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 ( y t +2 − y t − 2 ) Backus, Routledge, and Zin () Leads, lags, and logs 2 / 20

Leads and lags Stock prices and GDP S&P 500 1.00 1.00 Leads GDP Lags GDP Cross−Correlation with GDP 0.50 0.50 0.00 0.00 −0.50 −0.50 −1.00 −1.00 −10 −5 0 5 10 Lag Relative to GDP Backus, Routledge, and Zin () Leads, lags, and logs 3 / 20

Leads and lags Stock prices and GDP (year-on-year) S&P 500 (yoy) 1.00 1.00 Cross−Correlation with GDP 0.50 0.50 0.00 0.00 −0.50 −0.50 −1.00 −1.00 −10 −5 0 5 10 Lag Relative to GDP Backus, Routledge, and Zin () Leads, lags, and logs 4 / 20

Stock prices and GDP Backus, Routledge, and Zin () Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 Leads GDP NYSE Composite −5 −5 Lag Relative to GDP Lag Relative to GDP S&P 500 0 0 Lags GDP Leads and lags 5 5 Leads, lags, and logs 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 S&P 500 minus Short Rate Nasdaq Composite −5 −5 Lag Relative to GDP Lag Relative to GDP 0 0 5 5 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 5 / 20

Interest rates and GDP Backus, Routledge, and Zin () Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 Yield Spread (10y−3m) Short Rate (3m) −5 −5 Lag Relative to GDP Lag Relative to GDP 0 0 Leads and lags 5 5 Leads, lags, and logs 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 Yield Spread (GDP yoy) −5 −5 Lag Relative to GDP Lag Relative to GDP Real Rate 0 0 5 5 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 6 / 20

Consumption and GDP Backus, Routledge, and Zin () Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 −5 −5 Lag Relative to GDP Lag Relative to GDP Consumption Nondurables 0 0 Leads and lags 5 5 Leads, lags, and logs 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 −5 −5 Lag Relative to GDP Lag Relative to GDP Durables Services 0 0 5 5 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 7 / 20

Investment and GDP Backus, Routledge, and Zin () Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 Equipment and Software −5 −5 Lag Relative to GDP Lag Relative to GDP Investment 0 0 Leads and lags 5 5 Leads, lags, and logs 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −10 −5 −5 Lag Relative to GDP Lag Relative to GDP Residential Structures 0 0 5 5 10 10 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 8 / 20

Leads and lags Employment and GDP Employment (Nonfarm Payroll) Employment (Household Survey) Cross−Correlation with 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 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −5 0 5 10 −10 −5 0 5 10 Lag Relative to GDP Lag Relative to GDP Avg Weekly Hours (All) Avg Weekly Hours (Manuf) Cross−Correlation with GDP Cross−Correlation with GDP −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 −10 −5 0 5 10 −10 −5 0 5 10 Lag Relative to GDP Lag Relative to GDP 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 = V [ u t , µ t ( U t +1 )] U t c γ t (1 − n t ) 1 − γ u t = [(1 − β ) a ρ + β b ρ ] 1 /ρ V ( a , b ) = � 1 /α E t U α � µ t ( U t +1 ) = t +1 Interpretation = 1 / (1 − ρ ) IES CRRA = 1 − α α = ρ additive preferences ⇒ Backus, Routledge, and Zin () Leads, lags, and logs 12 / 20

The usual equations Technology Equations y t = f ( k t , z t n t ) [ ω k ν t + (1 − ω )( z t n t ) ν ] 1 /ν = y t = c t + i t = (1 − δ ) k t + i t k t +1 Interpretation Elast of Subst = 1 / (1 − ν ) ω ( y / k ) − ν Capital Share = Backus, Routledge, and Zin () Leads, lags, and logs 13 / 20

The usual equations Productivity process Equation ∞ � log z t +1 − log z t = ¯ g + χ j ε t +1 − j j =0 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 ∞ h ck ˆ � c ˆ = k + h cj ε t − j j =0 ∞ h nk ˆ � n ˆ = k + h nj ε t − j j =0 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 β ( c t +1 / c t ) ρ − 1 [ U t +1 /µ t ( U t +1 )] α − ρ = m t +1 Short rate = − log E t m t +1 r t 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