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
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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
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Stock price indexes
Interest rates and spreads
Consumption and employment
US data, quarterly, 1960 to present
Quarterly growth rates (first difference of logs), except
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Interest rates and spreads
Occasional year-on-year comparisons (yt+2 − yt−2 )
Backus, Routledge, and Zin ()
Leads, lags, and logs
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Leads and lags
Stock prices and GDP
Lags GDP
0.50
0.00
−0.50
−10
Backus, Routledge, and Zin ()
−5
0
Lag Relative to GDP
Leads, lags, and logs
5
10
−1.00
−1.00
Cross−Correlation with GDP
−0.50
0.00
0.50
Leads GDP
1.00
1.00
S&P 500
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Leads and lags
Stock prices and GDP
(year-on-year)
1.00
0.50
0.00
−0.50
−10
Backus, Routledge, and Zin ()
−5
0
Lag Relative to GDP
Leads, lags, and logs
5
10
−1.00
−1.00
Cross−Correlation with GDP
−0.50
0.00
0.50
1.00
S&P 500 (yoy)
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Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−5
0
5
Lag Relative to GDP
−5
0
5
Lag Relative to GDP
Backus, Routledge, and Zin ()
10
NYSE Composite
10
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−10
Leads, lags, and logs
−5
0
5
Lag Relative to GDP
−5
0
5
Lag Relative to GDP
10
Nasdaq Composite
10
−1.00−0.50 0.00 0.50 1.00
S&P 500 minus Short Rate
−1.00−0.50 0.00 0.50 1.00
−10
Lags GDP
−1.00−0.50 0.00 0.50 1.00
S&P 500
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
Leads GDP
−1.00−0.50 0.00 0.50 1.00
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
Leads and lags
Stock prices and GDP
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Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−5
0
5
Lag Relative to GDP
Backus, Routledge, and Zin ()
10
Short Rate (3m)
10
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−10
Leads, lags, and logs
−5
0
5
Lag Relative to GDP
−5
0
5
Lag Relative to GDP
10
Real Rate
10
−1.00−0.50 0.00 0.50 1.00
Yield Spread (GDP yoy)
−1.00−0.50 0.00 0.50 1.00
−5
0
5
Lag Relative to GDP
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−1.00−0.50 0.00 0.50 1.00
Yield Spread (10y−3m)
−1.00−0.50 0.00 0.50 1.00
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
Leads and lags
Interest rates and GDP
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Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−5
0
5
Lag Relative to GDP
Backus, Routledge, and Zin ()
10
Nondurables
10
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−10
Leads, lags, and logs
−5
0
5
Lag Relative to GDP
−5
0
5
Lag Relative to GDP
10
Durables
10
−1.00−0.50 0.00 0.50 1.00
Services
−1.00−0.50 0.00 0.50 1.00
−5
0
5
Lag Relative to GDP
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−1.00−0.50 0.00 0.50 1.00
Consumption
−1.00−0.50 0.00 0.50 1.00
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
Leads and lags
Consumption and GDP
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Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−5
0
5
Lag Relative to GDP
Backus, Routledge, and Zin ()
10
Equipment and Software
10
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−10
Leads, lags, and logs
−5
0
5
Lag Relative to GDP
−5
0
5
Lag Relative to GDP
10
Residential
10
−1.00−0.50 0.00 0.50 1.00
Structures
−1.00−0.50 0.00 0.50 1.00
−5
0
5
Lag Relative to GDP
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−1.00−0.50 0.00 0.50 1.00
Investment
−1.00−0.50 0.00 0.50 1.00
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
Leads and lags
Investment and GDP
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Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−5
0
5
Lag Relative to GDP
Backus, Routledge, and Zin ()
10
Avg Weekly Hours (All)
10
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−10
Leads, lags, and logs
−5
0
5
Lag Relative to GDP
−5
0
5
Lag Relative to GDP
Employment (Household Survey)
Avg Weekly Hours (Manuf)
10
−1.00−0.50 0.00 0.50 1.00
10
−1.00−0.50 0.00 0.50 1.00
−5
0
5
Lag Relative to GDP
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
−10
−1.00−0.50 0.00 0.50 1.00
Employment (Nonfarm Payroll)
−1.00−0.50 0.00 0.50 1.00
Cross−Correlation with GDP
−1.00−0.50 0.00 0.50 1.00
Leads and lags
Employment and GDP
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Leads and lags
Lead/lag summary
Things that lead GDP
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Stock prices
Yield curve and short rate
Consumption (a little)
Things that lag GDP
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Employment
Backus, Routledge, and Zin ()
Leads, lags, and logs
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The usual equations
(Almost) the usual equations
Basic real business cycle model except
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Recursive preferences (Kreps-Porteus/Epstein-Zin-Weil)
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CES production
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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
= ctγ (1 − nt )1−γ
V (a, b) = [(1 − β)aρ + βb ρ ]1/ρ
¡
¢1/α
α
µt (Ut+1 ) = Et Ut+1
Interpretation
IES
= 1/(1 − ρ)
CRRA = 1 − α
α = ρ
Backus, Routledge, and Zin ()
⇒
additive preferences
Leads, lags, and logs
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The usual equations
Technology
Equations
yt
= f (kt , zt nt )
= [ωktν + (1 − ω)(zt nt )ν ]1/ν
yt
= ct + i t
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
= ḡ +
∞
X
χj εt+1−j
j=0
Interpretation
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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
ĉ = hck k̂ +
n̂ = hnk k̂ +
∞
X
j=0
∞
X
hcj εt−j
hnj εt−j
j=0
Backus, Routledge, and Zin ()
Leads, lags, and logs
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Logs
Loglinear decision rules
Traditional methods
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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
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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
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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
Backus, Routledge, and Zin ()
= − log Et mt+1
Leads, lags, and logs
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Properties of the model
Impulse response of short rate
Backus, Routledge, and Zin ()
Leads, lags, and logs
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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
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recursive preferences
predictable component to productivity
Lots left to do
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equity
macro-based bond pricing
stochastic volatility
Backus, Routledge, and Zin ()
Leads, lags, and logs
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Related work
Related work
Leads and lags
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Ang-Piazzesi-Wei, Beaudry-Portier, Jaimovich-Rebelo, Stock-Watson
Predictable component
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Bansal-Yaron
Computation with recursive preferences
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Hansen-Sargent, Tallarini, Uhlig
Kreps-Porteus pricing kernel
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Hansen-Heaton-Li, Weil
Backus, Routledge, and Zin ()
Leads, lags, and logs
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