putting the cycle back into business cycle analysis

Putting the Cycle Back into Business CycleAnalysis

Paul Beaudry, Dana Galizia & Franck Portier

Vancouver School of Economics, Carleton University & Toulouse School ofEconomics

New Developments in MacroeconomicsADEMU conference

UCL, LondonNovember 2016

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0. IntroductionModern Approach to Business Cycles

I Equilibrium stochastic dynamic modeling in macro flowered inthe 1970’s

I At that point, postwar business cycles were at most 8 yearslong business cycles were defined as fluctuations atperiodicities of 8 years or less.

I 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

I This suggested that business cycle theory should not be aboutcycles, it should be about co-movements (see Sargent’stextbook)

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Figure 1: Sargent’s textbook

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Figure 2: Sargent’s textbook

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I Equilibrium stochastic dynamic modeling in macro flowered inthe 1970’s

I At that point, postwar business cycles were at most 8 yearslong business cycles were defined as fluctuations atperiodicities of 8 years or less.

I 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

I This suggested that business cycle theory should not be aboutcycles, it should be about co-movements (see Sargent’stextbook)

I All good!

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0. IntroductionWhat we do

I We question this focus in this workI We do three things

1. Show that the economy is “more cyclical than you think”, with40 more years of data.

2. Explain how “weak complementaries”, combined withaccumulation, favor cyclical behavior (recurrent booms andbusts) rather than indeterminacy

3. Estimate a New-Keynesian model extended to include the forcomplementarity forces we are highlighting, and see if the dataprefers 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 FactsLooking for Peaks

I If there are important cyclical forces in the economy ...

I ... this should show up as a distinct peak in the spectrum ofthe data

I It is well known that (detrended) output data does not displaysuch peaks (Granger (1969), Sargent (1987))

I However, output data may not be best placed to look, asthere is a marked non stationarity that one needs to get rid of.

I May be better to look at measures of factor use: ex: hoursworked, employment rates, unemployment rates, capitalutilization.

I Things might also have changed since the 70’s as we havenow 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

-7.85L

og

s

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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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2. General MechanismOverview

I Show a reduced form that is “dynamic Cooper & John(1988)”

I Understand the logic of the existence of strong cyclical forces(“limit cycle”) in our framework : weak strategiccomplementarities + dynamics

I Show a forward looking version with saddle path stable limitcycles

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2. General MechanismBasic environment

I N players (“firms”)

I Each firm accumulates a capital good in quantity Xi byinvesting Ii

I Decision rule and law of motion for X are

Xit+1 = (1− δ)Xit + Iit (1)

Iit = α0 − α1Xit + α2Iit−1 (2)

I all parameters are positive, δ < 1, α1 < 1, α2 < 1.

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2. General MechanismSymmetric allocations

I When all agent behave symmetrically:(ItXt

)=

(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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2. General MechanismSymmetric allocations

I When all agent behave symmetrically:(ItXt

)=

(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 demandcomplementarities

Xt

It

∆It = 0

∆Xt = 0

X s

I s

Here I have assumed that the two eigenvalues are real and positive.26 / 91

2. General MechanismIntroducing strategic interactions

I “Dynamic Cooper & John (1988)”

I Consider

Iit = α0 − α1Xit + α2Iit−1 + F

(∑Ijt

N

)I 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

)28 / 91

Figure 16: “Best response” rule for a given history

∑Ijt

N

Iit Iit=

∑Ijt

N

α0−α1Xit +α2Iit−1


(∑Ijt

N

)29 / 91

2. General MechanismRestrictions on F


(∑Ijt

N

)I We choose to be in a “non-pathological” case

I A steady state exists

I F ′(·) < 1 uniqueness of the period t equilibrium

I uniqueness of the steady state

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∑Ijt

N

Iit Iit=

∑Ijt

N

αt

Iit = αt + F(∑

IjtN

)


(∑Ijt

N

)31 / 91

2. General MechanismThe two sources of fluctuations

I We restrict to symmetric allocationsI Two forces of determination of allocations:

× static strategic interactions “multipliers” (Cooper &John (1988)) local instability

× History (accumulated I that shows up in X ) affects allocationsthrough the intercept of the “best response” function global stability

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∑Ijt

N

Iit Iit=

∑Ijt

N

αs

αt

Iit = αt + F(∑

IjtN

)

αt = α0 − α1∑∞

0 (1− δ)τ+1Ijt−1−τ + α2Iit−1

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2. General MechanismIntuition for the limit cycle


(∑Ijt

N

)I F

(∑Ijt

N

)(Strategic complementarities): centrifugal force that

pushes away from the steady state when close to.

I −α1Xit (Accumulation): centripetal force that pushes towardsthe steady state when away from.

I α2Iit−1 (Sluggishness) : avoid jumping back an forth aroundthe steady state (“flip”)

I The steady state locally unstable, but forces push the economysmoothly back to the steady state when it is further from it.

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Iit = α0−α1Xit + α2Iit−1 + F

(∑Ijt

N

)I F

(∑Ijt

N






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(∑Ijt

N

)I F

(∑Ijt

N






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(∑Ijt

N

)I F

(∑Ijt

N




I α2Iit−1 (Sluggishness) : avoid jumping back and forth aroundthe steady state (“flip”)


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2. General MechanismStable limit cycle

X s

I s

−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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2. General MechanismLooking for the limit cycle

I How do we prove the existence of a limit cyle?

I Limit cycles are typically ocurring when there are bifurcationsin non-linear dynamical systems

I Simply stated : the SS looses stability when a parameter (herethe degree of strategic complementarities) increases.

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2. General MechanismDynamics with strategic interactions

I MathI Dynamics is(

ItXt

)=

(α2 − α1 −α1(1− δ)

1 1− δ

)︸︷︷︸

ML

(It−1

Xt−1

)+

(α0+F (It)

0

)

I Local dynamics is(ItXt

)=

(α2−α1

1−F ′(I s) − α1(1−δ)1−F ′(I s)

1 1− δ

)︸︷︷︸

M

(It−1

Xt−1

)

+

( (1− α2−α1

1−F ′(I s)

)I s +

(α1(1−δ)1−F ′(I s)

)X s

0

)I When F ′(I s) varies from −∞ to 1, eigenvalues of M vary

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2. General MechanismStrategic substitutabilities – F ′ negative

Proposition 2

As F ′(I S) varies from 0 to −∞, the eigenvalues of M always staywithin the unit circle and therefore the system remains locallystable.

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2. General MechanismStrategic complementarities – F ′ positive

Proposition 3

As F ′(I s) varies from 0 towards 1, the dynamic system will becomelocally unstable.

(bifurcation)

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2. General MechanismBifurcations

I 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.

I We are interested in Hopf bifurcation because the limit cyclewill be “persistent”

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Proposition 4

As F ′(I s) varies from 0 towards 1, the dynamic system will becomeunstable and:

I No flip bifurcations

I If α2 > α1/(2− δ)2, then a Hopf (Neimark-Sacker)bifurcation will occur.

I If α2 < α1/(2− δ)2, then a flip bifurcation will occur.

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I In the case of flip and Hopf bifurcation, a limit cycle appears

I In case of Hopf, the cycle is “smooth”

I 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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2. General MechanismStability of the limit cycle

I In the case of the Hopf bifurcation, the limit cycle can beattractive (the bifurcation is supercritical) or repulsive (thebifurcation is subcritical)

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2. General MechanismStability of the limit cycle

I Proposition 5

If F ′′′(I s) is sufficiency negative, then the Hopf bifurcation will besupercritical. Therefore, the limit cycle is attractive.

I F ′′′ < 0 corresponds to an S − shaped reaction function

I

∑Ijt

N

Iit Iit=

∑Ijt

N

αt

Iit = αt + F(∑

IjtN

)

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2. General MechanismIntuition for genericity

Figure 18: Stable Steady State

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2. General MechanismIntuition for genericity

Figure 19: Hopf Supercritical bifurcation: Attractive Limit Cycle

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2. General MechanismNumerical example

I Let me show an arbitrary numerical example with thatreduced form model

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Figure 20: A particular F function

I

F (I )

a0slope β0

slope β

1

slope β2 F (I )

γ1I 1

γ2I 2

I 1 I 2I s47 / 91


Figure 21: Deterministic simulation

I , X , linear model I , X , nonlinear model

period50 100 150 200

0

2

4

6

8

10

12

14

X

I

period0 50 100 150 200 250

0

2

4

6

8

10

12

14

X

I

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Figure 22: Deterministic simulation of the nonlinear model

Spectral density

Period

4 8 12 20 32 40 60 800

0.2

0.4

0.6

0.8

1

1.2

1.4

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Figure 23: One stochastic simulation

Linear model Nonlinear model

period50 100 150 200

I

0

0.5

1

1.5

2

period0 50 100 150 200 250

I

0

0.5

1

1.5

2

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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

1.4I

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2. General MechanismWhat 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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2. General MechanismWhat 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

2. General MechanismWhat 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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2. General MechanismWhat 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

2. General MechanismAdding Forward looking elements

Iit = α0 − α1Xit + α2Iit−1 + α3Et [Iit+1] + F

(∑Ijt

N

)I with accumulation remaining the same

Xit = (1− δ)Xit + Iit

I Restrict attention to situations where this system is saddlepath stable absent of complementarities.

I Generally true when all three are between 0 and 1.

I In this case, more difficult to get analytical results.

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2. General MechanismSet of potential bifurcation with Forward looking elements

I Initial situation has two stable roots and one unstable

I Three types of bifurcations are possible:

1. The unstable root enters the unit circle: local indeterminacyarises

2. One stable root leaves the unit circle: instability arises with aflip or fold type bifurcation

3. Two stable roots leave the unit circle simultaneous becausethey 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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2. General MechanismSet 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.

I Hence, under quite simple conditions only relevant bifurcationis 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



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3. Empirical ExerciceExploring empirically whether US Business Cycles may reflect aStochastic Limit Cycles.

I Stylized NK model which is extended to allow for the forceshighlighted in our general structure.

I We add accumulation of durable-housing goods and habitpersistence : accumulation and sluggishness

I Financial frictions under the form of a counter-cyclical riskpremium : complementarities

I Estimate parameters based on spectrum observations andhigher moments. (use perturbation method and indirectinference)

I See whether model favors parameters the generate limit cycle.

I If so, explore nature of intrinsic limit cycle and perform somecounter factual exercises.

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3. Empirical ExerciceBasic Elements of the Model

1. Household buy consumption services to maximise utilitytaking prices as given

2. Firms supply consumption services to the market where theservices can come from existing durable goods or newproduction.

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 riskpremium, where the risk premium varies with the cycle.

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3. Empirical ExerciceExtending a 3 equation NK model

I The representative household who can buy/rent consumptionservices from the market.

I The households Euler will have the familiar form (assumingexternal habit)

U ′(Ct − γCt−1) = βtEtU(Ct+1 − γCt)(1 + rt)

I Allow that interest rate faced by household may include a riskpremium

(1 + rt) = (1 + it + rpt )

I where rpt can respond to activity, or unemploymentI Have Taylor rule of form

it = Φ1Etπt+1 + Φ2Et`t+1

I `t is (log) employment or output gap,I U(z) = − exp

− z

Ω

, Ω > 0 65 / 91

3. Empirical ExerciceAllowing for durable goods and accumulation

I Firms produce an intermediate factor

Mjt = BF (ΘtLjt) ,

I Mt is used to produce consumptions services or durablegoods, 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

I Households own the stock of durable-housing, rent it to firms,who supply back the consumption services as a compositegood.

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3. Empirical ExerciceRisk premium

I The interest rate which household face is assume to be equalto the policy rate plus a risk premium

(1 + rt) = (1 + it + rpt )

I where rpt is a premium over the back rate.

I The risk premium is assumed to be an decreasing function ofthe economic activity, where the output gap and employmentgap are interchangeable

rpt = g(Lt)

I Here we are allowing the interest premium to be a non-linearfunction of activity.

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3. Empirical ExerciceTaylor rule and dichotomy with inflation

I Modified Taylor rule:

it = φ0 + φ1Etπt+1 + φ2Et`t+1

I φ1 = 1 the Central Bank as setting the expected realinterest.

I This approach has the attractive feature of making the modelbloc recursive, where the inflation rate last is solved as afunction of the marginal cost.

I With this approach, we can explore the properties of the realvariables without need to to be specific about the source andduration of price stickiness (which is not our focus)

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3. Empirical ExerciceReduced Form

I Solution is

`t = µt − α1Xt + α2`t−1 + α3Et [`t+1] + F (`t)

I together with accumulation

Xt+1 = (1− δ)Xt + ψ`t

I Shocks

× AR(1) discount factor shock βt× TFP Θ is constant

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3. Empirical ExerciceEvidence on Rates

I Substantial evidence that interest rate spreads arecountercyclical

I But are movements in the spread at right frequency for ourstory?

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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

71 / 91

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

72 / 91

3. Empirical ExerciceEstimation

I Estimate parameters of model by SMMI 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)

I We will check whether model is also consistent with interestrate observations over this range.

I 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

9Data

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

9Data

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

2l t

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

3. Empirical ExerciceShocks

Table 1: Estimated Parameter Values

γ 0.5335Φ2 0.1906Ω 4.6178ρ -0.0000σ 0.8525

R1 -0.1626

R2 0.00076

R3 0.00027

I Shocks are important in our framework for explaining the dataI But they are iidI Hence, almost all dynamics are internal.

84 / 91

3. Empirical ExerciceNonlinearities

Table 2: Estimated Parameter Values

γ 0.5335Φ2 0.1906Ω 4.6178ρ -0.0000σ 0.8525

R1 -0.1626

R2 0.00076

R3 0.00027

I Nonlinearities are crucial for the existence of a stochastic limitcycle

I But they are small

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3. Empirical ExerciceNonlinearities

Figure 46: Sensitivity of the Real Interest Rate Faced by the Householdsto 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)(%

per

annum)

86 / 91

4.ConclusionI The dominant paradigm for explaining macro-economic

fluctuations focus on how different shocks perturb anotherwise stable system

I Such a perspective may be biased due to an excess focus onrather high frequency movements

I However, if we look at slightly lower frequencies – extendingfrom 32 to at least 40 quarters– there is strong signs ofcyclical behavior

I Contribution:

1. Shown that models with ’weak’ complementarities andaccumulation offer a promising framework to explain suchobservations. In particular, such framework can easily generatelimit cycles.

2. When extending a simple NK model to include these factorsand adopting a slightly lower frequency focus– 2-60 quarter–we find support for very strong endogenous cyclical mechanism.

I Would be interesting to extend such analysis to internationalcontext

87 / 91

88 / 91

Figure 47: Theoretical Spectral Density (Sum of three AR(2))

4 6 24 32 40 5060 80 100 200

Periodicity

0

2000

4000

6000

8000

10000

12000

14000

16000

89 / 91

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 50 600

5

10

15

20

25

30

35

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

30

35

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

30

35

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

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 50 600

5

10

15

20

25

30

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

30

35

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

30

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

30

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

30

35

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

20

40

60

80

100

120

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

10

20

30

40

50

60

70

80

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

Level

Various Bandpass

Log of Period

4 6 24 32 40 50 600

5

10

15

20

25

30

Level

Various Bandpass

91 / 91

putting the cycle back into business cycle analysis

Economy & Finance