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) Society for Economic Dynamics, July 2006

This version: July 11, 2006 Backus, Routledge, and Zin () Leads, lags, and logs 1 / 20

Overview

Leads and lags in business cycles (Almost) the usual equations Loglinear approximation Properties of the model Extensions

Backus, Routledge, and Zin () Leads, lags, and logs 1 / 20

Leads and lags

Leads and lags

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 (first difference of logs), except

◮ Interest rates and spreads ◮ Occasional year-on-year comparisons (yt+2 − yt−2) Backus, Routledge, and Zin () Leads, lags, and logs 2 / 20

Leads and lags

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, and Zin () Leads, lags, and logs 3 / 20

Leads and lags

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, and Zin () Leads, lags, and logs 4 / 20

Leads and lags

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, and Zin () Leads, lags, and logs 5 / 20

Leads and lags

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, and Zin () Leads, lags, and logs 6 / 20

Leads and lags

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, and Zin () Leads, lags, and logs 7 / 20

Leads and lags

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, and Zin () Leads, lags, and logs 8 / 20

Leads and lags

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, and Zin () Leads, lags, and logs 9 / 20

Leads and lags

Lead/lag summary

Things that lead GDP

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

Things that lag GDP

◮ Employment Backus, Routledge, and Zin () Leads, lags, and logs 10 / 20

The usual equations

(Almost) the usual equations

Basic real business cycle model except

◮ Recursive preferences (Kreps-Porteus/Epstein-Zin-Weil) ◮ CES production ◮ More complex shock process (predictable component in productivity) Backus, Routledge, and Zin () Leads, lags, and logs 11 / 20

The usual equations

Preferences

Equations Ut = V [ut, µt(Ut+1)] ut = cγ

t (1 − nt)1−γ

V (a, b) = [(1 − β)aρ + βbρ]1/ρ µt(Ut+1) =

• EtUα

t+1

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

Backus, Routledge, and Zin () Leads, lags, and logs 12 / 20

The usual equations

Technology

Equations yt = f (kt, ztnt) = [ωkν

t + (1 − ω)(ztnt)ν]1/ν

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

Backus, Routledge, and Zin () Leads, lags, and logs 13 / 20

The usual equations

Productivity process

Equation log zt+1 − log zt = ¯ g +

∞

• j=0

χjεt+1−j Interpretation

◮ Moving average allows predictable component Backus, Routledge, and Zin () Leads, lags, and logs 14 / 20

Logs

Loglinear decision rules

We’re looking for decision rules of the form ˆ c = hckˆ k +

∞

• j=0

hcjεt−j ˆ n = hnkˆ k +

∞

• j=0

hnjεt−j

Backus, Routledge, and Zin () Leads, lags, and logs 15 / 20

Logs

Loglinear decision rules

Traditional methods

◮ Dependence of decisions on capital independent of shocks ◮ Scale by z for stationarity ◮ Decision rules follow from loglinear approximation of derivatives of

value function

With recursive preferences

◮ Dependence of decisions on capital independent of shocks ◮ Scale by z for stationarity ◮ We need the value function, not just its derivatives ◮ Tallarini: logquadratic value function ◮ Us: loglinear value function ◮ Impact: risk aversion has no impact on quantities Backus, Routledge, and Zin () Leads, lags, and logs 16 / 20

Properties of the model

Asset prices

Pricing kernel mt+1 = β (ct+1/ct)ρ−1 [Ut+1/µt(Ut+1)]α−ρ Short rate rt = − log Etmt+1

Backus, Routledge, and Zin () Leads, lags, and logs 17 / 20

Properties of the model

Impulse response of short rate

Backus, Routledge, and Zin () Leads, lags, and logs 18 / 20

Bottom line

Bottom line

Asset prices contain information about business cycles Specifically: they lead the cycle Not true of traditional business cycle models We add

◮ recursive preferences ◮ predictable component to productivity

Lots left to do

◮ equity ◮ macro-based bond pricing ◮ stochastic volatility Backus, Routledge, and Zin () Leads, lags, and logs 19 / 20

Related work

Related work

Leads and lags

◮ Ang-Piazzesi-Wei, Beaudry-Portier, Jaimovich-Rebelo, Stock-Watson

Predictable component

◮ Bansal-Yaron

Computation with recursive preferences

◮ Hansen-Sargent, Tallarini, Uhlig

Kreps-Porteus pricing kernel

◮ Hansen-Heaton-Li, Weil Backus, Routledge, and Zin () Leads, lags, and logs 20 / 20