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International Macroeconomics


Devereux and Sutherland (2006) method - some a pplications

May 16, 2008
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International MacroeconomicsDevereux and Sutherland method

Intro and motivation

Financial transmission on shocks

Financial channels of international shocks transmissionrisk sharing

consumption smoothing

Transmission literature usually based on DSGE models witheither

complete mkts

trade in only one asset
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Drawbacks (II)

Models with trade in one assetgross and net foreign assets positions coincide

Lane and Milesi-Ferretti (2001, 2006): large gross portf olioholdings

what are the important macro determinants of size andcomposition of gross portfolio positions?

gross assets and liabilities can have important eect on macrodynamics

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Incomplete mkts and more than one asset

Should allow for

analyze the role of current account decits/surpluses inshaping international shocks transmission

analyze the implications of gross portfolio holdings

I i l M i
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Incomplete mkts and more than one asset - issues

1 With complete mkts it is possible to solve for the real macroallocation/dynamics independently of portfolio allocation

2 With incomplete mkts we have to solve jointly for macrooutcomes and portfolios

3 In a non-stochastic Steady State portfolio is indete rminate

4

The same applies in a rst order approximation tothe model5 In principle it is infeasible to apply standard approximat ion

methods to incomplete mkts models

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Devereux and Sutherland method - key aspects

At 1rst order of approximation only a solution for the StedyState portfolio is required to analyze dynamics

Time variations in portfolio shares are irrelevant for rst order

responses of macro variables

Characterize SS portfolio by a system of 2nd order approximation to portfolio selection condition1rts order approximation to the rest of the modelinterdependent as excess returns used for portfolio selectiondepend on GE model and macro outcomes depend on chosenportfolioDS solve this system to yield a closed form solution forportfolio holdings

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Basic framework

Basic model - goods and utility

Two countries, H and F, specialized in the produc tion of dierent goods, Y H and Y F , at prices P H and P ?F

Consumption is a bundle of H and F goods

Prices of H and F consumption bundle P and P ?

U = E t = t

t [u (C ) + ( . )]

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Basic framework

Basic model - assets

Assume n assetsAsset holdings of H households 1 , ..., n - denominated inunits of H consumption good

H households wealth

W t =n

i = 1

i , t

r are endogenous gross returns

r 0 = [ r 1t , . .. r nt ]

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Basic framework

H agents budget constraint

W t =n

i = 1

i , t =n

i = 1

i , t 1r it + Y t C t

where Y t = P H Y H / P

Dene excess returns on the n-th asset asr ix = (r i r n ) )

r 0xt = [ (r 1t r nt ) , ..., (r n 1t r nt )] and

0t = [ 1t , ..., n 1t ]

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Basic framework

H agents budget constraint

H hhs bc can be written as

W t =n

i = 1

i , t =n 1

i = 1

i , t 1r it + r nt W t 1 + Y t C t =

= 0t 1r it + r nt W t 1 + Y t C t

F agents bc is

1Q

W ?t =1Q

? 0t 1r it + r nt W ?

t 1 + Y t C t

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Devereux and Sutherland methodBasic framework

Optimality conditions

maxf c t , i g

. : U = E t

= t

t [u (C ) + ( . )]

st : W t =n

i = 1

i , t =n 1

i = 1

i , t 1r it + r nt W t 1 + Y t C t

u 0(C t ) = E t f u 0(C t + 1) r it + 1g =

= E t f u 0(C t + 1) r nt + 1g , 8i = 1, ... (n 1)

Q 1

t u 0

(C ?

t ) = E t Q 1

t + 1 u 0

C ?

t + 1 r it + 1 == E t Q 1t + 1 u 0 C

?

t + 1 r nt + 1

it + ?it = 0, 8i = 1, ... (n 1)

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Devereux and Sutherland methodBasic framework

Indeterminacy problem

Non-stochastic SS

1 = E

t

u 0(C t + 1)u 0(C t )

r it + 1 =

E t

u 0(C t + 1)u 0(C t )

r nt + 1 )

) r SS 1 = r SS 1 = (1/ )

First order approximation (assume u CRRA)

C t = E t f C t + 1g + E t f r 1t + 1g = E t f C t + 1g + E t f r 2t + 1g) E t f r 1t + 1g = E t f r 2t + 1g

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Devereux and Sutherland methodDS procedure

DS assumptions and notation

Shocks are AR(1) with variance/covariance matrix - timeinvariant

Innovations symmetrically distributed in interval [ , + ]

Assume symmetric non-stochastic SS with zero wealth:W = W ? = 0

CRRA utility u (C t ) = (1 ) 1 (C t )1

DeneW t = W t W C and r xt = r it r nt

= Y

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Basic principle

Goal: approximate the model at 1rst order around X and

DS show that only a solution for is required to analyze

dynamicsIn a rst order approximation to the non-portfoliopart of themodel the only aspect of the portfolio allocation that appearsis

W t =1

W t 1 + 0r xt + Y t C t

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Basic principle

Normal perturbation methodology: specify an approximationpoint and solve for approximate behaviour of variables aroundthat pointDS procedure reverses this principle: leave unspecied,derive it endogenously after having approximated the rest of the systemCan be thought of as a xed point problem (iterativealgorithm)

1 approximate non portfolio equations at 1rst order for a given 2 do implied linear decision rules satisfy (second order) portf olio

choice equations?

DS develop closed form solution for given the (1rst order)equations for the rest of the model

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Portfolio equations

Take 2nd order approximation of Euler Equations f or H and FSubtract them

E t C t + 1 C t + 1 Q t + 1 / ( r xt + 1) = 0

Sum them

E t f r xt + 1g = E t 12r 2xt + 1 + 12

C t + 1 C t + 1 Q t + 1 / ( r xt + 1)

Characterize equilibrium portfolio and expected excess returns

International MacroeconomicsD d S h l d h d
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Exploiting approximation properties

Up to 1rst order approximationE t f r 1t + 1g = E t f r 2t + 1g ) E t f r xt + 1g = 0

E t C t + 1

C t + 1

Q t + 1 / ( r xt + 1)

= Cov t C t + 1 C t + 1 Q t + 1 / ( r xt + 1)

time invariant ) second order (one-period-ahea d)

moments are time invariant

E t C t + 1 C t + 1 Q t + 1 / ( r xt + 1)= Cov C t + 1 C t + 1 Q t + 1 / ( r xt + 1)

International MacroeconomicsD d S th l d th d
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The procedure in practice (1)

Solve the 1rst order approximation to the non-port f olioequations of the model by treating 0 as givenE t f r xt + 1g = 0 ) t + 1 = 0r xt + 1 is a zero mean i.i.d randomvariable

W t = 1 W t 1 + t + Y t C t

Full model linearized to 1rst order

A1 s t + 1E t c t + 1 = A2 s t

c t + A3x t + B t

s t + 1 = F 1x t + F 2s t + F 3 t

c t = P 1x t + P 2s t + P 3 t

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Derive 0 as the one that satises 2nd order approximation of portfolio equationsExtract r xt + 1 vector from the solution

r xt + 1 = R 1 t + 1 + R 2t + 1

Recall is endogenous

xt + 1

= 0r xt + 1

Combining the equations above

r xt + 1 = R 10R 2

1 0R 1

+ R 2 t + 1

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Extract from the solution

C t + 1

C t + 1

Q t + 1

/ = D 1 t + 1 + D 2t + 1 + D 3

s t + 1x t

D 10R 2

1 0R 1+ D 2 t + 1 + D 3

s t + 1x t

E t C t + 1 C t + 1 Q t + 1 / ( r xt + 1) =

E t D 10R 2

1 0R 1+ D 2 t + 1 r xt + 1 = R D 0

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Derive 0 as portfolio holdings satisfying

E t C t + 1

C t + 1

Q t + 1

/ (

r xt + 1) =

R

D

0

= 0

Closed form solution for 0 in terms of vectors derived f romlinear solver

= R 2 D 02R 01 D 1R 2 R 021 R 2 D 02

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Devereux and Sutherland methodAn example

Model setup

One good, H and F country

u (C t ) = 1(1 ) (C t )1

log Y t = Y log Y t 1 + t , Y , log Y ?

t = Y log Y ?

t 1 + ?

t , Y

Var = 2Y 00 2Y

Two bonds. H bc:W t = B , t + ?B , t = B , t 1r Bt +

?

B , t 1r ?

Bt + Y t C t

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An example

Model setup

Dene R B and R ?B to be the nominal payos of the bonds

r Bt = R B , t P t 1 / P t and r ?

Bt = R ?

B , t P ?

t 1 / P ?

t

LOP ) P = SP ?

) Q = 1Money demands

M t = P t Y t and M ?t = P ?

t Y ?

t

Monetary shocks - uncorrelated, variance 2M

log M t = M log M t 1 + t , M , log M ?

t = M log M ?

t 1 + ?

t , M

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An example

Focs and resource constraint

(C t ) = E t n(C t + 1) r Bt + 1o= E t n(C t + 1) r ?Bt + 1o(C ?t )

= E t nC ?t + 1 r Bt + 1o= E t nC ?t + 1 r ?Bt + 1oC t + C ?t = Y t + Y

?

t

Two assets and four shocks ) incomplete asset mkts

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An example

First order approximation - excess returns

r Bt = R Bt + P t 1 P t r ?Bt = R ?

Bt + P ?

t 1 P ?

t

E t 1 f r Bt r ?Bt g = 0 ) (R Bt R ?

Bt ) =E t 1 (P t ) P t 1 E t 1 (P

?

t ) + P ?

t 1

( r Bt r ?Bt ) = R Bt R ?

Bt + P t 1 P t P ?

t 1 + P ?

t =

= E t 1P t

P t 1 E t 1

P

?

t +P

?

t 1 +P t 1

P t

P

?

t 1 +P

?

t

) r xt = E t 1 P t E t 1 P ?t P t + P ?

t

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An example

First order approximation - excess returns

Dene P E t 1 = E t 1 P t P E ?

t 1 = E t 1 P ?

t

) r xt = P E t 1 P t P E ?

t 1 P ?

t

The realized excess return is given by the dierence betweenH and F price surprise

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An example

First order approximation - expectation al equat ions

(C t ) = E t (C t + 1) + E t (r Bt + 1)

(C ?t ) = E t C ?

t + 1 + E t r ?

Bt + 1

DeneE t (r Bt + 1) = E t r ?Bt + 1 = r E t

P E t = E t P t + 1

P E ?t = E t P ?

t + 1

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An example

First order approximation - non-expecta tional equations

C t + C ?t = Y t + Y ?

t

W t = (

1/ )

W t 1 +

Y t

C t + t

M t = P t + Y t

M ?t = P ?

t + Y ?

t

r xt = P E t 1 P t P E ?

t 1 P ?

t

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An example

Variables

Endogenous states = [ P E t , P E ?

t , W t , r E t ]

Jumps = [ C t , C ?t , P t , P ?

t , r x t ]

Exogenous states = [ Y t , Y ?t , M t , M ?

t , t ]


A l
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An example

Steady State Portfolio

Extract appropriate rows from the solution to the (1rst order)system

= R 2 D 02R 01 D 1R 2 R 02 1 R 2 D 02

B = ?

B = 2Y

2 ( 2M + 2Y ) (1 Y )

H households sell the bond in their own currency and buy thebond in the foreign currency
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An example


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An example

Optimal portfolio dependance on variances

B = ?

B = 2Y

2 ( 2M + 2Y ) (1 Y )

Demand for foreign currency denominated bonds is increasingin output variance

decreasing in monetary shocks variance

Although prices are fully exible, monetary volatility is costlybecause it reduces the usefulness of nominal bonds as arisk-sharing instrument


An example
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An example

Matrix from the linearized system

PP: Recursive equilibrium law of motion for x(t) on x(t -1):0 0 0 0

0 0 0 00 0 1 00 0 0 0

Not stationary!


An example
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An example

Impulse responses - H endowment shock

20 40 60 80 1000

0.5

1H techn shock

20 40 60 80 1000

0.5

1F techn shock

20 40 60 80 1000

0.5

1H monetary shock

20 40 60 80 1000

0.5

1F monetary shock


An example
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An example


20 40 60 80 100-1

-0.5

0

0.5

1 H consumption

20 40 60 80 100-1

-0.5

0

0.5

1 F consumption

20 40 60 80 100-1

-0.5

0

0.5

1 H price

20 40 60 80 100-1

-0.5

0

0.5

1F price

20 40 60 80 100-1

-0.5

0

0.5

1return differential


An example
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p


20 40 60 80 100-1

-0.5

0

0.5

1H expected price

20 40 60 80 100-1

-0.5

0

0.5

1F expected price

20 40 60 80 100-10

-5

0

5

10wealth

20 40 60 80 100

-0.2

-0.1

0

0.1

0.2

expected returns


An example
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10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1 nominal exchange rate

10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1real exchange rate

International MacroeconomicsDevereux and Sutherland methodAn example
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Impulse responses - H monetary shock

20 40 60 80 1000

0.5

1H techn shock

20 40 60 80 1000

0.5

1F techn shock

20 40 60 80 1000

0.5

1H monetary shock

20 40 60 80 1000

0.5

1F monetary shock

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20 40 60 80 100-1

-0.5

0

0.5

1 H consumption

20 40 60 80 100-1

-0.5

0

0.5

1 F consumption

20 40 60 80 100-1

-0.5

0

0.5

1 H price

20 40 60 80 100-1

-0.5

0

0.5

1F price

20 40 60 80 100-1

-0.5

0

0.5

1return differential

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20 40 60 80 100-1

-0.5

0

0.5

1 H expected price

20 40 60 80 100-1

-0.5

0

0.5

1 F expected price

20 40 60 80 100-10

-5

0

5

10wealth

20 40 60 80 100

-0.2

-0.1

0

0.1

0.2

expected returns

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10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1 nominal exchange rate

10 20 30 40 50 60 70 80 90 100

-1

-0.5

0

0.5

1real exchange rate

International MacroeconomicsDevereux and Sutherland methodSome issues about DS method
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When are asset markets (eectively) co mplete?

Focus on 1rst order approximation: is it sucient to have asmany assets as shocks?

Characteristics of assets available matter

Endowment economies, Y H , Y F shocks, H and F equities )(up to 1rst order) full risk sharing

Endowment economies, Y H , Y F shocks, H equity and H bond) incomplete mkts

Order of approximation matters

International MacroeconomicsDevereux and Sutherland methodSome issues about DS method
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Other issues

The model is non-stationary

stationarity inducing featuresgross asset holdings vary wrt non-stationary model and acrossdierent parametrization of stationarity features

Can retrieve closed form solution for shares only if model fairlysimple
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Documents

DS Class 2008 [EDocFind.com][1]