Learning, Confidence, and Business Cycles
Cosmin Ilut

Hikaru Saijo

Duke & NBERUC Santa Cruz

CIGS Conference on Macroeconomic Theory and Policy
May 2016

Ilut, Saijo

Learning, Confidence, and Business Cycles

1 / 27

Parsimonious mechanism for business cycle dynamics

Propose: Endogenous idiosyncratic uncertaintyI

firms learn about own profitability prospects

Behaves as if linear RBC model with endogenously determined
1

Countercyclical labor wedge and spreads (from excess returns)

2

Co-movement from demand shocks

3

Amplification, propagation and hump-shaped dynamics

Ilut, Saijo

Learning, Confidence, and Business Cycles

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Parsimonious mechanism for business cycle dynamics

Propose: Endogenous idiosyncratic uncertaintyI

firms learn about own profitability prospects

Do not require additional shocks or rigidities such as
1

Wedge shocks (countercyclical labor wedge and spreads)

2

Nominal rigidities (co-movement)

3

Habit, adjustment cost (internal propagation)

Ilut, Saijo

Learning, Confidence, and Business Cycles

3 / 27

Countercyclical endogenous idiosyncratic uncertainty

Firms face Knightian uncertainty about own profitability
1

Learning through production: lower scale → more uncertainty

2

Uncertainty affects input choice: more uncertainty → lower scale

Feedback arises from any shock that moves activity

Ilut, Saijo

Learning, Confidence, and Business Cycles

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Countercyclical idiosyncratic uncertainty shows up

As countercyclical wedges: labor and asset prices move ’too much’compared to what econometrician measuresI

rationalize ’excess volatility’

In linear decision rules at firm level
In the cross-sectional average through aggregation

Ilut, Saijo

Learning, Confidence, and Business Cycles

5 / 27

Model: Preferences

Representative household: recursive multiple priors utility
Ut (C ; s t ) = ln Ct − ϕ

Ht1+η
+ β min E p [Ut+1 (C ; s t , st+1 )]1+ηp∈Pt (s t )Pt (s t ): one-stead-ahead set of probability distributions
Larger set Pt (s t ) → less confidence

Ilut, Saijo

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Production
Firms: continuum, indexed by l ∈ [0, 1], perfectly competitive
1−ααYl,t = At {zl,t Kl,t−1Hl,t+ νl,t }Ilut, Saijo

Learning, Confidence, and Business Cycles

7 / 27

Production
Firms: continuum, indexed by l ∈ [0, 1], perfectly competitive
1−ααYl,t = At {zl,t Kl,t−1Hl,t+ νl,t }Aggregate TFP shock
ln At = ρA ln At−1 + A,t ,Ilut, Saijo

A,t ∼ N(0, σA2 )Learning, Confidence, and Business Cycles

7 / 27

Production
Firms: continuum, indexed by l ∈ [0, 1], perfectly competitive
1−ααYl,t = At {zl,t Kl,t−1Hl,t+ νl,t }Aggregate TFP shock
ln At = ρA ln At−1 + A,t ,A,t ∼ N(0, σA2 )Idiosyncratic TFP shockzl,t = (1 − ρz )z̄ + ρz zl,t−1 + z,l,t ,Ilut, Saijo

Learning, Confidence, and Business Cycles

z,l,t ∼ N(0, σz2 )7 / 27

Production
Firms: continuum, indexed by l ∈ [0, 1], perfectly competitive
1−ααYl,t = At {zl,t Kl,t−1Hl,t+ νl,t }Aggregate TFP shock
ln At = ρA ln At−1 + A,t ,A,t ∼ N(0, σA2 )Idiosyncratic TFP shockzl,t = (1 − ρz )z̄ + ρz zl,t−1 + z,l,t ,Idiosyncratic additive shock,Ilut, Saijo

z,l,t ∼ N(0, σz2 )νl,t ∼ N(0, σν2 )Learning, Confidence, and Business Cycles

7 / 27

Information
1−ααYl,t = At {zl,t Kl,t−1Hl,t+ νl,t }zl,t and νl,t unobservable to agents → learning
Non-invertibility problem: path of output and input not fully revealing
about the unobservable shocks
Interpretations of additive shock
1

Aggregation of production units with common and idiosyncratic shocks

2

Sale is signal on unobservable persistent demand shock

Details
Ilut, Saijo

Learning, Confidence, and Business Cycles

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Heterogeneous-firm RBC model
Firms: choose {Kl,t , Hl,t , Il,t } to maximize
E0∗

∞
X

M0t Dl,tt=0
IM0t : prices of contingent claims, under worst case probabilities
Dl,t = Yl,t − Wt Hl,t − Il,tResource constraint:
Yt = Ct + It + Gt
ln Gt = (1 − ρg )G + ρg ln Gt−1 + g ,t ,Ilut, Saijo

Learning, Confidence, and Business Cycles

g ,t ∼ N(0, σg2 )9 / 27

Timeline of events within a period
Stage 1Aggregate shocks(observable)

Stage 2
Idiosyncratic shocks(unobservable)

Given forecasts and variance ofhidden TFP, choose inputs,
labor market clearing

Investment,
consumption,asset purchase

Production, update
forecasts and variance ofhidden TFP

RCEIlut, Saijo

Learning, Confidence, and Business Cycles

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Learning and ambiguity about idiosyncratic productivity
Estimate zl,t from observables: linear + Gaussian → Kalman filter
1−ααObservation : Yl,t /At = Kl,t−1Hl,tzl,t + νl,tTransition : zl,t = (1 − ρz )z̄ + ρz zl,t−1 + z,l,tIlut, Saijo

Learning, Confidence, and Business Cycles

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Learning and ambiguity about idiosyncratic productivity
Estimate zl,t from observables: linear + Gaussian → Kalman filter
1−ααObservation : Yl,t /At = Kl,t−1Hl,tzl,t + νl,tTransition : zl,t = (1 − ρz )z̄ + ρz zl,t−1 + z,l,t1−αα
Low production input Kl,t−1
Hl,t→ high Mean Square Error Σl,t|t

Ilut, Saijo

Learning, Confidence, and Business Cycles

11 / 27

Learning and ambiguity about idiosyncratic productivity
Estimate zl,t from observables: linear + Gaussian → Kalman filter
1−ααObservation : Yl,t /At = Kl,t−1Hl,tzl,t + νl,tTransition : zl,t = (1 − ρz )z̄ + ρz zl,t−1 + z,l,t1−αα
Low production input Kl,t−1
Hl,t→ high Mean Square Error Σl,t|t

Not confident in the Kalman filter estimate: set of distributions
Et zl,t+1 = (1 − ρz )z̄ + ρz z̃l,t|t + µl,t ;Ilut, Saijo

Learning, Confidence, and Business Cyclesµl,t ∈ [−al,t , al,t ]11 / 27

Learning and ambiguity about idiosyncratic productivity
Estimate zl,t from observables: linear + Gaussian → Kalman filter
1−ααObservation : Yl,t /At = Kl,t−1Hl,tzl,t + νl,tTransition : zl,t = (1 − ρz )z̄ + ρz zl,t−1 + z,l,t1−αα
Low production input Kl,t−1
Hl,t→ high Mean Square Error Σl,t|t

Not confident in the Kalman filter estimate: set of distributions
Et zl,t+1 = (1 − ρz )z̄ + ρz z̃l,t|t + µl,t ;µl,t ∈ [−al,t , al,t ]Confidence lower when estimation uncertainty is higherq−al,t = −ηa Σl,t|t

I

Distributions “close” to filter estimate (relative entropy distance)

Ilut, Saijo

Learning, Confidence, and Business Cycles

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Ambiguity and the law of large numbers
Each firm’s expected zl,t+1 under worst-case probability
Et∗ zl,t+1 = (1 − ρz )z̄ + ρz z̃l,t|t−al,t| {z }√=−ηaΣl,t|tI

Household acts as if conditional mean of each zl,t+1 is lower

I

First-order effect of uncertainty

Cross-sectional average given by a setZZz̄ − al,t dl, z̄ + al,t dl
I

Epstein & Schneider (2003): formal treatment of LLN with ambiguity

Ilut, Saijo

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Linearized solution

1

Filtering problem is linear → analytic law of motion for Σl,t|t
I

Inputs have first-order effect on the level of posterior variance

2First-order feedback from uncertainty to decision rules through −al,t3

In turn, linear decision rules → easy aggregation
I

Cross-sectional mean: sufficient statistic for tracking distributions

Ilut, Saijo

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Implication: comovement and countercyclical labor wedge
Standard model
ϕHtη = λt MPLt
→ H and C move in opposite direction unless TFP or ‘ϕ’ shock

Ilut, Saijo

Learning, Confidence, and Business Cycles

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Implication: comovement and countercyclical labor wedge
Standard model
ϕHtη = λt MPLt
→ H and C move in opposite direction unless TFP or ‘ϕ’ shock
Our model: labor chosen under worst case expectation
ϕHtη = Et∗ [λt MPLt ]Low confidence → C low → standard effect is H high→ choose H as if productivity low → H low

Ilut, Saijo

Learning, Confidence, and Business Cycles

14 / 27

Implication: comovement and countercyclical labor wedge
Standard model
ϕHtη = λt MPLt
→ H and C move in opposite direction unless TFP or ‘ϕ’ shock
Our model: labor chosen under worst case expectation
ϕHtη = Et∗ [λt MPLt ]Low confidence → C low → standard effect is H high→ choose H as if productivity low → H low
Labor wedge: implicitly define labor tax
ϕHtη = (1 − τt )λt MPLt⇒

Et∗ [λt MPLt ]= 1 − τtλt MPLt

Low confidence → econometrician rationalizes ‘surprisingly low’ H byhigh labor tax
Ilut, Saijo

Learning, Confidence, and Business Cycles

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Implication: countercyclical ex-post excess return
Euler conditions for capital and risk-free assetsK
λt = βEt∗ [λt+1 Rt+1]

λt = βEt∗ [λt+1 Rt ]K −R =0→ under linearization, Et∗ Rt+1
t

Pricing based on worst case 6= econometrician’s DGP
During low confidence times, demand for capital ‘surprisingly low’
K − R high→ ex-post excess return Rt+1
t
Implication extends to defaultable corporate bonds→ countercyclical excess bond premia (Gilchrist & Zakrajsek 2012)
Ilut, Saijo

Learning, Confidence, and Business Cycles

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Calibration
Magnitude of feedback loop determined by
1

Variability of inputs
I
I

2

Inverse Frisch elasticity η = 0
Capital utilization

Size and variability of posterior varianceI

Idiosyncratic TFP shock ρz = 0.5, σz = 0.4
FI

SS posterior variance Σ = 0.1
F3

establishment-level data (Bloom et al. 2014, Kehrig 2015)
estimated posterior variance of firm-specific shocks (David et al. 2015)

Size of entropy constraint
Reasonable theoretical upper bound ηa = 2 (Ilut & Schneider 2014)
I Empirical: firm-level capital return forecasts across analysts
IF

Set ηa = 0.4 to get average dispersion of 39% (vs 43% in Senga 2014)

Parameters

Ilut, Saijo

Learning, Confidence, and Business Cycles

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IRF to aggregate TFP shock

Percent deviation

OutputInvestment
8

2

6
4

1

2
0

0
5

10
15
Consumption

20

0.16

5

10
Hours

15

20

5

10

15

20

2

0.14
1

0.12
0.1
0.08

0
5

0
−0.02

10
15
Labor wedge

20

Ex−post excess return0

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0.12

IRF0.1to aggregate TFP shock
0.08

5

10
15
Labor wedge

1
0

20

5

10

15

20

Ex−post excess return

0

0

−0.02
−0.04

−0.5

−0.06
−0.08

−1
5

10

15

20

Cross−sectional mean of worst−case TFP
0.06
0.04

5

10

15

20

Cross−sectional mean of realized TFP
1Baseline
Passive
0
RE

0.02
0
5

Ilut, Saijo

10

15

20

−1

Learning, Confidence, and Business Cycles

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10

15

20

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IRF to government spending shock

Percent deviation

OutputInvestment
1.5

0.6

1

0.4

0.5
0.2
0
5

10
15
Consumption

20

0.01

0.6

0

0.4

−0.01

0.2
5

10
15
Labor wedge

200Ilut, Saijo

Learning, Confidence, and Business Cycles

5

10
Hours

15

20

5

10

15

20

Ex−post excess return
19 / 27

0.4

0

IRF to government spending shock
0.2
−0.01

5

10
15
Labor wedge

20

5

10

15

20

Ex−post excess return

0
−0.1

−0.01

−0.2

−0.02

−0.3
5

10

15

20

Cross−sectional mean of worst−case TFP
0.02
0.01
0

5

Ilut, Saijo

10

15

20

5

10

15

20

Cross−sectional mean of realized TFP
1Baseline
Passive
0
RE−1

Learning, Confidence, and Business Cycles

5

10

15

20

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Bayesian estimation on US aggregate data
Linearization ⇒ estimation using standard Kalman filter
Quantitative model with additional rigidities (CEE, 2005)
I
I

real: habit formation, investment adjustment costs
nominal: sticky prices and wages

Shocks: TFP, G, mon. policy and ’financial wedge’ shockk∆kt ' Et∗ Rt+1
− Rt

US Data: Yt , Ht , It , Ct , πt , Rt , Spreadt (on BAA corporate bond)Spreadt ≡ Rtk − Rt−1




∗
∗
= Et−1
Rtk − Rt−1 + Rtk − Et−1
Rtk
{z
}
{z
}
|
|
wedge shockI
I

endogenous uncertainty

estimate both flex and sticky price versions
stochastic singularity ⇒ iid measurement error

Ilut, Saijo

Learning, Confidence, and Business Cycles

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Results
1

Endogenous uncertainty: parsimonious friction ⇒ reduce otherrigidities
Model
RE
BaselineIlut, Saijo

ηa
0
1.3

Pr(price ∆)
0.24
0.44

Pr(wage ∆)
0.04
0.98

Inv. adj. cost
0.3
0.06

Learning, Confidence, and Business Cycles

Habit
0.62
0.47

22 / 27

Results
1

Endogenous uncertainty: parsimonious friction ⇒ reduce otherrigidities
Model
RE
Baseline

2

ηa
0
1.3

Pr(price ∆)
0.24
0.44

Pr(wage ∆)
0.04
0.98

Inv. adj. cost
0.3
0.06

Habit
0.62
0.47

Endogenous uncertainty model fits data better
marginal data density is higher (both flex and sticky price versions)
under RE: observed spread is mostly just measurement error
I but well fitted under model with endogenous uncertainty
I
I

Ilut, Saijo

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Spread: data vs. models

1.3
Data

Quarterly percentage points

1.2
1.1

BaselineRE
1
0.9

Baseline(ambiguity only)

0.8
0.7
0.6
0.5
0.4
1985

Ilut, Saijo

1990

1995

2000

2005

Learning, Confidence, and Business Cycles

2010
23 / 27

Endogenous uncertainty: countercyclical spread ⇒ bus.
cycle comovement
Percent deviation

OutputInvestment

−0.5

−2

−1
−4
−1.5

Baseline
RE counterfactual

−6
−2
5

10

15

20

5

Consumption
0

15

20

15

20

Hours
−0.2

−0.2

−0.4

−0.4

−0.6

−0.6
−0.8

−0.8

−1

−1

Ilut, Saijo

10

5

10

15 Confidence,
20 and Business Cycles
5
Learning,

10

24 / 27

Results
1

Endogenous uncertainty: parsimonious friction ⇒ reduce otherrigidities
Model
RE
Baseline

2

ηa
0
1.3

Pr(price ∆)
0.24
0.44

Pr(wage ∆)
0.04
0.98

Inv. adj. cost
0.3
0.06

Habit
0.62
0.47

Endogenous uncertainty model fits data better
marginal data density is higher (both flex and sticky price versions)
under RE: observed spread is mostly just measurement error
I but well fitted under model with endogenous uncertainty
I
I

3

Variance decomposition: financial shock more important with learning
Model (sticky price)
Y
H
I
C
π
R
RE
0.15 0.23 0.12 0.22 0.88 0.90
Baseline
0.73 0.81 0.76 0.61 0.88 0.84
Ilut, Saijo

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Policy implication of endogenous uncertainty
Endogenous uncertainty ⇒ Policy matters
Policy experiment:I

modify Taylor rule to include adjustment to credit spread φspreadI

lower output growth variation: from stabilizing endogenous uncertainty

φspread
0
-0.5
-1.0
-1.5

Ilut, Saijo

Std. of output growthBaseline Fixed uncertainty
0.60
0.60
0.59
0.60
0.57
0.60
0.52
0.63

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Conclusion
Heterogeneous-firm business cycle model where firms face Knightian
uncertainty about their own profitability
Feedback loop between uncertainty and economic activity produces
I

Countercyclical labor wedge and ex-post excess return on capitalI

Co-movement in response to non-TFP shocks

I

Strong internal propagation with amplified and hump-shaped dynamics

Estimation: inference on rigidities and shocks
Policy implications

Ilut, Saijo

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Interpreting the additive shock (νl,t )1

At the aggregate level, observationally equivalent to model wherefirms face unobservable demand shock
I

Each unit of good l : provides sum of good specific and idiosyncratic
qualityYl,tX(zl,t + ν̃l,j,t )Yel,t =
j=1

I
I

I
2

1−α
α
where units produced Yl,t = Kl,t−1
Hl,tNoisy signal about persistent quality zl,t : procyclical precision


σν̃2
e
Yl,t /Yl,t = zl,t + νl,t , νl,t ∼ N 0,
Yl,tdemand is a function of estimate of quality zl,tAggregation of production units with common and idio shocks

ReturnIlut, Saijo

Learning, Confidence, and Business Cycles

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Kalman filter
Estimatez̃l,t|t = z̃l,t|t−1 + Gainl,t (Yl,t /At − z̃l,t|t−1 Fl,t )Kalman gainGainl,t =2 ΣFl,tl,t|t−12 Σ2Fl,tl,t|t−1 + σν,t



−1
Fl,t

Mean square error
Σl,t|t = (1 − Gainl,t Fl,t )Σl,t|t−12 Σ
σν,tl,t|t−1= 2
2
Fl,t Σl,t|t−1 + σν,tReturnIlut, Saijo

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Illustration: distinguishing distributions

ReturnIlut, Saijo

Learning, Confidence, and Business Cycles

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Relative entropy distanceAgents consider the conditional means µ∗l,t+1 that are sufficiently close to
the long run average of zero in the sense of relative entropy:(µ∗l,t+1 )22ρ2z Σl,t|t

1
≤ ηa2
2

LHS: relative entropy between two normal distributions that share thesame variance ρ2z Σl,t|t but have different means (µ∗l,t+1 and zero)
ReturnIlut, Saijo

Learning, Confidence, and Business Cycles

4 / 16

Linearized solution
1

Filtering problem is linear → analytic law of motion for Σl,t|t
I

Inputs have first-order effect on the level of posterior varianceΣ̂l,t−1|t−1 = εΣ,Σ Σ̂l,t−2|t−2 − εΣ,F F̂l,t−1 ,2First-order feedback from uncertainty to decision rules through −al,tEt∗ ẑl,t = εz,z z̃ˆl,t−1|t−1 − εz,Σ Σ̂l,t−1|t−1 ,

3

(1)

(2)

In turn, linear decision rules → easy aggregation
I

Cross-sectional mean: sufficient statistic for tracking distributions

Et∗ ẑt = εz,z z̃ˆt−1|t−1 −εz,Σ εΣ,Σ Σ̂t−2|t−2 + εz,Σ εΣ,F F̂t−1
{z
}
|

(3)

=0

where x̂t ≡R

x̂l,t dl

ReturnIlut, Saijo

Learning, Confidence, and Business Cycles

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Recursive competitive equilibrium
Household’s problem at stage 1 : hats RVs resolved at stage 2


→
−
H 1+η
+ E ∗ [V2h (m̂; ξˆ2 , X )]V1h ( θ l , B; ξ1 , X ) = max − ϕH1+η
Z
s.t. m̂ = WH + RB + (D̂l + P̂l )θl dl − G(4)Household’s problem at stage 2:


Z
−

0
h →
0
0 0
0
dF
(X
|X
)
V2h (m; ξ2 , X ) = max
θ
,
B
;
ξ
,
X
ln
C
+
β
V
l
1
1
−
→
C , θl 0 ,B 0s.t. m ≥ C + B 0 +Ilut, SaijoZ

Pl θl0 dl;

ξ10 = Γ(ξ2 , X )Learning, Confidence, and Business Cycles(5)

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Recursive competitive equilibrium
Firm l’s problem at stage 1
v1f (z̃l , Σl , Kl ; ξ1 , X ) = max E ∗ [v2f (z̃ˆl0 , Σ0l , Kl ; ξˆ2 , X )]
Hl

s.t.

(6)

Updating rules of Kalman filter

Firm l’s problem at stage 2: v2f (z̃l0 , Σ0l , Kl ; ξ2 , X ) equals


Z

f
0
0
0
0
0
max λ (Yl − WHl − Il ) + β v1 z̃l , Σl , Kl ; ξ1 , X dF (X |X )Il

s.t. Kl0 = (1 − δ)Kl + Il ; ξ10 = Γ(ξ2 , X )(7)

ReturnIlut, Saijo

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Parameters
γ
α
β
η
δ0
δ2 /δ1
ηa
Σ̄
ḡ
ρz
σz
ρA
ρg
ρσ

Labor augmenting tech growth
Capital share
Discount factor
Inverse Frisch elasticity
SS depreciation
Convexity of depreciation
Size of entropy constraint
SS posterior variance(Kalman gain)
SS share of gov spending
Idiosyncratic TFPIdiosyncratic TFPAggregate TFPGovernment spending
Firm-level dispersion

1.004
0.3
0.99
0
0.025
0.15
0.4
0.1
0.47
0.2
0.5
0.4
0.95
0.95
0.85

ReturnIlut, Saijo

Learning, Confidence, and Business Cycles

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HP-filtered moments (TFP shock only)σ(y )σ(c)/σ(y )σ(i)/σ(y )σ(h)/σ(y )σ(c, y )σ(i, y )σ(h, y )σ(y , τl )σ(h, τl )σ(yt , yt−1 )σ(ht , ht−1 )σ(∆yt , ∆yt−1 )σ(∆ht , ∆ht−1 )

Data
1.11
0.72
3.57
1.64
0.86
0.92
0.88
-0.83
-0.97
0.89
0.95
0.39
0.71

Our model
1.11
0.11
2.95
1.02
0.72
0.99
0.99
-0.95
-0.95
0.87
0.88
0.44
0.52

RE
0.49
0.17
3.23
0.86
0.85
0.99
0.99
0
0
0.66
0.66
-0.06
-0.06

Note: We choose the st. dev of aggregate TFP shock so that the output st. dev in themodel matches the data.
ReturnIlut, Saijo

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Government spending multiplier

BaselineRE

2.2
2dYt/dGt

1.8
1.6
1.4
1.2
1
0.8
0.6
2

4

6

8

10

12

14

16

18

20

ReturnIlut, Saijo

Learning, Confidence, and Business Cycles

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Law of large numbers for risky random variables
firm 1

firm 2?

?

....

firm l?

....

firm N?

cross-sectional average:z̄
Ilut, Saijo

Learning, Confidence, and Business Cycles

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Law of large numbers for ambiguous random variables
firm 1

firm 2....

firm l....

firm N√2ηa ρz Σlcross-sectional average:???
Ilut, Saijo

Learning, Confidence, and Business Cycles

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Law of large numbers for ambiguous random variables
firm 1?

firm 2?

....

firm l?

....

firm N?

cross-sectional
R ∗ average:z̄ + µl,t dl
Ilut, Saijo

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Law of large numbers for ambiguous random variables
firm 1

firm 2?

?

....

firm l?

....

firm N?

cross-sectional
R ∗ average:z̄ + µl,t dl
Ilut, Saijo

Learning, Confidence, and Business Cycles

14 / 16

Law of large numbers for ambiguous random variables
firm 1

firm 2

?

?

....

....

firm l?

firm N?

cross-sectional
average:RR[z̄ − al,t dl, z̄ + al,t dl]

Ilut, Saijo

Learning, Confidence, and Business Cycles

15 / 16

Law of large numbers for ambiguous random variables
firm 1

firm 2?

?

....

firm l?

....

firm N?

cross-sectional
average:RR[z̄ − al,t dl, z̄ + al,t dl]

Ilut, Saijo

Learning, Confidence, and Business Cycles

16 / 16