Download Frequency, Duration, and Numerical Convergence

T HE Z ERO L OWER B OUND :
F REQUENCY, D URATION ,
AND N UMERICAL C ONVERGENCE
Alexander W. Richter
Auburn University
Nathaniel A. Throckmorton
DePauw University
I NTRODUCTION
• Popular monetary policy rule due to Taylor (1993)
r̂t = φπ̂t + εt ,
εt bounded support
• Taylor principle requires φ > 1 (active monetary policy)
◮ Necessary and sufficient for unique bounded equilibrium
• Three key assumptions
1. Fiscal policy is passive
2. Policy parameters are fixed
3. Zero lower bound (ZLB) never binds
• Leeper (1991) relaxes the first assumption and
Davig and Leeper (2007) relaxes the second assumption
• This paper relaxes the third assumption
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M AIN F INDINGS
• We adopt a textbook New Keynesian model with two
alternative stochastic processes:
1. 2-state Markov process governing monetary policy
2. Persistent discount factor or technology shocks
• Convergence is not guaranteed even if the Taylor principle
is satisfied when the ZLB does not bind.
• The boundary of the convergence region imposes a clear
tradeoff between the expected frequency and average
duration of ZLB events
◮
◮
Household can expect frequent—but brief—ZLB events or
infrequent—but prolonged—ZLB events
Parameters of the stochastic process affect convergence
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A L ITTLE B ACKGROUND
• Davig and Leeper (2007): Fisherian Economy
φ(st )πt = Et πt+1 + νt ,
ν ∼ AR(1)
pij = Pr[st = j|st−1 = i] and φ(st = j) = φj , st ∈ {1, 2}
• Integration over st
−s
−s
E[πt+1 |st = i, Ω−s
t ] = pi1 E[π1t+1 |Ωt ] + pi2 E[π2t+1 |Ωt ],
where Ω−s
t = {νt , νt−1 , . . . , st−1 , st−2 , . . .}
• Define πjt = πt (st = j, νt ). The system is
φ1 0 π1t
p11 p12 Et π1t+1
ν
=
+ t
0 φ2 π2t
p21 p22 Et π2t+1
νt
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D ETERMINACY: F ISHERIAN E CONOMY
• The existence of a unique bounded MSV solution requires
p11 (1 − φ2 ) + p22 (1 − φ1 ) + φ1 φ2 > 1 (LRTP)
• Example determinacy/convergence regions
p11 = 0.8; p22 = 0.95
p11 = 0.5; p22 = 0.95
1.4
φ2
φ2
1.4
1.2
1
1
0.8
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1.2
1
AND
1.2
φ1
1.4
0.5
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φ1
AND
1.5
N UMERICAL C ONVERGENCE
D ETERMINACY: NK E CONOMY
• Example determinacy regions
p11 = 0.5; p22 = 0.95
p11 = 0.8; p22 = 0.95
1.4
φ2
φ2
1.4
1.2
1
1.2
1
0.5
1
φ1
1.5
−1
0
1
φ1
• ZLB is similar to DL with φ1 = 0 and φ2 > 1, but with a
truncated distribution on the nominal interest rate
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N ONLINEAR F ISHERIAN E CONOMY
A second-order approximation of the Euler equation around the
deterministic steady state implies
r̂t + (r̂t − Et [π̂t+1 ] + Et [β̂t+1 ])2 =
{z
}
|
=0 (First Order)
Et [π̂t+1 ] − Et [β̂t+1 ] − (Et [(π̂t+1 − β̂t+1 )2 ] − (Et [π̂t+1 − β̂t+1 ])2 )
|
{z
}
=0 (First Order, Jensen’s Inequality)
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N ONLINEAR F ISHERIAN E CONOMY
1.5
Nonlinear
(ρ = 0.95)
1.4
φ2
1.3
Nonlinear
(ρ = 0.85)
1.2
1.1
Fixed Regime
Convergence
Region
Linear
1
0.8
0.9
1
1.1
φ1
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N UMERICAL P ROCEDURE
• We compute global nonlinear solutions to each setup using
policy function iteration on a dense grid
◮
◮
◮
Linear interpolation and Gauss-Hermite quadrature
Duration of ZLB events is stochastic
Expectational effects of hitting and leaving ZLB
• Algorithm is non-convergent whenever
◮ a policy function continually drifts from steady state
◮ the iteration step (max distance between policy function
values on successive iterations) diverges for 100+ iterations
• Algorithm is convergent whenever
◮ the iteration step is less than 10−13 for 10+ iterations
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N UMERICAL P ROCEDURE
• Our algorithm yields the same determinacy regions Davig
and Leeper analytically derive in their Fisherian economy
and New Keynesian economy
◮
When the LRTP is satisfied (not satisfied), our algorithm
converges (diverges)
• Within the class of MSV solutions, there is a link between
the convergent solution and determinate equilibrium
◮
◮
Non-MSV solutions with fundamental or non-fundamental
components may still exist
Finding locally unique MSV solutions with a ZLB is helpful
since most research is based on MSV solutions.
• Not a proof, but it provides confidence that our algorithm
accurately captures MSV solutions
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L ITERATURE
• Linearized models with a singular ZLB event
◮ Eggertsson and Woodford (2003), Christiano (2004),
Braun and Waki (2006), Eggertsson (2010, 2011),
Erceg and Linde (2010), Christiano et al. (2011),
Gertler and Karadi (2011), and many others
• Nonlinear models with recurring ZLB events
◮ Judd et al. (2011), Fernández Villaverde et al. (2012), Gust
et al. (2012), Basu and Bundick (2012), Mertens and Ravn
(2013), Aruoba and Schorfheide (2013), Gavin et al. (2014)
• Determinacy in Markov-switching models
◮ Davig and Leeper (2007), Farmer et al. (2009,2010),
Cho (2013), Barthélemy and Marx (2013)
• Determinacy in models with a ZLB constraint
◮ Benhabib et al (2001a), Alstadheim and Henderson (2006)
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K EY M ODEL F EATURES
• Representative Household
◮ Values consumption and leisure with preferences
E0
∞
X
t=0
◮
◮
βet {log(ct ) − χn1+η
/(1 + η)}
t
Cashless economy and bonds are in zero net supply
No capital accumulation
• Intermediate and final goods firms
◮ Monopolistically competitive intermediate firms produce
differentiated inputs
◮ Rotemberg (1982) quadratic costs to adjusting prices
◮ A competitive final goods firm combines the intermediate
inputs to produce the consumption good
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E XOGENOUS ZLB E VENTS
• The monetary authority follows
(
r̄(πt /π ∗ )φ
rt =
1
for
for
st = 1
st = 2
Baseline: r̄ = 1.015, π ∗ = 1.005, and φ ∈ {1.3, 1.5, 1.7}
• st follows a 2-state Markov chain with transition matrix
Pr[st = 1|st−1 = 1] Pr[st = 2|st−1 = 1]
p11 p12
=
Pr[st = 1|st−1 = 2] Pr[st = 2|st−1 = 2]
p21 p22
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C ONVERGENCE (S HADED ) R EGIONS
0.5
Non-Convergence
p22
0.4
0.3
0.2
0.1
φ = 1.7
0
0.6
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0.65
AND
φ = 1.5
0.7
0.75
0.8
p11
φ = 1.3
0.85
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0.9
AND
0.95
1
N UMERICAL C ONVERGENCE
C ONVERGENCE (S HADED ) R EGIONS
Prob. of Going to ZLB (p 1 2)
0.4
Non-Convergence
0.35
0.3
φ = 1.7
0.25
0.2
φ = 1.5
0.15
0.1
0.05
0
φ = 1.3
1
1.2
1.4
1.6
1.8
2
2.2
Avg. Duration of ZLB Event (1/p 2 1)
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E NDOGENOUS ZLB E VENTS
• The monetary authority follows
rt = max{1, r̄(πt /π ∗ )φ }
• Discount Factor (β) or Technology (a) follows
zt = z̄(zt−1 /z̄)ρz exp(εt ),
εt ∼ N(0, σε2 )
• Let zt−1 ∈ {z1 , . . . , zN }. Probability of going to or staying at
the ZLB given zt−1 is
Pr{st = 2|zt−1 = zi } = π −1/2
X
φ(εj |0, σε )
j∈J2,t (i)
J2,t (i) is the set of indices where the ZLB binds given the
state zt−1 = zi . φ are the Gauss-Hermite weights.
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Prob. of Going to/Staying at ZLB
Z ERO L OWER B OUND P ROBABILITIES
1
(ρ a , σε ) = (0.70, 0.0180)
(ρ a , σε ) = (0.75, 0.0155)
(ρ a , σε ) = (0.80, 0.0133)
0.8
0.6
0.4
0.2
0
−2
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AND
2
4
Technology
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8
C ONVERGENCE (S HADED ) R EGIONS
Non-convergence
0.03
0.025
φ = 1.7
σε
0.02
φ = 1.5
0.015
0.01
φ = 1.3
0.005
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
ρa
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E XPECTATIONAL E FFECT
π̄ = 0.5
Inflation Rate
1
0
−1
(ρa , σε , σz ) = (0.70, 0.0187, 0.0262)
(ρa , σε , σz ) = (0.90, 0.0091, 0.0209)
−2
−4
−2
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0
2
Technology
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AND
8
N UMERICAL C ONVERGENCE
Prob. of Going to/Staying at ZLB
Z ERO L OWER B OUND P ROBABILITIES
1
(ρ β , σε ) = (0.80, 0.0036)
(ρ β , σε ) = (0.82, 0.0029)
(ρ β , σε ) = (0.84, 0.0022)
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
Discount Factor
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C ONVERGENCE (S HADED ) R EGIONS
Non-convergence
0.025
σε
0.02
φ = 1.7
0.015
φ = 1.5
0.01
φ = 1.3
0.005
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
ρβ
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E XPECTATIONAL E FFECT
2.5
π̄ = 0.5
Inflation Rate
1.25
0
−1.25
(ρβ , σε , σβ ) = (0.80, 0.0035, 0.0058)
(ρβ , σε , σβ ) = (0.90, 0.0011, 0.0025)
−2.5
−1.25 −1 −0.75 −0.5 −0.25 0 0.25
Discount Factor
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0.75
1
1.25
N UMERICAL C ONVERGENCE
C ONCLUSION
• Convergence region boundary imposes tradeoff between
the frequency and duration of ZLB events
• This is important because
◮
◮
◮
The central bank can pin down prices when the interest rate
is at its ZLB
Knowing parameter restrictions is important for estimation
and policy analysis
Small changes in the parameters impact decision rules and
where ZLB first binds
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