The Natural Rate of Interest in a Nonlinear DSGE Model · 2017-07-29 · INTRODUCTION THE MODEL ESTIMATION STRATEGY RESULTS CONCLUSION Objective Estimate the natural rate of interest

INTRODUCTION THE MODEL ESTIMATION STRATEGY RESULTS CONCLUSION

The Natural Rate of Interestin a Nonlinear DSGE Model

Yasuo Hirose Takeki SunakawaKeio University Kobe University

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Outline

1 INTRODUCTION

2 THE MODEL

3 ESTIMATION STRATEGY

4 RESULTS

5 CONCLUSION

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Background

Since Wicksell (1898), the natural rate of interest has been one ofthe key concepts in monetary policy analyses.

Characterized as the real interest rate that yields price stability.

Modern New Keynesian (NK) framework makes the concept of thenatural rate relevant for welfare (Woodford, 2003; Galí, 2008).

Provides a benchmark for measuring the stance of monetary policy.

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Background (cont’d)

The natural rate is unobservable and must be estimated.

Short-run estimates based on DSGE models

Andres, Lopez-Salido, and Nelson (2009); Barsky, Justiniano, andMelosi (2014); Curdia (2015), Curdia, Ferrero, Ng, and Tambalotti(2015); Del Negro, Giannone, Giannoni, and Tambalotti (2017);Justiniano and Primiceri (2010); Edge, Kiley, and Laforte (2008);Neiss and Nelson (2003)

Employ linearized DSGE models that abstract from the ZLB.

Exception: Iiboshi, Shintani, and Ueda (2017)

Long-run estimates based on semi-structural/reduced-form models

Holston, Laubach, and Williams (2016); Johannsen and Mertens(2016); Kiley (2015); Laubach and Williams (2003, 2016); Lubik andMatthes (2015); Pescatori and Turunen (2015); Williams (2015)

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Objective

Estimate the natural rate of interest in the US using a nonlinearNew Keynesian model with the ZLB.

Examine how and to what extent nonlinearity affects the estimatesof the natural rate and its driving forces.

1 Parameter estimates based on a linearized model can be differentfrom those based on its original nonlinear model.

Fernández-Villaverde and Rubio-Ramírez (2005);Fernández-Villaverde, Rubio-Ramírez, and Santos (2006)

2 The recent experience of the low interest rates has led researchersto conduct an empirical analysis that takes account of the ZLB.

Gust, Herbst, López-Salido, and Smith (2017); Richter andThrockmorton (2016a, 2016b); Iiboshi, Shintani, and Ueda (2017);Aruoba, Cuba-Borda, and Schorfheide (2017)

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Estimation Strategy

Two-step approach as in Aruoba, Cuba-Borda, and Schorfheide(2017)

1 Estimate a linearized model using data prior to the date when thenominal interest rate is bounded at zero.

2 Given estimated parameters, we solve a fully nonlinear model withthe ZLB and apply a nonlinear filter to extract the sequence of thenatural rate for a full sample.

For comparison, the natural rate based on its linear counterpart isalso computed.

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Findings

The estimated natural rate based on the nonlinear model is higherthan that based on the linearized model in the periods when thenominal interest rate is close to or bounded at zero.

This difference is mainly ascribed to a contractionary effect of theZLB which is considered only in the nonlinear model.

Other nonlinearities, such as price and wage dispersion, have alimited effect on the estimate of the natural rate.

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Outline

1 INTRODUCTION

2 THE MODEL


4 RESULTS

5 CONCLUSION

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Overview

A standard NK model with sticky wages1 Households (labor packer, labor unions)2 Final-good firm3 Intermediate-good firms4 Central Bank

Calvo (1983)-type stickiness in prices and wages

Stochastic trend in TFP

The natural rate of interest is defined as the real interest rate thatwould prevail if prices and wages were fully flexible with nocost-push shocks.

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HouseholdsEach household h ∈ [0, 1] maximizes the utility function

E0

∞∑t=0

D0,t

{log (Ch,t − γCt−1)−

l1+ηh,t

1 + η

},

subject to the budget constraint

PtCh,t +Bh,t ≤Wnh,tlh,t +Rnt−1Bh,t−1 + Th,t,

where Dt,t+j =∏t+jk=t+1 β/dk is the cumulative discount factor.

Discount factor shock:

log dt = ρd log dt−1 + εd,t,

where εd,t ∼ N(0, σ2d).

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Labor Packer

A labor packer resells the labor package lf,t to intermediate-goodfirm f ∈ [0, 1] by collecting each differentiated labor {lf,h,t} fromhouseholds so as to maximize profit

Wnt lf,t −

∫ 1

0Wnh,tlf,h,tdh,

subject to a CES aggregator

lf,t ≡(∫ 1

0lθw−1θw

f,h,t dh

) θwθw−1

.

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Labor UnionsLabor unions set nominal wages on a staggered basis as in Erceg,Henderson and Levin (2000).

In each period, a fraction 1− ξw ∈ (0, 1) of labor unions reoptimizesnominal wages, while the remaining fraction ξw indexes nominalwages to the economy’s trend growth γa and a weighted average ofpast inflation Πt−1 and steady-state inflation Π.

The unions that reoptimize their nominal wages then maximizeexpected utility

Et

∞∑j=0

ξjwDt,t+j

(γjaWn

h,t

Pt+j

j∏k=1

(Πιwt+k−1Π

1−ιw)Λh,t+jlh,t+j −l1+ηh,t+j

1 + η

),

where lh,t ≡∫ 10 lf,h,tdf , subject to the labor demand

lf,h,t+j =

[γjaWn

h,t

Wnt+j

j∏k=1

(Πιwt+k−1Π

1−ιw)]−θw lf,t+j .12 / 39


Final-Good Firm

The representative final-good firm produces output Yt by choosingintermediate inputs {Yf,t} so as to maximize profit

PtYt −∫ 1

0Pf,tYf,tdf,

subject to a CES production technology

Yt =

(∫ 1

0Y

θp−1

θp

f,t df

) θpθp−1

.

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Intermediate-Good Firms

Each intermediate-good firm f produces differentiated good Yf,tby cost-minimizing labor input lf,t (purchased from labor packer f )subject to the production function

Yf,t = Atlf,t.

At is represents total factor productivity which follows

logAt = log γa + logAt−1 + at,

where at = ρaat−1 + εa,t and εa,t ∼ N(0, σ2a).

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Intermediate-Good Firms (cont’d)Intermediate-good firms set prices as in Calvo (1983).

In each period, a fraction 1− ξp ∈ (0, 1) of firms reoptimizes theirprices while the remaining fraction ξp indexes prices to a weightedaverage of past inflation Πt−1 and steady-state inflation Π.

The firms that reoptimize their prices then maximize

Et∞∑j=0

ξjpDt,t+jΛt+jΛt

[Pf,tPt+j

j∏k=1

(Πιpt+k−1Π

1−ιp)− Wt+j

At+jzt+j

]Yf,t+j ,

subject to the final-good firm’s demand

Yf,t+j =

[Pf,tPt+j

j∏k=1

(Πιpt+k−1Π

1−ιp)]−θp Yt+j .Cost-push shock: log zt = ρz log zt−1 + εz,t, εz,t ∼ N(0, σ2z).

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Market Clearing Conditions

The final-good market clearing condition is

Yt = Ct.

The labor market clearing condition is

lt =∆p,t∆w,tYt

At,

where lt =∫ 10

∫ 10 lf,h,tdfdh is aggregate labor input.

∆p,t =∫ 1

0(Pf,t/Pt)

−θp df > 1: price dispersion

∆w,t =∫ 1

0(Wh,t/Wt)

−θw dh > 1: wage dispersion

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Natural Rate of Interest

The natural output and the natural rate are defined as the onesthat would prevail both wages and prices were perfectly flexiblewith no cost-push shocks.

Given by assuming ξw = ξp = 0 and zt = 1 for all t:

(Y ∗t − γY ∗t−1

)(Y ∗tAt

)η= µAt,

R∗t = β−1dt

(Et{Y ∗t − γY ∗t−1Y ∗t+1 − γY ∗t

})−1.

Y ∗t is determined by the sequence of At.

R∗t is determined by the sequences of Y ∗

t and dt.

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Central Bank

Monetary policy rule is given by

Rnt = max[Rnt , 1

],

Rnt =(Rnt−1

)φr [RΠ

(Πt

Π

)φπ ( YtY ∗t

)φy]1−φrexp(εr,t),

where εr,t ∼ N(0, σ2r ) is a monetary policy shock.

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Outline

1 INTRODUCTION

2 THE MODEL


4 RESULTS

5 CONCLUSION

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Estimation Strategy

Two-step approach as in Aruoba, Cuba-Borda, and Schorfheide(2017)

1 Estimate a linearized model using data prior to the date when thenominal interest rate is bounded at zero.

2 Given estimated parameters, we solve a fully nonlinear model withthe ZLB and apply a nonlinear filter to extract the sequence of thenatural rate for a full sample.

For comparison, the natural rate based on its linear counterpart isalso computed.

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Log-Linearized Equilibrium Conditions

yt =γa

γa + γ(Etyt+1 + at+1) +

γ

γa + γ(yt−1 − at)

− γa − γγa + γ

(Rt − EtΠt+1 − dt

),

Πt =β

1 + βιpEtΠt+1 +

ιp1 + βιp

Πt−1 + κp (wt + zt) ,

Πw,t = β(EtΠw,t+1 + Etat+1 − ιwΠt

)+ ιwΠt−1 − at

+ κw

[(η +

γaγa − γ

)yt −

γ

γa − γyt−1 +

γ

γa − γat − wt

],

wt = Πw,t − Πt + wt−1,

y∗t =γ

γa(1 + η)− γη(y∗t−1 − at),

Rnt = φrRnt−1 + (1− φr)

[φπΠt + φy (yt − y∗t )

]+ εr,t,

where κp ≡ (1−ξp)(1−ξpβ)ξp(1+βιp)

and κw ≡ (1−ξw)(1−βξw)ξw(1+ηθw)

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Parameter EstimationData: The growth rate of per capita real GDP (∆ logGDPt); theinflation rate of GDP deflator (∆ logPGDPt); the federal fundsrate (FFt); the log of hours worked (logHt).

Sample: 1983:I–2007:IV, up to the date when the nominal interestrate is bounded at zero.

Observation equations:100∆ logGDPt

100∆ logPGDPtFFt

100 logHt

=

aπ

r + πl

+

yt − yt−1 + at

Πt

Rtlt

,Estimated by Bayesian methods.

In the subsequent analysis, parameters are fixed at the posteriormode.

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Nonlinear Solution

The policy functions:S = h(S−1, τ),

where S−1 = (y−1,Π−1, w−1, Rn−1,∆p,−1,∆w,−1, y

∗−1);

τ = (d, a, z, εr).

Computed using a projection method with a Smolyak algorithm.

Adopt more efficient algorithm developed by Judd, Maliar, Maliar,and Valero (2014).

Very accurate, albeit with the reduced number of grid points.

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Nonlinear Solution (cont’d)

According to samples simulated from our nonlinear solution, theeconomy is at the ZLB for 11.8% of quarters, and the averageduration of ZLB spells is 4.3 quarters.

Fernández-Villaverde, Gordon, Guerrón-Quintana, andRubio-Ramírez (2015): ZLB prob. = 5.5%; ZLB duration = 2.1quarters

Gust, Herbst, López-Salido, and Smith (2017): ZLB prob. = 4%;ZLB duration = 3.5 quarters

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Nonlinear Filtering

Apply a generic particle filter as described in Fernández-Villaverdeand Rubio-Ramírez (2007) and Herbst and Schorfheide (2015).

The data used for filtering is the same as those used for theparameter estimation, but the sample period is extended to2016:III.

Measurement errors of output growth, inflation, the nominalinterest rate, and hours worked are respectively set to be 20%,20%, 10%, and 5% of their standard deviations in the data.

The number of particles: N = 20000.

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Data vs. Smoothed Estimates

1990 2000 2010

-2

-1

0

1

2

Output growth

DataSmoothed

1990 2000 2010-0.5

0

0.5

1

1.5Inflation

1990 2000 2010-10

-5

0

5

Hours

1990 2000 20100

1

2

3Nominal interest rate


Outline

1 INTRODUCTION

2 THE MODEL


4 RESULTS

5 CONCLUSION

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Natural rate of interest

1985 1990 1995 2000 2005 2010 2015-25

-20

-15

-10

-5

0

5

10

15

20Natural rate of interest

Nonlinear modelLinear model


Identification of Natural Interest Rate

The natural output and the natural rate are determined by:

(Y ∗t − γY ∗t−1

)(Y ∗tAt

)η= µAt,

R∗t = β−1dt

(Et{Y ∗t − γY ∗t−1Y ∗t+1 − γY ∗t

})−1.

Y ∗t is determined by the sequence of At.

R∗t is determined by the sequences of Y ∗

t and dt.

Therefore, the natural rate is pinned down by identifying at and dt.

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Identification of Productivity Shock at

In the linear model, at is identified by the data on output and hoursworked.

Labor market clearing condition: yt = lt

Observation equations: 100∆ logGDPt = a+ yt − yt−1 + at;logHt = l + lt

In the nonlinear model, the labor market clearing condition is

lt = ∆p,t∆w,tyt,

where ∆p,t ≥ 1 and ∆w,t ≥ 1.

yt becomes smaller. ⇒ at is identified to be larger.

Higher productivity results in the higher natural rate.

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Identification of Discount Factor Shock dt

In the nonlinear model, the ZLB has a contractionary effect on theeconomy

not only when the nominal interest is already binding at zero

but also when there is uncertainty about whether the ZLB will bindor not in the future.

Hills, Nakata, and Schmidt (2016) quantify such an uncertainty effecton inflation.

While such an effect lowers expected output and inflation, actualoutput and inflation are pegged to the data in the filtering process.

dt must increase to satisfy the optimality conditions of householdsand firms.

As a result, the natural rate rises.

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Estimated Shocks

1985 1990 1995 2000 2005 2010 2015-6

-4

-2

0

2

4Discount factor shock


1985 1990 1995 2000 2005 2010 2015-1

-0.5

0

0.5

1Productivity shock

Expected Inflation and Output

1985 1990 1995 2000 2005 2010 2015-1.5

-1

-0.5

0

0.5

1Expected inflation


1985 1990 1995 2000 2005 2010 2015

-5

0

5

Expected output

Price and Wage Dispersions

1985 1990 1995 2000 2005 2010 20150

0.01

0.02

0.03

0.04Price Dispersion

1985 1990 1995 2000 2005 2010 20150

0.02

0.04

0.06

0.08

0.1Wage Dispersion


Quasi-Linear Model

Consider a quasi-linear model, in which the ZLB constraint isimposed but all the equilibrium conditions are linearized.

Monetary policy rule is replaced with

Rnt = max[

˜Rn

t ,− log RΠ],

˜Rn

t = φr˜Rn

t−1 + (1− φr)(φπΠt + φy (yt − y∗t )

)+ εr,t.

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Natural Rate of Interest: Nonlinear vs. Quasi-Linear

1985 1990 1995 2000 2005 2010 2015-15

-10

-5

0

5

10

15

20Natural rate of interest

Nonlinear modelQuasi-linear model

Estimated Shocks: Nonlinear vs. Quasi-Linear

1985 1990 1995 2000 2005 2010 2015-5

0

5Discount factor shock

Nonlinear modelQuasi-linear model

1985 1990 1995 2000 2005 2010 2015-1

-0.5

0

0.5

1Productivity shock


Outline

1 INTRODUCTION

2 THE MODEL


4 RESULTS

5 CONCLUSION

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Conclusion

Estimated the natural rate of interest in a nonlinear NK model.

The estimated natural rate is very different from the one estimatedwith its linear counterpart.

The difference is mainly explained by a contractionary effect of theZLB.

Other nonlinearities, such as price and wage dispersions, have alimited effect on the estimate of the natural rate.

Our analysis could be extended to exploit a medium-scale model.

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Prior and Posterior Distributions of ParametersPrior Posterior

Parameter Distribution Mean S.D. Mode S.D.γ Beta 0.500 0.050 0.684 0.032η Gamma 2.000 0.250 1.782 0.252ξw Beta 0.500 0.050 0.618 0.069ιw Beta 0.500 0.050 0.526 0.053ξp Beta 0.500 0.050 0.737 0.036ιp Beta 0.500 0.050 0.514 0.052φπ Gamma 1.500 0.250 2.075 0.201φy Gamma 0.125 0.050 0.096 0.025φr Beta 0.500 0.050 0.784 0.020a Normal 0.540 0.100 0.484 0.062π Normal 0.624 0.100 0.634 0.049r Gamma 0.699 0.100 0.675 0.057l Normal 0.000 0.100 -0.011 0.095ρd Beta 0.500 0.050 0.691 0.034ρa Beta 0.500 0.050 0.402 0.041ρz Beta 0.500 0.050 0.664 0.056100σd Inv. Gamma 0.500 2.000 0.635 0.103100σa Inv. Gamma 0.500 2.000 0.520 0.040100σz Inv. Gamma 0.500 2.000 1.901 0.460100σr Inv. Gamma 0.500 2.000 0.145 0.012

Documents

The Natural Rate of Interest in a Nonlinear DSGE Model · 2017-07-29 · INTRODUCTION THE MODEL ESTIMATION STRATEGY RESULTS CONCLUSION Objective Estimate the natural rate of interest