Risk and Ambiguity in Models of Business Cycles Dave Backus, Axelle - PowerPoint PPT Presentation

Introduction Risk Ambiguity Last thoughts Annex 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 23, 2014 Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex The “Great Recession” and its aftermath Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex 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, too ◮ But... measured productivity didn’t fall very much More What’s missing? Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex What happened? Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex 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

Introduction Risk Ambiguity Last thoughts Annex What happened? VIX IQR Sales Growth 0.6 0.35 0.55 0.5 0.3 0.45 0.4 0.25 0.35 0.3 0.2 0.25 0.2 0.15 0.15 0.1 0.1 2006 2008 2010 2012 2006 2008 2010 2012 Quarter Quarter Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex 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? ◮ How persistent are the effects of fluctuations in uncertainty? Potential issues: Barro-King, Tallarini Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex 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

Introduction Risk Ambiguity Last thoughts Annex Preview of results Uncertainty shocks have a limited impact Separation property ◮ The internal dynamics of capital accumulation are independent of risk and risk aversion ◮ Up to a loglinear approximation Business cycle properties ◮ Risk aversion magnifies exposure to uncertainty shocks ◮ But business cycle fluctuations are primarily driven by IES Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex Risk and uncertainty Recursive references U t = V [ c t , µ t ( U t +1 )] [(1 − β ) c ρ t + βµ t ( U t +1 ) ρ ] 1 /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

Introduction Risk Ambiguity Last thoughts Annex Scaling 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 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 ) = max V ˜ k t +1 , x t +1 , v t +1 )] c t ˜ g t +1 ˜ k t +1 = f (˜ s.t. k t , n ) − ˜ c t Numerical solution More Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex Benchmark calibration 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

Introduction Risk Ambiguity Last thoughts Annex A model essentially loglinear 2.24 3.75 0.15 numerical solution numerical solution 3.7 2.235 3.65 0.12 2.23 3.6 2.225 density measure 3.55 0.09 2.22 log k(t+1) log V(t) 3.5 2.215 3.45 0.06 2.21 3.4 2.205 3.35 0.03 2.2 3.3 2.195 3.25 0 3.3 3.4 3.5 3.6 3.3 3.4 3.5 3.6 log k(t) log k(t) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex Insights from loglinearization I Goal: loglinear decision rule for capital log ˜ h k log ˜ k t + h ⊤ k t +1 = x x t + h v v t − log g t +1 Dynamic programming version of Campbell (JME, 1994) ◮ Using Euler equation and envelope condition Loglinearization around the stochastic steady-state Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex Insights from loglinearization II Guess loglinear value function p k log ˜ k t + p ⊤ log J t = x x t + p v v t + p 0 log J ρ − 1 q k log ˜ k t + q ⊤ J k , t = x x t + q v v t + q 0 t 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 where ( λ k , λ c , λ r ) are steady-state objects. More Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex Separation property Claim In the approximated law of motion for capital, log ˜ h 0 + h k log ˜ k t + h ⊤ k t +1 = x x t + h v v t − log g t +1 1 h k is independent of parameters that describe exogenous shocks 2 h k , h x are independent of risk aversion (as long as the steady-state is kept unchanged) h ⊤ x = σλ c q ⊤ = λ k + σλ c ( q k − λ r ) , h k x � � = λ k + σλ c ( q k − λ r ) + λ r q k q k � − 1 ( ρ − 1 − q k ) e T A � q x = (1 − σ q k λ c ) I − A Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex An informative property 3.7 0.15 numerical solution loglinear approximation 3.65 0.12 3.6 density measure 0.09 3.55 log k(t+1) 3.5 0.06 3.45 0.03 3.4 3.35 0 3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.58 3.6 log k(t) Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex Risk aversion magnifies uncertainty shocks Shock in volatility (+1std) −4 1.6 x 10 Volatility 1.55 1.5 5 10 15 20 25 30 35 40 Response in capital 0.1 0 % −0.1 5 10 15 20 25 30 35 40 Response in consumption 0.2 RA=2 0 RA=10 % RA=50 −0.2 5 10 15 20 25 30 35 40 Quarter Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex But business cycle properties essentially depend on IES US Data Benchmark Cst. vol. Risk Aversion 2 10 50 10 Standard deviations (%) Output growth 1.04 0.82 0.82 0.82 0.82 Consumption growth 0.55 0.75 0.75 0.76 0.75 Investment growth 2.79 1.03 1.04 1.06 1.02 Correlations with output growth Consumption growth 0.52 0.99 0.99 0.97 0.99 Investment growth 0.65 0.98 0.97 0.93 0.98 Intertemporal elasticity of substitution: 0.5 Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex But business cycle properties essentially depend on IES US Data Benchmark IES 0.5 1.5 Standard deviations (%) Output growth 1.04 0.82 0.82 Consumption growth 0.55 0.75 0.39 Investment growth 2.79 1.04 1.92 Correlations with output growth Consumption growth 0.52 0.99 0.98 Investment growth 0.65 0.97 0.93 Risk aversion: 10 Backus, Ferriere, & Zin (NYU) Risk & Ambiguity

Introduction Risk Ambiguity Last thoughts Annex What did we learn? Separation property ◮ The internal dynamics of capital accumulation are independent of risk and risk aversion Business cycle properties ◮ Are primarily driven by IES The quantitative effect of stochastic volatility is limited Backus, Ferriere, & Zin (NYU) Risk & Ambiguity