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) Zicklin School of Business, Baruch College October 24, 2007 This version: October 23, 2007 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 46

2 Overview of recursive preferences Time preference Risk Preference Chew-Dekel Risk premiums Applications of Kreps-Porteus preferences Pricing kernels Risk sharing Business cycles (the paper in the title) Extensions? Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 1 / 46

3 Time preference Time preference Time aggregator V U t = V (u t, U t+1 ) Additive preferences U t = (1 β)u t + βu t+1 = (1 β) β j u t+j j=0 Why don t we care about this? Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 2 / 46

4 Risk preference Risk preference overview Certainty equivalent functions Chew-Dekel preferences Small, lognormal, and extreme risks Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 3 / 46

5 Risk preference Risk preference Basics: states s {1,..., S}, consumption c(s), probabilities p(s) Certainty equivalent function: µ satisfying U(µ,..., µ) = U[c(1),..., c(s)] Risk aversion: µ(c) E(c) Chew-Dekel preferences (risk aggregator M) µ = s p(s)m[c(s), µ] Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 4 / 46

6 Risk preference Chew-Dekel examples Expected utility M(c, m) = c α m 1 α /α + m(1 1/α) Weighted utility M(c, m) = (c/m) γ c α m 1 α /α + m[1 (c/m) γ /α]. Disappointment aversion M(c, m) = c α m 1 α /α + m(1 1/α) + δi (m c)(c α m 1 α m)/α I (x) = 1 if x > 0, 0 otherwise Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 5 / 46

7 Risk preference Chew-Dekel as adjusted probabilities Expected utility µ = ( ) 1/α p(s)c(s) α s Weighted utility: ditto with ˆp(s) = p(s)c(s) γ u p(u)c(u)γ, Disappointment aversion: ditto with ˆp(s) = p(s)(1 + δi [µ c(s)]) p(s)(1 + δi [µ c(s)]), u Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 6 / 46

8 Risk preference Small risks Two states (1 + σ, 1 σ), equal probs, Taylor series around σ = 0 Expected utility Weighted utility µ(eu) 1 (1 α)σ 2 /2 µ(wu) 1 [1 (α + 2γ)]σ 2 /2 Disappointment aversion ( ) ( ) δ 4 + 4δ µ(da) 1 σ (1 α) 2 + δ 4 + 4δ + δ 2 σ 2 /2 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 7 / 46

9 Risk preference Lognormal risks Let: log c N(κ 1, κ 2 ), rp = log[e(c)/µ] Expected utility rp(eu) = (1 α)κ 2 /2 Weighted utility rp(wu) = [1 (α + 2γ)]κ 2 /2 Disappointment aversion rp(wu) = E2C2E Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 8 / 46

10 Risk preference Extreme risks Let: log E exp(log c) = κ 1 + κ 2 /2! + κ 3 /3! + κ 4 /4! Expected utility rp(eu) = (1 α)κ 2 /2 + (1 α 2 )κ 3 /3! + (1 α 3 )κ 4 /4! Weighted utility rp(wu) = [1 (α + 2γ)]κ 2 /2 + [1 (α + 2γ) 2 + γ(α + γ)]κ 3 /3! + [1 (α + 2γ) 3 + 2γ(α + γ)(α + 2γ)]κ 4 /4! Disappointment aversion rp(da) = Another E2C2E Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 9 / 46

11 Kreps-Porteus preferences Kreps-Porteus preferences Recursive preferences U t = V [u t, µ t (U t+1 )] Kreps-Porteus/Epstein-Zin-Weil V (u t, µ t ) = [(1 β)ut ρ + βµ ρ t ] 1/ρ µ t (U t+1 ) = ( E t Ut+1 α IES = 1/(1 ρ) CRRA = 1 α ) 1/α α = ρ additive preferences Invariant to monotonic transformations: eg, Ût = Ut ρ /ρ Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 10 / 46

12 Kreps-Porteus preferences Kreps-Porteus overview Pricing kernels Risk sharing Business cycles Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 11 / 46

13 Kreps-Porteus preferences Kreps-Porteus pricing kernels Marginal rate of substitution m t+1 = β(c t+1 /c t ) ρ 1 [U t+1 /µ t (U t+1 )] α ρ Note role of future utility Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 12 / 46

14 Kreps-Porteus preferences Kreps-Porteus pricing kernels (continued) Example: let consumption growth follow log x t = log x + χ j w t j j=0 Pricing kernel log m t+1 = constant + [(ρ 1)χ 0 + (α ρ)(χ 0 + X 1 )]w t+1 + (ρ 1) χ j+1 w t j X 1 = β j χ j j=1 j=0 ( Bansal-Yaron term) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 13 / 46

15 Kreps-Porteus preferences Kreps-Porteus risk sharing Pareto problem with two recursive agents Bryan did this a few weeks ago Issues Time-varying pareto weights Representative agent may look different from individuals Possible nonstationary consumption distribution Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 14 / 46

16 Kreps-Porteus preferences Kreps-Porteus business cycle overview 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 15 / 46

17 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 16 / 46

18 Leads and lags in data Stock prices and GDP Cross Correlation with GDP Leads GDP S&P 500 Lags GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 17 / 46

19 Leads and lags in data Stock prices and GDP (year-on-year) Cross Correlation with GDP S&P 500 (yoy) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 18 / 46

20 Leads and lags in data Stock prices and GDP Cross Correlation with GDP Leads GDP S&P 500 Lags GDP Cross Correlation with GDP S&P 500 minus Short Rate Cross Correlation with GDP NYSE Composite Cross Correlation with GDP Nasdaq Composite Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 19 / 46

21 Leads and lags in data Interest rates and GDP Cross Correlation with GDP Yield Spread (10y 3m) Cross Correlation with GDP Yield Spread (GDP yoy) Cross Correlation with GDP Short Rate (3m) Cross Correlation with GDP Real Rate Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 20 / 46

22 Leads and lags in data Consumption and GDP Cross Correlation with GDP Consumption Cross Correlation with GDP Services Cross Correlation with GDP Nondurables Cross Correlation with GDP Durables Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 21 / 46

23 Leads and lags in data Investment and GDP Cross Correlation with GDP Investment Cross Correlation with GDP Structures Cross Correlation with GDP Equipment and Software Cross Correlation with GDP Residential Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 22 / 46

24 Leads and lags in data Employment and GDP Cross Correlation with GDP Employment (Nonfarm Payroll) Cross Correlation with GDP Employment (Household Survey) Cross Correlation with GDP Avg Weekly Hours (All) Cross Correlation with GDP Avg Weekly Hours (Manuf) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 23 / 46

25 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 Why? Maybe employment (a little) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 24 / 46

26 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 25 / 46

27 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 25 / 46

28 The usual suspects Preferences Equations Interpretation U t = V [u t, µ t (U t+1 )] u t = c t (1 n t ) λ V (u t, µ t ) = [(1 β)u ρ t + βµ ρ t ] 1/ρ µ t (U t+1 ) = ( E t U α t+1 IES = 1/(1 ρ) CRRA = 1 α ) 1/α α = ρ additive preferences Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 26 / 46

29 The usual suspects Technology: production Equations Interpretation y t = f (k t, z t n t ) = [ωk ν t + (1 ω)(z t n t ) ν ] 1/ν y t = c t + i t Elast of Subst = 1/(1 ν) Capital Share = ω(y/k) ν Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 27 / 46

30 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 28 / 46

31 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 1t+1 (first element) Interpretation A = [0] no predictable component Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 29 / 46

32 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 30 / 46

33 Logs Scaling the Bellman equation Key input: (V, µ, f, g) are hd1 Natural version { } J(k t, x t, z t ) = max V c t (1 n t ) λ, µ t [J(k t+1, x t+1, z t+1 ] c t,n t 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, etc] J( k t, x t, 1) = max c t,n t V { } c t (1 n t ) λ, µ t [x 1t+1 J( k t+1, x t+1, 1)] subject to: k t+1 = g[f ( k t, n t ) c t, k t ]/x 1t+1 plus productivity process & initial conditions Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 31 / 46

34 Logs Necessary conditions First-order conditions Envelope condition Massive expression (1 β) c t ρ 1 (1 n t ) ρλ = M t g it λ(1 β) c t ρ (1 n t ) ρλ 1 = M t g it f nt J kt = J 1 ρ t M t (g it f kt + g kt ) M t = β µ t (x 1t+1 J t+1 ) ρ α E t [(x 1t+1 J t+1 ) α 1 J kt+1 ] Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 32 / 46

35 Logs Loglinear approximation Objective: loglinear decision rules [ˆk t log k t log k, etc] ĉ t = h ck ˆkt + hcx ˆx t ˆn t = h nk ˆkt + hnx ˆx t Key input: log J( k t, x t ) = p 0 + p k log k t + px 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 33 / 46

36 Leads and lags in models Leads and lags in the model: overview Growth model: no labor or adjustment costs Three processes for productivity growth Random walk (A = 0) Two-period lead Small predictable component The challenge Barro and King Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 34 / 46

37 Leads and lags in models Random walk: impulse responses Consumption 1 Productivity Investment Quarters after Shock Interest Rate10 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 35 / 46

38 Leads and lags in models Random walk: cross correlations Consumption Investment Interest Rate Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 36 / 46

39 Leads and lags in models Two-period lead: cross correlations Consumption Investment Interest Rate Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 37 / 46

40 Leads and lags in models Predictable component: cross correlations Consumption Investment Interest Rate Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 38 / 46

41 Summary and extensions Summary Data: interest rates lead the cycle Model: ditto from predictable component in productivity growth Extensions Labor dynamics: Gali s result? Stochastic volatility Could this result from endogenous dynamics? Monetary policy? Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 39 / 46

42 Extra slides 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 40 / 46

43 Extra slides Autocorrelations of quarterly growth rates Autocorrelations of Growth Rates GDP Lag Bartlett s formula for MA(q) 95% confidence bands Consumption Lag Bartlett s formula for MA(q) 95% confidence bands Investment Lag Bartlett s formula for MA(q) 95% confidence bands Government Purchases Lag Bartlett s formula for MA(q) 95% confidence bands Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 41 / 46

44 Extra slides Random walk: autocorrelations 1 GDP 0.5 Consumption Investment Interest Rate Lag Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 42 / 46

45 Extra slides Predictable component: autocorrelations 1 GDP 0 Consumption Investment Interest Rate Lag Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 43 / 46

46 Extra slides Predictable component: impulse responses Productivity Consumption Investment Quarters after Shock Interest Rate10 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 44 / 46

47 Extra slides Approximation methods Problem: find decision rule u t = h(x t ) satisfying E t F (x t, u t, w t+1 ) = 1, w t N(0, v) Judd + many others Taylor series expansion of F nth moment shows up in nth-order term Us + much of modern finance Taylor series expansion of f = log F in E t exp[f (x t, u t, w t+1 )] = 1 All moments show up even in linear approximation Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 45 / 46

48 Extra slides Approximation methods: linear example Linear perturbation method Linear approximation of F F (x t, u t, w t+1 ) = F + F x (x t x) + F u (u t u) + F w w t+1 E t F = 1 u t u = (1 F )/F u (F x /F u )(x t x) Decision rule doesn t depend on variance of w (or higher moments) Affine finance method Linear approximation of f = log F f (x t, u t, w t+1 ) = f + f x (x t x) + f u (u t u) + f w w t+1 E t f = 1 u t u = (f + f w v/2)/f u (f x /f u )(x t x) Note impact of variance v (higher moments would show up, too) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 46 / 46