Risks For The Long Run And The Real Exchange...

Risks For The Long Run And The Real Exchange Rate

Riccardo Colacito, Mariano M. Croce

Presented by Steven Laufer: October 2, 2007

Riccardo Colacito, Mariano M. Croce () Risks For The Long Run And The Real Exchange RatePresented by Steven Laufer: October 2, 2007 1

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Overview

“International Equity Premium Puzzle”

Model with long-run risks

Calibration Exercises

Estimation Attempts & Proposed Extensions

Discussion


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Asset returns imply a highly volatile stochastic discount factorgoverning returns on domestic and foreign currency denominatedassets.

Currency exchange rates should adjust to adjust to prevent arbitrageopportunities but variance of depreciation rate is low.

Implication is a high correlation between stochastic discount factors,but cross-country correlation of consumption is low.

Need highly correlated component of consumption growth. . .


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Two countries: home (h) and foreign (f) with mit = log(M i

t):

Et [exp(mft+1)R

ft+1] = 1 = Et [exp(mh

t+1)Rht+1]

If the foreign asset is traded in the home country, with exchange rateet ,

Et [exp(mft+1)R

ft+1] = 1 = Et [exp(mh

t+1)et+1

etR f

t+1].

If markets are complete:

πt+1 = mft+1 −mh

t+1

where πt+1 = log(et+1/et). (Backus, Foresi and Telmer 1996)


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Var(πt+1) = Var(mft+1 −mh

t+1)

ρmh,mf =σ2

mh + σ2mf − σ2

π

2σmhσmf

σ2mi > .20 from Hansen-Jagannathan bounds

σ2π ∼ .11− .15

Implies that ρmh,mf > .96 but covariance in consumption is only 27%.

“In a one county model, consumption growth does not vary enough toexplain the excess return over the risk free rate. In a two countrymodel, consumption growth does not co-vary enough to keep track ofthe movements in the exchange rate.”


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Model: Preferences

Two countries (i ∈ {h, f }), each with a representative consumer, eachwith a single county-specific good.

Complete home bias- each country derives utility only from its ownendowment. Epstein-Zin preferences:

U it = {(1− δ)(C i

t )1−γ

θ + δ[Et [(Uit+1)

1−γ ]]1/θ}θ

1−γ

δ is discount rate, γ is CRRA, ψ is IES, θ = 1−γ1−1/ψ


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Model: Assets

Et [Mit+1R

ij ,t+1] = 1

mit+1 = log M i

t+1 = θ log δ − θ

ψ∆c i

t+1 + (θ − 1) log R ic,t+1

where R it+1 is the return on the asset that pays dividend stream {C i

t}.

R it+1 =

v ic,t+1 + 1

v ic,t

exp∆ct+1

where v ic,t+1 is the price-consumption ratio in country i .

Complete markets: exchange rate adjusts to equalize returns acrosscountries.

In equilibrium, autarky in goods and asset holdings. (Not a model oftrade.)


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Model: Consumption Process

Consumption follows an exogenous law of motion:

∆c it = x i

t−1 + εic,t

x it = ρxx

it−1 + εix ,t

Shocks are i.i.d. with correlations governed by Σ:

[εhc,tεfc,tε

hx ,tε

fx ,t ] ∼ N(0,Σ)

Σ = σ2

[Hc 00 φ2

eHx

]Hc =

[1 ρhf

c

ρhfc 1

]Hx =

[1 ρhf

x

ρhfx 1

]


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Model: First-Order Approximations

First-order Taylor approximation for price-consumption ratio

v ic,t = v i

c

(1 +

(ψ − 1)

ψ(1− ρxδ)x it

)Solutions to first-order approximation of model:

mit+1 = log δ − 1

ψx it + δ

1− 1ψ

1− ρhfx

εix ,t+1 + εic,t+1

et+1

et= mf

t+1 −mht+1

r ic,t+1 = rc +

1

ψx it + δ

1− 1ψ

1− ρxδεix ,t+1 + εic,t+1

r if ,t+1 = rf +

1

ψx it

where r if ,t+1 is the log risk-free rate and rj is the average log return

on asset j .


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Model: Two Propositions

Proposition 1. For a given choice of parameters, and provided thatρh,fx > ρh,f

c , the lowest cross country correlation of the stochasticdiscount factor is achieved (uniquely) by (ψ, ρx) = ( 1

γ δ̃, 0) where

δ̃ = 1−2ρxδ+δ2

δ2(1−ρ2x )

.

Proposition 2. For a given choice of parameters, the lowest volatilityof the depreciation rate is achieved for ρhf

x = 1.


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Calibration: Baseline


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Calibration: Varying Paramters


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Calibration: Matching Other Moments

Introduce dividend growth process:

∆d it = µd + λxt−1 + εid ,t

[εhc,tεfc,tε

hx ,tε

fx ,tε

hd ,tε

fd ,t ] ∼ N(0, Σ̃)

Σ̃ =

[Σ 00 σ2φ2

dHd

]Hd = σ2

[1 ρhf

d

ρhfd 1

]


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Calibration: Matching other Moments


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Calibration: Matching other Moments

Match most moments fairly well including correlation of excessreturns.

Correlation of risk-free rate is one: depends only on x it which are

perfectly correlated.

Add stochastic volatility: Doesn’t change results, correlation ofrisk-free rate drops to .98


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Estimation of Long Run Risks

Attempts to estimate the persistent component

Compare frequency spectra with and without predictable component.Can’t distinguish two cases.

Use Kalman filter to estimate state-space system. Estimatesreasonably close to calibrated values but very large standard errors.

Estimate state space system with additional data from Germany andJapan. Doesn’t tighten confidence intervals.

Simulated Method of Moments estimation with additional moments.Get tighter estimates for consumption laws of motion but not forpreference parameters or dividend process.


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Estimation: MLE Estimates


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Estimation: Additional Countries


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Estimation: Additional Moments


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Extensions

Additional factors to match yield curve evidence. Yield curves havesharper features at low frequencies but can’t be identified inone-factor model.


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Extensions

Economic interpretations of xt .

Include wider set of moments.

Relaxing home bias assumption to study traded and non-tradedgoods.


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Discussion

Paper shows how long run risk can be used to solve internationalequity premium puzzle.

Results are very sensitive to high persistence and high cross-countrycorrelation.

Attempt to estimate the model but are unable to properly identifyparameters or to provide evidence that the persistent component ispresent.

No discussion of approximation errors.


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Discussion

Is there hope of identifying model?

Look at country pairs where would expect less correlation inconsumption process. Is there more freedom from the data to lowerthe correlation?

Relax home bias assumption. Implications for bi-lateral trade?


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Risks For The Long Run And The Real Exchange...

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