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

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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,

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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

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Introduction Risk Ambiguity Last thoughts Annex

The “Great Recession” and its aftermath

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The “Great Recession” and its aftermath

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The “Great Recession” and its aftermath

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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?

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What happened?

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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.

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What happened?

2006 2008 2010 2012 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Quarter VIX

2006 2008 2010 2012 0.1 0.15 0.2 0.25 0.3 0.35

Quarter IQR Sales Growth

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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

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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

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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

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Risk and uncertainty

Recursive references Ut = V [ct, µt(Ut+1)] = [(1 − β)cρ

t + βµt(Ut+1)ρ]1/ρ

µt(Ut+1) = [Et(Uα

t+1)]1/α

V , µt homogeneous of degree one, α, ρ < 1, σ = 1/(1 − ρ) Stochastic structure of productivity at log gt = log(at/at−1) = log g + e⊤xt (“productivity growth”) xt+1 = Axt + v 1/2

t

Bw1t+1 (“news”) vt+1 = (1 − ϕv)v + ϕvvt + τw2t+1 (“risk”) (w1t, w2t) = iid standard normals

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Scaling

Bellman equation J(kt, xt, vt, at) = max

ct

V

  • ct, µt[J(kt+1, xt+1, vt+1, at+1)]
  • s.t.

kt+1 = f (kt, atn) − ct Assume f hd1: f (k, an) = kω(an)1−ω + (1 − δ)k Rescaled Bellman equation [˜ kt = kt/at, ˜ ct = ct/at] J(˜ kt, xt, vt) = max

˜ ct

V

  • ˜

ct, µt[gt+1J(˜ kt+1, xt+1, vt+1)]

  • s.t.

gt+1˜ kt+1 = f (˜ kt, n) − ˜ ct Numerical solution

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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) e 1 normalization A no predictable component (“news”) B 1 normalization v 1/2 0.015 Tallarini (2000, Table 4), rounded off ϕv 0.95 arbitrary τ 0.74 × 10−5 makes v three standard deviations from zero

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A model essentially loglinear

3.3 3.4 3.5 3.6 2.195 2.2 2.205 2.21 2.215 2.22 2.225 2.23 2.235 2.24 log k(t) log V(t) numerical solution 3.3 3.4 3.5 3.6 3.25 3.3 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75 log k(t) log k(t+1) numerical solution 0.03 0.06 0.09 0.12 0.15 density measure

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Insights from loglinearization I

Goal: loglinear decision rule for capital

log ˜ kt+1 = hk log ˜ kt + h⊤

x xt + hvvt − log gt+1

Dynamic programming version of Campbell (JME, 1994)

◮ Using Euler equation and envelope condition

Loglinearization around the stochastic steady-state

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Insights from loglinearization II

Guess loglinear value function log Jt = pk log ˜ kt + p⊤

x xt + pvvt + p0

log Jρ−1

t

Jk,t = qk log ˜ kt + q⊤

x xt + qvvt + q0

Loglinearize capital’s marginal product and law of motion log fkt = λr log ˜ kt + λ0 log ˜ kt+1 = λk log ˜ kt − λc log ˜ ct + λ1 − log gt+1 where (λk, λc, λr) are steady-state objects.

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Separation property

Claim In the approximated law of motion for capital, log ˜ kt+1 = h0 + hk log ˜ kt + h⊤

x xt + hvvt − log gt+1

1 hk is independent of parameters that describe exogenous

shocks

2 hk, hx are independent of risk aversion (as long as the

steady-state is kept unchanged)

hk = λk + σλc(qk − λr), h⊤

x = σλcq⊤ x

qk = qk

  • λk + σλc(qk − λr)
  • + λr

qx = (ρ − 1 − qk)eTA

  • (1 − σqkλc)I − A

−1

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An informative property

3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.58 3.6 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 log k(t) log k(t+1) numerical solution loglinear approximation 0.03 0.06 0.09 0.12 0.15 density measure

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Risk aversion magnifies uncertainty shocks

5 10 15 20 25 30 35 40 1.5 1.55 1.6 x 10

−4

Volatility Shock in volatility (+1std) 5 10 15 20 25 30 35 40 −0.1 0.1 Response in capital % 5 10 15 20 25 30 35 40 −0.2 0.2 Quarter % Response in consumption RA=2 RA=10 RA=50

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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

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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

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What did we learn?

Separation property

◮ The internal dynamics of capital accumulation are independent

  • f risk and risk aversion

Business cycle properties

◮ Are primarily driven by IES

The quantitative effect of stochastic volatility is limited

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Risk and ambiguity

Divide the state in two: st = (s1t, s2t) Ambiguity (Klibanoff, Marinacci, & Mukerji; Ju & Miao) risk = p1t(s1t+1|s2t+1, It) ambiguity = p2t(s2t+1|It) Two-part certainty equivalent µ1t(Ut+1) =

  • E1t(Uα

t+1)

1/α µ2t[µ1t(Ut+1)] =

  • E2t[µ1t(Ut+1)]γ)

1/γ α controls risk aversion, γ < α controls ambiguity aversion

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A second certainty equivalent

Consider three stochastic processes

  • xt the mean growth rate (unobservable)
  • gt the realized growth rate (observable)
  • vt the stochastic volatility

log gt = log g + xt + σ1w1,t xt+1 = σ2w2,t+1 vt+1 = ϕ¯ v + (1 − ϕ)vt + σ2w3,t The separation property holds.

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Calibration

Parameter Value Comment Preferences ρ −1 arbitrary α −9 arbitrary γ −29 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) e 1 normalization A no predictable component (“news”) B 1 normalization v 1/2 0.015 Tallarini (2000, Table 4), rounded off ϕv 0.95 arbitrary τ 0.74 × 10−5 makes v three standard deviations from zero

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The model is still essentially loglinear

3.35 3.4 3.45 3.5 3.55 3.6 3.3 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7 log k(t) log k(t+1) numerical solution loglinear approximation 0.03 0.06 0.09 0.12 0.15 density measure

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Learning

Add stochastic volatility vt to the signal

xt+1 = ϕxt + σ2w2,t+1 vt+1 = (1 − ϕv)¯ v + ϕvvt + σ3w3,t+1 log gt+1 = log g + xt+1 + v 1/2

t

w1,t+1

Learning: Kalman filter

ˆ xt+1 = ϕ vt−1 At + vt−1 ˆ xt + ϕ At At + vt−1 log (gt/g) At+1 = σ2

2 +

ϕ2At At + vt−1 vt−1 Fluctuating uncertainty

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What did we learn?

What does risk + ambiguity add to a RBC model?

◮ Affects responses of variables to shocks ◮ But not internal dynamics of capital (separation property)

Feedback from endogenous variables to uncertainty? + We did not talk about asset pricing - everybody else does it

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Related work (some of it)

Recursive preferences

◮ Kreps & Porteus; Epstein & Zin; Weill

Recursive business cycles

◮ Tallarini; Campanale, Castro, & Clementi; Rubio & Villaverde;

Liu & Miao

Ambiguity and business cycles

◮ Klibanoff, Marinacci, & Mukerji; Jahan-Parvar & Miao; Ju &

Miao; Ilut & Schneider

Approximation methods

◮ Hansen & Sargent; Anderson, Hansen, McGrattan, & Sargent;

Campbell; Kaltenbrunner and Lochstoer; Malkhozov

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Scaling

Back

Rescaled Bellman equation J(˜ kt, xt, vt) = max

˜ ct

V

  • ˜

ct, µt[gt+1J(˜ kt+1, xt+1, vt+1)]

  • s.t.

gt+1˜ kt+1 = f (˜ kt, n) − ˜ ct plus shocks & initial conditions Let K ≡ gt+1˜ kt+1. Then,

J(˜ kt, xt, vt) = max

K

V

  • f (˜

kt, n) − K

  • , µt
  • gt+1J

K gt+1 , xt+1, vt+1, 1

  • plus shocks & initial conditions

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Productivity

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