NMonetary reaction function forward looking and non linear behaviours

Department of Economics Professorship of Macroeconomics Christian Albrechts University of Kiel Monetary Reaction Function Forward looking and Non linear Behaviours Master’s thesis in Master of Science[.]

Department of EconomicsProfessorship of MacroeconomicsChristian-Albrechts-University of Kiel

Monetary Reaction Function:

Forward-looking and Non-linear

Behaviours

Master’s thesis in Master of Science Economics

Supervisor: Prof Dr Kai Carstensen

WS 2016/2017

Abstract This paper investigates forward-looking and non-linear teristics of monetary reaction funciton using Fed’s data By employing hybridNew Keynesian with non-linear Phillips curve and asymmetric preferences ofFed, a “hybrid-type” non-linear reaction function is derived The estimationresults provide more empirical evidence for a non-linear monetary policy rule

charac-in the period between 1983 and 2008, as well as a notable role of looking behavior of the policy rule toward output gap

forward-Name: Nguyen Thi Truc Ngan

Field of study: Economics

Semester: 8

E-mail: trucngan.19@gmai.com Deadline: 01.06.2017

Matriculation Number: 1020472

2.1 Instrument rule and Targeting rule 2

2.2 Forward-looking behaviours 5

2.3 Non-linearity of the Reaction Function 6

2.3.1 Convex Aggregate Supply curve 7

2.3.2 Asymmetric Preferences 8

2.4 Zero Lower Bound 8

3 Monetary Reaction Function model 9 3.1 Case I: Linear Aggregate Supply (τ = 0) 13

3.2 Case II: Quadratic Loss fucntion (γ → 0) 13

3.3 Case III: Linear Rule 14

3.4 With Zero Lower Bound 14

4 Estimation 14 4.1 Preliminary analysis 14

4.1.1 Generated regressor 15

4.1.2 Measurement error 16

4.1.3 Multicollinearity 17

4.1.4 Inflation Target in the U.S 18

4.2 Estimation 18

4.3 Estimation results 19

List of Abbreviations

1 Introduction

Literatures on monetary reaction function provide us learnings about teristics of policy as well as the priority target of the central bank With theintroducting of New Keynesian model as the monetary transmission mecha-nism (Clarida, Gali, and Gertler 1999) and inflation targeting (Svensson 1997,Svensson 2003), the most used derivations of optimal rules is base on a linear-quadratic framework This framework involves the central bank minimizing itsquadratic-form objectives function subject to a linear structure of the econ-omy Derivations of this framework produce a linear reaction function, ortargeting rule (Svensson 1997, Svensson 2003).1

charac-This linear reaction function means the Federal Reserve of the UnitedStates’s adjustment of Federal fund rate is a straight line of inflation and out-put However, in recent literatures, this linear-quadratic framework has beenchallenged, either by considering a non-linear Phillips curve (Orphanides andWieland 2000; Dolado, Marıa-Dolores, and Naveira 2005); or by abandoningthe quadratic loss function assumption and adopting asymmetric preferencesinstead (Dolado and Pedrero 2002; Cukierman et al 1999); or both (Surico2003; Surico 2007; Dolado, Pedrero, and Ruge-Murcia 2004)

This paper studies the non-linearity of monetary reaction function bining both channels: non-linear Phillips curve and asymmetric loss function

com-of central bank as conducted by Dolado, Pedrero, and Ruge-Murcia (2004)

We would like to engage in a quasi-convex Phillip curve as in Dolado et al.(2004) Next, our asymmetric objective function suggested biased preference

in inflation only, output gap is also included, but in a quadratic form2 Thissetup will be discussed more in Section 3

Several literatures using forward monetary policy rule augmenting purelyNew Keynesian model have been conducted (Surico 2003; Clarida, Gali, andGertler 1999) Some of them have showed that a purely-forward lookingmodel may sometime misspecified due to the lack of history dependence (Gal´ı,Gertler, and Lopez-Salido 2005) Thus, in this paper, the hybrid New Key-nesian model, which covers both backward and forward-looking variables, isemployed instead

1

In most papers, including this study, the term monetary policy rule, monetary reaction function or Taylor rule are used interchargeable.

2 This asymmetric loss function with the entering of output gap has also been mentioned

by Dolado, Pedrero, and Ruge-Murcia (2004), section 2.6, and a reaction function has been derived Nonetheless, in their studies, an empirical application based on this function was not done.

For empirical studies, estimation of the non-linear rule is done using theU.S data According to Belke and Klose (2013), some evidence for a structuralbreak in the time of 2008-financial-crisis has been found Thus, we wouldemploy two periods of time: the first period is after Volcker and Greenspanbecame the chairman of Fed excluding the period when Fed targeted non-borrowed reserves, until the financial crisis in 2008 (1983Q1 to 2008Q2); andthe second period is after the crisis until recent time (2008Q3 to 2016Q4).The rest of the paper is structured as follows In section 2, we will discusssome theories regardings monetary policy rules as well as reasons for consid-ering a non-linear reaction function Section 3 is the derivation of a non-linearreaction funciton under asymmetric preferences and hybrid non-linear Keyne-sian model, with the consideration of some linear cases and zero lower bound.Section 4 concludes estimation methodology and results using two subsam-ples Conclusion of this studies is covered in section 5 Two appendices arederivation of the general reaction rule and description of the data.

2.1 Instrument rule and Targeting rule

First introduced by John Taylor (1993), Taylor rule (instrument rule) hasbeen widely used to monitor the policy rule of central banks because of itssimpliness yet effectiveness At the time, this simple rule can well explainedresponds of instrument rates to the “price stability” target of central banks

Ever since, several studies have been done to examined this simple strument rules, which included some variations of the Taylor rules Levin,Wieland, and Williams (1999) has empirically proven the relatively robust-ness of the simple Taylor reaction function with a combination of smoothinginterest rate Belke and Klose (2013) modified Taylor rules using Fisher equa-tion to estimate reaction functions in the presence of zero-lower bound (ZLB)

in-3 from Taylor (1993), i t is the instrument rate in period t, f is a constant, π t is inflation rate at time t, π* is inflation target and x t is the output gap In the original Taylor rule, f π

is 1.5 and f x is 0.5

Their empirical results provide more evidence for a structural break in reactionfucntion of ECB and Fed between before and after 2008 crisis.4

In another study, Kim and Nelson (2006) used a forward-looking Taylorrule with interest rate smoothing to estimate Fed’s monetary policy Theirwork focuses on the change of instrument rate over time and effectively findevidence for time-varying response of Fed toward future macroeconomic con-ditions Not only for US data, this simple rule is also applied in differrentmacro models.5

In the context of policy design, for the monetary policy rule to work andstimulate the economy, a highly-creditable announcement from central banks

is desirable In other words, central bank’s commitmet must be trustworthy

In our study, data from the United States is used, so we believe we can safelyassume that creditibility from FED is relatively high Following the mini-mizing the objective function approach, the matter related to creditibility ofpolicy design is the distinctions between discretion and commitment policy.There are several reasons that commitment is more preferable in recentstudies As stated in Svensson (1997), discretion policy could lead to toomuch inflation variability and too little output variability Clarida, Gali, andGertler (1999) argue that monetary policy under commitment could be used

to manipulate private sector expectation about the future Morever, in thepresence of Zero lower bound, commitment optimal monetary policy rule hasbeen proven to be effective to stimulate the economy (Nakov, 2005) Problemsrelated to ZLB would be further discussed in section 2.4

With the expansion of rational-expectation and development of nomics theory, the idea that central banks’ decision making process is morefocused on optimization and forward-looking behaviors starts to gain moresupports Stated that credibility of central bank should highly relevant toexpectation of pivate sector, Clarida, Gali, and Gertler (1999) combined NewKeynesian perspective with “instrument rule” of monetary policy and de-rive an optimal instrument rule which emphasizes targeting objects of centralbanks

macroeco-4

Belke and Klose (2013) estimated ECB and Fed’s reaction function using real interest rate rather the nominal one In their model, even when nominal rate is close to zero, central banks still have power to stimulate the economy via quantitative easing.

5 Br¨ uggemann and Riedel (2011) use a non-linear Taylor rule to estimate the United Kingdom’s monetary reaction funtion; Cukierman, Muscatelli, et al (2008) also use non- linear instrument rule for the U.K data but with asymmetric preference; Salgado, Garcia, and Medeiros (2005) examined Brazil’s instrument rule with both linear and non-linear Taylor rule;

Even being heavily examined in various studies, this simple instrument rulewas intepreted in a very narrow view of “policy rule” (Svensson, 2003) Fromthe definition of Taylor (2003) about monetary policy rule, the instrumentrule should not be followed strictly but be seen as a “guidelines” and should

be viewed looser Following the ideas of inflation targeting rule 6, he thenpropose to broaden the approach of monetary policy rule as “a precribedguide for monetary-policy conduct” that include both “instrument rules” and

“targeting rule” Accordingly, the endogenous variables that enter the centralbank’s loss function are “target variables”, and “targeting” means minimizingthis loss function From latest studies, this “targeting rule” seem to be betterfit to modern theory (Dolado et al 2003, 2004; Gal´ı 2015; Cacciatore, Ghironi,Turnovsky, et al 2015)

Inherit the ideas of “targeting rule” and the optimal monetary policy der commitment, Svensson (2003) argues that commiting to follow even asimple transmission mechanism under direct optimal-control approach7 will

un-be impracticable and hardly verifiable.8

While committing to a direct optimal monetary policy is not practical,committing to a simple instrument rule or a simple Taylor-rule type is notoptimal either9 Instead, in his study, Svensson (2003) has proven the superi-ority of “targeting rule” over instrument rule Accordingly, inflation targetingrule could be understood as commiment to a targeting rule10 and he stated,

“ Targeting rules have the important advantage that they allow the use ofjudgment and extra-model information They are also more robust and easier

to verify than optimal instrument rules, but they can nevertheless bring theeconomy close to the socially optimal equilibrium ”

Thereafter, the idea of interpreting monetary policy rule as inflation geting has gained more attention and have been adopted by various countries

tar-It is also widely applied to estimate the reaction function (Svensson and

Wood-6

from Svensson, 1997, 1999; Rudebusch and Svensson, 1999

7

Clarida, Gali and Getler (1999) use New-Keynesian model and follow Woodford (1998)

to minimize the objective function using Lagrange with with forward-looking aggregate ply and aggregate demand relations contraints From their model, they stated that under commitment, a global optimal monetary policy could be attained.

sup-8

see Svensson (2003), section 3 for more details

9 e.g Even the instrument rule is robust, it may not be optimal in some case (e.g Zero lower bound), there is no room for extra-model information or adjustments, lack of history dependence, no improvement of instrument rule is allowed if new information arrives, (Svensson, 2003, section 4)

10

this could be a general targeting rule (commit to minimizing a specific central bank’s objective function) or a specific targeting rule In his paper, Svensson also stated that specific targeting rule could be more helpful and most literature, both support and againts targeting rule, focused this rule.

ford, 2004; Dolado, Pedredo and Ruge-Murcia, 2004) It is not until 2012 thatthe U.S started adopting this approach, however, some empirical studies havealso proven that the inflation targeting rule could also well describe the be-havior of central bank in the period prior-2012 (Dolado et al, 2004; Dolado,Maria-Dolores and Naveira, 2005; Goodfriend, 2004).

2.2 Forward-looking behaviours

By definition, a variable is referred as forward-looking if it depends on pectation of future values From the 90s of last century, the importance offorward-looking dimension toward economic policy (or monetary policy) washighly admitted, as Donald Kohn (1995) stated: “Policymakers cannot avoidlooking into future” Several empirical studies has been conducted and provethat monetary policy is driven more by future values rather than by laggedvariables (Clarida et al., 1999; Belke and Klose, 2013)

ex-Using a simple forecast-based rule with inflation forecasts as ate targets in the operation of forward-looking behavior, Batini and Haldane(1999) theoretically and empirically have debated benefits of including thisaspect Accordingly, this rule can express:

intermedi-• Transmission lags Transmission lags are the most straightforward son for the need of forward-looking behaviors There are some lagsneeded for the monetary policy start to affect inflation and output.Without demonstrating this characteristic, as Batini and Haldane said,policymakers will always be behind its target

rea-• Information encompassing Expected inflation is an indicator “mostclosely related with future value of interest variables”, therefore, it shouldcontains all information that could have effect on future path of inflation

• Output encompassing Ouput actually does not enter this rule, but theyargue that with a wise choice of targeting horizon, inflation forecast-based rules can secure the output stabilizing

Another method to implement the forward-looking dimension is by ifying the economic model that describes practices of private sectors (a.k.aNew Keynesian model) (Clarida, Gali, and Gertler 1999, Clarida, Gali, andGertler 2000) or by letting expectation of future values enter the central bank’sobjective function (Dolado, Pedrero, and Ruge-Murcia 2004) or both (Surico

mod-2007, Surico 2003)

Applying advances from methodology of macroeconomic modelling, ida, Gali and Getler (1999) adopted the New Keynesian perspectives implying

Clar-the forward-looking dimension into Clar-their reaction function FurClar-thermore, inanother study, they11 applied forward-looking model to estimate response ofinstrument rates to expected inflation They found that after Volker-Greespanera, the interest rate is highly sensitive to expected inflation, again confirmthat the Federal Reserve did focus more on controlling inflation after Volkertime This conclusion is highly recognized, their model as well as method isalso further applied in various studies.

The method of combining the forward-looking model with commitment totargeting rule is explained by Svensson (2003) and was followed by severalreseachers (Surico, 2003, 2007; Svensson and Woodford, 2004).This forward-looking monetary reaction function seems to be the most used among re-searchers (Svensson, 2003; Surico, 2003, 2007; Dolado et al., 2005)

However, while forward looking behaviours is dominant in most researches,some recent tests for backward and forward-looking Taylor-based rule havebeen conducted and it seems that the there is no strong evidence supports apurely-forward-looking behaviors (Sooreea 2008; Dolado, Pedrero, and Ruge-Murcia 2004; Svensson and Woodford 2004) The reason for it could be thelack of “history dependence”, even using interest rate smoothing is also apopular way to account for this matter, the role of backward-looking in themodel may be underestimated Thus, in our setup, we would like to use

a hybrid New Keynesian model which includes both forward and looking variables

backward-2.3 Non-linearity of the Reaction Function

Traditionally, combination between quadratic loss function and a linear namic system has been used to acquire a linear reaction function12, or to de-velop an optimal monetary policy rule The dynamic system could be forward-looking or backward-looking as we discussed above, but in this framework, thenominal interest rate will be a linear function of current or expectation of infla-tion and output gap This straight-forward traditional rule could also provide

dy-a resondy-ably good description of the monetdy-ary policy Nonetheless, it is able to enable non-linearity in the setup of optimal monetary policy rule due

desir-to some reasons (Dolado et al., 2005)

11 Clarida, Gali, and Gertler 2000.

12

Svensson, 1997; Clarida et al., 1999.

• Nonlinear trade-off between inflation and output gap Under traditionalKeynesian assumption, nominal wages are elastic upward but rigorousdownward, leading to a quasi-convex aggregate supply curve13.

• Central bank’s asymmetric preferences From recent literarutes, thereare more evidences for the belief that central bank’s preference should

be non-linear This asymmetry means that the central bank not onlycare about the level of deviation of inflation/output from its target butalso the sign of deviations

• Uncertainty about the NAIRU As mentioned in Meyer et al (2001), highuncertainty about the NAIRU would leads to a non-linear interest-ratepolicy, which implies cautions of policy-makers when adjusting interestrate in response to small output gaps

To the extent of this paper, achieving non-linearity of monetary reactionfunction through a quasi-convex aggregate supply curve and asymmetry ofcentral bank’s preference is the main focus Furthermore, following Dolado

et al., (2004), a general model combining both non-linear Phillips curve andcentral bank’s asymmetric preferences is preferable in this study

As mentioned above, by tranditional Keynesian assumption, inflation is plied to be a decreasing and convex function of employment rate Which meansinflation will be driven down much more by an increase of unemployment ratewhen this unemployment rate is high than when it is low14 Then by Okun’slaw, the co-movement between output gap and unemployment rate leads to theconvex relationship between output gap and inflation (Schaling,1999; Nobayand Peel, 2000; Dolado et al., 2005)

im-In particular, Dolado, Mar´ıa-Dolores and Naveira (2005) studied the metry of monetary policy reaction function through a non-linear Phillips curve.They found empirical support for asymmetric reaction function that, theweights for positive inflation deviation or output gap is larger than negativeones This non-linear behavior is spotted from European central banks butnot from the Federal reserve

2.3.2 Asymmetric Preferences

The second approach to add non-linearity in monetary reaction function isthrough central bank’s objective function (Nobay and Peel, 1998; Gerlach,2000; Surico, 2003, 2007) This setup allows different weights for positive andnegative output gap and/or inflation deviation from its target For instances,Cukierman (1999) showed that, in gerneral, the policy-makers would preferexpansions to recessions, thus implies inflation bias This findings was exam-inied by Ruge-Murica (2002) using cross-section data from OECD Contradict

to this finding, Mishkin and Posen (1998) argues deflation bias would be amore reasonable outcome

More evidences support the idea of asymmetric preferences was certified(Orphanides and Wieland, 2000; Surico, 2003, 2007) Dolado et al (2002)used data from Spain and found that, the central bank of Spain faces a biggerloss to possitive than to negative deviation from inflation target This findingwas agained confirmed by Dolado et al (2004) and Surico (2007) Using anon-linear Taylor rule with asymmetric preferences, Dolado et al (2004) havegiven more empirical supports for non-linearity after 1983 In contrast, Surico(2007) has found evidence for non-linear interest rate reaction function before

1979 only and with respect to ouput gap

Analysis on non-linear Taylor rule due to asymmetric preferences was cussed theoretically and empirically by Cukierman and Muscatelli (2008) Us-ing data from the U.K and the U.S, their emprirical results support the ideathat asymmetric preferences play an important role in monetary policy, andthe nature of asymmetry would change alongside with the macroeconomicproblem at the time Particularly, they found that, inflation targeting in the

dis-UK showed a recession avoidance preferences, while in the U.S, inflation ance preferences dominant during the Vietnam war

avoid-2.4 Zero Lower Bound

Non-negativity constraint on nominal interest rate is usually called Zero LowerBound restriction This restriction means that the government is not able

to reduce the nominal interest rate below zero because private agents wouldsidestep from this negative rate and choose to hold money instead Thisrational constraint and its effect on the monetary policy however has notgained much attention until last decades, when more discussions related thismatter are conducted (Coenen and Wieland 2003; Coenen, Orphanides, andWieland 2004; Eggertsson and Woodford 2003)

The dilemma of this zero lower bound is that sometimes the monetarypolicy requires the interest rates to be negative to boost the economy, it meanswhen zero bound is reached, the monetary policy is no longer effective This

is called liquidity trap

When liquidity trap happens, the interest rate is not low enough to ulate the economy, thus monetary expansion policy is not working, the pes-simists believe that central bank has no power and they suggested that thesolution should be even higher inflation (Eggertsson and Woodford, 2003)

stim-In particular, Eggertsson and Woodford (2003) have throughoutly discussedThe zero bound on interest rates and conluded from their formulary, that theopen-market operations do not have sufficient effects, either instantly or at alater date

In their papers, Reifschneider and Williams (2000) tried to quantify theeffects of this constraint on macroeconomic stabilization and indicated a mod-ification to Taylor rule which reduces the consequence of this zero bound.Receiving more attention, the Zero Lower Bound constraint on nominal in-terest rate has been deeply reconsidered and in some newly researches, thecentral banks appear to be not powerless at all (Belke and Klose, 2013; Sugoand Teranishi, 2005; Gerlach and Lewis, 2014) In particular, Belke and Klose(2013) modified Taylor reaction function with the presence of zero lower bound

by using real interest rate instead of the nominal one The real interest rate isnot binded by zero lower bound thus negative in crisis time Sugo and Teran-ishi 2005, on the other hand, suggest that when zero bound is bind, the bestpolicy would be commitment optimal policy which has zero lower bound being

a constraint in optimization conditions, then using Kuhn-Tucker conditions tosolve for optima

We may notice that the nominal interest rate from late 2008 until recenttime has been very close to zero, implying a different monetary policy rule.Nevertheless, as finding optimal policy with zero lower bound is not my mainresearch, so I would use is FED’s fund rates from 1983 until 2008, in whichperiod that interest rates are always above zero Then in extension of thisstudy, we would partly estimate the monetary policy in the post-2008-crisisperiod using the approach of modifying the Taylor rule as proposed by Belkeand Klose (2013)

The derivation of the model used in this paper follows closely the one posed by Surico (2007), Dolado et al (2004) and Svensson (1997, 1999) with

pro-The dilemma of this zero lower bound is that sometimes the monetarypolicy requires the interest rates to be negative to boost the economy, it meanswhen zero bound is reached, the monetary policy is no longer effective This

is called liquidity trap

When liquidity trap happens, the interest rate is not low enough to ulate the economy, thus monetary expansion policy is not working, the pes-simists believe that central bank has no power and they suggested that thesolution should be even higher inflation (Eggertsson and Woodford, 2003)

stim-In particular, Eggertsson and Woodford (2003) have throughoutly discussedThe zero bound on interest rates and conluded from their formulary, that theopen-market operations do not have sufficient effects, either instantly or at alater date

In their papers, Reifschneider and Williams (2000) tried to quantify theeffects of this constraint on macroeconomic stabilization and indicated a mod-ification to Taylor rule which reduces the consequence of this zero bound.Receiving more attention, the Zero Lower Bound constraint on nominal in-terest rate has been deeply reconsidered and in some newly researches, thecentral banks appear to be not powerless at all (Belke and Klose, 2013; Sugoand Teranishi, 2005; Gerlach and Lewis, 2014) In particular, Belke and Klose(2013) modified Taylor reaction function with the presence of zero lower bound

by using real interest rate instead of the nominal one The real interest rate isnot binded by zero lower bound thus negative in crisis time Sugo and Teran-ishi 2005, on the other hand, suggest that when zero bound is bind, the bestpolicy would be commitment optimal policy which has zero lower bound being

a constraint in optimization conditions, then using Kuhn-Tucker conditions tosolve for optima

We may notice that the nominal interest rate from late 2008 until recenttime has been very close to zero, implying a different monetary policy rule.Nevertheless, as finding optimal policy with zero lower bound is not my mainresearch, so I would use is FED’s fund rates from 1983 until 2008, in whichperiod that interest rates are always above zero Then in extension of thisstudy, we would partly estimate the monetary policy in the post-2008-crisisperiod using the approach of modifying the Taylor rule as proposed by Belkeand Klose (2013)

The derivation of the model used in this paper follows closely the one posed by Surico (2007), Dolado et al (2004) and Svensson (1997, 1999) with

pro-some modification for aggregate supply curve and central bank’s preferences

as mentioned above

In many literatures, the US has been considered as a not so opened omy, adopting this assumption, we would like to use a hybrid New Keynesianmodel to describe behaviors of private sectors In this setup, function (3.1) isthe hybrid New Keynesian Phillips curve, which is similar to the one used byGal´ı and Getler (1999) and Gal´ı, Getler and L´opez-Salido (2005) This modelbased on Calvo’s (1983) staggered price setting framework, which assumesthat at each period t, a fraction (1 − ω) of all firms are able to reoptimize theirprice while the fraction ω ∈ (0, 1) of firms keep prices unchange It could beinterpreted that a firm has a chance to reoptimize their price with probabilty(1 − ω), the parameter ω is the degree of nominal regidity or price stickiness.Nevertheless, in fact, only a fraction of those firms, whose are able to ad-just prices, have forward-looking expectaion and reoptimize based on futuremaginal cost, the remaining firms are backward-looking and set prices accord-ing to rule-of-thumb (Gal´ı and Gertler 1999), there rule-of-thumb-firms areexpressed as the term γbπt−1in the function (3.1) Thus, past inflation is also

econ-an arguement in the Phillips curve If there is no firm adjust prices based onthe past level of aggregate price, γf = θ (θ is discount factor) and the NKPC

Equation (3.2) is an IS relationship where output gap depends on bothone-period forward and backwad values, interest rate and future inflation;

νt and εt is error term with zero means and variances σ2ν, σε2 respectively.Similar to Dolado et al (2004), we also allow the possibility that these shocksare conditional heteroscedastic Function (3.3) embodies the quasi-convexaggregate supply curve17, if τ = 0 then the AS curve is linear, otherwise it isconvex

Assume that policy-makers choose short-term interest rate which mizes the objective function In many literatures, a linex function, proposed

mini-by Varian (1974), was used for the central bank’s asymmetric preference Forinstance, funtion (3.4) was used by Surico (2003) as loss function18

• First, it allows different weights for positive and negative deviation fromthe targets

• Second, it states that the central bank’s loss depends on both size andsign of deviations

• Finally, quadratic form of loss function is a special case of linex functionwhich α and γ tend to zero

Dolado et al (2004) also used a similar loss function which depends only

on inflation, since this is the main target of central bank19 Nevertheless, asDolado et al (2004) argued, eventhough with inflation being the only target,Fed’s objective funciton should include both ouptut gap and inflation Yet

the inverse of the labor supply elasticity, θ is discount factor (in the book, β is used instead) Details in “Manuscript for the Lecture Macroeconomic Dynamics and Optimal Monetray Policy” by Wohltmann, 2014.

17 This form was also used in Dolado et al (2003, 2004); Schaling (1999)

18

see Surico 2007 for details

19 Dolado, Mar´ıa-Dolores and Ruge-Murcia (2004) assume the central bank’s problem is

from a recent study of Surico (2007), it seems that the asymmetric ence of FED for output gap has disappear after 1982 Following debates fromDolado et al (2004), under convex Phillips curve assumption, the output ratesare always below the value predicted from a linear model Thus, instead ofminimizing deviation of output gap, the central bank would minimize devia-tions of F(xt) from zero Besides, the usual quadratic form of loss functioncan well approximate the loss associated with deviation of F (xt) from zero,FED’s problem becomes:

where 0 < β < 1 is the discount rate, π∗ is optimal target inflation rate,

α captures the asymmetry in the objective function, λ ≥ 0 represents centralbank’s aversion towards output fluctuations

The derivation of the Taylor rule, following Svensson (2003) and Dolado

et al (2005), yielded from first-order condition for minimizing (3.5) substract

(3.7)where σ2π,t denotes the conditional variance of inflation rate The supscript tindicates σπ,t may be a time-variant variable ζt is idd innovation where σπ,t2follows a GARCH(1,1) process This will be better discussed in the followingpart 4.1 and Appendix A

The forward-term in this Taylor rule is the expected value of inflationitself and expected value of output gap through function (3.3) Besides, thelagged value of output gap and deviation of inflation from target from lastperiod is also determinants of interest rate, which could account for “historydependence”

Non-linearity in the reaction function (3.7) is expressed through F (xt+1)and ησ2π,t terms This reaction function (3.7) also nests cases when Phillipscurve is linear (τ = 0) or contral bank’s preference is asymmetric (γ → 0) orboth The following sections consider three special cases of linearity and onecase where zero lower bound is binding

3.1 Case I: Linear Aggregate Supply (τ = 0)

When τ = 0, the function F(.) becomes F (xt) = γxxt, the Phillips curve

is linear If this happens, the non-linearity in reaction function comes fromasymmetric preferences only, and the non-linear Taylor rule is:

(3.8)

In this function (3.8), the non-linearity is expressed through ησπ,t2 term only.Under asymmetric preferences, the conditional variance of inflation is also adeterminants of interest rate When σπ,t2 depends on lagged inflation andoutput, the lagged values of inflation and output will have effect on interestrate in a non-linear way

In particular, if η > 0, then positive relation between conditional variance

of inflation and interest rate implies α > 0, which in turn means that thecentral banks adhere a bigger loss to positive than negative inflation deviationsfrom the targets

3.2 Case II: Quadratic Loss fucntion (γ → 0)

This case is contradict with the previous case, when γ → 0 happens, thecentral bank’s preference could be well approximate by the usual quadraticfunction With γ → 0 and τ > 0, the non-linear Taylor rule takes the form:

3.3 Case III: Linear Rule

This case implies both γ → 0 and τ = 0, this is the usual linear optimalmonetary reaction function with quadratic loss function, which has the form:

3.4 With Zero Lower Bound

Belle and Klose (2013) showed that there should be a structural break betweenbefore and after 2008 crisis Following their approach, we also used a realinterest rate rt to avoid the zero lower bound constraint on the Taylor rule.Under rational expectation assumption, we can assume that πet = πt with πte

is inflation expectation Using the Fisher equation it= rt+ πte, we can rewritefunction (3.7) as:

4.1 Preliminary analysis

In this study, our non-linear reaction function is estimated using quarterly U.S.data The sample period is from 1983Q1 to 2016Q4, but we divided it intotwo sub-sample: 1983Q1 to 2008Q2, right before the crisis period, and from2008Q3 till 2016Q4 Following Dolado et al (2004), we also used two proxiesfor output gap, detrended IPI index and minus detrended Unemployment rate.Details about measurement of variables are described in appendix B Followingparts are some discussions about problems we may encouter when estimating

4.1.1 Generated regressor

From the reaction function (3.7), conditional variance of inflation is also aregressor, therefore it should be time-variant Many other researchers alsostate that conditional variance of inflation should be heteroskedastic (Engle,1983) To deal with this problem, a two-stage procedure sugessted by Dolado,Pedrero, and Ruge-Murcia (2004) is used First, the conditional variance

of inflation is measured from estimating the aggregate supply function (3.1).Then, σ2

π,tis used in the reaction function and estimated by Generalize Method

of Moments (GMM)

In the first stage, there are two main issues in estimation, testing for theconvexity of Phillips curve and whether conditional variance varibable is time-variant must be conducted As we discussed above, the specific form of non-linear reaction function depends on non-linearity of Phillips curve, thus it iscrucial to define whether τ 6= 0 is true Next, to check conditional variancefor inflation is not a constant over time, we need to test for conditional het-erescedasticity of inflation

We used the same estimation and testing method as described by Dolado

et al (2004) Accordingly, we would estimate aggregate supply curve (3.1)

by non-linear least square assuming that inflation is homoscedastic Then wecould test for H0 : τ = 0 using t-test, and H0 : no conditional heteroscedastic-ity using LM test20

Using quarterly data, the hypothesis that AS curve is quasi-convex is jected, linear relation can also well characterize the Phillips curve For thesecond test, the null hypothesis could be rejected at a standard level whichimplies that σ2π,t is indeed a time-variant variable However, misspecifying ofARCH-type fitting model for conditional variance might lead to consequences

re-of biased and inconsistent estimation Therefore, a specification test for ity of ARCH model proposed by Pagan and Ullah (1988) is used Accrodingly,

valid-if ARCH model is well specvalid-ified, the squared standardize residuals should not

be autocorrelated Again, using LM test, we can test for H0 : no relation for squared standardized residuals21 The test results displayed inTable 1 states that we can not reject the null hypothesis, which implies thatGARCH(1,1) could be a better captures for conditional heteroscedasticity ofthe U.S inflation

autocor-20 LM = T R 2 where T is the number of observations and R 2 is the uncenter R 2 of the OLS regression of the squared unemployment residuals on a constant and six of its lags.

Table 1: LM Test for Neglected ARCH and non-linear AS curve

Note: LM = T R 2 where T is number of observations and R 2 is the uncentered R 2 of the OLS regressions of the squared unemployment residual on a constant and six of its lags *** means the null hypothesis can be rejected at the significant level of 1%.

Following the results of tests above, we estimate a aggregate supply curve(3.1) with τ = 0 restriction and conditional variance of inflation followsGARCH(1,1) model The results are demonstrated in Table 2 In line withthese results, the reaction function (3.8) is used in the second stage

Measurement error happens when the assumption that all observations arecorrectly measured is violated There could be several reasosn for it: somevariables could not be measured; or some elements are well defined but couldnot be correctly measured and a proxy is used instead,etc In those cases, thetrue value is not possible to capture but being oberved with error

Measurement error could be a trouble sometimes, especially when the nitude effect is large In this case, the consequences might be biased and eveninconsistent estimation Over years, method of dealing with this problem

mag-is dmag-iscussed, and one of the most popular approach mag-is regression calibration(Caroll and Stefanki, 1990; Glesser, 1990) The main idea of this design is asfollows, when a regressor (X) can not be correctly captured but is measured

by another variable (X2) with errors In estimation, the regressor (X) would

be replaced by an estimate Xr which is a function of X2 to find a consistentestimator for coefficients of the model

However, even being widely used, some drawbacks of this approach havebeen recorded Thus Freedman et al (2004) introduced another method fordealing with measurement errors Similar to regression calibration, they alsosubstitute an estimated value for regressor X, “ but in which the first andsecond moments of the substituted value are consistent estimates of the firstand second moments of X ”22 Thus this design is calll “moment reconstruc-tion”

In the function (3.8), ˜ηζt is a classical measurement error and becomes apart of error term in regression23 But as we showed that the GARCH(1,1)model could well approximate the conditional variance of inflation, we canassume ζt to be an independent identical distributed innovation thus avoidmeasurement error problems, even if there is, the magnitude effect is verysmall and would not cause endogeneity bias

Multicollinearity is the situation where the explanatory regressor are highlyintercorrelated, which might result in biased estimation However, in practice,especially in economic analysis, it is difficult to propose a model with non-collinearity In most cases, the explanatory variables would be correlated, aslong as the correlation is not too high, the unbiasness of the estimation shouldnot be affected

By looking at our reaction rule (3.8), there is potential severe multicollinearproblem in inflation In particular, the first and second term (πt−1− π∗) and(πt− π∗) are highly correlated with πt and intercorrelate with each others Infact, estimating the function (3.8) would lead to inflation πt is omitted Toadjust the rule, we would choose to keep the inflation πt itself and excludethe deviation from targeting as explanatory variable Then we could rewrite(3.8) as:

There are still potential multicollinear between (πt−1− π∗), πt and πt+2, so

we also include one extra estimation model which does not inclue πt Theestimation results are report in Table 3 and Table 4, model (3)