Leads, Lags, and Logs: Asset Prices in Business Cycle Analysis - - PowerPoint PPT Presentation

Leads, Lags, and Logs: Asset Prices in Business Cycle Analysis

David Backus (NYU), Bryan Routledge (CMU), and Stanley Zin (CMU) Zicklin School of Business, Baruch College October 24, 2007

This version: October 23, 2007 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 46

Overview of recursive preferences

Time preference Risk Preference

◮ Chew-Dekel ◮ Risk premiums

Applications of Kreps-Porteus preferences

◮ Pricing kernels ◮ Risk sharing ◮ Business cycles (the paper in the title)

Extensions?

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 46

Time preference

Time preference

Time aggregator V Ut = V (ut, Ut+1) Additive preferences Ut = (1 − β)ut + βUt+1 = (1 − β)

∞

• j=0

βjut+j Why don’t we care about this?

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 2 / 46

Risk preference

Risk preference overview

Certainty equivalent functions Chew-Dekel preferences Small, lognormal, and extreme risks

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 3 / 46

Risk preference

Risk preference

Basics: states s ∈ {1, . . . , S}, consumption c(s), probabilities p(s) Certainty equivalent function: µ satisfying U(µ, . . . , µ) = U[c(1), . . . , c(S)] Risk aversion: µ(c) ≤ E(c) Chew-Dekel preferences (risk aggregator M) µ =

• s

p(s)M[c(s), µ]

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 4 / 46

Risk preference

Chew-Dekel examples

Expected utility M(c, m) = cαm1−α/α + m(1 − 1/α) Weighted utility M(c, m) = (c/m)γcαm1−α/α + m[1 − (c/m)γ/α]. Disappointment aversion M(c, m) = cαm1−α/α + m(1 − 1/α) + δI(m − c)(cαm1−α − m)/α I(x) = 1 if x > 0, 0 otherwise

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 5 / 46

Risk preference

Chew-Dekel as adjusted probabilities

Expected utility µ =

• s

p(s)c(s)α 1/α Weighted utility: ditto with ˆ p(s) = p(s)c(s)γ

• u p(u)c(u)γ ,

Disappointment aversion: ditto with ˆ p(s) = p(s)(1 + δI[µ − c(s)])

• u p(s)(1 + δI[µ − c(s)]),

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 6 / 46

Risk preference

Small risks

Two states (1 + σ, 1 − σ), equal probs, Taylor series around σ = 0 Expected utility µ(EU) ≈ 1 − (1 − α)σ2/2 Weighted utility µ(WU) ≈ 1 − [1 − (α + 2γ)]σ2/2 Disappointment aversion µ(DA) ≈ 1 −

• δ

2 + δ

• σ − (1 − α)
• 4 + 4δ

4 + 4δ + δ2

• σ2/2

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 7 / 46

Risk preference

Lognormal risks

Let: log c ∼ N(κ1, κ2), rp = log[E(c)/µ] Expected utility rp(EU) = (1 − α)κ2/2 Weighted utility rp(WU) = [1 − (α + 2γ)]κ2/2 Disappointment aversion rp(WU) = E2C2E

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 8 / 46

Risk preference

Extreme risks

Let: log E exp(log c) = κ1 + κ2/2! + κ3/3! + κ4/4! Expected utility rp(EU) = (1 − α)κ2/2 + (1 − α2)κ3/3! + (1 − α3)κ4/4! Weighted utility rp(WU) = [1 − (α + 2γ)]κ2/2 + [1 − (α + 2γ)2 + γ(α + γ)]κ3/3! + [1 − (α + 2γ)3 + 2γ(α + γ)(α + 2γ)]κ4/4! Disappointment aversion rp(DA) = Another E2C2E

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 9 / 46

Kreps-Porteus preferences

Kreps-Porteus preferences

Recursive preferences Ut = V [ut, µt(Ut+1)] Kreps-Porteus/Epstein-Zin-Weil V (ut, µt) = [(1 − β)uρ

t + βµρ t ]1/ρ

µt(Ut+1) =

• EtUα

t+1

1/α IES = 1/(1 − ρ) CRRA = 1 − α α = ρ ⇒ additive preferences Invariant to monotonic transformations: eg, Ut = Uρ

t /ρ

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 10 / 46

Kreps-Porteus preferences

Kreps-Porteus overview

Pricing kernels Risk sharing Business cycles

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 11 / 46

Kreps-Porteus preferences

Kreps-Porteus pricing kernels

Marginal rate of substitution mt+1 = β(ct+1/ct)ρ−1[Ut+1/µt(Ut+1)]α−ρ Note role of future utility

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 12 / 46

Kreps-Porteus preferences

Kreps-Porteus pricing kernels (continued)

Example: let consumption growth follow log xt = log x +

∞

• j=0

χjwt−j Pricing kernel log mt+1 = constant + [(ρ − 1)χ0 + (α − ρ)(χ0 + X1)]wt+1 + (ρ − 1)

∞

• j=0

χj+1wt−j X1 =

∞

• j=1

βjχj (“Bansal-Yaron” term)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 13 / 46

Kreps-Porteus preferences

Kreps-Porteus risk sharing

Pareto problem with two recursive agents Bryan did this a few weeks ago Issues

◮ Time-varying pareto weights ◮ Representative agent may look different from individuals ◮ Possible nonstationary consumption distribution Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 14 / 46

Kreps-Porteus preferences

Kreps-Porteus business cycle overview

Pictures: leads and lags in US data Equations: the usual suspects + bells & whistles Computations: loglinear approximation More pictures: leads and lags in the model Extensions

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 15 / 46

Leads and lags in data

Leads and lags in US data

Cross-correlation functions of GDP with

◮ Stock price indexes ◮ Interest rates and spreads ◮ Consumption and employment

US data, quarterly, 1960 to present Quarterly growth rates (log xt − log xt−1) except

◮ Interest rates and spreads (used as is) ◮ Occasional year-on-year comparisons (log xt+2 − log xt−2) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 16 / 46

Leads and lags in data

Stock prices and GDP

Leads GDP Lags GDP −1.00 −0.50 0.00 0.50 1.00 −1.00 −0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

S&P 500

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 17 / 46

Leads and lags in data

Stock prices and GDP (year-on-year)

−1.00 −0.50 0.00 0.50 1.00 −1.00 −0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

S&P 500 (yoy)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 18 / 46

Leads and lags in data

Stock prices and GDP

Leads GDP Lags GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

S&P 500

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

S&P 500 minus Short Rate

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

NYSE Composite

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Nasdaq Composite

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 19 / 46

Leads and lags in data

Interest rates and GDP

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Yield Spread (10y−3m)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Yield Spread (GDP yoy)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Short Rate (3m)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Real Rate

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 20 / 46

Leads and lags in data

Consumption and GDP

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Consumption

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Services

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Nondurables

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Durables

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 21 / 46

Leads and lags in data

Investment and GDP

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Investment

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Structures

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Equipment and Software

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Residential

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 22 / 46

Leads and lags in data

Employment and GDP

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Employment (Nonfarm Payroll)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Employment (Household Survey)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Avg Weekly Hours (All)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Avg Weekly Hours (Manuf)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 23 / 46

Leads and lags in data

Lead/lag summary

Things that lead GDP

◮ Stock prices ◮ Yield curve and short rate ◮ Maybe consumption (a little)

Things that lag GDP

◮ Maybe employment (a little)

Why?

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 24 / 46

The usual suspects

(Almost) the usual equations

Streamlined Kydland-Prescott except

◮ Recursive preferences (Kreps-Porteus/Epstein-Zin-Weil) ◮ CES production ◮ Adjustment costs ◮ Unit root in productivity ◮ Predictable component in productivity growth Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 25 / 46

The usual suspects

(Almost) the usual equations

Streamlined Kydland-Prescott except

◮ Recursive preferences (Kreps-Porteus/Epstein-Zin-Weil) ◮ CES production ◮ Adjustment costs ◮ Unit root in productivity ◮ Predictable component in productivity growth Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 25 / 46

The usual suspects

Preferences

Equations Ut = V [ut, µt(Ut+1)] ut = ct(1 − nt)λ V (ut, µt) = [(1 − β)uρ

t + βµρ t ]1/ρ

µt(Ut+1) =

• EtUα

t+1

1/α Interpretation IES = 1/(1 − ρ) CRRA = 1 − α α = ρ ⇒ additive preferences

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 26 / 46

The usual suspects

Technology: production

Equations yt = f (kt, ztnt) = [ωkν

t + (1 − ω)(ztnt)ν]1/ν

yt = ct + it Interpretation Elast of Subst = 1/(1 − ν) Capital Share = ω(y/k)−ν

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 27 / 46

The usual suspects

Technology: capital accumulation

Equations kt+1 = g(it, kt) = (1 − δ)kt + kt[(it/kt)η(i/k)1−η − (1 − η)(i/k)]/η Interpretation No adjustment costs if η = 1

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 28 / 46

The usual suspects

Productivity

Equations log xt+1 = (I − A) log x + A log xt + Bwt+1 {wt} ∼ NID(0, I) log zt+1 − log zt = log x1t+1 (first element) Interpretation A = [0] ⇒ no predictable component

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 29 / 46

Logs

Computation overview

Scaling

◮ Recast as stationary problem in “scaled” variables

Loglinear approximation

◮ Loglinearize value function (not log-quadratic) ◮ Loglinearize necessary conditions ◮ With constant variances, recursive preferences irrelevant to quantities

(but not asset prices)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 30 / 46

Logs

Scaling the Bellman equation

Key input: (V , µ, f , g) are hd1 Natural version J(kt, xt, zt) = max

ct,nt V

• ct(1 − nt)λ, µt[J(kt+1, xt+1, zt+1]
• subject to:

kt+1 = g[f (kt, ztnt) − ct, kt) plus productivity process & initial conditions Scaled version [˜ kt = kt/zt, ˜ ct = ct/zt, etc] J(˜ kt, xt, 1) = max

˜ ct,nt V

• ˜

ct(1 − nt)λ, µt[x1t+1J(˜ kt+1, xt+1, 1)]

• subject to:

˜ kt+1 = g[f (˜ kt, nt) − ˜ ct, ˜ kt]/x1t+1 plus productivity process & initial conditions

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 31 / 46

Logs

Necessary conditions

First-order conditions (1 − β)˜ cρ−1

t

(1 − nt)ρλ = Mtgit λ(1 − β)˜ cρ

t (1 − nt)ρλ−1

= Mtgitfnt Envelope condition Jkt = J1−ρ

t

Mt(gitfkt + gkt) “Massive expression” Mt = β µt(x1t+1Jt+1)ρ−αEt[(x1t+1Jt+1)α−1Jkt+1]

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 32 / 46

Logs

Loglinear approximation

Objective: loglinear decision rules [ˆ kt ≡ log ˜ kt − log ˜ k, etc] ˆ ct = hckˆ kt + h⊤

cxˆ

xt ˆ nt = hnkˆ kt + h⊤

nxˆ

xt Key input: log J(˜ kt, xt) = p0 + pk log ˜ kt + p⊤

x log xt

Solution

◮ Brute force loglinearization of necessary conditions ◮ Riccati equation separable: first pk, then px ◮ Lots of algebra, but separability allows you to do it by hand Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 33 / 46

Leads and lags in models

Leads and lags in the model: overview

Growth model: no labor or adjustment costs Three processes for productivity growth

◮ Random walk (A = 0) ◮ Two-period lead ◮ Small predictable component

The challenge

◮ Barro and King Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 34 / 46

Leads and lags in models

Random walk: impulse responses

2 4 6 8 10 12 14 16 18 20 1 2 Productivity 2 4 6 8 10 12 14 16 18 20 0.4 0.6 0.8 Consumption 2 4 6 8 10 12 14 16 18 20 1 Investment 2 4 6 8 10 12 14 16 18 20 5 10 Interest Rate Quarters after Shock Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 35 / 46

Leads and lags in models

Random walk: cross correlations

−6 −4 −2 2 4 6 −1 1 Consumption −6 −4 −2 2 4 6 −1 1 Investment −6 −4 −2 2 4 6 −1 1 Interest Rate Lag Relative to GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 36 / 46

Leads and lags in models

Two-period lead: cross correlations

−6 −4 −2 2 4 6 −1 1 Consumption −6 −4 −2 2 4 6 −1 1 Investment −6 −4 −2 2 4 6 −1 1 Interest Rate Lag Relative to GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 37 / 46

Leads and lags in models

Predictable component: cross correlations

−6 −4 −2 2 4 6 −1 1 Consumption −6 −4 −2 2 4 6 −1 1 Investment −6 −4 −2 2 4 6 −1 1 Interest Rate Lag Relative to GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 38 / 46

Summary and extensions

Summary

◮ Data: interest rates lead the cycle ◮ Model: ditto from predictable component in productivity growth

Extensions

◮ Labor dynamics: Gali’s result? ◮ Stochastic volatility ◮ Could this result from endogenous dynamics? Monetary policy? Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 39 / 46

Extra slides

Related work

Leads and lags in data

◮ Ang-Piazzesi-Wei, Beaudry-Portier, King-Watson, Stock-Watson

Predictable components in models

◮ Bansal-Yaron, Jaimovich-Rebelo

(Log)linear approximation

◮ Campbell, Hansen-Sargent, Lettau, Tallarini, Uhlig

Kreps-Porteus pricing kernels

◮ Hansen-Heaton-Li, Weil Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 40 / 46

Extra slides

Autocorrelations of quarterly growth rates

−0.200.00 0.20 0.40 0.60 GDP 5 10 15 20 25 Lag

Bartlett’s formula for MA(q) 95% confidence bands

−0.200.00 0.20 0.40 0.60 Consumption 5 10 15 20 25 Lag

Bartlett’s formula for MA(q) 95% confidence bands

−0.200.00 0.20 0.40 0.60 Investment 5 10 15 20 25 Lag

Bartlett’s formula for MA(q) 95% confidence bands

−0.200.00 0.20 0.40 0.60 Government Purchases 5 10 15 20 25 Lag

Bartlett’s formula for MA(q) 95% confidence bands

Autocorrelations of Growth Rates

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 41 / 46

Extra slides

Random walk: autocorrelations

1 2 3 4 5 6 0.5 1 GDP 1 2 3 4 5 6 0.5 1 Consumption 1 2 3 4 5 6 −1 1 Investment 1 2 3 4 5 6 0.8 0.9 1 Interest Rate Lag Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 42 / 46

Extra slides

Predictable component: autocorrelations

1 2 3 4 5 6 −1 1 GDP 1 2 3 4 5 6 0.5 1 Consumption 1 2 3 4 5 6 −1 1 Investment 1 2 3 4 5 6 0.6 0.8 1 Interest Rate Lag Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 43 / 46

Extra slides

Predictable component: impulse responses

2 4 6 8 10 12 14 16 18 20 0.5 1 Productivity 2 4 6 8 10 12 14 16 18 20 0.2 0.3 0.4 Consumption 2 4 6 8 10 12 14 16 18 20 1 2 Investment 2 4 6 8 10 12 14 16 18 20 5 10 Interest Rate Quarters after Shock Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 44 / 46

Extra slides

Approximation methods

Problem: find decision rule ut = h(xt) satisfying EtF(xt, ut, wt+1) = 1, wt ∼ N(0, v) Judd + many others

◮ Taylor series expansion of F ◮ nth moment shows up in nth-order term

Us + much of modern finance

◮ Taylor series expansion of f = log F in

Et exp[f (xt, ut, wt+1)] = 1

◮ All moments show up even in linear approximation Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 45 / 46

Extra slides

Approximation methods: linear example

Linear “perturbation” method

◮ Linear approximation of F

F(xt, ut, wt+1) = F + Fx(xt − x) + Fu(ut − u) + Fwwt+1 EtF = 1 ⇒ ut − u = (1 − F)/Fu − (Fx/Fu)(xt − x)

◮ Decision rule doesn’t depend on variance of w (or higher moments)

“Affine” finance method

◮ Linear approximation of f = log F

f (xt, ut, wt+1) = f + fx(xt − x) + fu(ut − u) + fwwt+1 Etf = 1 ⇒ ut − u = −(f + fwv/2)/fu − (fx/fu)(xt − x)

◮ Note impact of variance v (higher moments would show up, too) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 46 / 46