Putting the cycle back into business cycle analysis

Putting the Cycle Back into Business Cycle
Analysis
Paul Beaudry, Dana Galizia & Franck Portier
Vancouver School of Economics, Carleton University & Toulouse School of
Economics
New Developments in Macroeconomics
ADEMU conference
UCL, London
November 2016
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0. Introduction
Modern Approach to Business Cycles
Equilibrium stochastic dynamic modeling in macro flowered in
the 1970’s
At that point, postwar business cycles were at most 8 years
long business cycles were defined as fluctuations at
periodicities of 8 years or less.
Spectral densities of main macro variables were showing the
“Typical Spectral Shape of an Economic Variable”
(Granger 1969) look like a persistent AR(1) spectrum
This suggested that business cycle theory should not be about
cycles, it should be about co-movements (see Sargent’s
textbook)
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0. Introduction
Figure 1: Sargent’s textbook
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0. Introduction
Figure 2: Sargent’s textbook
4 / 91

0. Introduction
Equilibrium stochastic dynamic modeling in macro flowered in
the 1970’s
At that point, postwar business cycles were at most 8 years
long business cycles were defined as fluctuations at
periodicities of 8 years or less.
Spectral densities of main macro variables were showing the
“Typical Spectral Shape of an Economic Variable”
(Granger 1969) look like a persistent AR(1) spectrum
This suggested that business cycle theory should not be about
cycles, it should be about co-movements (see Sargent’s
textbook)
All good!
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0. Introduction
What we do
We question this focus in this work
We do three things
1. Show that the economy is “more cyclical than you think”, with
40 more years of data.
2. Explain how “weak complementaries”, combined with
accumulation, favor cyclical behavior (recurrent booms and
busts) rather than indeterminacy
3. Estimate a New-Keynesian model extended to include the for
complementarity forces we are highlighting, and see if the data
prefers a “recurring cycle approach”
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Roadmap
1. Motivating Facts
2. General Mechanism
3. Empirical Exercice
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Roadmap
1. Motivating Facts
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1. Motivating Facts
Looking for Peaks
If there are important cyclical forces in the economy ...
... this should show up as a distinct peak in the spectrum of
the data
It is well known that (detrended) output data does not display
such peaks (Granger (1969), Sargent (1987))
However, output data may not be best placed to look, as
there is a marked non stationarity that one needs to get rid of.
May be better to look at measures of factor use: ex: hours
worked, employment rates, unemployment rates, capital
utilization.
Things might also have changed since the 70’s as we have
now 40 more years of observation
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1. Motivating Facts
Figure 3: Non Farm Business Hours per Capita
1950 1960 1970 1980 1990 2000 2010
Date
-8.15
-8.1
-8.05
-8
-7.95
-7.9
-7.85Logs
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1. Motivating Facts
Figure 4: Non Farm Business Hours per Capita Spectrum
4 6 24 32 40 50 60 80
Periodicity
0
5
10
15
20
25
30
35
Level
Various High-Pass
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1. Motivating Facts
Figure 5: Non Farm Business Hours per Capita, Highpass (50) Filtered
1950 1960 1970 1980 1990 2000 2010
-8
-6
-4
-2
0
2
4
6%
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1. Motivating Facts
Figure 6: Total Hours per Capita Spectrum
4 6 24 32 40 50 60 80
Periodicity
0
5
10
15
20
25
30
Level
Various High-Pass
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1. Motivating Facts
Figure 7: Unemployment Rate Spectrum
4 6 24 32 40 50 60 80
Periodicity
0
1
2
3
4
5
6
Level
Various High-Pass
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1. Motivating Facts
Figure 8: Capacity Utilization Spectrum
4 6 24 32 40 50 60 80
Periodicity
0
5
10
15
20
25
30
35
40
Level
Various High-Pass
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1. Motivating Facts
Figure 9: Hours Spectrum in Smets & Wouters’ Model
4 6 24 32 40 50 60 80
Periodicity
0
2
4
6
8
10
12
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1. Motivating Facts
Figure 10: Output per Capita Spectrum
4 6 24 32 40 50 60
Periodicity
0
50
100
150
200
250
300
350
400
Level
Various High-Pass
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1. Motivating Facts
Figure 11: Output per Capita Spectrum
4 6 24 32 40 50 60
Periodicity
0
5
10
15
20
25
30
35
Various High-Pass
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1. Motivating Facts
Figure 12: TFP Spectrum
4 6 24 32 40 50 60
Periodicity
0
5
10
15
Various High-Pass
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1. Motivating Facts
Figure 13: Non Farm Business Hours per Capita Spectrum
4 6 24 32 40 50 60 80
Periodicity
0
5
10
15
20
25
30
35
Level
Various High-Pass
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Roadmap
1. Motivating Facts
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Roadmap
1. Motivating Facts
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Overview
Show a reduced form that is “dynamic Cooper & John
(1988)”
Understand the logic of the existence of strong cyclical forces
(“limit cycle”) in our framework : weak strategic
complementarities + dynamics
Show a forward looking version with saddle path stable limit
cycles
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Basic environment
N players (“ﬁrms”)
Each ﬁrm accumulates a capital good in quantity Xi by
investing Ii
Decision rule and law of motion for X are
Xit+1 = (1 − δ)Xit + Iit (1)
Iit = α0 − α1Xit + α2Iit−1 (2)
all parameters are positive, δ < 1, α1 < 1, α2 < 1.
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Symmetric allocations
When all agent behave symmetrically:
It
Xt
=
α2 − α1 −α1(1 − δ)
1 1 − δ
ML
It−1
Xt−1
+
α0
0
Both eigenvalues of the matrix ML lie within the unit circle.
Therefore, the system is stable.
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Symmetric allocations
When all agent behave symmetrically:
It
Xt
=
α2 − α1 −α1(1 − δ)
1 1 − δ
ML
It−1
Xt−1
+
α0
0
Proposition 1
Both eigenvalues of the matrix ML lie within the unit circle.
Therefore, the system is stable.
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Figure 14: “Phase diagram” in the model without demand
complementarities
Xt
It
∆It = 0
∆Xt = 0
Xs
Is
Here I have assumed that the two eigenvalues are real and positive.
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Introducing strategic interactions
“Dynamic Cooper & John (1988)”
Consider
Iit = α0 − α1Xit + α2Iit−1 + F
Ijt
N
F > 0 : strategic complementarities
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Figure 15: “Best response” rule for a given history - Multiple Equilibria
Ijt
N
Iit Iit=
Ijt
N
α0 −α1Xit +
α2Iit−1
Ijt
N
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Figure 16: “Best response” rule for a given history
Ijt
N
Iit Iit=
Ijt
N
α0 −α1Xit +
α2Iit−1
Ijt
N
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Restrictions on F
Ijt
N
We choose to be in a “non-pathological” case
A steady state exists
F (·) < 1 uniqueness of the period t equilibrium
uniqueness of the steady state
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Ijt
N
Iit Iit=
Ijt
N
αt
Iit = αt + F
Ijt
N
Ijt
N
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The two sources of ﬂuctuations
We restrict to symmetric allocations
Two forces of determination of allocations:
× static strategic interactions “multipliers” (Cooper &
John (1988)) local instability
× History (accumulated I that shows up in X) aﬀects allocations
through the intercept of the “best response” function
global stability
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Ijt
N
Iit Iit=
Ijt
N
αs
αt
Iit = αt + F
Ijt
N
αt = α0 − α1
∞
0 (1 − δ)τ+1Ijt−1−τ + α2Iit−1
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Intuition for the limit cycle
Ijt
N
F
Ijt
N (Strategic complementarities): centrifugal force that
pushes away from the steady state when close to.
−α1Xit (Accumulation): centripetal force that pushes towards
the steady state when away from.
α2Iit−1 (Sluggishness) : avoid jumping back an forth around
the steady state (“ﬂip”)
The steady state locally unstable, but forces push the economy
smoothly back to the steady state when it is further from it.
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Iit = α0−α1Xit + α2Iit−1 + F
Ijt
N
F
Ijt
α2Iit−1 (Sluggishness) : avoid jumping back an forth around
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Ijt
N
F
Ijt
α2Iit−1 (Sluggishness) : avoid jumping back and forth around
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Stable limit cycle
Xs
Is
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Xt
It
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Looking for the limit cycle
How do we prove the existence of a limit cyle?
Limit cycles are typically ocurring when there are bifurcations
in non-linear dynamical systems
Simply stated : the SS looses stability when a parameter (here
the degree of strategic complementarities) increases.
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Dynamics with strategic interactions
Math
Dynamics is
It
Xt
=
α2 − α1 −α1(1 − δ)
1 1 − δ
ML
It−1
Xt−1
+
α0+F(It)
0
Local dynamics is
It
Xt
=
α2−α1
1−F (Is ) − α1(1−δ)
1−F (Is )
1 1 − δ
M
It−1
Xt−1
+
1 − α2−α1
1−F (Is ) Is + α1(1−δ)
1−F (Is ) Xs
0
When F (Is) varies from −∞ to 1, eigenvalues of M vary
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Strategic substitutabilities – F negative
Proposition 2
As F (IS ) varies from 0 to −∞, the eigenvalues of M always stay
within the unit circle and therefore the system remains locally
stable.
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Strategic complementarities – F positive
Proposition 3
As F (Is) varies from 0 towards 1, the dynamic system will become
locally unstable.
(bifurcation)
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Bifurcations
3 types of bifurcation
× Fold bifurcation: appearance of an eigenvalue equal to 1,
× Flip bifurcation: appearance of an eigenvalue equal to -1
× Hopf bifurcation: appearance of two complex conjugate
eigenvalues of modulus 1.
We are interested in Hopf bifurcation because the limit cycle
will be “persistent”
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Bifurcations
Proposition 4
As F (Is) varies from 0 towards 1, the dynamic system will become
unstable and:
No ﬂip bifurcations
If α2 > α1/(2 − δ)2, then a Hopf (Neimark-Sacker)
bifurcation will occur.
If α2 < α1/(2 − δ)2, then a ﬂip bifurcation will occur.
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Bifurcations
In the case of ﬂip and Hopf bifurcation, a limit cycle appears
In case of Hopf, the cycle is “smooth”
Condition for an Hopf bifurcation:
Ijt
N
× as δ approaches 0, α2 > 1/4
× in general, what is needed is δ not too large and α2 large
enough
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Stability of the limit cycle
In the case of the Hopf bifurcation, the limit cycle can be
attractive (the bifurcation is supercritical) or repulsive (the
bifurcation is subcritical)
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Stability of the limit cycle
Proposition 5
If F (Is) is suﬃciency negative, then the Hopf bifurcation will be
supercritical. Therefore, the limit cycle is attractive.
F < 0 corresponds to an S − shaped reaction function
Ijt
N
Iit Iit=
Ijt
N
αt
Iit = αt + F
Ijt
N
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Intuition for genericity
Figure 18: Stable Steady State
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Intuition for genericity
Figure 19: Hopf Supercritical bifurcation: Attractive Limit Cycle
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Numerical example
Let me show an arbitrary numerical example with that
reduced form model
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Numerical example
Figure 20: A particular F function
I
F(I)
a0
slope β0
slope β1
slope β2 F(I)
γ1I1
γ2I2
I1 I2Is
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Numerical example
Figure 21: Deterministic simulation
I, X, linear model I, X, nonlinear model
period
50 100 150 200
0
2
4
6
8
10
12
14
X
I
period
0 50 100 150 200 250
0
2
4
6
8
10
12
14
X
I
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Numerical example
Figure 22: Deterministic simulation of the nonlinear model
Spectral density
Period
4 8 12 20 32 40 60 80
0
0.2
0.4
0.6
0.8
1
1.2
1.4
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Numerical example
Figure 23: One stochastic simulation
Linear model Nonlinear model
period
50 100 150 200
I
0
0.5
1
1.5
2
period
0 50 100 150 200 250
I
0
0.5
1
1.5
2
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Numerical example
Figure 24: Deterministic and Stochastic simulation of the nonlinear model
Spectral density of I
4 8 12 20 32 40 60 80
Period
0
0.2
0.4
0.6
0.8
1
1.2
1.4
I
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What the results are not
Figure 25: xt = sin(ωt)
0 50 100 150 200
t
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
xt
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What the results are not
Figure 26: xt = sin(ωt) + ut
0 50 100 150 200
t
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
xt
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What the results are
Figure 27: Reduced Form Model
0 20 40 60 80 100 120
period
0
0.5
1
1.5
2
I
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What the results are
Figure 28: Reduced Form Model
0 20 40 60 80 100 120
period
0
0.5
1
1.5
2
I
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Adding Forward looking elements
Iit = α0 − α1Xit + α2Iit−1 + α3Et[Iit+1] + F
Ijt
N
with accumulation remaining the same
Xit = (1 − δ)Xit + Iit
Restrict attention to situations where this system is saddle
path stable absent of complementarities.
Generally true when all three are between 0 and 1.
In this case, more diﬃcult to get analytical results.
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Set of potential bifurcation with Forward looking elements
Initial situation has two stable roots and one unstable
Three types of bifurcations are possible:
1. The unstable root enters the unit circle: local indeterminacy
arises
2. One stable root leaves the unit circle: instability arises with a
ﬂip or fold type bifurcation
3. Two stable roots leave the unit circle simultaneous because
they are complex: this is a Hopf bifurcation
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Figure 29: Eigenvalues of the Reduced Form Model
-1 -0.5 0 0.5 1 1.5 2
Re(λ)
-1
-0.5
0
0.5
1
Im(λ)
α1 = 0.5, α2 = 0.45, α3 = -0.1, δ = 0.5
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-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1
Re(λ)
-1.5
-1
-0.5
0
0.5
1
1.5
Im(λ)
α1 = -0.3, α2 = -0.2, α3 = -0.5, δ = 0.05
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-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
Re(λ)
-1.5
-1
-0.5
0
0.5
1
1.5
Im(λ)
α1 = 0.3, α2 = 0.6, α3 = -0.3, δ = 0.05
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Figure 32: A Saddle Limit Cycle
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Set of results
Proposition 6
If unique steady state, then no indeterminacy nor Fold bifurcations
Proposition 7
If α2 (sluggishness) suﬃciently large, then no Flip Bifurcations.
Hence, under quite simple conditions only relevant bifurcation
is a determinate Hopf.
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Figure 33: Eigenvalues Conﬁguration At First Bifurcation (δ = .05)
red = indeterminacy, yellow = unstable, gray = saddle
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Roadmap
1. Reduced Form Model
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Roadmap
1. Reduced Form Model
62 / 91

Exploring empirically whether US Business Cycles may reﬂect a
Stochastic Limit Cycles.
Stylized NK model which is extended to allow for the forces
highlighted in our general structure.
We add accumulation of durable-housing goods and habit
persistence : accumulation and sluggishness
Financial frictions under the form of a counter-cyclical risk
premium : complementarities
Estimate parameters based on spectrum observations and
higher moments. (use perturbation method and indirect
inference)
See whether model favors parameters the generate limit cycle.
If so, explore nature of intrinsic limit cycle and perform some
counter factual exercises.
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Basic Elements of the Model
1. Household buy consumption services to maximise utility
taking prices as given
2. Firms supply consumption services to the market where the
services can come from existing durable goods or new
production.
3. These ﬁrms have sticky prices.
4. Central Bank set policy rate according to a type of Taylor rule
5. Interest rate faced by households is the policy rate plus a risk
premium, where the risk premium varies with the cycle.
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Extending a 3 equation NK model
The representative household who can buy/rent consumption
services from the market.
The households Euler will have the familiar form (assuming
external habit)
U (Ct − γCt−1) = βtEtU(Ct+1 − γCt)(1 + rt)
Allow that interest rate faced by household may include a risk
premium
(1 + rt) = (1 + it + rp
t )
where rp
t can respond to activity, or unemployment
Have Taylor rule of form
it = Φ1Etπt+1 + Φ2Et t+1
t is (log) employment or output gap,
U(z) = − exp − z
Ω , Ω > 0 65 / 91

Allowing for durable goods and accumulation
Firms produce an intermediate factor
Mjt = BF (ΘtLjt) ,
Mt is used to produce consumptions services or durable
goods, that are sold to the households:
Ct = sXt + (1 − ϕ)Mt
where Xt is the stock of durable goods,
Xt+1 = (1 − δ)Xt + ϕMt
Households own the stock of durable-housing, rent it to ﬁrms,
who supply back the consumption services as a composite
good.
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Risk premium
The interest rate which household face is assume to be equal
to the policy rate plus a risk premium
(1 + rt) = (1 + it + rp
t )
where rp
t is a premium over the back rate.
The risk premium is assumed to be an decreasing function of
the economic activity, where the output gap and employment
gap are interchangeable
rp
t = g(Lt)
Here we are allowing the interest premium to be a non-linear
function of activity.
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Taylor rule and dichotomy with inflation
Modified Taylor rule:
it = φ0 + φ1Etπt+1 + φ2Et t+1
φ1 = 1 the Central Bank as setting the expected real
interest.
This approach has the attractive feature of making the model
bloc recursive, where the inflation rate last is solved as a
function of the marginal cost.
With this approach, we can explore the properties of the real
variables without need to to be specific about the source and
duration of price stickiness (which is not our focus)
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Reduced Form
Solution is
t = µt − α1Xt + α2 t−1 + α3Et [ t+1] + F( t)
together with accumulation
Xt+1 = (1 − δ) Xt + ψ t
Shocks
× AR(1) discount factor shock βt
× TFP Θ is constant
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Evidence on Rates
Substantial evidence that interest rate spreads are
countercyclical
But are movements in the spread at right frequency for our
story?
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Figure 34: Spectral Density, Spread (BBA bonds-FFR, Moody’s)
4 6 24 32 40 50 60
Periodicity
0
1
2
3
4
5
6
7
8
9
Level
Various High-Pass
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Figure 35: Spectral Density, Real Policy Rate
4 6 24 32 40 50 60
Periodicity
0
2
4
6
8
10
12
Level
Various High-Pass
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Estimation
Estimate parameters of model by SMM
Targets
× spectrum of hours worked on the frequencies 2-50
× spectrum of interest rate spread on the frequencies 2-50
× Set of other higher moments (kurtosis and skewness of hours
and spread)
We will check whether model is also consistent with interest
rate observations over this range.
We calibrate δ = .05.
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Figure 36: Spectrum ﬁt for Hours
4 6 24 32 40 50 60
Periodicity
0
5
10
15
20
Data
Model
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Figure 37: Hours Spectrum in Smets & Wouters’ Model
4 6 24 32 40 50 60 80
Periodicity
0
2
4
6
8
10
12
75 / 91

Figure 38: Spectrum ﬁt for Spread
4 6 24 32 40 50 60
Periodicity
0
1
2
3
4
5
6
7
8
9
Data
Model
76 / 91

Figure 39: Spectrum ﬁt for Real Policy Rate
4 6 24 32 40 50 60
Periodicity
0
1
2
3
4
5
6
7
8
9
Data
Model
77 / 91

Figure 40: Fit for Target Moments
-1 -0.5 0 0.5 1 1.5 2 2.5 3
corr(l, rp
)
skew(l)
skew(rp
)
kurt(l)
kurt(rp
)
Data
Model
78 / 91

Figure 41: Sample Draw for Hours
0 50 100 150 200 250
Period #
-6
-5
-4
-3
-2
-1
0
1
2
3
Hours(%)
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Figure 42: Sample Draw for Hours, no shocks
0 50 100 150 200 250
Period #
-5
-4
-3
-2
-1
0
1
2
Hours(%)
80 / 91

Figure 43: Spectrum for Hours, no shocks
4 6 24 32 40 50 60
Periodicity
0
10
20
30
40
50
60 Data
Model
81 / 91

Figure 44: Sample Draw for Hours no Complementarities
0 50 100 150 200
t
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
ˆlt
82 / 91

Figure 45: Spectrum for Hours, no Complementarities
4 6 24 32 40 50 60
Periodicity
0
5
10
15
20
Data
Model
83 / 91

Shocks
Table 1: Estimated Parameter Values
γ 0.5335
Φ2 0.1906
Ω 4.6178
ρ -0.0000
σ 0.8525
R1 -0.1626
R2 0.00076
R3 0.00027
Shocks are important in our framework for explaining the data
But they are iid
Hence, almost all dynamics are internal.
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Nonlinearities
Table 2: Estimated Parameter Values
γ 0.5335
Φ2 0.1906
Ω 4.6178
ρ -0.0000
σ 0.8525
R1 -0.1626
R2 0.00076
R3 0.00027
Nonlinearities are crucial for the existence of a stochastic limit
cycle
But they are small
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Nonlinearities
Figure 46: Sensitivity of the Real Interest Rate Faced by the Households
to Economic Activity
-6 -4 -2 0 2 4
Hours (%)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Φ2l+˜R(l)(%perannum)
86 / 91

4.Conclusion
The dominant paradigm for explaining macro-economic
fluctuations focus on how different shocks perturb an
otherwise stable system
Such a perspective may be biased due to an excess focus on
rather high frequency movements
However, if we look at slightly lower frequencies – extending
from 32 to at least 40 quarters– there is strong signs of
cyclical behavior
Contribution:
1. Shown that models with ’weak’ complementarities and
accumulation offer a promising framework to explain such
observations. In particular, such framework can easily generate
limit cycles.
2. When extending a simple NK model to include these factors
and adopting a slightly lower frequency focus– 2-60 quarter–
we find support for very strong endogenous cyclical mechanism.
Would be interesting to extend such analysis to international
context
87 / 91

Figure 47: Theoretical Spectral Density (Sum of three AR(2))
4 6 24 32 40 5060 80100 200
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
16000
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4 6 24 32 40 50 60
Periodicity
0
0.5
1
1.5
2
×10
4
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
0.5
1
1.5
2
2.5
×10
4
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
0.5
1
1.5
2
2.5
×10
4
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
0.5
1
1.5
2
2.5
×10
4
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
16000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
0.5
1
1.5
2
×10
4
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
0.5
1
1.5
2
×10
4
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
5000
10000
15000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
Theoretical
Average
4 6 24 32 40 50 60
Periodicity
0
2000
4000
6000
8000
10000
12000
14000
16000
Theoretical
Average
90 / 91

Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
35
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
35
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
35
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
35
Level
Various Bandpass
4 6 24 32 40 50 60
Log of Period
0
5
10
15
20
25
30
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
35
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
35
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
20
40
60
80
100
120
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
10
20
30
40
50
60
70
80
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
Level
Various Bandpass
Log of Period
4 6 24 32 40 50 60
0
5
10
15
20
25
30
Level
Various Bandpass
91 / 91

Putting the cycle back into business cycle analysis

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