no joint signi…cance of the independent variables (not available under GMM). Jstatistic: null hypothesis that the instrument set is valid (only available if there are overidentifying restrictions). DurbinWatson statistic: null hypothesis that successive error terms are serially uncorrelated against an AR(1) alternative. [ ] reports the number of statistically signi…cant coe¢ cient estimates, () the number of Fstatistic rejections, {} the number of Jstatistic nonrejections and <> the number of times the DW statistic exceeds its upper critical value (all at the 5% level of signi…cance).
Table 3: Taylor Condition Estimation, HP Filtered Data, Ravn and Uhlig (2002) Smoothing Parameter, 100 Years Simulated, 1000 Estimations Average.
20
Unrestricted OLS
2SLS
GMM
OLS
2SLS
GMM
Standard error
7.57E07 [0] 1.65E05
6.54E06 [0] 3.24E04
7.10E07 [3] 2.09E05
7.73E07 [0] 1.45E05
2.86E06 [0] 4.37E05
1.51E06 [1] 2.68E05
Adjusted mean


1.83E07


9.81E08
2.166 [998]
2.484 [724]
2.423 [1000]
0.633 [969]
2.417 [965]
0.682 [974]
Standard error
0.391
32.906
0.298
0.195
1.125
0.222
Adjusted mean
2.166
2.141
2.423
0.628
2.291
0.679
0.283 [1000] 0.043
0.304 [623] 7.324
0.314 [1000] 0.027
0.155 [1000] 0.029
0.175 [834] 0.074
0.168 [1000] 0.030
0.283 0.237 [827]
0.231 0.573 [430]
0.314 0.312 [982]
0.155 0.222 [685]
0.160 0.595 [666]
0.168 0.268 [870]
Standard error
0.131
10.199
0.099
0.133
0.361
0.135
Adjusted mean
0.229 0.152 [982] 0.043
0.193 0.453 [351] 15.068
0.312 0.174 [998] 0.039
0.195 0.174 [976] 0.052
0.455 0.604 [973] 0.252
0.259 0.194 [984] 0.057
0.151 2.026 [994]
0.128 1.532 [424]
0.174 2.289 [1000]
0.173
0.578
0.193
Standard error
0.435
116.012
0.339
Adjusted mean
2.024
1.211
2.289
0.789 0.778 4.75E10 (1000) N/A
<0 <0 0.077 (874) N/A
0.842 0.833 N/A 0.213 {1000}
0.576 0.558 2.08E07 (1000) N/A
<0 <0 0.005 (981) 0.344 {848}
0.590 0.572 N/A 0.249 {1000}
1.568 <54> 99
1.653 <330> 98
1.517 <49> 96
1.728 <333> 99
1.728 <378> 98
1.715 <306> 96
0
Et
t+1
Et gc;t+1 Standard error Adjusted mean
Et gl;t+1
Et gV;t+1 Standard error Adjusted mean
Et Rt+1
Assumed
=0
BP Filter, 38 Window
5
M ean; Rsquare Adjusted Rsquare Pr(Fstatistic) Pr(Jstatistic) DurbinWatson Sample size (1000x) Notes: ‘Standard error’measures the variation in the coe¢ cient estimates. ‘Adjusted mean’assigns a value of zero to non statistically signi…cant estimates. Fstatistic: null hypothesis of no joint signi…cance of the independent variables (not available under GMM). Jstatistic: null hypothesis that the instrument set is valid (only available if there are overidentifying restrictions). DurbinWatson statistic: null hypothesis that successive error terms are serially uncorrelated against an AR(1) alternative. [ ] reports the number of statistically signi…cant coe¢ cient estimates, () the number of Fstatistic rejections, {} the number of Jstatistic nonrejections and <> the number of times the DW statistic exceeds its upper critical value (all at the 5% level of signi…cance).
Table 4: Taylor Condition Estimation, Band Pass Filtered Data (38 years), 100 Years Simulated, 1000 Estimations Average.
21
Unrestricted
215 Window
OLS
2SLS
GMM
OLS
2SLS
GMM
Standard error
2.26E06 [0] 4.01E05
4.22E05 [0] 0.001
2.54E07 [22] 4.82E05
1.82E06 [0] 3.27E05
2.46E06 [0] 4.83E05
3.44E06 [21] 6.26E05
Adjusted mean


4.25E07


9.03E07
2.179 [1000]
3.816 [580]
2.306 [1000]
0.614 [999]
1.127 [763]
0.964 [999]
Standard error
0.195
51.040
0.272
0.108
0.640
0.169
Adjusted mean
2.179
1.402
2.306
0.614
0.936
0.963
0.277 [1000] 0.016
0.570 [730] 5.546
0.302 [1000] 0.025
0.170 [1000] 0.017
0.262 [851] 0.103
0.207 [1000] 0.026
0.277 0.295 [997]
0.265 0.737 [526]
0.302 0.359 [999]
0.170 0.210 [732]
0.230 0.263 [405]
0.207 0.277 [935]
Standard error
0.067
8.208
0.085
0.088
0.203
0.111
Adjusted mean
0.294 0.196 [1000] 0.024
0.242 0.347 [807] 0.271
0.359 0.269 [1000] 0.031
0.182 0.158 [998] 0.032
0.152 0.307 [944] 0.077
0.272 0.236 [1000] 0.042
0.196 1.761 [1000]
0.273 5.586 [335]
0.269 1.729 [1000]
0.158
0.292
0.236
Standard error
0.201
114.905
0.322
Adjusted mean
1.761
0.712
1.729
0.830 0.821 2.50E24 (1000) N/A
<0 <0 0.051 (907) N/A
0.782 0.770 N/A 0.315 {1000}
0.625 0.609 5.04E10 (1000) N/A
<0 <0 0.003 (985) 0.298 {757}
0.522 0.501 N/A 0.298 {1000}
1.558 <141> 99
2.059 <972> 98
2.040 <881> 96
1.954 <864> 99
2.052 <998> 98
2.205 <977> 96
0
Et
t+1
Et gc;t+1 Standard error Adjusted mean
Et gl;t+1
Et gV;t+1 Standard error Adjusted mean
Et Rt+1
Assumed
=0
BP Filter,
5
M ean; Rsquare Adjusted Rsquare Pr(Fstatistic) Pr(Jstatistic) DurbinWatson Sample size (1000x) Notes: ‘Standard error’measures the variation in the coe¢ cient estimates. ‘Adjusted mean’assigns a value of zero to non statistically signi…cant estimates. Fstatistic: null hypothesis of no joint signi…cance of the independent variables (not available under GMM). Jstatistic: null hypothesis that the instrument set is valid (only available if there are overidentifying restrictions). DurbinWatson statistic: null hypothesis that successive error terms are serially uncorrelated against an AR(1) alternative. [ ] reports the number of statistically signi…cant coe¢ cient estimates, () the number of Fstatistic rejections, {} the number of Jstatistic nonrejections and <> the number of times the DW statistic exceeds its upper critical value (all at the 5% level of signi…cance).
Table 5: Taylor Condition Estimation, Band Pass Filtered Data (215 years), 100 Years Simulated, 1000 Estimations Average.
22
35, there has been a decline in the precision with which the coe¢ cients are estimated, a decline in the magnitude of the Rsquare and adjusted Rsquare statistics and a decline in the number of rejections of the null hypothesis of the Fstatistic for joint signi…cance. This is not surprising given that an element of misspeci…cation has been introduced into the estimating equation. The number of rejections of the null hypothesis of the DW test statistic also tends to increase although the GMM procedure applied to 2 15 …ltered data still fails to reject the null for 94:5% of the simulated samples. The estimated in‡ation coe¢ cients are now found to be substantially greater than the coe¢ cients obtained from the ‘correctly speci…ed’estimating equation (26) and hence substantially greater than the predicted value. For instance, the GMM estimate for the unrestricted estimating equation rises from 2:306 in Table 5 to 5:274 in Table 6 (or 5:235 according to the adjusted mean). Similarly, the OLS estimate increases from 2:179 to 4:219 (or 4:185 adjusted).19 . The estimates clearly diverge further from the theoretical prediction of = 2:125 under this particular form of misspeci…cation. The incorrectly speci…ed estimating equation also induces a substantial decrease in the estimated coe¢ cients for the productive time growth rate and the forward nominal interest rate. The estimated coe¢ cient on the productive time growth rate decreases from 0:294 to 2:073 (both adjusted means) between Table 5 and Table 6 according to the OLS estimator and from 0:359 to 2:790 for the GMM estimator, hence the estimates diverge further from their predicted value of 3 = 0. The GMM estimates of the forward interest rate term also decrease from 1:729 in Table 5 to 4:372 in Table 6 (adjusted means where appropriate). Again, the estimates diverge further from the theoretical prediction of 1:125. The estimated coe¢ cients for output growth in Table 6 are comparable to those for consumption growth presented in Table 5, despite the e¤ect that the misspeci…cation has on the other estimates. For example, the OLS estimate for 3 is 0:300 (adjusted mean) in Table 6 compared to the corresponding estimate of 0:277 in Table 5. For the GMM estimator the coe¢ cient on output growth is 0:402 (adjusted mean) in Table 6 compared to the corresponding estimate of 0:302 reported in Table 5. The velocity growth term is estimated precisely by the GMM estimator even after the modi…cation to the estimating equation. Estimates of 4 retain the correct sign and are of a similar magnitude as under the correctly speci…ed estimating equation; for example, a GMM estimate of 0:190 in Table 6 compared to a corresponding estimate of 0:269 in Table 5. For the restricted speci…cation ( 5 = 0), the estimates undergo similar changes as those obtained from the restricted version of the ‘correct’ estimating equation (26). The OLS and GMM estimators generate in‡ation coe¢ cients which often fall below unity in a manner incompatible with the theoretical model 1 9 Corresponding upward shifts in the estimated in‡ation coe¢ cient are found for the 3 8 band pass …lter results (results not reported) and even larger increases are found for the HP …ltered data (results not reported).
23
from which the Taylor condition is derived, although the GMM estimator provides a notable exception (Table 6). In short, the results obtained from applying equation (27) to the simulated data show that adapting the estimating equation in a seemingly minor way can have a substantial impact upon the coe¢ cient estimates obtained. The erratic results produced by this misspeci…ed estimating equation provide an illustration of the fundamental di¤erence between the Taylor condition and a conventional interest rate rule. Unlike a Taylor rule, the Taylor condition cannot be modi…ed in an ad hoc manner.20 In order to make the progression from (26) to (27) in a legitimate manner, one would need to alter the underlying model by excluding physical capital, for example. A new set of arti…cial data would then need to be simulated from this alternative model prior to reestimation.
5.4
A Conventional Interest Rate Rule
The estimation procedure is now reapplied to the following estimating equation: Rt =
0
+
1 Et t+1
+
2 Et gy;t+1
+
5 Et Rt+1
+ "t :
(28)
This estimating equation corresponds to the misspeci…ed representation of the Taylor condition with output growth plus further restrictions on the terms in productive time and the velocity of money; see equation (25). Equation (28) can be interpreted as a ‘dynamic forwardlooking Taylor rule’ for 5 6= 0 or a ‘static forwardlooking Taylor rule’ under the restriction 5 = 0: Notably, the term in velocity growth is absent from this expression. This omission might be expected to have a bearing on the estimates because equations (26) and (27) produced a large number of statistically signi…cant estimates for the velocity growth coe¢ cient. The results are again similar across the HP and band pass …lters so only the 2 15 band pass results are presented (Table 7).21 The estimates are generally found to be poor in terms of the number of statistically signi…cant estimates produced and in terms of mean Rsquare and adjusted Rsquare statistics. This is not surprising given that yet another source of misspeci…cation has been added to the estimating equation. The mean coe¢ cient on in‡ation does not exceed unity for any of the three estimators considered. The results are also comparatively weak in terms of the frequency with which the null hypothesis of the Fstatistic is rejected and in terms of the number of nonrejections of the null hypothesis of the Hansen Jtest. The latter …nding calls into question the validity of the instrument set used for the GMM estimator for equation (28). 2 0 In contrast, conventional interest rate rules are exogenously speci…ed and thus amenable to arbitrary modi…cations. Clarida et al. (1998), for example, add the exchange rate to the standard Taylor rule and Cecchetti et al. (2000) and Bernanke and Gertler (2001) consider whether policymakers should react to asset prices. 2 1 The instrument set now comprises of four lags of expected future in‡ation, four lags of expected future output growth, the second, third and fourth lags of the nominal interest rate and a constant term for the GMM estimator or just the shortest lag of each and a constant term for the exactly identi…ed 2SLS estimator.
24
Unrestricted OLS
2SLS
GMM
OLS
2SLS
GMM
3.66E06 [0] 6.19E05
0.001 [0] 0.031
3.07E06 [16] 8.58E05
9.34E07 [0] 2.43E05
4.46E06 [0] 9.65E05
2.80E06 [5] 6.42E05
0
Standard error Adjusted mean
Et
t+1
Assumed
=0
BP Filter, 215 Window
5


9.38E07


4.23E07
4.219 [971]
25.003 [189]
5.274 [961]
0.541 [941]
2.338 [882]
1.101 [990]
Standard error
1.715
1290.303
2.582
0.170
1.151
0.264
Adjusted mean
4.185
2.522
5.235
0.532
2.010
1.100
0.303 [967] 0.125
2.019 [206] 122.643
0.406 [940] 0.204
0.038 [563] 0.020
0.211 [262] 0.304
0.082 [956] 0.029
0.300 2.098 [959]
0.244 14.698 [211]
0.402 2.815 [954]
0.029 0.284 [424]
0.077 1.462 [353]
0.081 0.610 [941]
Et gy;t+1 Standard error Adjusted mean
Et gl;t+1 Standard error
0.892
825.197
1.417
0.189
1.740
0.232
Adjusted mean
2.073 0.118 [884] 0.042
1.672 0.069 [310] 16.979
2.790 0.191 [970] 0.064
0.189 0.095 [733] 0.043
0.655 0.246 [587] 0.164
0.598 0.158 [923] 0.061
0.113 3.878 [907]
0.117 29.503 [128]
0.190 4.498 [849]
0.084
0.161
0.156
Et gV;t+1 Standard error Adjusted mean
Et Rt+1 Standard error
1.812
1437.910
2.802
Adjusted mean
3.767
1.757
4.372
0.361 0.327 0.001 (995) N/A
<0 <0 0.379 (411) N/A
0.127 0.079 N/A 0.226 {1000}
0.246 0.214 0.020 (924) N/A
<0 <0 0.055 (829) 0.260 {682}
<0 <0 N/A 0.264 {1000}
1.882 <699> 99
1.982 <868> 98
2.342 <945> 96
2.295 <1000> 99
2.145 <997> 98
2.728 <999> 96
M ean; Rsquare Adjusted Rsquare Pr(Fstatistic) Pr(Jstatistic) DurbinWatson Sample size (1000x) Notes: ‘Standard error’measures the variation in the coe¢ cient estimates. ‘Adjusted mean’assigns a value of zero to non statistically signi…cant estimates. Fstatistic: null hypothesis of no joint signi…cance of the independent variables (not available under GMM). Jstatistic: null hypothesis that the instrument set is valid (only available if there are overidentifying restrictions). DurbinWatson statistic: null hypothesis that successive error terms are serially uncorrelated against an AR(1) alternative. [ ] reports the number of statistically signi…cant coe¢ cient estimates, () the number of Fstatistic rejections, {} the number of Jstatistic nonrejections and <> the number of times the DW statistic exceeds its upper critical value (all at the 5% level of signi…cance).
Table 6: Output Growth instead of Consumption Growth, Band Pass Filtered data (215 years), 100 Years Simulated, 1000 Estimations Average.
25
The in‡ation coe¢ cients are estimated surprisingly precisely under the restriction on 5 . However, these estimates di¤er quite substantially between estimating procedures for the 2 15 …lter; 0:317 (adjusted mean) for OLS compared to 0:892 for GMM (adjusted mean). In short, imposing a ‘conventional Taylor rule’restricts the true estimating equation to such an extent that the theoretical prediction that the coe¢ cient on expected in‡ation exceeds unity is not veri…ed. An estimated in‡ation coe¢ cient of this magnitude might erroneously be interpreted to signify that the Taylor principle is violated but this result is simply a product of a misspeci…ed estimating equation in the present context. Only if the model excluded physical capital and set velocity to one, by excluding exchange credit for example, would such an estimating equation be appropriate.
6
Alternative Interpretations of the Taylor Condition
Consider two alternative representations of the Taylor condition; a backwardlooking version and an alternative version which features credit.
6.1
Backward Looking Taylor Condition
The Taylor condition can be reformulated to feature a lagged dependent variable on the right hand side instead of the lead dependent variable which appears in equation (22). This yields a similar expression written in terms of Rt+1 instead of Rt :
Rt+1
R
=
(
1)
Et (
t+1
(
1) 1 l l
(
1)
)+
Et g l;t+1
(
1)
Et g c;t+1
V
(
1)
Et g V;t
g
(29) 1
(
1)
Rt
R :
While equation (29) compares better to interest rate rules which feature a lagged dependent variable on the right hand side as an ‘interest rate smoothing’term, the lead nominal interest rate is now the dependent variable. Such an expression is more akin to a forecasting equation for the nominal interest rate than an interest rate rule. Such a transformation also raises the fundamental issue discussed by McCallum (2010). He argues that the equilibrium conditions of a structural model stipulate whether any given di¤erence equation is forwardlooking ("expectational") or backwardlooking ("inertial") and that the researcher is not free to alter the direction of causality implied by the model as is convenient. The forward looking representation of the Taylor condition (22) is the long accepted rational expectations version; for example, Lucas (1980) suggests that the forward looking "…lters" suit models which feature an optimizing consumer. 26
Unrestricted
215 Window
OLS
2SLS
GMM
OLS
2SLS
GMM
0
6.42E07 [0]
7.62E05 [0]
2.48E06 [8]
7.00E07 [0]
5.16E06 [0]
1.85E07 [12]
Standard error
2.57E05
0.006
1.15E04
2.58E05
2.12E04
1.13E04
Adjusted mean


1.12E06


4.92E07
0.310 [239] 0.448
0.671 [22] 162.564
0.132 [344] 0.955
0.326 [926] 0.099
1.472 [416] 3.431
0.894 [981] 0.315
0.185 0.020 [277]
0.017 0.185 [40]
0.092 0.019 [237]
0.317 0.021 [408]
0.826 0.035 [163]
0.892 0.031 [460]
Standard error
0.017
16.569
0.027
0.012
0.582
0.027
Adjusted mean
0.012 4.225 [27] 88.304
0.012 0.918 [524] 1.002
0.013 N/A
0.041 N/A
0.024 N/A
Standard error
0.011 0.008 [147] 0.475
Adjusted mean
0.012
0.104
0.788
0.169
<0
<0
0.153
<0
<0
0.142 0.029 (887) N/A 1.828 <850> 99
<0 0.527 (162) N/A 2.012 <974> 98
<0 N/A 0.050 {339} 2.236 <991> 96
0.136 0.024 (891) N/A 1.817 <783> 99
<0 0.149 (599) 0.352 {679} 2.047 <996> 98
<0 N/A 0.058 {440} 2.186 <990> 96
Et
t+1
Standard error Adjusted mean
Et gy;t+1
Et Rt+1
Assumed
=0
BP Filter,
5
M ean; Rsquare Adjust Rsquare Pr(Fstatistic) Pr(Jstatistic) DurbinWatson Sample size (1000x)
Notes: ‘Standard error’measures the variation in the coe¢ cient estimates. ‘Adjusted mean’assigns a value of zero to non statistically signi…cant estimates. Fstatistic: null hypothesis of no joint signi…cance of the independent variables (not available under GMM). Jstatistic: null hypothesis that the instrument set is valid (only available if there are overidentifying restrictions). DurbinWatson statistic: null hypothesis that successive error terms are serially uncorrelated against an AR(1) alternative. [ ] reports the number of statistically signi…cant coe¢ cient estimates, () the number of Fstatistic rejections, {} the number of Jstatistic nonrejections and <> the number of times the DW statistic exceeds its upper critical value (all at the 5% level of signi…cance).
Table 7: Output Growth in a Standard Taylor Rule, Band Pass Filtered Data (215 years), 100 Years Simulated, 1000 Estimations Average.
27
In fact, we would argue that the timing of the cashinadvance economy is such that our forwardlooking rule in equation (22) is the correct model, while equation (29) is consistent with the alternative "cashwhenI’mdone" timing which we do not employ (see Carlstrom and Fuerst, 2001).
6.2
Credit Interpretation of the Taylor Condition
Christiano et al. (2010) have considered how the growth rate of credit might be included as part of a Taylor rule so that "allowing an independent role for credit growth (beyond its role in constructing the in‡ation forecast) would reduce the volatility of output and asset prices." The term in velocity growth can be rewritten as the growth rate of credit in the following way: Since Vt = dt ct 1 m m b and g V;t = m 1 m mt = 1 (1 mt ) ; then V Vt = 1 c ct c 1 c g (1 m c );t ct where g (1 m );t is the growth rate of normalized credit. The Taylor condition is c now rewritten as: Rt
R
=
Et ( (1
t+1 m c
)+
) Et g (1
m c
Et g c;t+1 R Et
);t+1
g +
(1
Rt+1
R :
)
l 1
l
Et g l;t+1 (30)
2
m m (1 )(1 c t ) (R 1)(1 )( c t ) t t The credit coe¢ cient can be derived as (1 m ) 2 (1 ) R 1 (1 ) 1 [ ( mctt )] c 0: A positive expected credit growth rate decreases the current net nominal interest rate Rt : With velocity set at one as in a standard cashinadvance economy, neither credit nor velocity would enter the Taylor condition since the credit service does not exist and velocity does not vary over time.
7
Discussion
Expressing the monetary policy process in terms of the nominal interest rate carries the advantage of reconciling the language of economists, who have traditionally depicted the money supply as the instrument of monetary policy, with the language of central bankers, who are more accustomed to conducting policy deliberations in terms of a shortterm interest rate (Mehrling, 2006). Alvarez et al. (2001) caution that modelling monetary policy solely in terms of a nominal interest rate rejects the quantity theory in spite of the strong empirical link between money growth, in‡ation and, interest rates. Schabert (2003), for example, uses the equilibrium conditions of a standard cashinadvance model to derive the conditions under which a money supply rule and an interest rate rule are ‘equivalent’, while Fève and Auray (2002) generate simulated data from a similar model and demonstrate that an interest rate rule can be spuriously recovered from this data even though monetary policy is modelled in terms of a money growth rule. This paper has derived an expression similar to a conventional interest rate rule as an equilibrium condition of an endogenous growth model with endoge28
nous velocity in which monetary policy is characterized as a stochastic money supply rule. The theoretical model underpinning this expression implies that the coe¢ cient on in‡ation exceeds unity in general, takes a value of unity as a special case at the Friedman (1969) optimum, but that it may not fall below unity. Simulation exercises support the theoretical restriction placed on this coe¢ cient, so long as the estimating equation accurately re‡ects the equilibrium condition. The Lucas critique perspective does apply strictly within this paper in that the variance distribution of the money supply process is invariant across the whole period of the simulated data. This means that the average long run money supply growth rate is a "structural" but "policy" parameter that the consumer takes as given and includes in the coe¢ cient on the in‡ation rate in the equilibrium condition as expressed in ‘Taylor rule form’. The in‡ation coe¢ cient is invariant as all other parameters in this coe¢ cient are also structurally given utility and technology parameters. Comparing this to reality and to results such as Leeper and Zha (2003) that suggest potential changes in the variance distribution of the money supply is a complex task. For example, Orlik and Veldkamp (2012) …nd that "uncertainty shocks" result from insu¢ ciently complex models of the forecaster, which they reconcile by adding further degrees of state, model and parameter uncertainty within the forecasting process. Their results might be interpreted as allowing the unforecasted aggregate risk to be a result of rare crisis events that are hard to forecast and which require a fat tailed distribution rather than normal distributions, as assumed in our model above.22 Their unpredicted aggregate risk for U.S. data (Figure 2) appears to be correlated with unexpected in‡ation during periods when there were major wars or bank crises that demanded high …scal expenditure. In other words, "abnormally" high in‡ation can follow after unexpectedly large government de…cits due to wars or crises, after some of this debt gets "monetized" through the central bank buying some of the newly issued debt. If we could assume high kurtosis in order to include such rare events, then it may be that the in‡ation coe¢ cient of a rederived Taylor condition would in this environment be invariant over the entire sample period. In other words, rare events could seem to cause shifts in variance which are better represented by high kurtosis. This would mean for actual data that the Lucas critique might hold for one regime of money supply policy over the whole period rather than …nding shifts in regime, a speculation that quali…es the broad interpretation of our results but remains a task for future research. Our results can be interpreted in several other ways. First, the derivation could be said to represent an ‘equivalence proposition’between the money supply process modelled and an ‘interest rate rule’, which actually represents an equilibrium condition of the model. This would be similar to the interpretation adopted in Alvarez et al. (2001), Végh (2002) and Schabert (2003). 2 2 Fat tails typicallly can embody low probability but highrisk events such that the tails are nonnegligible; in quantitative …nance, "econophysics", and commodity markets these tails have long been identi…ed as being of importance (for example Mandelbrot, 1963, Fama, 1965, Haas and Pigorsch, 2009).
29
Second, the Taylor condition can be interpreted as the interest rate rule which results from the money supply process in the context of the Benk et al. (2010) model. Woodford similarly derives the interest rate rule which "implements" strict in‡ation targeting in the New Keynesian model (Woodford, 2003, pp.290295). However, the money supply does not enter that model. Changes in the velocity of money therefore play no role and thus cannot be used to help explain why traditional Taylor rules might be using misspeci…ed estimating equations in …nding an in‡ation coe¢ cient of less than one in empirical applications. The fact that our framework assigns a central role to money potentially implies that the money growth rule can o¤er guidance to policymakers at times when the conventional monetary policy instrument encounters the zero lower bound, as is the case at the present time. Third the Taylor condition contrasts with the equilibrium condition for the nominal interest rate derived from a standard Euler equation: Canzoneri et al. (2007, p.1866), for example, derive an expression for the nominal interest rate from a conventional Euler equation in which the coe¢ cient on the in‡ation term is one.23 For post1966 U.S. data, they show that the Eulerequationimplied nominal interest rate …ts poorly to the observed nominal interest rate. On the other hand, a conventional Taylor rule with a coe¢ cient on in‡ation in excess of unity has often been found to …t the observed nominal interest rate rather well (for example, Taylor, 1993). Clarida et al. (2000) estimate such an in‡ation coe¢ cient for ‘postVolcker’subsamples of U.S. data but not for a preVolcker subsample. The Taylor condition (22) therefore represents an equilibrium condition which contains a coe¢ cient on in‡ation consistent with empirical results which …nd evidence for a ‘Taylor principle’, while suggesting that results which fail to …nd support for the Taylor principle may omit potentially important variables such as the velocity of money. Our preliminary estimation results on actual U.S. data that include velocity growth …nd an in‡ation coe¢ cient above one for both subsamples, part of our future work. The Taylor condition derivation here also resonates with Hetzel (2000) who warns that empirical correlations between a shortterm interest rate and macroeconomic variables such as output and in‡ation cannot be interpreted to reveal the behavior of policymakers (i.e. their policy rule) unless the relationship obtained can be declared as structural. It is also consistent with Cochrane (2011), who argues that the Taylor rule su¤ers from an identi…cation problem in the New Keynesian model. Our contribution has been to o¤er one very particular explanation based on a neoclassical monetary model extended to include endogenous growth and endogenous velocity in order to shed light on the structural relationships which might underpin the reduced form expressions to which Hetzel (2000) refers. Finally, the Taylor condition here captures bank crises. A decrease in the productivity of banking causes a more costly supply of exchange credit and substitution towards a higher demand for real money balances. Velocity goes 2 3 Their expression is a lognormal approximation to a standard Euler equation and is written in terms of the inverse of the gross nominal interest rate. Therefore, it also contains second moments and the coe¢ cient on in‡ation has a theoretical coe¢ cient of minus one.
30
down. Velocity has in fact decreased signi…cantly during the current period of bankcrisiswithprolonged unemployment, and this is captured in the Taylor condition through a negative expected growth rate of money velocity. Following a narrow Taylor rule is not the point then in this environment. Behavior could still be described by the Taylor condition once the velocity and employment growth terms are included as in this model. The point is to rein in …scal expenditure during crisis periods without printing so much money as to cause high in‡ation. Liquidity e¤ects involved in driving the interest rate to zero are unlikely to be surprises in the current crisis due to the ‘forward guidance’issued by the U.S. Federal Reserve. They have been printing money which …lls the void created by the drop in bank’s exchange credit, thus avoiding de‡ation in a way many have felt is appropriate. This means that equilibrium conditions like the Taylor condition of this model need not be departed from. The Taylor rule without velocity or employment terms only applies when these variables are not signi…cantly varying, which has not been the case in the recent crisis or in the in‡ation runup from the late 1960s to the early 1980s, nor during the Great Depression. Signi…cant cycles in velocity from trend are the rule rather than the exception as shown in Benk et al. (2010) who document only two long cycles downward in U.S. monetary history since 1919: during the banking crisis of the Great Depression and during the …rst decade of the 21st century that has included the …rst major banking crisis since the Depression. Perhaps a more e¢ cient comprehensive …nancial intermediary deposit insurance scheme, with riskbased premiums, would be more e¢ cient government policy than the current amalgam, but this involves bank productivity which this model and its Taylor condition take as given exogenously.
8
Conclusion
The paper has derived a general equilibrium dynamic Taylor condition for a constant relative risk aversion economy with leisure, Lucas (1988) endogenous growth, and with endogenous velocity through production of exchange credit in a …nancial intermediary. The importance of a ‡uctuating velocity in replicating the ‘Taylor principle’is consistent with the role for velocity reported by Reynard (2004, 2006) and with Benk et al. (2010). While providing a theoretical means to overview the empirical literature relating to the Taylor rule, as reviewed by Siklos and Wohar (2005), here the focus is …rst to show that estimation of a Taylor rule may result in the spurious inference that the central bank is engaged in Taylor rule reaction behavior rather than simply supplying money. This is established here by generating arti…cial data as simulated from the model and then successfully estimating a theoretical Taylor condition. This condition is simply an equilibrium condition in the economy in which the central bank stochastically makes changes in the money supply growth rate to …nance government spending. For example, such money supply changes tend to occur whenever the government needs to depart from its stationary money supply growth rate and resort to the ‘…scal in‡ation
31
tax’. This typically can occur during banking crisis, recession, or war. Money velocity growth itself enters as a variable and ends up playing a potentially signi…cant role; in particular this occurs when velocity is changing, such as during the recent banking crisis and during the 1930s when U.S. velocity cycled downwards, as identi…ed in Benk et al. (2010), and in the "preVolcker" U.S. high in‡ation of the 1970s. Velocity is endogenized in the model following the banking …nancial intermediation microeconomic literature, where …nancial services are produced according to a CobbDouglas production function that includes deposited funds as an input. This approach implies a bank service sector valueadded that is consistent with the U.S. national income accounting treatment of the bank service sector, as emphasized by Inklaar and Wang (2013). The paper shows how the banking production of exchange credit is surprisingly crucial to the derivation of the ‘Taylor principle’. This result emanates from an endogenous velocity of money; a simple (cashonly) cashinadvance constraint with a constant velocity of one is shown to provide an in‡ation coe¢ cient of unity. Through endogenous growth, we can derive an output gap measure not inconsistent with Taylor and Wieland’s (2010) emphasis on changes in output as a measure for the output gap. In our model, the output growth term does not enter directly unless we also include an investment growth term; otherwise the consumption growth is the ‘output gap’term of the model’s Taylor condition. Estimation results are also provided for two misspeci…ed models using the simulated data from the correct model. One includes output growth without including investment growth. The second is a standard Taylor rule which is appropriate for the model economy only if there is no physical capital and if exchange credit does not exist as an alternative to money (AF = 0): Omitted variables cause signi…cant misspeci…cation bias in the reported results. The implication is that the results hold promise for explaining disparate estimated rules across di¤erent periods and countries, as well as during bank crises, sudden …nancial deregulation, or times of other signi…cant shifts in money velocity. This could help organize and show greater robustness for this literature. By simulating data of the model and estimating successfully a ‘Taylor rule’ from the data, the paper shows that such a rule can be identi…ed econometrically from the economy’s asset pricing behavior when the central bank simply prints money stochastically. In that case it would be spurious to claim that an estimated Taylor rule reveals how the central bank actually conducts policy through interest rate targeting. Rather, the central bank simply satis…es the government’s …scal needs via direct and indirect taxes, including the in‡ation tax. Put di¤erently, if this economy were representative of the actual economy then estimating a standard Taylor (1993) model using actual data would be expected to produce an in‡ation coe¢ cient above one in keeping with the Taylor principle only if velocity (or exchange credit), employment and investment did not change over the sample period. Reynard (2004, 2006) and Benk et al. (2010), for example, show the importance of timevarying velocity while signi…cant business cycle ‡uctuations in employment and investment are a welldocumented feature of business cycle research. An important quali…cation is that the paper does not claim regime changes 32
do not occur. Rather it analyzes the equilibrium within a regime of given …rst and second moments of the stochastic money supply process. Regime changes in our framework could cause di¤erent magnitudes of the Taylor condition in‡ation coe¢ cient but it would remain above one for a positive nominal interest rate.
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[2]
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