From the ADEMU Project: Paul Beaudry, Dana Galizia and Franck Portier's presentation from the New Developments in Macroeconomics lecture held at UCL on 9 November 2016.
Putting the cycle back into business cycle analysis
1. 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
1 / 91
2. 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)
2 / 91
5. 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)
All good!
5 / 91
6. 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”
6 / 91
9. 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
9 / 91
10. 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
10 / 91
11. 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
11 / 91
23. 2. General Mechanism
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
23 / 91
24. 2. General Mechanism
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.
24 / 91
25. 2. General Mechanism
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.
25 / 91
26. 2. General Mechanism
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.
25 / 91
27. 2. General Mechanism
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.
26 / 91
28. 2. General Mechanism
Introducing strategic interactions
“Dynamic Cooper & John (1988)”
Consider
Iit = α0 − α1Xit + α2Iit−1 + F
Ijt
N
F > 0 : strategic complementarities
27 / 91
29. Figure 15: “Best response” rule for a given history - Multiple Equilibria
Ijt
N
Iit Iit=
Ijt
N
α0 −α1Xit +
α2Iit−1
Iit = α0 − α1Xit + α2Iit−1 + F
Ijt
N
28 / 91
30. Figure 16: “Best response” rule for a given history
Ijt
N
Iit Iit=
Ijt
N
α0 −α1Xit +
α2Iit−1
Iit = α0 − α1Xit + α2Iit−1 + F
Ijt
N
29 / 91
31. 2. General Mechanism
Restrictions on F
Iit = α0 − α1Xit + α2Iit−1 + 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
30 / 91
32. Figure 17: “Best response” rule for a given history
Ijt
N
Iit Iit=
Ijt
N
αt
Iit = αt + F
Ijt
N
Iit = α0 − α1Xit + α2Iit−1 + F
Ijt
N
31 / 91
33. 2. General Mechanism
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
32 / 91
34. Figure 17: “Best response” rule for a given history
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
32 / 91
35. 2. General Mechanism
Intuition for the limit cycle
Iit = α0 − α1Xit + α2Iit−1 + F
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.
33 / 91
36. 2. General Mechanism
Intuition for the limit cycle
Iit = α0 − α1Xit + α2Iit−1 + F
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.
33 / 91
37. 2. General Mechanism
Intuition for the limit cycle
Iit = α0−α1Xit + α2Iit−1 + F
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.
33 / 91
38. 2. General Mechanism
Intuition for the limit cycle
Iit = α0 − α1Xit + α2Iit−1 + F
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.
33 / 91
39. 2. General Mechanism
Intuition for the limit cycle
Iit = α0 − α1Xit + α2Iit−1 + F
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 and 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.
33 / 91
44. 2. General Mechanism
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.
35 / 91
45. 2. General Mechanism
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
36 / 91
46. 2. General Mechanism
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.
37 / 91
47. 2. General Mechanism
Strategic complementarities – F positive
Proposition 3
As F (Is) varies from 0 towards 1, the dynamic system will become
locally unstable.
(bifurcation)
38 / 91
48. 2. General Mechanism
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”
39 / 91
49. 2. General Mechanism
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.
40 / 91
50. 2. General Mechanism
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:
Iit = α0 − α1Xit + α2Iit−1 + F
Ijt
N
× as δ approaches 0, α2 > 1/4
× in general, what is needed is δ not too large and α2 large
enough
41 / 91
51. 2. General Mechanism
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)
42 / 91
52. 2. General Mechanism
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
43 / 91
56. 2. General Mechanism
Numerical example
Figure 20: A particular F function
I
F(I)
a0
slope β0
slope β1
slope β2 F(I)
γ1I1
γ2I2
I1 I2Is
47 / 91
57. 2. General Mechanism
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
48 / 91
58. 2. General Mechanism
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
49 / 91
59. 2. General Mechanism
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
50 / 91
60. 2. General Mechanism
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
51 / 91
61. 2. General Mechanism
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
51 / 91
62. 2. General Mechanism
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
52 / 91
63. 2. General Mechanism
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
53 / 91
64. 2. General Mechanism
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
54 / 91
65. 2. General Mechanism
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
55 / 91
66. 2. General Mechanism
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.
56 / 91
67. 2. General Mechanism
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
57 / 91
68. 2. General Mechanism
Set of potential bifurcation with Forward looking elements
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
57 / 91
69. 2. General Mechanism
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
57 / 91
70. 2. General Mechanism
Set of potential bifurcation with Forward looking elements
Figure 30: Eigenvalues of the Reduced Form Model
-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
57 / 91
71. 2. General Mechanism
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
57 / 91
72. 2. General Mechanism
Set of potential bifurcation with Forward looking elements
Figure 31: Eigenvalues of the Reduced Form Model
-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
57 / 91
73. 2. General Mechanism
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
57 / 91
75. 2. General Mechanism
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.
59 / 91
76. Figure 33: Eigenvalues Configuration At First Bifurcation (δ = .05)
red = indeterminacy, yellow = unstable, gray = saddle
60 / 91
79. 3. Empirical Exercice
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.
63 / 91
80. 3. Empirical Exercice
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.
64 / 91
81. 3. Empirical Exercice
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
82. 3. Empirical Exercice
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.
66 / 91
83. 3. Empirical Exercice
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.
67 / 91
84. 3. Empirical Exercice
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)
68 / 91
85. 3. Empirical Exercice
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
69 / 91
86. 3. Empirical Exercice
Evidence on Rates
Substantial evidence that interest rate spreads are
countercyclical
But are movements in the spread at right frequency for our
story?
70 / 91
89. 3. Empirical Exercice
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.
73 / 91
90. 3. Empirical Exercice
Figure 36: Spectrum fit for Hours
4 6 24 32 40 50 60
Periodicity
0
5
10
15
20
Data
Model
74 / 91
99. 3. Empirical Exercice
Figure 45: Spectrum for Hours, no Complementarities
4 6 24 32 40 50 60
Periodicity
0
5
10
15
20
Data
Model
83 / 91
100. 3. Empirical Exercice
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.
84 / 91
101. 3. Empirical Exercice
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
85 / 91
102. 3. Empirical Exercice
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
103. 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