Risk and Ambiguity in Models of Business Cycles Dave Backus, Axelle Ferriere, and Stan Zin Carnegie-Rochester-NYU Conference April 18, 2014 This version: April 20, 2014 Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
What happened? What we see ◮ Much deeper recession than usual ◮ Longer recovery — maybe slower, too Like Kydland-Prescott with productivity shocks? ◮ Relative magnitudes look right ◮ Comovements look right ◮ But... measured productivity didn’t fall very much More What’s missing? Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
What happened? Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
What happened? Marco Buti, Director General of the European Commission Economic theory suggests that uncertainty has a detrimental effect on economic activity by giving agents the incentive to postpone investment, consumption and employment decisions until uncertainty is resolved, and by pushing up the cost of capital through increased risk premia. Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
What happened? Nick Bloom The onset of the Great Recession was accompanied by a massive surge in uncertainty . The size of this uncertainty shock was so large it potentially accounted for around one third of the 9% drop in GDP versus trend during 2008-2009. Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
What we do Take a streamlined business cycle model Ask: How does uncertainty affect the dynamics of output, consumption, and investment? ◮ How are business cycle properties affected by fluctuations in uncertainty? ◮ Can “uncertainty shocks” magnify downturns or produce slow recoveries? Potential issues: Barro-King, Tallarini Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Modeling ingredients Streamlined business cycle model - Recursive preferences - Unit root in productivity - Fixed labor supply With fluctuations in uncertainty - Risk (stochastic volatility) - Ambiguity (unobservable long-term growth) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Preview of results Solution method Transparent loglinear approximation Checked numerically Business cycle properties Separation property Intertemporal elasticity of substitution vs. risk aversion Risk vs. ambiguity Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Risk and uncertainty Recursive references U t = V [ c t , µ t ( U t +1 )] t + βµ t ( U t +1 ) ρ ] 1 /ρ [(1 − β ) c ρ = [ E t ( U α t +1 )] 1 /α µ t ( U t +1 ) = V , µ t homogeneous of degree one, α, ρ < 1, σ = 1 / (1 − ρ ) Stochastic structure of productivity a t log ( a t / a t − 1 ) = log g + e ⊤ x t (“productivity growth”) log g t = Ax t + v 1 / 2 x t +1 = Bw 1 t +1 (“news”) t v t +1 = (1 − ϕ v ) v + ϕ v v t + τ w 2 t +1 (“risk”) ( w 1 t , w 2 t ) = iid standard normals Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Stationarity Bellman equation � � J ( k t , x t , v t , a t ) = max V c t , µ t [ J ( k t +1 , x t +1 , v t +1 , a t +1 )] c t s.t. k t +1 = f ( k t , a t n ) − c t plus shocks & initial conditions Assume f hd 1: f ( k , an ) = k ω ( an ) 1 − ω + (1 − δ ) k Rescaled Bellman equation [˜ k t = k t / a t , ˜ c t = c t / a t ] J (˜ c t , µ t [ g t +1 J (˜ � � k t , x t , v t , 1) = max V ˜ k t +1 , x t +1 , v t +1 , 1)] ˜ c t g t +1 ˜ k t +1 = f (˜ s.t. k t , n ) − ˜ c t plus shocks & initial conditions More Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Recursive business cycle model Bellman equation t + βµ t ( g t +1 J t +1 ) ρ ] 1 /ρ c ρ J t = max [(1 − β )˜ ˜ c t k t +1 = [ f (˜ ˜ s.t. k t , n ) − ˜ c t ] / g t +1 plus shocks & initial conditions First-order and envelope conditions c ρ − 1 βµ t ( g t +1 J t +1 ) ρ − α E t [( g t +1 J t +1 ) α − 1 J kt +1 ] (1 − β )˜ = t J 1 − ρ βµ t ( g t +1 J t +1 ) ρ − α E t [( g t +1 J t +1 ) α − 1 J kt +1 ] f kt J kt = t c ρ − 1 J ρ − 1 ⇒ (1 − β )˜ = J kt / f kt t t Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Loglinearization strategy Goal: loglinear decision rule and (controlled) law of motion h ck log ˜ k t + h ⊤ log ˜ c t = cx x t + h cv v t log ˜ h kk log ˜ k t + h ⊤ k t +1 = kx x t + h kv v t − log g t +1 Dynamic programming version of Campbell (JME, 1994) Loglinearization around the stochastic steady-state Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Loglinear approximation Guess loglinear value function p k log ˜ k t + p ⊤ log J t = x x t + p v v t + p 0 Loglinearize capital’s marginal product and law of motion λ r log ˜ log f kt = k t + λ 0 log ˜ λ k log ˜ k t +1 = k t − λ c log ˜ c t + λ 1 − log g t +1 More Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Loglinearized policies 1 The law of motion for capital is log ˜ h k 0 + h kk log ˜ k t + h ⊤ k t +1 = kx x t + h kv v t − log g t +1 h kk = λ k − λ c [ σ + (1 − σ ) p k + σλ r ] h ⊤ kx = − λ c (1 − σ ) p ⊤ x h kv = − λ c (1 − σ ) p v Tallarini ( σ = 1) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Loglinearized policies 2 The law of motion for capital is log ˜ h k 0 + h kk log ˜ k t + h ⊤ k t +1 = kx x t + h kv v t − log g t +1 h kk = λ k − λ c [ σ + (1 − σ ) p k + σλ r ] h ⊤ kx = − λ c (1 − σ ) p ⊤ x h kv = − λ c (1 − σ ) p v Separation property ◮ h kk is independent of risk and risk aversion. � �� � ρ p k − 1 − λ r = ρ p k − 1 λ k − λ c [(1 − σ ) p k + σ (1 + λ r )] . Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Loglinearized policies 3 The law of motion for capital is log ˜ h k 0 + h kk log ˜ k t + h ⊤ k t +1 = kx x t + h kv v t − log g t +1 h kk = λ k − λ c [ σ + (1 − σ ) p k + σλ r ] h ⊤ kx = − λ c (1 − σ ) p ⊤ x h kv = − λ c (1 − σ ) p v Exposure to shocks ( x t , v t ) ◮ If σ < 1, an increase in v t lowers consumption ◮ Magnitude depends on α ( p v ∝ α ) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Loglinear approximation 3.8 0.15 numerical solution 3.7 0.12 3.6 3.5 density measure 0.09 log k(t+1) 3.4 0.06 3.3 3.2 0.03 3.1 3 0 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 log k(t) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Loglinear approximation 3.8 0.15 numerical solution loglinear approximation 3.7 0.12 3.6 3.5 density measure 0.09 log k(t+1) 3.4 0.06 3.3 3.2 0.03 3.1 3 0 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 log k(t) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Recursive business cycle model: benchmark parameters Parameter Value Comment Preferences ρ − 1 arbitrary α − 9 arbitrary β — chosen to hit k / y = 10 (quarterly) Technology ω 1/3 Kydland and Prescott (1982, Table I), rounded off δ 0.025 Kydland and Prescott (1982, Table I) Productivity growth log g 0.004 Tallarini (2000, Table 4) 1 normalization e A 0 no predictable component (“news”) B 1 normalization v 1 / 2 0.015 Tallarini (2000, Table 4), rounded off ϕ v 0.95 arbitrary 0 . 74 × 10 − 5 τ makes v three standard deviations from zero Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Recursive business cycle model: properties US Data Additive Recursive Recursive Parameter changes ρ = − 1 ρ = − 1 ρ = 1 / 3 α = − 1 α = − 9 α = − 9 Standard deviations (%) Output growth 1.04 1.00 1.00 1.01 Consumption growth 0.55 0.69 0.71 1.34 Investment growth 2.79 1.91 1.61 4.54 Correlations with output growth Consumption growth 0.52 0.99 0.96 –0.23 Investment growth 0.65 0.99 0.94 0.35 Asset returns Maximum risk premium > 0 . 01 ?? Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
What did we learn? Separation property - Responses to endogenous state variables are independent of risk aversion IES vs. risk-aversion - IES changes signs and correlations - Risk aversion changes magnitudes Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Risk and ambiguity Divide the state in two: s t = ( s 1 t , s 2 t ) Ambiguity (Klibanoff, Marinacci, & Mukerji; Ju & Miao) risk = p 1 t ( s 1 t +1 | s 2 t +1 , I t ) ambiguity = p 2 t ( s 2 t +1 |I t ) Two-part certainty equivalent � 1 /α E 1 t ( U α � µ 1 t ( U t +1 ) = t +1 ) � 1 /γ E 2 t [ µ 1 t ( U t +1 )] γ ) � µ 2 t [ µ 1 t ( U t +1 )] = α controls risk aversion, γ < α controls ambiguity aversion Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
Loglinear structure of consumption equivalents Log of CE with risk log c ∼ N ( κ 1 , κ 2 ) log E ( c α ) ακ 1 + α 2 κ 2 / 2 = log µ ( c ) = κ 1 + ακ 2 / 2 Log of CE with ambiguity about mean log c ∼ N ( ν, κ 2 ) (risk) ν ∼ N ( κ 1 , κ 3 ) (ambiguity) log µ ( c ) = κ 1 + ακ 2 / 2 + γκ 3 / 3 Backus, Ferriere, & Zin (NYU) Risk & Ambiguity
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