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
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
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
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, & Zin (NYU & CMU) Leads, lags, and logs 3 / 24
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, & Zin (NYU & CMU) Leads, lags, and logs 4 / 24
Backus, Routledge, & Zin (NYU & CMU) Stock prices and GDP 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 / 24
Backus, Routledge, & Zin (NYU & CMU) Interest rates and GDP 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 / 24
Backus, Routledge, & Zin (NYU & CMU) Consumption and GDP 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 / 24
Backus, Routledge, & Zin (NYU & CMU) Investment and GDP 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 / 24
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, & Zin (NYU & CMU) Leads, lags, and logs 9 / 24
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
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
The usual equations Preferences Equations = V [ u t , µ t ( U t +1 )] U t c t (1 − n t ) θ 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, & Zin (NYU & CMU) Leads, lags, and logs 12 / 24
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 + y t − c t k t +1 = g ( k t , z t n t ) − c t Interpretation Elast of Subst = 1 / (1 − ν ) ω ( y / k ) − ν Capital Share = Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 13 / 24
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
The usual equations 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, & Zin (NYU & CMU) Leads, lags, and logs 15 / 24
The usual equations Bellman equations Natural version � � c t (1 − n t ) θ , µ t [ J ( k t +1 , z t +1 , w t +1 )] J ( k t , z t , w t ) = max c t , n t V 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 (˜ c t (1 − n t ) θ , µ t [ x t +1 J (˜ k t +1 , 1 , w t +1 )] k t , 1 , w t ) = max ˜ c t , n t V ˜ k t +1 = [ g (˜ ˜ subject to: 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
Logs Loglinear approximation What we have so far: θ = 0, n = 1 Goal: loglinear dynamics [ˆ x ≡ log ˜ x − log ¯ x ] h ck ˆ ˆ = k t + h cw ( L ) w t c t ˆ h kk ˆ ⇒ k t +1 = k t + h kw ( L ) w t +1 Challenges ◮ Standard LQ methods don’t apply with recursive preferences ◮ Infinite-dimensional state space Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 17 / 24
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 ◮ r t = constant + σ − 1 E t log( c t +1 / c t ) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 18 / 24
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
Properties Impulse response: current shock Productivity (magenta), Consumption (red), Capital (blue) 1 0.5 0 0 5 10 15 20 Growth Rates of Consumption (solid) and GDP (dashed) 0.1 0.08 0.06 0.04 0.02 0 5 10 15 20 Time in Quarters Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 20 / 24
Properties Impulse response: future shock Productivity (magenta), Consumption (red), Capital (blue) 1 0.5 0 −0.5 0 5 10 15 20 Growth Rates of Consumption (solid) and GDP (dashed) 3 2 1 0 −1 0 5 10 15 20 Time in Quarters Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 21 / 24
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