Intro
Model
Solution and Estimation
Results
The Empirical Implications of the Interest-Rate
Lower Bound
Christopher Gust, David López-Salido, and Matthew E. Smith
Federal Reserve Board
March 1, 2013
Conclusions
Intro
Model
Solution and Estimation
Results
Conclusions
Disclaimer
The views expressed here are solely the responsibility of the authors
and should not be interpreted as re‡ecting the views of the Board
of Governors of the Federal Reserve System or of any other person
associated with the Federal Reserve System.
Intro
Model
Solution and Estimation
Results
Conclusions
Motivation
The ZLB can limit the corrective response of monetary policy
to adverse shocks. A key question is how much did the ZLB
constrain policy during the last recession.
We address this question by estimating a nonlinear version of
a New Keynesian model widely used in monetary economics.
Our methodology can be applied in other contexts in which
occasionally binding constraints have important economic
consequences (e.g., …nancial constraints).
Intro
Model
Solution and Estimation
Results
Conclusions
Preview of Main Results
Our estimates suggest that 20% of the drop in U.S. GDP
during the last recession was due to the ZLB.
The credibility of this …nding depends upon the empirical
plausibility of our model. We …nd that:
The model provides a reasonably good …t of the dynamics for
output, in‡ation, and the nominal interest rate.
The model matches up well with evidence from …nancial
markets and professional forecasters regarding the expected
duration of the ongoing ZLB spell. Matches this evidence even
though it was not used to estimate the model.
Intro
Model
Solution and Estimation
Results
Conclusions
Key Features of the Model
A nonlinear New Keynesian model featuring:
Representative household with habits for consumption.
Monopolistically-competitive …rms that face adjustment costs
in changing prices.
Monetary policy operates according to an interest rate rule
that is constrained by the lower bound.
The sources of uncertainty are a unit root shock to technology,
and AR(1) shock to the intertemporal allocation of demand,
and an iid monetary policy surprise.
Intro
Model
Solution and Estimation
Results
Conclusions
Nonlinear Equilibrium Conditions
An intertemporal Euler equation:
λt =
β
Rt Et λt +1 π t +11 ,
ηt
where λt = [ct
γct
1]
1
A price-setting relationship:
λt +1 yt +1 π t +1
π t +1
e 1
e 1
(
-1)
+
λt yt
π
π
ϕ
ϕ λt
ϕ πt 2
yt = ct + ( -1) yt
yt = Zt ht
2 π
An interest rate rule for the desired or notional interest rate:
(
πt
πt
β
-1)
= Et
π
π
ηt
Rt = (
πG 1
)
β
ρR
(Rt
1)
ρR
πt
π
γπ
Lower bound constraint:
Rt = max[1, Rt ]
γy
yt
yt
1
exp(εR ,t )
Intro
Model
Solution and Estimation
Results
Conclusions
Solution Method
We could have imposed the ZLB but log-linearized the
remaining equilibrium conditions (i.e., constrained linear). We
found this solution method inaccurate at the lower bound.
Instead, we use a projection method: collocation along with
Chebychev polynomials.
We do not approximate the decision rules with polynomials
directly. Our approximation takes a piecewise approach to
account for the kink associated with the lower bound.
We …nd a large improvement in accuracy from this approach
relative to the constrained linear approach.
Intro
Model
Solution and Estimation
Results
Conclusions
Estimation Procedure
We use the particle …lter within a Metropolis-Hastings (MH)
algorithm.
As observables, we use U.S. data on GDP growth, in‡ation,
and the nominal interest rate from 1983Q1-2011Q4.
We follow Smith (2011) and introduce a surrogate into the
MH algorithm. The surrogate is used to prescreen proposed
values to avoid expensive likelihood evaluations. For our
surrogate, we use linearization with the Kalman …lter.
Intro
Model
Solution and Estimation
Results
Conclusions
Posterior Distribution of Selected Parameters
parameter
γ
ϕ
γπ
γy
ρR
ρη
mean
0.47
94.2
0.79
0.25
0.86
0.88
stdev
0.06
14.9
0.11
0.049
0.035
0.018
prior type
Beta
Gamma
Normal
Normal
Beta
Beta
prior mean
0.50
85.0
0.0
0.0
0.50
0.50
prior stdev
0.22
15.0
1.0
1.0
0.28
0.28
Model
Solution and Estimation
Results
The Path of Estimated Shocks
Discount Rate Shock
2
Great Recession
0
1
Percent
−0.5
−1
0
−1.5
−1
Median
68% Band
−2
1980
1985
−2
2005 2010 2015
1990
1995
2000
2005
2010
Productivity Innovation
4
Great Recession
2
Percent
2
1
0
0
−1
−2
−2
−3
2005 2010 2015
−4
1980
1985
1990
1995
2000
2005
2010
Monetary Policy Innovation
1
Great Recession
0.4
0.5
Percent
Intro
0.2
0
0
−0.2
−0.5
−0.4
2005 2010 2015
−1
1980
1985
1990
1995
2000
2005
2010
Conclusions
Model
Solution and Estimation
Results
Conclusions
What pushed the economy to the ZLB?
Inflation
2
5
1
4
Annual Percent
Quarterly Percent
Output Growth
0
−1
−2
−3
2008
3
2
1
2009
2010
2011
0
2008
2012
Nominal Interest Rate
8
6
6
5
4
4
3
2
Fitted data (median)
Only Beta (εη )
Only Tech (εz )
Only Monetary (εR )
1
0
2008
2009
2010
2011
2009
2010
2011
2012
Notional Interest Rate
7
Annual Percent
Annual Percent
Intro
2012
2
0
−2
−4
−6
2008
2009
2010
2011
2012
Model
Solution and Estimation
Results
Conclusions
Contribution of the ZLB to the Great Recession
Output Level
Inflation
101
5
100
4
Annual Percent
99
98
97
3
2
1
96
0
95
94
2008
2009
2010
2011
−1
2004
2012
2006
Nominal Interest Rate
6
4
4
2
2
0
−2
−4
−6
2008
2010
2012
2010
2012
0
−6
2006
2010
−2
−4
−8
2004
2008
Notional Rate
6
Annual Percent
Annual Percent
Intro
2012
−8
2004
Data
Fitted
No ZLB Counter
95% Bands
2006
2008
Model
Solution and Estimation
Results
Conclusions
Distribution of the Probability of Hitting the ZLB
The CDF of the Prob(Rt = R)
1
Probability
0.8
0.6
50% of samples outside ZLB
0.4
Average: 3.1%
0.2
0
0
2
4
Population
Sample
6
8
10
12
14
16
18
20
Percent
Percent of Observations
Right Tail of Population Histogram of Prob(Rt = R)
20
15
97.5th Percentile
10
99.5th Percentile
5
0
3.5
4
4.5
5
5.5
Percent
Right Tail of Sample Histogram of Prob(Rt = R)
12
% of Observations
Percent of Observations
Intro
10
8
6
0.05
0
30 32 34 36 38 40 42 44 46 48 50
Percent
97.5th Percentile
4
99.5th Percentile
2
0
Far−Right Tail of the Histogram
2
4
6
8
10
12
14
16
Percent
18
20
22
24
26
28
30
Model
Solution and Estimation
Results
Conclusions
Distribution of the Duration of a ZLB Spell
Estimated CDF of Duration of ZLB Spells
1
0.9
0.8
Probability
0.7
Pr(Duration>12)=2.6%
0.6
0.5
0.4 Pr(Duration<4)=78%
0.3
0.2
0.1
0
5
10
15
20
25
Right Tail of Histogram of Duration of ZLB Spells
8
Far−Right Tail of the Histogram
0.08
Percent of Spells
7
6
Percent of Spells
Intro
5
0.06
0.04
0.02
0
20
25
4
30
Quarters
35
40
3
97.5th Percentile
2
99.5th Percentile
1
0
5
10
15
20
Quarters
25
30
Model
Solution and Estimation
Results
Expected Duration of U.S. Lower Bound Spell
Financial Market Federal Funds−Rate Expectations
6
6
68% confidence bands
5
5
4
Percent
Percent
4
median
3
3
2
2
1
1
0
2009
2010
2011
2012
2013
0
2010
2014
Expectations as of 2009:Q1
2011
2012
2013
2014
Expectations as of 2010:Q2
Blue Chip 3−month Treasury Bill Expectations
6
6
5
5
4
4
Percent
Percent
Intro
3
3
2
2
1
1
0
2009
2010
2011
2012
2013
2014
Expectations as of 2009:Q1
0
2010
2011
2012
2013
2014
Expectations as of 2010:Q2
Conclusions
Intro
Model
Solution and Estimation
Results
Conclusions
Conclusions
The ZLB constraint was a sign…cant factor in exacerbating
the last recession; accounted for 20% of the drop in output.
Duration of the ZLB spell implied by the model in line with
…nancial markets and Blue-Chip expectations: the protracted
nature of the ongoing ZLB spell was largely unexpected.
Ongoing work: Extend the model to incorporate endogenous
capital and …nancial frictions to better account for the nature
of the shocks that push the economy to the ZLB and their
e¤ects on the composition of aggregate demand.
Apply our methodology to the Great Depression and Japan.
Intro
Model
Solution and Estimation
Thank you for your attention.
Results
Conclusions
Model
Solution and Estimation
Results
Conclusions
Figure 1: Smoothed Estimates of Model Objects
Inflation
Great Recession
4
3
10
2
8
2
6
0
1
Annual Percent
Quarterly Percent
Output Growth
0
−1
2008 2010 2012
4
2
−2
0
−3
1980
1990
2000
2010
2020
1980
Nominal Interest Rate
1990
2000
2010
2020
Notional Interest Rate
12
12
Data
Median
68% Confidence Bands
10
10
8
8
Annual Percent
Annual Percent
Intro
6
4
6
4
2
0
−2
2
−4
0
1980
1990
2000
2010
2020
−6
1980
1990
2000
2010
2020
Model
Solution and Estimation
Results
Conclusions
Figure 2: The Dynamic Path of Alternative Initial
Conditions
Nominal Rate
Notional Rate
6
6
4
Percent
Percent
4
2
2
0
−2
0
2
4
6
8
10
−4
12
2
4
Quarter
6
8
10
12
10
12
Quarter
Output Growth
Inflation
4
4
Percent
Percent
2
0
−2
3
2
−4
−6
2
4
6
8
10
1
12
2
4
Quarter
Percent
Percent
0
1
0.5
−0.5
−1
0
2
4
6
8
10
12
−1.5
2
4
Quarter
6
8
10
12
Quarter
Monetary Shock
Slope of Phillips Curve
0.1
0.056
0.05
0.054
0
0.052
−0.05
−0.1
8
Discount Rate Shock
0.5
1.5
−0.5
6
Quarter
Technology Shock
2
Percent
Intro
0.05
2
4
6
8
10
12
0.048
2
Quarter
4
6
Quarter
2006:Q1 Baseline
2009:Q2 Baseline
8
10
12
Model
Solution and Estimation
Results
Conclusions
Figure 3: The E¤ects of a Discount Rate Shock
Nominal Rate
Notional Rate
2
Percentage Point (Deviation from Baseline)
Percentage Point (Deviation from Baseline)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
2
4
6
8
10
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
12
2
4
Quarter
Output
8
10
12
10
12
Inflation
Percentage Point (Deviation from Baseline)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
6
Quarter
1.4
Percent (Deviation from Baseline)
Intro
2
4
6
8
10
12
1.2
1
0.8
0.6
0.4
0.2
0
2
Quarter
4
6
Quarter
2006:Q1 Baseline
2009:Q2 Baseline
8
Model
Solution and Estimation
Results
Conclusions
Figure 4: The E¤ects of a Producitivity Shock
Nominal Rate
Notional Rate
0.2
Percentage Point (Deviation from Baseline)
Percentage Point (Deviation from Baseline)
0.12
0.1
0.08
0.06
0.04
0.02
0
2
4
6
8
10
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
12
2
4
Quarter
6
8
10
12
10
12
Quarter
Output
Inflation
0
Percentage Point (Deviation from Baseline)
1
Percent (Deviation from Baseline)
Intro
0.9
0.8
0.7
0.6
0.5
0.4
2
4
6
8
10
12
−0.05
−0.1
−0.15
−0.2
−0.25
−0.3
−0.35
2
Quarter
4
6
Quarter
2006:Q1 Baseline
2009:Q2 Baseline
8
Model
Solution and Estimation
Results
Figure 5: The Path of the Estimated Shocks
Discount Rate Shock
2
Great Recession
0
1
Percent
−0.5
−1
0
−1.5
−1
Median
68% Band
−2
1980
1985
−2
2005 2010 2015
1990
1995
2000
2005
2010
Productivity Innovation
4
Great Recession
2
Percent
2
1
0
0
−1
−2
−2
−3
2005 2010 2015
−4
1980
1985
1990
1995
2000
2005
2010
Monetary Policy Innovation
1
Great Recession
0.4
0.5
Percent
Intro
0.2
0
0
−0.2
−0.5
−0.4
2005 2010 2015
−1
1980
1985
1990
1995
2000
2005
2010
Conclusions
Intro
Model
Solution and Estimation
Results
Conclusions
Figure 11: The Dynamic of Short and Long Lower Bound
Spells
Output
Inflation
115
3
2.5
Percent
Percent
110
105
100
1
5
10
15
0.5
20
10
15
Quarter
Notional Interest Rate
5
6
4
4
3
2
1
0
5
Quarter
Nominal Interest Rate
Percent
Percent
95
2
1.5
20
2
0
−2
5
10
15
−4
20
5
10
15
Quarter
Quarter
Discount Factor Shock
Technology Shock
0
20
0.5
0
Percent
Percent
−0.5
−0.5
−1
−1
−1.5
−1.5
5
10
15
20
−2
5
Quarter
10
Quarter
Long Duration Spells
Short Duration Spells
15
20
Model
Solution and Estimation
Results
Conclusions
Figure B.1: Comparison of Model Solutions (2009:Q2
Initial Conditions)
Nominal Rate
Notional Rate
5
5
0
3
Percent
Percent
4
2
0
−5
−10
1
5
10
15
20
−15
Inflation
3
2
2
0
1
−2
−4
15
20
15
20
0
−1
5
10
15
20
−2
5
Quarter
10
Quarter
Consumption Euler Error
−3
0.025
5
0.02
4
0.015
3
0.01
2
0.005
0
10
Quarter
Output Growth
4
−6
5
Quarter
Percent
Percent
Intro
x 10
Sticky Price Euler Error
1
5
10
15
20
0
5
Quarter
10
Quarter
Projection Method
Constrained−Linear Method
15
20