Asset Prices in Business Cycle Analysis David Backus (NYU), Bryan - PowerPoint PPT Presentation

Asset Prices in Business Cycle Analysis David Backus (NYU), Bryan Routledge (CMU), and Stanley Zin (CMU) New York Fed | November 16, 2007 This version: November 15, 2007 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 34

Outline Pictures: leads and lags in US data Equations: the usual suspects + bells & whistles Computations: loglinear approximation More pictures: leads and lags in the model Extensions Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 34

Leads and lags in data Leads and lags in US data 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 (used as is) ◮ Occasional year-on-year comparisons (log x t +2 − log x t − 2 ) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 2 / 34

Leads and lags in data 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 / 34

Leads and lags in data 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 / 34

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 Leads and lags in data Lags GDP 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 / 34

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 in data 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 / 34

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 in data 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 / 34

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 in data 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 / 34

Leads and lags in data 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 / 34

Leads and lags in data Lead/lag summary Things that lead GDP ◮ Stock prices ◮ Yield curve and short rate ◮ Maybe consumption (a little) Things that lag GDP ◮ Maybe employment (a little) Why? Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 10 / 34

The usual suspects (Almost) the usual equations Streamlined Kydland-Prescott except ◮ Recursive preferences (Kreps-Porteus/Epstein-Zin-Weil) ◮ CES production ◮ Adjustment costs ◮ Unit root in productivity ◮ Predictable component in productivity growth Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 11 / 34

The usual suspects (Almost) the usual equations Streamlined Kydland-Prescott except ◮ Recursive preferences (Kreps-Porteus/Epstein-Zin-Weil) ◮ CES production ◮ Adjustment costs ◮ Unit root in productivity ◮ Predictable component in productivity growth Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 11 / 34

The usual suspects Preferences Equations U t = V [ u t , µ t ( U t +1 )] c t (1 − n t ) λ = u t [(1 − β ) u ρ t + βµ ρ t ] 1 /ρ V ( u t , µ t ) = � 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 / 34

The usual suspects Technology: production 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 Interpretation Elast of Subst = 1 / (1 − ν ) ω ( y / k ) − ν Capital Share = Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 13 / 34

The usual suspects Technology: capital accumulation Equations k t +1 = g ( i t , k t ) (1 − δ ) k t + k t [( i t / k t ) η ( i / k ) 1 − η − (1 − η )( i / k )] /η = Interpretation No adjustment costs if η = 1 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 14 / 34

The usual suspects Productivity Equations log x t +1 = ( I − A ) log x + A log x t + Bw t +1 { w t } ∼ NID(0 , I ) log z t +1 − log z t = log x 1 t +1 (first element) Interpretation A = [0] ⇒ no predictable component Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 15 / 34

Logs Computation overview Scaling ◮ Recast as stationary problem in “scaled” variables Loglinear approximation ◮ Loglinearize value function ( not log-quadratic) ◮ Loglinearize necessary conditions ◮ With constant variances, recursive preferences irrelevant to quantities (but not asset prices) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 16 / 34

Logs Scaling the Bellman equation Key input: ( V , µ, f , g ) are hd1 Natural version � � c t (1 − n t ) λ , µ t [ J ( k t +1 , x t +1 , z t +1 ] J ( k t , x t , z t ) = max c t , n t V subject to: k t +1 = g [ f ( k t , z t n t ) − c t , k t ) plus productivity process & initial conditions Scaled version [˜ k t = k t / z t , ˜ c t = c t / z t ] � � J (˜ c t (1 − n t ) λ , µ t [ x 1 t +1 J (˜ k t , x t , 1) = max c t , n t V ˜ k t +1 , x t +1 , 1)] ˜ ˜ k t +1 = g [ f (˜ c t , ˜ subject to: k t , n t ) − ˜ k t ] / x 1 t +1 plus productivity process & initial conditions Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 17 / 34

Logs Loglinear approximation Objective: loglinear decision rules [ˆ k t ≡ log ˜ k t − log ˜ k , etc] h ck ˆ k t + h ⊤ ˆ = cx ˆ c t x t h nk ˆ k t + h ⊤ n t ˆ = nx ˆ x t Key input: log J (˜ p 0 + p k log ˜ k t + p ⊤ k t , x t ) = x log x t Solution ◮ Brute force loglinearization of necessary conditions ◮ Riccati equation separable: first p k , then p x ◮ Lots of algebra, but separability allows you to do it by hand Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 18 / 34

Logs Necessary conditions First-order conditions c ρ − 1 (1 − n t ) ρλ (1 − β )˜ = M t g it t c ρ t (1 − n t ) ρλ − 1 λ (1 − β )˜ = M t g it f nt Envelope condition J 1 − ρ J kt = M t ( g it f kt + g kt ) t “Massive expression” β µ t ( x 1 t +1 J t +1 ) ρ − α E t [( x 1 t +1 J t +1 ) α − 1 J kt +1 ] M t = Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 19 / 34