Beauty Contests and the Term Structure Martin Ellison 1 Andreas - - PowerPoint PPT Presentation

Beauty Contests and the Term Structure

Martin Ellison1 Andreas Tischbirek2

1University of Oxford, NuCamp & CEPR 2HEC Lausanne, University of Lausanne

13 June 2019 3rd MMCN Conference, Goethe University Frankfurt

Motivation

Can information frictions help to solve the bond premium puzzle?

• Generally approaches in the DSGE literature
• take risk as given and manipulate preferences
• take preferences as given and add sources of fundamental risk
• What we do
• take fundamental risk and preferences as given and explore

importance of information and expectations

• Key ingredients of our preferred model
• Households observe a noisy signal rather than a fundamental
• Strategic complementarity in forecast formation

Literature

Bond premium puzzle

• Recursive preferences

Epstein and Zin (1989), Rudebusch and Swanson (2012), van Binsbergen et al. (2012)

• Model uncertainty

Barillas et al. (2009), Collard et al. (2018)

• Long-run risk

Bansal and Yaron (2004), Croce (2014)

• Rare disasters

Rietz (1988), Barro (2006)

• Habit Formation

Jermann (1998), Abel (1999), Rudebusch and Swanson (2008) Information in strategic settings and volatility

• Use of public information

Morris and Shin (2002), Angeletos and Pavan (2007)

• Volatility from information frictions

Angeletos and La’O (2013), Bergemann et al. (2015), Angeletos et al. (2018)

Overview

1 Decomposing term premia—the role of information 2 Simplified model with different information structures 3 More quantitative RBC model and estimation

Decomposing the Term Premium

Household side of generic DSGE model

• Representative household maximises

Et

∞

• s=t

βs−tu(cs, ls) subject to Ptct +

N

• n=1

p(n)

t

b(n)

t

= wtlt +

N

• n=1

p(n−1)

t

b(n)

t−1 + Pt + Tt

• b(n)

t

—non-contingent default-free zero-coupon bonds with maturity n = 1, 2, . . . , N

• p(n)

t

—bond price (note p(0)

t

= 1)

Decomposing the Term Premium

• Interior solution

p(n)

t

= Etmt+1p(n−1)

t+1 ,

n ∈ {1, 2, . . . , N} with stochastic discount factor (SDF) mt+1 ≡ β uc(ct+1, lt+1) uc(ct, lt) 1 Πt+1

• Hypothetical “risk-neutral price”

˜ p(n)

t

= e−itEt ˜ p(n−1)

t+1 ,

n ∈ {1, 2, . . . , N}

• Term premium (in per-period terms)

ψ(n)

t

= 1 n

• ˜

p(n)

t

− p(n)

t

Decomposing the Term Premium

Example – Two-period bond

• Term premium

ψ(2)

t

= −1 2Covt(mt+1, p(1)

t+1)

• Mean term premium (by law of total covariance)

Eψ(2)

t

= 1 2 [−Cov(mt+1, mt+2) + Cov(Etmt+1, Et+1mt+2)]

ψ(n)

t

Eψ(n)

t

Simple Economy

• Production function of representative firm

yt = At¯ l1−α

• Technology at ≡ ln At follows

at = xt + ηt, ηt ∼ N(0, σ2

η)

xt = ρxt−1 + εt, εt ∼ N(0, σ2

ε)

• Household utility is logarithmic
• SDF can be expressed as

mt+1 = β ct+1 ct −1 = β At+1¯ l1−α At¯ l1−α −1 ≈ β (1 + at − at+1)

Simple Economy - Full Information

0.2 0.4 0.6 0.8 1

;

1 2 3 4 5 6 7 8 9 #10-5 !Cov(mt+1; mt+2) Cov # E(mt+1jI$

t ); E(mt+2jI$ t+1)

$ EA(2)

F I

0.2 0.4 0.6 0.8 1

Var(xt)=Var(at)

1 2 3 4 5 6 7 8 9 #10-5

Figure: Components of the mean term premium (n = 2).

β = 0.99, Var(at) = 0.012. Left panel: Var(xt)/Var(at) = 0.9. Right panel: ρ = 0.8.

Simple Economy - Partial Information

0.2 0.4 0.6 0.8 1

;

1 2 3 4 5 6 #10-5 !Cov(mt+1; mt+2) Cov # E(mt+1jI$

t ); E(mt+2jI$ t+1)

$ EA(2)

F I

Cov # E(mt+1jI0

t); E(mt+2jI0 t+1)

$ EA(2)

P I

0.2 0.4 0.6 0.8 1

Var(xt)=Var(at)

1 2 3 4 5 6 #10-5

Figure: Components of the mean term premium (n = 2).

β = 0.99, Var(at) = 0.012. Left panel: Var(xt)/Var(at) = 0.9. Right panel: ρ = 0.8.

Simple Economy - Noisy Information

0.2 0.4 0.6 0.8 1

;

1 2 3 4 5 6 #10-5 !Cov(mt+1; mt+2) Cov # E(mt+1jI$

t ); E(mt+2jI$ t+1)

$ EA(2)

F I

Cov # E(mt+1jI00

t ); E(mt+2jI00 t+1)

$ EA(2)

NI

0.2 0.4 0.6 0.8 1

Var(xt)=Var(at)

1 2 3 4 5 6 #10-5

Figure: Components of the mean term premium (n = 2).

β = 0.99, Var(at) = 0.012, σ2

ξ = Var(at)/2,

Left panel: Var(xt)/Var(at) = 0.9. Right panel: ρ = 0.8.

Simple Economy - Beauty Contest

Information structure

• Agent i ∈ [0, 1] receives signal si,t allowing them to deduce

xn

i,t = xt + nt + ni,t

→ Agents heterogeneously informed now

• Noise persistent so that

xn

i,t = ρxn i,t−1 + εn i,t

where εn

i,t = εt + ξt + ζi,t

• Forming expectation about mt+1 requires inferring xt from xn

i,t

Simple Economy - Beauty Contest

Strategic complementarity

• Strategic forecast formed by minimising

(1−ω)E

• (ˆ

E (xt|Ii,t) − xt)2|Ii,t

• −ωE

1 ˆ E (xt|Ij,t) dj

• Ii,t
• ˆ

E (xt|Ii,t)

• First-order condition of i

ˆ E (xt|Ii,t) = E (xt|Ii,t) + ω 2(1 − ω)E 1 ˆ E (xt|Ij,t) dj

• Ii,t
• where

E(xt|Ii,t) =

• σ2

ε

σ2

ε + σ2 ξ + σ2 ζ

• xn

i,t

Equilibrium

Simple Economy - Beauty Contest

0.2 0.4 0.6 0.8 1

;

1 2 3 4 5 6 #10-5 0.2 0.4 0.6 0.8 1

Var(xt)=Var(at)

1 2 3 4 5 6 #10-5 !Cov(mt+1; mt+2) Cov # E(mt+1jI$

t ); E(mt+2jI$ t+1)

$ EA(2)

F I

Cov h ^ E(mt+1jIi;t); ^ E(mt+2jIi;t+1) i EA(2)

BC

Figure: Components of the term premium (n = 2).

β = 0.99, Var(at) = 0.012, σ2

ξ = σ2 ζ = 0.05σ2 ε and ω = 0.5.

Left panel: Var(xt)/Var(at) = 0.9. Right panel: ρ = 0.8.

More General Business Cycle Model

• As simple model, but now
• Labour supply endogenous (competitive labour market)

yt = Atl1−α

t

• Household utility of more general form

u(ct, lt) = 1 1 − σ

• ct − χ0

l1+χ

t

1 + χ 1−σ

• Solution based on exact SDF rather than an approximation

thereof (model solved numerically now) mt+1 = β  ct+1 − χ0

l1+χ

t+1

1+χ

ct − χ0

l1+χ

t

1+χ

 

−σ

More General Business Cycle Model

• Bond prices

p(n)

i

(xn

i,t, ηt)

= E

• mt+1p(n−1)

i

(xn

i,t+1, ηt+1)|Ii,t

• =
• R
• R
• R
• R

β    γ1(ext+1+ηt+1)γ2 −

χ0 1+χ

(1−α)

χ0

ext+1+ηt+1 γ2 γ1(ext+ηt )γ2 −

χ0 1+χ

(1−α)

χ0

ext+ηt γ2   

−σ

p(n−1)

i

(xn

i,t+1, ηt+1) × dFx,x′,(xn)′|xn(xt, xt+1, xn i,t+1|xn i,t) dFη(ηt+1)

where

xt|xn

i,t ∼ N

• θxn

i,t,

σ2

ε

1 − ρ2

• 1 −

σ2

ε

σ2

ε + σ2 ξ + σ2 ζ

• Num. quadrature

Precision

Quantitative Analysis

Parameter Value Description Target (Data) β 0.9997 Discount factor i(4) = 0.0205 − 0.0191 (Treasury yields, Adrian et al. (2013), 4/1/99 - 30/6/17; Inflation expectations, SPF, 1999q1-2017q2) α 0.384 1 - Labour share 1 − α = 0.6160 (Share of labour compensation in GDP, Penn World Table, 1999-2014) χ 0.708 Inverse Frisch elasticity Var(ln lt)/Var(ln ct) = 0.3428 (Consumption of nondurables and services, BEA; Population and hours, BLS, 1999q1-2017q2) χ0 2.04 Labour utility weight l = 1/3

Table: Calibrated parameters

Quantitative Analysis

1995 2000 2005 2010 2015

Year

0.01 0.02 0.03 0.04 0.05

Forecasted productivity growth

Figure: Forecasted annual productivity growth over next ten years (SPF). Solid line: Median. Dashed lines: 25th and 75th percentile.

Quantitative Analysis

Parameter Estimate 95% Confidence Interval Description ρ 0.90 [0.81, 0.99] Shock persistence σε 2.0 × 10−3 [9.7 × 10−4, 3.1 × 10−3] SD of innovation to persistent tech. component ση 8.0 × 10−4 [0, 2.4 × 10−3] SD of i.i.d. transitory tech. component σξ 9.9 × 10−5 [9.8 × 10−5, 1.0 × 10−4] SD of innovation to common noise component σζ 2.2 × 10−3 [1.9 × 10−3, 2.5 × 10−3] SD of innovation to idiosyncratic noise component ω 0.80 [0.78, 0.82] Strategic complementarity σ 6.0 [5.7, 6.3] Coefficient of relative risk aversion

Table: Estimated parameters

Quantitative Analysis

Moment US data Estimated Model with full Model with 1999q1-2017q2 model information ω = 0 Targeted Var ˆ γ50

t

• 1.52 × 10−5

1.42 × 10−5 2.11 × 10−7 4.68 × 10−8 Cov ˆ γ50

t , ˆ

γ50

t−4

• 1.25 × 10−5

9.29 × 10−6 1.38 × 10−7 3.06 × 10−8 E ˆ γ75

t

− ˆ γ25

t

• 5.32 × 10−3

5.40 × 10−3 3.10 × 10−4 Eψ(4)

t

8.2 bps 8.2 bps 2.6 bps 1.0 bps Not targeted Eψ(8)

t

21.2 bps 16.0 bps 5.4 bps 2.0 bps Eψ(12)

t

34.5 bps 21.1 bps 7.6 bps 2.7 bps Eψ(16)

t

46.7 bps 24.4 bps 9.3 bps 3.3 bps Eψ(20)

t

57.2 bps 26.7 bps 10.7 bps 3.7 bps

Table: Data and model moments

Quantitative Analysis

4 6 8 10 12 14 16 18 20

Quarters

1 2 3 4 5 6

Average Term Premium (on prices)

#10-3 Data (1999q1 - 2017q2) Model (Full information) Model (Estimated model)

Figure: Term Premia in the data and in the models

Wrapping Up

• Term premia can be broken down into autocovariances of

realised and expected SDFs

• Standard signal extraction problems in representative agent

framework generally do not contribute to sizeable term premia

• Term premia arise through expectational components in a

model with

• autocorrelated noise
• a motive not to discount the noisy signal (beauty contest)
• Significant improvement in quantitative fit of small-scale

model through information frictions

Bond yield estimates

1999 2003 2007 2011 2015 0.01 0.02 0.03 0.04 0.05 0.06 0.07 i(1) i(5) i(10)

Figure: Nominal Treasury Zero-Coupon Yields (Adrian, Crump & Moench, 2013)

Bond yield estimates

1970 1980 1990 2000 2010 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 i(1) i(5) i(10)

Figure: Nominal Treasury Zero-Coupon Yields (Adrian, Crump & Moench, 2013)

Bond yield estimates

2 4 6 8 10

Maturity (years)

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

• Avrg. nominal yield (ACM, 2013)
• Avrg. nominal risk premium (ACM, 2013)
• Avrg. real risk premium (AACMY, 2016)

Figure: Zero-Coupon Treasury Yield Curve (4/1/1999 - 30/6/2017)

Decomposing the Term Premium

Proposition

For n ∈ {2, 3, . . .}, ψ(n)

t

is given by

ψ(n)

t

= 1 n

n−2

• k=0

ιt(k)  −Covt  mt+k+1,

n−2

• j=k

mt+j+2   +Covt  Et+kmt+k+1,

n−2

• j=k

Et+j+1mt+j+2    

where

ιt(k) ≡

• 1

for k = 0 k−1

j=0 Ete−it+j

• therwise

Decomposing the Term Premium

Lemma

For n ∈ {2, 3, . . .}, Eψ(n)

t

satisfies

Eψ(n)

t

= 1 n

n−2

• k=0

  E [ιt(k)]  −Cov  mt+k+1,

n−2

• j=k

mt+j+2   + Cov  Etmt+k+1, Et

n−2

• j=k

mt+j+2   + Cov  Et+kmt+k+1,

n−2

• j=k

Et+j+1mt+j+2   − Cov  Etmt+k+1, Et

n−2

• j=k

Et+j+1mt+j+2     + Γι   

where

Γι = Cov  ιt(k), −Covt  mt+k+1,

n−2

• j=k

mt+j+2   + Covt  Et+kmt+k+1,

n−2

• j=k

Et+j+1mt+j+2    

Simple Economy - Full Information

• Technology process observed, especially xt, ηt ⊂ I∗

t

• Conditional expectation of SDF

E(mt+1|I∗

t ) = β(1 + at − ρxt)

• Expectational component of term premium

Cov

• E(mt+1|I∗

t ), E(mt+2|I∗ t+1)

• = β2 ρ(1 − ρ)

1 + ρ σ2

ε

Simple Economy - Partial Information

• at ⊂ I′

t but xt and ηt unobserved at all times

• Conditional expectation of SDF

E(mt+1|I′

t) = β

• 1 + at − ρE(xt|I′

t)

• xt has to be inferred from “noisy signal” at = xt + ηt
• Kalman filter

ρE(xt|I′

t) = ρE(xt|I′ t−1) + Kt

• at − E(xt|I′

t−1)

• Σt+1 = ρ(ρ − Kt)Σt + σ2

ε

Kt = ρΣt Σt + σ2

η

where Σt ≡ Var(xt|I′

t−1), x0 ∼ N(0, Σ)

Simple Economy - Noisy Information

• Signal about technology observed, but as /

∈ I

′′

t ∀s, t

st = at + ξt, ξt ∼ N(0, σ2

ξ)

• Conditional expectation of SDF

E

• mt+1|I

′′

t

• = β
• 1 + (1 − ρ)E
• xt|I

′′

t

• xt has to be inferred from signal st = xt + ηt + ξt
• Kalman gain now

K s

t =

ρΣs

t

Σs

t + σ2 η + σ2 ξ + 2σηξ

Simple Economy - Beauty Contest

• Symmetric linear equilibrium

ˆ E (xt|Ii,t) = θxn

i,t,

θ = (1 − ω)σ2

ε

(1 − ω)(σ2

ε + σ2 ξ + σ2 ζ) − ω 2 (σ2 ε + σ2 ξ)

0.1 0.2 0.3 0.4 0.5

<2

9=(<2 " + <2 9 + <2 1)

0.5 0.6 0.7 0.8 0.9 1

3 ! = 0 ! = 0:1 ! = 0:2 ! = 0:3 ! = 0:4 ! = 0:5

Figure: θ against fraction of variance in xn

i,t from public noise.

σ2

ε/(σ2 ε + σ2 ξ + σ2 ζ) = 0.5

More General Business Cycle Model - Computation

• Exogenous processes {xt}, {ηt}, {εt},
• xn

i,t

• , {ξt + ζi,t}

discretised into Markov chains with ˜ N = 15 nodes

• Probability weights approximating distribution of xt|xn

i,t at

each node of

• xn

i,t

• found via moment matching procedure

⇒ comparable to Gauss-Hermite quadrature

min

πi

• 

 

˜ N

• j=1

πi,j

• ˜

xx,j − θ˜ xxn

i ,i

k   

˜ N−1 k=0

−

• E
• (xt − θ˜

xxn

i ,i)k|xn

i,t

˜

N−1 k=0

• 2

such that πi,j ∈ [0, 1] ∀j

• Calculate bond valuation up to desired order at each point in

state space by recursive integration using discretised distributions (linear interpolation for xn

i,t+1)

• Analogous procedure for “risk-neutral price”
• Simulate model for T = 1, 000, 000 periods and approximate

unconditional term premium using sample mean

More General Business Cycle Model - Computation

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02

xn

i

10-8 10-7 10-6 10-5 10-4 10-3 10-2

Error(xn

i ; !)

! = 0 ! = 0:3 ! = 0:6

Figure: Approximation error in distribution of xt|xn

i,t