Risk and Ambiguity in Models of Business Cycles

Transcription

1 Risk and Ambiguity in Models of Business Cycles Dave Backus, Axelle Ferriere, and Stan Zin North American Meetings of the Econometric Society June 20, 2014 This version: August 2, 2014

2 The Great Recession and its aftermath

3 The Great Recession and its aftermath

4 The Great Recession and its aftermath

5 What happened? What we see Magnitude: deeper recession than usual Persistence: longer recovery maybe slower, too Like Kydland-Prescott with productivity shocks? Relative magnitudes look right Comovements look right, too But... measured productivity didn t fall very much More

6 What happened? What we see Magnitude: deeper recession than usual Persistence: longer recovery maybe slower, too Like Kydland-Prescott with productivity shocks? Relative magnitudes look right Comovements look right, too But... measured productivity didn t fall very much More What s missing?

7 What we do Take a streamlined business cycle model Ask: How does uncertainty affect the dynamics of output, consumption, and investment? Magnitude: Does uncertainty magnify fluctuations? Persistence: Can it reduce the speed of recovery? Compute solutions with Transparent loglinear approximation Acurate numerical method

8 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)

9 What we find Fluctuations in uncertainty have little impact Persistence Separation property: internal dynamics independent of risk and risk aversion Persistence must be in the shock Magnitude Impact typically small, but magnified by risk aversion Business cycle properties governed by IES

10 Risk Recursive references U t = V [c t, µ t (U t+1 )] = [(1 β)ct ρ + βµ t (U t+1 ) ρ ] 1/ρ µ t (U t+1 ) = [E t (Ut+1)] α 1/α V, µ t homogeneous of degree one, RA = 1 α, IES σ = 1/(1 ρ) Productivity a t log g t = log(a t /a t 1 ) = log g + e x t x t+1 = Ax t + v 1/2 t Bw 1t+1 ( news ) v t+1 = (1 ϕ v )v + ϕ v v t + τw 2t+1 ( risk ) (w 1t, w 2t ) = iid standard normals

11 Scaling Bellman equation J(k t, x t, v t, a t ) = max c t V { c t, µ t [J(k t+1, x t+1, v t+1, a t+1 )] } s.t. k t+1 = f (k t, a t n) c t f hd1: eg, 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( k t, x t, v t ) = max c t V { c t, µ t [g t+1 J( k t+1, x t+1, v t+1 )] } s.t. g t+1 kt+1 = f ( k t, n) c t

12 Parameter values Parameter Value Comment Preferences ρ 1 intertemporal substitution = σ = 1/(1 ρ) = 1/2 α 9 risk aversion = 1 α = 10 β chosen to hit k/y = 10 (quarterly) Technology ω 1/3 Kydland and Prescott (1982, Table I), rounded off δ Kydland and Prescott (1982, Table I) Productivity growth log g Tallarini (2000, Table 4) e 1 normalization A 0 no predictable component ( news ) B 1 normalization v 1/ Tallarini (2000, Table 4), rounded off ϕ v 0.95 arbitrary τ makes v three standard deviations from zero

13 Model is essentially loglinear numerical solution numerical solution log V(t) log k(t+1) density measure log k(t) log k(t) 0

14 Loglinearization I Goal: loglinear decision rule for capital log k t+1 = h k log k t + hx x t + h v v t log g t+1 Dynamic programming version of Campbell (JME, 1994) Loglinearization around the stochastic steady-state

15 Loglinearization II Loglinearize capital s marginal product and law of motion log f kt = λ r log k t + λ 0 log k t+1 = λ k log k t λ c log c t + λ 1 log g t+1 where (λ k, λ c, λ r ) are steady-state objects. Guess loglinear value function and derivative log J t = p k log k t + px x t + p v v t + p 0 log Jt ρ 1 J kt = q k log k t + qx x t + q v v t + q 0 More

16 Separation property Claim (Tallarini) Consider the loglinear approximation of capital s law of motion, log k t+1 = h 0 + h k log k t + h x x t + h v v t log g t+1 If we hold constant the stochastic steady state: 1 h k is independent of properties of all shocks and risk aversion 2 h x is independent of properties of uncertainty shocks and risk aversion h k = λ k + σλ c(q k λ r ), h x = σλ cq x q k = q k [ λk + σλ c(q k λ r ) ] + λ r q x = (σ 1 + q k )e T A [ (1 σq k λ c)i A ] 1

17 Loglinearization III numerical solution loglinear approximation log k(t+1) density measure log k(t) 0

18 Risk aversion magnifies uncertainty More Volatility 1.6 x 10 4 Shock in volatility (+1std) Response in consumption % 0 % Quarter Response in capital RA=2 RA=10 RA=

19 Business cycles and risk aversion US Data Model w/ RA = Cst. vol. Risk Aversion Standard deviations (%) Output growth Consumption growth Investment growth Correlations with output growth Consumption growth Investment growth Intertemporal elasticity of substitution: 0.5

20 Business cycles and IES US Data Model IES Standard deviations (%) Output growth Consumption growth Investment growth Correlations with output growth Consumption growth Investment growth Risk aversion: 10

21 Risk and ambiguity Divide state in two: s t = (s 1t, s 2t ) (ask about Stan s story) Smooth ambiguity risk = p 1t (s 1t+1 s 2t+1, I t ) ambiguity = p 2t (s 2t+1 I t ) Two-part certainty equivalent µ 1t (U t+1 ) = [ E 1t (Ut+1) α ] 1/α ( risk ) µ 2t [µ 1t (U t+1 )] = { E 2t [µ 1t (U t+1 )] γ ) } 1/γ ( ambiguity ) α controls risk aversion, γ < α controls ambiguity aversion

22 Ambiguity about what? Rule of thumb Risk about observables Ambiguity about unobservables Example: observe productivity growth g t but not its mean x t Risk: log g t+1 x t+1 N (log g + x t+1, b) Ambiguity: x t+1 AR(1) Filtering gives us (say) x t+1 I t N ( x t+1, h t+1 ), I t = g t

23 Ambiguity about what? Rule of thumb Risk about observables Ambiguity about unobservables Example: observe productivity growth g t but not its mean x t Risk: log g t+1 x t+1 N (log g + x t+1, b) Ambiguity: x t+1 AR(1) Filtering gives us (say) x t+1 I t N ( x t+1, h t+1 ), I t = g t But: none of this has much impact

24 Summary Uncertainty fluctuations have intuitive appeal But they add little to standard business cycle model Magnitude: impact is small with common parameter values Persistence: they add nothing to internal dynamics, just the persistence of the shocks themselves

25 Summary Uncertainty fluctuations have intuitive appeal But they add little to standard business cycle model Magnitude: impact is small with common parameter values Persistence: they add nothing to internal dynamics, just the persistence of the shocks themselves Where next? Uncertainty about parameters? Endogenous uncertainty? (Veldkamp, Schaal) Micro uncertainty with financial frictions? (Arellano, Bai, & Kehoe) Cause or effect? (Alessandria, Choi, Kaboski, & Midrigan)

26 Related work (some of it) Recursive business cycles Campanale, Castro, & Clementi; Tallarini Approximation methods Anderson, Hansen, McGrattan, & Sargent; Campbell; Kaltenbrunner and Lochstoer; Malkhozov Risk and business cycles Basu & Bundick; Caldara, Fernandez-Villaverde, Rubio-Ramirez, & Wen; Justiniano & Primiceri; Liu & Miao Ambiguity and business cycles Klibanoff, Marinacci, & Mukerji; Ju & Miao; Ilut & Schneider; Jahan-Parvar & Miao

27 Intetemporal substitution and uncertainty Back Volatility 1.6 x 10 4 Shock in volatility (+1std) Response in consumption % 0 % Quarter Response in capital IES=0.5 IES=

28 Productivity