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

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asset prices in business cycle analysis

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

May 20, 2023 35 likes •407 views

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

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SLIDE 1

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

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SLIDE 2

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

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SLIDE 3

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 xt − log xt−1) except

◮ Interest rates and spreads (used as is) ◮ Occasional year-on-year comparisons (log xt+2 − log xt−2) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 2 / 34

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SLIDE 4

Leads and lags in data

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 / 34

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SLIDE 5

Leads and lags in data

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 / 34

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SLIDE 6

Leads and lags in data

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 / 34

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SLIDE 7

Leads and lags in data

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 / 34

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SLIDE 8

Leads and lags in data

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 / 34

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SLIDE 9

Leads and lags in data

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 / 34

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SLIDE 10

Leads and lags in data

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 / 34

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SLIDE 11

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

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SLIDE 12

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

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SLIDE 13

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

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SLIDE 14

The usual suspects

Preferences

Equations Ut = V [ut, µt(Ut+1)] ut = ct(1 − nt)λ V (ut, µt) = [(1 − β)uρ

t + βµρ t ]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 / 34

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SLIDE 15

The usual suspects

Technology: production

Equations yt = f (kt, ztnt) = [ωkν

t + (1 − ω)(ztnt)ν]1/ν

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

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 13 / 34

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SLIDE 16

The usual suspects

Technology: capital accumulation

Equations kt+1 = g(it, kt) = (1 − δ)kt + kt[(it/kt)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

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SLIDE 17

The usual suspects

Productivity

Equations log xt+1 = (I − A) log x + A log xt + Bwt+1 {wt} ∼ NID(0, I) log zt+1 − log zt = log x1t+1 (first element) Interpretation A = [0] ⇒ no predictable component

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 15 / 34

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SLIDE 18

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

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SLIDE 19

Logs

Scaling the Bellman equation

Key input: (V , µ, f , g) are hd1 Natural version J(kt, xt, zt) = max

ct,nt V

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

kt+1 = g[f (kt, ztnt) − ct, kt) plus productivity process & initial conditions Scaled version [˜ kt = kt/zt, ˜ ct = ct/zt] J(˜ kt, xt, 1) = max

˜ ct,nt V

˜

ct(1 − nt)λ, µt[x1t+1J(˜ kt+1, xt+1, 1)]

subject to:

˜ kt+1 = g[f (˜ kt, nt) − ˜ ct, ˜ kt]/x1t+1 plus productivity process & initial conditions

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 17 / 34

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SLIDE 20

Logs

Loglinear approximation

Objective: loglinear decision rules [ˆ kt ≡ log ˜ kt − log ˜ k, etc] ˆ ct = hckˆ kt + h⊤

cxˆ

xt ˆ nt = hnkˆ kt + h⊤

nxˆ

xt Key input: log J(˜ kt, xt) = p0 + pk log ˜ kt + p⊤

x log xt

Solution

◮ Brute force loglinearization of necessary conditions ◮ Riccati equation separable: first pk, then px ◮ Lots of algebra, but separability allows you to do it by hand Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 18 / 34

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SLIDE 21

Logs

Necessary conditions

First-order conditions (1 − β)˜ cρ−1

t

(1 − nt)ρλ = Mtgit λ(1 − β)˜ cρ

t (1 − nt)ρλ−1

= Mtgitfnt Envelope condition Jkt = J1−ρ

t

Mt(gitfkt + gkt) “Massive expression” Mt = β µt(x1t+1Jt+1)ρ−αEt[(x1t+1Jt+1)α−1Jkt+1]

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 19 / 34

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SLIDE 22

Logs

Expectations and certainty equivalents

Example: let log x ∼ N(κ1, κ2) Expectations and certainty equivalents for lognormals E(x) = exp(κ1 + κ2/2) E(xα) = exp(ακ1 + α2κ2/2) µ(x) = [E(xα)]1/α = exp(κ1 + ακ2/2). Effect of risk is multiplicative (like β)

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 20 / 34

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SLIDE 23

Logs

Approximation: two flavors

Problem: find decision rule ut = h(xt) satisfying EtF(xt, ut, wt+1) = 1, wt ∼ N(0, κ2) 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

Et exp[f (xt, ut, wt+1)] = 1

◮ All moments show up even in linear approximation Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 21 / 34

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SLIDE 24

Logs

Approximation: example

Linear “perturbation” method

◮ Linear approximation of F

F(xt, ut, wt+1) = F + Fx(xt − x) + Fu(ut − u) + Fwwt+1 EtF = 1 ⇒ ut − u = (1 − F)/Fu − (Fx/Fu)(xt − x)

◮ Decision rule doesn’t depend on variance of w (or higher moments)

“Affine” finance method

◮ Linear approximation of f = log F

f (xt, ut, wt+1) = f + fx(xt − x) + fu(ut − u) + fwwt+1 Et exp(f ) = 1 ⇒ ut − u = −(f + fwκ2/2)/fu − (fx/fu)(xt − x)

◮ Note impact of variance v (higher moments would show up, too) Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 22 / 34

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SLIDE 25

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 23 / 34

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SLIDE 26

Leads and lags in models

Random walk: impulse responses

2 4 6 8 10 12 14 16 18 20 1 2 Productivity 2 4 6 8 10 12 14 16 18 20 0.4 0.6 0.8 Consumption 2 4 6 8 10 12 14 16 18 20 1 Investment 2 4 6 8 10 12 14 16 18 20 5 10 Interest Rate Quarters after Shock Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 24 / 34

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SLIDE 27

Leads and lags in models

Random walk: cross correlations

−6 −4 −2 2 4 6 −1 1 Consumption −6 −4 −2 2 4 6 −1 1 Investment −6 −4 −2 2 4 6 −1 1 Interest Rate Lag Relative to GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 25 / 34

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SLIDE 28

Leads and lags in models

Two-period lead: cross correlations

−6 −4 −2 2 4 6 −1 1 Consumption −6 −4 −2 2 4 6 −1 1 Investment −6 −4 −2 2 4 6 −1 1 Interest Rate Lag Relative to GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 26 / 34

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SLIDE 29

Leads and lags in models

Predictable component: cross correlations

−6 −4 −2 2 4 6 −1 1 Consumption −6 −4 −2 2 4 6 −1 1 Investment −6 −4 −2 2 4 6 −1 1 Interest Rate Lag Relative to GDP Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 27 / 34

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SLIDE 30

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 28 / 34

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SLIDE 31

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 29 / 34

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SLIDE 32

Extra slides

Autocorrelations of quarterly growth rates

−0.200.00 0.20 0.40 0.60 GDP 5 10 15 20 25 Lag

Bartlett’s formula for MA(q) 95% confidence bands

−0.200.00 0.20 0.40 0.60 Consumption 5 10 15 20 25 Lag

Bartlett’s formula for MA(q) 95% confidence bands

−0.200.00 0.20 0.40 0.60 Investment 5 10 15 20 25 Lag

Bartlett’s formula for MA(q) 95% confidence bands

−0.200.00 0.20 0.40 0.60 Government Purchases 5 10 15 20 25 Lag

Bartlett’s formula for MA(q) 95% confidence bands

Autocorrelations of Growth Rates

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 30 / 34

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SLIDE 33

Extra slides

Random walk: autocorrelations

1 2 3 4 5 6 0.5 1 GDP 1 2 3 4 5 6 0.5 1 Consumption 1 2 3 4 5 6 −1 1 Investment 1 2 3 4 5 6 0.8 0.9 1 Interest Rate Lag Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 31 / 34

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SLIDE 34

Extra slides

Predictable component: autocorrelations

1 2 3 4 5 6 −1 1 GDP 1 2 3 4 5 6 0.5 1 Consumption 1 2 3 4 5 6 −1 1 Investment 1 2 3 4 5 6 0.6 0.8 1 Interest Rate Lag Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 32 / 34

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SLIDE 35

Extra slides

Predictable component: impulse responses

2 4 6 8 10 12 14 16 18 20 0.5 1 Productivity 2 4 6 8 10 12 14 16 18 20 0.2 0.3 0.4 Consumption 2 4 6 8 10 12 14 16 18 20 1 2 Investment 2 4 6 8 10 12 14 16 18 20 5 10 Interest Rate Quarters after Shock Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 33 / 34

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SLIDE 36

Extra slides

Theory and reality (circa 1964)

Realist MacLeod:

◮ Mundell’s article [ignores] complications associated with speculation in

the forward market. It can only bring discredit on the economics profession to leave unchallenged his attempt to draw from the model policy conclusions that are applicable in the real world.

Theorist Mundell:

◮ Theory is the poetry of science. It is simplification, abstraction, the

exaggeration of truth, a caricature of reality. Dr McLeod calls my assumptions unrealistic. I certainly hope he is right. I left out a million variables that made my caricature of reality unrealistic. At the same time, it enabled me to find fruitful, but refutable, empirical generalizations.

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 34 / 34

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SLIDE 37

Extra slides

Theory and reality (circa 1964)

Realist MacLeod:

◮ Mundell’s article [ignores] complications associated with speculation in

the forward market. It can only bring discredit on the economics profession to leave unchallenged his attempt to draw from the model policy conclusions that are applicable in the real world.

Theorist Mundell:

◮ Theory is the poetry of science. It is simplification, abstraction, the

exaggeration of truth, a caricature of reality. Dr McLeod calls my assumptions unrealistic. I certainly hope he is right. I left out a million variables that made my caricature of reality unrealistic. At the same time, it enabled me to find fruitful, but refutable, empirical generalizations.

Backus, Routledge, & Zin (NYU & CMU) Leads, lags, and logs 34 / 34