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

Transcription

1 Leads, Lags, and Logs: Asset Prices in Business Cycle Analysis David Backus (NYU), Bryan Routledge (CMU), and Stanley Zin (CMU) NYU Macro Lunch December 7, 2006 This version: December 7, 2006 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 24

2 Outline Pictures: leads and lags in business cycle indicators Equations: (almost) the usual ones Computation: loglinear approximation Properties: leads and lags in the model [under construction] Extensions Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 24

3 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 (log x t log x t 1 ), except Interest rates and spreads Occasional year-on-year comparisons (log x t+2 log x t 2 ) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 2 / 24

4 Leads and lags Stock prices and GDP Leads GDP S&P 500 Lags GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 3 / 24

5 Leads and lags Stock prices and GDP (year-on-year) S&P 500 (yoy) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 4 / 24

6 Leads and lags Stock prices and GDP Leads GDP S&P 500 Lags GDP S&P 500 minus Short Rate NYSE Composite Nasdaq Composite Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 5 / 24

7 Leads and lags Interest rates and GDP Yield Spread (10y 3m) Yield Spread (GDP yoy) Short Rate (3m) Real Rate Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 6 / 24

8 Leads and lags Consumption and GDP Consumption Services Nondurables Durables Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 7 / 24

9 Leads and lags Investment and GDP Investment Structures Equipment and Software Residential Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 8 / 24

10 Leads and lags Employment and GDP Employment (Nonfarm Payroll) Employment (Household Survey) Avg Weekly Hours (All) Avg Weekly Hours (Manuf) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 9 / 24

11 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 (a little) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 10 / 24

12 The usual equations (Almost) the usual equations Basic real business cycle model except Recursive preferences (Kreps-Porteus/Epstein-Zin-Weil) CES production Predictable component in productivity growth Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 11 / 24

13 The usual equations Preferences Equations U t = V [u t, µ t (U t+1 )] u t = c t (1 n t ) θ V (a, b) = [(1 β)a ρ + βb ρ ] 1/ρ µ t (U t+1 ) = ( E t U α 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 12 / 24

14 The usual equations Technology Equations y t = f (k t, z t n t ) = [ωkt ν + (1 ω)(z t n t ) ν ] 1/ν y t = c t + i t k t+1 = (1 δ)k t + y t c t = g(k t, z t n t ) c t Interpretation Elast of Subst = 1/(1 ν) Capital Share = ω(y/k) ν Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 13 / 24

15 The usual equations Productivity growth Equation log z t+1 log z t = log x t+1 = log x + χ j w t+1 j j=0 = log x + χ(l)w t+1 {w t } NID(0, 1) Interpretation Predictability if χ j 0 for j 1 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 14 / 24

16 The usual equations Asset prices Pricing kernel m t+1 = β (c t+1 /c t ) ρ 1 [U t+1 /µ t (U t+1 )] α ρ Short rate r t = log E t m t+1 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 15 / 24

17 The usual equations Bellman equations Natural version J(k t, z t, w t ) = max c t,n t V { } c t (1 n t ) θ, µ t [J(k t+1, z t+1, w t+1 )] subject to: k t+1 = g(k t, z t n t ) c t z t+1 = z t x exp[χ(l)w t+1 ] Scaled version [ k t = k t /z t, c t = c t /z t, etc] J( k t, 1, w t ) = max c t,n t V { } c t (1 n t ) θ, µ t [x t+1 J( k t+1, 1, w t+1 )] subject to: kt+1 = [g( k t, n t ) c t ]/x t+1 x t+1 = x exp[χ(l)w t+1 ] Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 16 / 24

18 Logs Loglinear approximation What we have so far: θ = 0, n = 1 Goal: loglinear dynamics [ˆx log x log x] Challenges ĉ t = h ck ˆk t + h cw (L)w t ˆk t+1 = h kk ˆkt + h kw (L)w t+1 Standard LQ methods don t apply with recursive preferences Infinite-dimensional state space Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 17 / 24

19 Logs Properties of the solution Quantities independent of risk aversion (α) Fix steady state, let β adjust Why this is a good thing Persistence governed by IES (σ) h kk 1 as σ 0 Predictability affects consumption dynamics (h cw ) h cj X j = i=1 λ i χ j+i Interest rate rt = constant + σ 1 E t log(c t+1 /c t ) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 18 / 24

20 Logs Identifying the productivity process Standard approach: univariate properties (acf) How do we distinguish these choices for {χ 0, χ 1,...}? {1, 0, 0, 0,...} v. {0, 0, 1, 0,...}? {1,.1, 0, 0,...} v. {.1, 1, 0, 0,...}? Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 19 / 24

21 Properties Impulse response: current shock Productivity (magenta), Consumption (red), Capital (blue) Growth Rates of Consumption (solid) and GDP (dashed) Time in Quarters Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 20 / 24

22 Properties Impulse response: future shock Productivity (magenta), Consumption (red), Capital (blue) Growth Rates of Consumption (solid) and GDP (dashed) Time in Quarters Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 21 / 24

23 Properties Impulse response: persistent component Productivity (magenta), Consumption (red), Capital (blue) Growth Rates of Consumption (solid) and GDP (dashed) Time in Quarters Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 22 / 24

24 Summary and extensions Asset prices lead the cycle Therefore: predictable component in productivity growth Next on the to-do list macro-based affine bond pricing equity Possibilities labor multiple sources of information stochastic volatility Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 23 / 24

25 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 24 / 24