Leads, Lags, and Logs: Asset Prices in Business Cycle Analysis - - PowerPoint PPT Presentation

▶

leads lags and logs asset prices in business cycle

Leads, Lags, and Logs: Asset Prices in Business Cycle Analysis - - PowerPoint PPT Presentation

Feb 10, 2023 23 likes •275 views

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)

slide-1

SLIDE 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

slide-2

SLIDE 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

slide-3

SLIDE 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 xt − log xt−1), except

◮ Interest rates and spreads ◮ Occasional year-on-year comparisons (log xt+2 − log xt−2) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 2 / 24

slide-4

SLIDE 4

Leads and lags

Stock prices and GDP

Leads GDP Lags 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 −10 −5 5 10 Lag Relative to GDP

S&P 500

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 3 / 24

slide-5

SLIDE 5

Leads and lags

Stock prices and GDP (year-on-year)

−1.00 −0.50 0.00 0.50 1.00 −1.00 −0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

S&P 500 (yoy)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 4 / 24

slide-6

SLIDE 6

Leads and lags

Stock prices and GDP

Leads GDP Lags 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 −10 −5 5 10 Lag Relative to GDP

S&P 500

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

S&P 500 minus Short Rate

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

NYSE Composite

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Nasdaq Composite

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 5 / 24

slide-7

SLIDE 7

Leads and lags

Interest rates and 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 −10 −5 5 10 Lag Relative to GDP

Yield Spread (10y−3m)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Yield Spread (GDP yoy)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Short Rate (3m)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Real Rate

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 6 / 24

slide-8

SLIDE 8

Leads and lags

Consumption and 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 −10 −5 5 10 Lag Relative to GDP

Consumption

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Services

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Nondurables

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Durables

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 7 / 24

slide-9

SLIDE 9

Leads and lags

Investment and 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 −10 −5 5 10 Lag Relative to GDP

Investment

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Structures

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Equipment and Software

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Residential

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 8 / 24

slide-10

SLIDE 10

Leads and lags

Employment and 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 −10 −5 5 10 Lag Relative to GDP

Employment (Nonfarm Payroll)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Employment (Household Survey)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Avg Weekly Hours (All)

−1.00−0.50 0.00 0.50 1.00 −1.00−0.50 0.00 0.50 1.00 Cross−Correlation with GDP −10 −5 5 10 Lag Relative to GDP

Avg Weekly Hours (Manuf)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 9 / 24

slide-11

SLIDE 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

slide-12

SLIDE 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

slide-13

SLIDE 13

The usual equations

Preferences

Equations Ut = V [ut, µt(Ut+1)] ut = ct(1 − nt)θ V (a, b) = [(1 − β)aρ + βbρ]1/ρ µt(Ut+1) =

EtUα

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

slide-14

SLIDE 14

The usual equations

Technology

Equations yt = f (kt, ztnt) = [ωkν

t + (1 − ω)(ztnt)ν]1/ν

yt = ct + it kt+1 = (1 − δ)kt + yt − ct = g(kt, ztnt) − ct Interpretation Elast of Subst = 1/(1 − ν) Capital Share = ω(y/k)−ν

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 13 / 24

slide-15

SLIDE 15

The usual equations

Productivity growth

Equation log zt+1 − log zt = log xt+1 = log ¯ x +

∞

j=0

χjwt+1−j = log ¯ x + χ(L)wt+1 {wt} ∼ NID(0, 1) Interpretation

◮ Predictability if χj = 0 for j ≥ 1 Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 14 / 24

slide-16

SLIDE 16

The usual equations

Asset prices

Pricing kernel mt+1 = β (ct+1/ct)ρ−1 [Ut+1/µt(Ut+1)]α−ρ Short rate rt = − log Etmt+1

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 15 / 24

slide-17

SLIDE 17

The usual equations

Bellman equations

Natural version J(kt, zt, wt) = max

ct,nt V

ct(1 − nt)θ, µt[J(kt+1, zt+1, wt+1)]
subject to:

kt+1 = g(kt, ztnt) − ct zt+1 = zt¯ x exp[χ(L)wt+1] Scaled version [˜ kt = kt/zt, ˜ ct = ct/zt, etc] J(˜ kt, 1, wt) = max

˜ ct,nt V

˜

ct(1 − nt)θ, µt[xt+1J(˜ kt+1, 1, wt+1)]

subject to:

˜ kt+1 = [g(˜ kt, nt) − ˜ ct]/xt+1 xt+1 = ¯ x exp[χ(L)wt+1]

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 16 / 24

slide-18

SLIDE 18

Logs

Loglinear approximation

What we have so far: θ = 0, n = 1 Goal: loglinear dynamics [ˆ x ≡ log ˜ x − log ¯ x] ˆ ct = hckˆ kt + hcw(L)wt ⇒ ˆ kt+1 = hkkˆ kt + hkw(L)wt+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

slide-19

SLIDE 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)

◮ hkk → 1 as σ → 0

Predictability affects consumption dynamics (hcw)

◮ hcj ∝ Xj = ∞

i=1 λ−iχj+i

Interest rate

◮ rt = constant + σ−1Et log(ct+1/ct) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 18 / 24

slide-20

SLIDE 20

Logs

Identifying the productivity process

Standard approach: univariate properties (acf) How do we distinguish these choices for {χ0, χ1, . . .}? {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

slide-21

SLIDE 21

Properties

Impulse response: current shock

5 10 15 20 0.5 1 Productivity (magenta), Consumption (red), Capital (blue) 5 10 15 20 0.02 0.04 0.06 0.08 0.1 Time in Quarters Growth Rates of Consumption (solid) and GDP (dashed)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 20 / 24

slide-22

SLIDE 22

Properties

Impulse response: future shock

5 10 15 20 −0.5 0.5 1 Productivity (magenta), Consumption (red), Capital (blue) 5 10 15 20 −1 1 2 3 Time in Quarters Growth Rates of Consumption (solid) and GDP (dashed)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 21 / 24

slide-23

SLIDE 23

Properties

Impulse response: persistent component

5 10 15 20 0.5 1 1.5 2 Productivity (magenta), Consumption (red), Capital (blue) 5 10 15 20 0.05 0.1 0.15 0.2 0.25 Time in Quarters Growth Rates of Consumption (solid) and GDP (dashed)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 22 / 24

slide-24

SLIDE 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

slide-25

SLIDE 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