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

Beauty Contests and the Term Structure Martin Ellison 1 Andreas Tischbirek 2 1 University of Oxford, NuCamp & CEPR 2 HEC Lausanne, University of Lausanne 13 June 2019 3 rd 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 ∞ � β s − t u ( c s , l s ) E t s = t subject to N N � p ( n ) b ( n ) � p ( n − 1) b ( n ) P t c t + = w t l t + t − 1 + P t + T t t t t n =1 n =1 • b ( n ) —non-contingent default-free zero-coupon bonds with t maturity n = 1 , 2 , . . . , N • p ( n ) —bond price (note p (0) = 1) t t

Decomposing the Term Premium • Interior solution p ( n ) = E t m t +1 p ( n − 1) t +1 , n ∈ { 1 , 2 , . . . , N } t with stochastic discount factor (SDF) m t +1 ≡ β u c ( c t +1 , l t +1 ) 1 u c ( c t , l t ) Π t +1 • Hypothetical “risk-neutral price” p ( n ) p ( n − 1) = e − i t E t ˜ ˜ t +1 , n ∈ { 1 , 2 , . . . , N } t • Term premium (in per-period terms) = 1 � � ψ ( n ) p ( n ) − p ( n ) ˜ t t t n

Decomposing the Term Premium Example – Two-period bond • Term premium = − 1 ψ (2) 2 Cov t ( m t +1 , p (1) t +1 ) t • Mean term premium (by law of total covariance) = 1 E ψ (2) 2 [ − Cov ( m t +1 , m t +2 ) + Cov ( E t m t +1 , E t +1 m t +2 )] t ψ ( n ) E ψ ( n ) t t

Simple Economy • Production function of representative firm y t = A t ¯ l 1 − α • Technology a t ≡ ln A t follows η t ∼ N (0 , σ 2 a t = x t + η t , η ) ε t ∼ N (0 , σ 2 x t = ρ x t − 1 + ε t , ε ) • Household utility is logarithmic • SDF can be expressed as � − 1 � c t +1 m t +1 = β c t � − 1 � A t +1 ¯ l 1 − α = β A t ¯ l 1 − α ≈ β (1 + a t − a t +1 )

Simple Economy - Full Information # 10 -5 # 10 -5 ! Cov(m t+1 ; m t+2 ) 9 9 # $ Cov E(m t+1 jI $ t ) ; E(m t+2 jI $ t+1 ) E A (2) 8 8 F I 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ; Var(x t ) = Var(a t ) Figure: Components of the mean term premium ( n = 2). β = 0 . 99, Var ( a t ) = 0 . 01 2 . Left panel: Var ( x t ) / Var ( a t ) = 0 . 9. Right panel: ρ = 0 . 8.

Simple Economy - Partial Information 6 # 10 -5 6 # 10 -5 ! Cov(m t+1 ; m t+2 ) # $ Cov E(m t+1 jI $ t ) ; E(m t+2 jI $ t+1 ) 5 E A (2) 5 F I # $ Cov E(m t+1 jI 0 t ) ; E(m t+2 jI 0 t+1 ) E A (2) P I 4 4 3 3 2 2 1 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ; Var(x t ) = Var(a t ) Figure: Components of the mean term premium ( n = 2). β = 0 . 99, Var ( a t ) = 0 . 01 2 . Left panel: Var ( x t ) / Var ( a t ) = 0 . 9. Right panel: ρ = 0 . 8.

Simple Economy - Noisy Information 6 # 10 -5 6 # 10 -5 ! Cov(m t+1 ; m t+2 ) # $ E(m t+1 jI $ t ) ; E(m t+2 jI $ Cov t+1 ) 5 5 E A (2) F I # $ E(m t+1 jI 00 t ) ; E(m t+2 jI 00 Cov t+1 ) E A (2) NI 4 4 3 3 2 2 1 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Var(x t ) = Var(a t ) ; Figure: Components of the mean term premium ( n = 2). β = 0 . 99, Var ( a t ) = 0 . 01 2 , σ 2 ξ = Var ( a t ) / 2, Left panel: Var ( x t ) / Var ( a t ) = 0 . 9. Right panel: ρ = 0 . 8.

Simple Economy - Beauty Contest Information structure • Agent i ∈ [0 , 1] receives signal s i , t allowing them to deduce x n i , t = x t + n t + n i , t → Agents heterogeneously informed now • Noise persistent so that x n i , t = ρ x n i , t − 1 + ε n i , t where ε n i , t = ε t + ξ t + ζ i , t • Forming expectation about m t +1 requires inferring x t from x n i , t

Simple Economy - Beauty Contest Strategic complementarity • Strategic forecast formed by minimising �� 1 � � � � (ˆ E ( x t |I i , t ) − x t ) 2 |I i , t ˆ ˆ � (1 − ω ) E − ω E E ( x t |I j , t ) dj � I i , t E ( x t |I i , t ) � 0 • First-order condition of i �� 1 � ω � ˆ ˆ � E ( x t |I i , t ) = E ( x t |I i , t ) + 2(1 − ω ) E E ( x t |I j , t ) dj � I i , t � 0 where � � σ 2 ε x n E ( x t |I i , t ) = i , t σ 2 ε + σ 2 ξ + σ 2 ζ Equilibrium

Simple Economy - Beauty Contest 6 # 10 -5 6 # 10 -5 ! Cov(m t+1 ; m t+2 ) # $ E(m t+1 jI $ t ) ; E(m t+2 jI $ Cov t+1 ) 5 5 E A (2) h i F I ^ E(m t+1 jI i ; t ) ; ^ Cov E(m t+2 jI i ; t+1 ) E A (2) 4 4 BC 3 3 2 2 1 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ; Var(x t ) = Var(a t ) Figure: Components of the term premium ( n = 2). β = 0 . 99, Var ( a t ) = 0 . 01 2 , σ 2 ξ = σ 2 ζ = 0 . 05 σ 2 ε and ω = 0 . 5. Left panel: Var ( x t ) / Var ( a t ) = 0 . 9. Right panel: ρ = 0 . 8.

More General Business Cycle Model • As simple model, but now • Labour supply endogenous (competitive labour market) y t = A t l 1 − α t • Household utility of more general form � 1 − σ � l 1+ χ 1 t u ( c t , l t ) = c t − χ 0 1 − σ 1 + χ • Solution based on exact SDF rather than an approximation thereof (model solved numerically now) − σ l 1+ χ    c t +1 − χ 0 t +1 1+ χ m t +1 = β  l 1+ χ c t − χ 0 t 1+ χ

More General Business Cycle Model • Bond prices p ( n ) ( x n i , t , η t ) i � m t +1 p ( n − 1) � ( x n = E i , t +1 , η t +1 ) |I i , t i � (1 − α ) � � � � � γ 2 − σ  χ 0  γ 1 ( e x t +1 + η t +1 ) γ 2 − e x t +1 + η t +1  1+ χ χ 0  = β � (1 − α ) � γ 2 χ 0 γ 1 ( e x t + η t ) γ 2 − e x t + η t   1+ χ χ 0 R R R R p ( n − 1) ( x n i , t +1 , η t +1 ) × dF x , x ′ , ( x n ) ′ | x n ( x t , x t +1 , x n i , t +1 | x n i , t ) dF η ( η t +1 ) i where � � �� σ 2 σ 2 ε ε x t | x n θ x n i , t ∼ N i , t , 1 − ε + σ 2 ξ + σ 2 1 − ρ 2 σ 2 ζ Num. quadrature Precision

Quantitative Analysis Parameter Value Description Target (Data) i (4) = 0 . 0205 − 0 . 0191 β 0.9997 Discount factor (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 l t ) / Var (ln c t ) = 0 . 3428 (Consumption of nondurables and services, BEA; Population and hours, BLS, 1999q1-2017q2) χ 0 2.04 Labour utility weight l = 1 / 3 Table: Calibrated parameters

Quantitative Analysis 0.05 Forecasted productivity growth 0.04 0.03 0.02 0.01 0 1995 2000 2005 2010 2015 Year 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 2 . 0 × 10 − 3 [9 . 7 × 10 − 4 , 3 . 1 × 10 − 3 ] σ ε SD of innovation to persistent tech. component 8 . 0 × 10 − 4 [0 , 2 . 4 × 10 − 3 ] σ η SD of i.i.d. transitory tech. component 9 . 9 × 10 − 5 [9 . 8 × 10 − 5 , 1 . 0 × 10 − 4 ] σ ξ SD of innovation to common noise component 2 . 2 × 10 − 3 [1 . 9 × 10 − 3 , 2 . 5 × 10 − 3 ] σ ζ 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 γ 50 1 . 52 × 10 − 5 1 . 42 × 10 − 5 2 . 11 × 10 − 7 4 . 68 × 10 − 8 Var � � ˆ t Cov � γ 50 γ 50 � 1 . 25 × 10 − 5 9 . 29 × 10 − 6 1 . 38 × 10 − 7 3 . 06 × 10 − 8 ˆ t , ˆ t − 4 E � γ 75 γ 25 � 5 . 32 × 10 − 3 5 . 40 × 10 − 3 3 . 10 × 10 − 4 ˆ − ˆ 0 t t E ψ (4) 8.2 bps 8.2 bps 2.6 bps 1.0 bps t Not targeted E ψ (8) 21.2 bps 16.0 bps 5.4 bps 2.0 bps t E ψ (12) 34.5 bps 21.1 bps 7.6 bps 2.7 bps t E ψ (16) 46.7 bps 24.4 bps 9.3 bps 3.3 bps t E ψ (20) 57.2 bps 26.7 bps 10.7 bps 3.7 bps t Table: Data and model moments