Endogenous Money and Exchange Rates Nelson C. Mark IMF Institute - - PowerPoint PPT Presentation

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Endogenous Money and Exchange Rates

Nelson C. Mark IMF Institute Seminar, 21 March 2008

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1 A set of exchange rate puzzles

• 1. The PPP Puzzle:

Why is the real exchange rate both highly persistent and volatile? Why is the nominal exchange rate so highly correlated with the real exchange rate? A successful model will explain these facts. Chari, Kehoe, McGrattan do with price stickiness lasting 4 quarters and relative risk aversion coe¢cient of 5. Flexible price models have di¢culty explaining volatility (without high risk aversion). Sticky price models have di¢culty explaining persis- tence (without unrealistically sticky prices).

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From Chari, Kehoe and McGrattan 2002, RE Stud.

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• 2. Traditional monetary approachs

The PPP theory: Higher in‡ation should lead to dollar weakening. s = p p ! s =

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Monetary models – Lucas JME 1982 s = (m m) (y y) – Obstfeld–Rogo¤ JPE 1995. s = (m m) 1 (c c) – Original monetary model. mt pt = + yt it + vt m

t p t

= + y

t i t + v t

qt = st + p

t pt

it i

t

= Et (st+1) st

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– Substitute into UIP for the di¤erence equation st = 1 1 +

B @(mt m

t) (yt y t ) + qt

| {z }

ft

(vt v

t )

| {z }

zt

1 C A +

• 1 + Et (st+1)

ft = (mt m

t) (yt y t ) + qt

– Iterate forward, st = 1 1 + Et

1

X

j=0

• 1 +

j

ft+j + zt+j

• – If ft is AR(1) with coe¢cient ;

st = 1 1 + (1 ) [(mt m

t) (yt y t )]

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Low-brow econometric results. (a)

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• 3. Relatively new empirical evidence on news and the exchange rate.

Real-time (5 minute sampling) exchange rate and announcement news (errors). Anderson et. al. AER 2003.

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Their …ndings: – Only unanticipated shocks to the fundamentals have a statistically signi…cant e¤ect on the exchange rate. – Good real news good news for the dollar: Unexpectedly strong real indicator is followed by a stengthening of the dollar. Consistent with most theory. – Bad news about in‡ation is good news for the dollar: Unexpectedly high in‡ation is followed by a strengthening of the dollar. This is contrary to most theory.

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In‡ation Di¤erentials and Real Dollar–DM Rate

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In‡ation Di¤erentials and Real Dollar–DM Rate

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In‡ation Di¤erentials and Real Dollar–DM Rate

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In‡ation Di¤erentials wutg Structural Break and Real Dollar–DM Rate

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• 4. Simpli…ed Taylor-rule model with uncovered interest parity. Taylor rule

(without interest rate smoothing). Let it be the target rate, be targeted in‡ation, xt = yt yp

t be the output gap,

{ be the natural nominal rate, st = pt p

t be an exchange rate target.

Then the deviation from the targeted exchange rate is qt = st + p

t pt.

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In Germany, it = (Ett+1 ) + xxt + qqt In the U.S., no exchange rate feedback i

t =

• Et

t+1

• + xx

t

Important for > 1: This is called the Taylor Principle. – Take Germany to be the home country. Ignore constant terms. Assume identical targets and coe¢cients across countries.

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– Exploit UIP Etst+1 st = it i

t

Etst+1 Et

• pt+1 p

t+1

• st (pt p

t)

= it i

t Et

• t+1

t+1

• Etqt+1qt = ( 1) Et
• t+1

t+1

• +x (xt x

t)+qqt

qt = (1 ) 1 + q

1

X

j=0

1 1 + q

!j

Et

• t+1+j

t+1+j

• x

1 + q

1

X

j=0

1 1 + q

!j

Et

• xt+j x

t+j

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– Assume in‡ation di¤erentials and output gap follow AR(1) processes . qt = (1 ) 1 + q (t

t)

xx 1 + q x (xt x

t)

Signi…cantly di¤erent character of fundamentals. Good news for home output is good news for the exchange rate. Bad news about in‡ation means what about the exchange rate? Depends on

SLIDE 19

2 The existence of interest rate rules

2.1 Clarida, Gali, Gertler QJE: For the Fed.

r is the nominal rate (FFR) rr = r

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2.2 Other central banks

Germany, Japan, US, UK, France, Italy. CGG 1998 EER. Data 1979-1993

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3 Taylor rules in New Keynesian (New Open Economy) Models

Benigno, JME PPP puzzle: Flexible price models have di¢culty explaining volatility (without high risk aversion). Sticky price models have di¢culty explaining persistence (without unrealistically sticky prices). Benigno breaks the link between real exchange rate persistence and the speed of nominal price adjustment.

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Some useful calculations: Typical household prob- lem in new Keynesian international macroeco- nomics

Household wants to maximize lifetime utility de…ned over consumption, leisure, and real money balances. Assume that period utility is separable. Et

1

X

j=t

j

"

U

• Ct+j
• + N

Mt+j

Pt+j

!

V

• Lt+j

#

SLIDE 28

where home goods are indexed as i 2 [0; n) and foreign goods are i 2 [n; 1] with consumption indices, C = (a1x + a2y)

1

• x

= (b1)

(1)

• Z n

0 c (i) di

• y

= (b2)

(1)

• Z 1

n c (i) di

!

• and

1 1 is the elasticity of substitution between home and foreign good

indices,

1 1 is the elasticity of substitution between within country varieties.

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Obtaining the price index: Cost minimization prob- lem I.

For a two-good CES consumption index C = (a1x + a1y)

1

• the elasticity of substitution between x and y is

1 1:

The corresponding price index is P = a

1 1

1

(p1)

• 1 + a

1 1

2

p

• 1

2

!1

• Proof:

For a given expenditure PC, we seek the best way to allocate it across x and y: Form the Lagrangian L = pxx + pyy +

• C (a1x + a1y)

1

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First-order conditions p1 = a1x1 (a1x + a2y)

1 1

p2 = a2y1 (a1x + a2y)

1 1

Eliminate the multiplier p1 p2 = a1 a2 x y

!1

From this relation, we can write x = p1 p2

!

a2 a1

!

1 1

y

• r

y = p2 p1

!

a1 a2

!

1 1

x

SLIDE 31

Take the expression for x and plug into the consumption index, C = (a1x + a1y)

1

• C

=

@a1 @ @

p1 p2

!

1

a2 a1

!

1

1 A y 1 A + a2y 1 A

1

• =

a

1 1

2

p

1 1

2

y

"

a

1 1

1

(p1)

• 1 + (a2)

1 1 p

• 1

2

!#1

• By symmetry of the consumption index, it must also be the case that

C = a

1 1

1

p

1 1

1

x

"

a

1 1

1

(p1)

• 1 + (a2)

1 1 p

• 1

2

!#1

• : Thus, we have

C = a

1 1

2

p

1 1

2

y

"

a

1 1

1

(p1)

• 1 + (a2)

1 1 p

• 1

2

!#1

• C

= a

1 1

1

p

1 1

1

x

"

a

1 1

1

(p1)

• 1 + (a2)

1 1 p

• 1

2

!#1

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which gives x = Ca

1 1

1

p

1 1

1

"

a

1 1

1

(p1)

• 1 + (a2)

1 1 p

• 1

2

!#1

• y

= Ca

1 1

2

p

1 1

2

"

a

1 1

1

(p1)

• 1 + (a2)

1 1 p

• 1

2

!#1

• Use these expressions of x and y in the budget constraint and after some

simpli…cation, one obtains, p1x + p2y = a

1 1

1

(p1)

• 1 + a

1 1

2

p

• 1

2

!1

• C

Therefore P = a

1 1

1

(p1)

• 1 + a

1 1

2

p

• 1

2

!1

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Demand functions: x = Ca

1 1

1

p

1 1

1

"

a

1 1

1

(p1)

• 1 + (a2)

1 1 p

• 1

2

!#1

• = a

1 1

1

P p1

!

1 1

C y = a

1 1

2

P p2

!

1 1

C

Obtaining the price sub-indices for home and for- eign goods: Cost minimization problem II.

For the home goods consumption index Ch = b

(1)

• 1

Z n

0 c (i) di

1

SLIDE 34

the price sub-index is Ph = b

(1)

• 1

Z n

0 (p (i))

• 1 di

11

• Proof: The problem is how to allocate a given expenditure PC across

i 2 [0; n); where Ch = b

(1)

• 1

Z n

0 c (i) di

1

• – Note here that b

(1)

• 1

R n

0 c (i) di

1

is shorthand for

P1

i c (i)1

:

Since

@

P1

i c(i)1

• @c(i)

= c (i)(1) P1

i c (i)1

1 ; by analogy, we

have the di¤erentian rule, @b

(1)

• 1

R n

0 c (i) di

1

• @c (i)

= b

(1)

• 1

c (i)(1)

Z n

0 c (i) di

1

= c (i)(1) b(1)

1

C(1

h

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– Form the Lagrangian L =

Z n

0 p (i) c (i) di +

@Ch b

(1)

• 1

Z n

0 c (i) di

1

• 1

A

First-order conditions are p (i) = c (i)(1) b(1)

1

C(1)

h

which we use to write p (i) p (j) = c (i) c (j)

!1

c (i) = p (i) p (j)

! 1

1

c (j)

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– Use the expression for c (i) in the sub-index, Ch = b

(1)

• 1

Z n

0 c (i) di

1

= b (1)

• 1

B @ Z n

p (i) p (j)

!

1

c (j) di

1 C A

1

• =

b

(1)

• 1

p (j)

1 1 c (j)

Z n

0 (p (i))

• 1 di

1

• !

c (j) = Chb

(1)

• 1

p (j)

1 1

Z n

0 (p (i))

• 1 di

1

• – Now use the expression for c (j) in the budget constraint.

PhCh =

Z n

0 p (j) c (j) dj =

Z n

0 (p (i))

• 1 di

1

Chb (1)

• 1

Z n

0 p (j)

• 1 dj

Ph = b

(1)

• 1

Z n

0 (p (i))

• 1 di

1

Z n

0 p (j)

• 1 dj

! Ph = b

(1)

• 1

Z n

0 (p (i))

• 1 di

11

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– Use the price index back in the c (j) expression to write the good’s demand c (j) = p (j)

1 1 b (1)

• 1

Z n

0 (p (i))

• 1 di

11

Z n

0 (p (i))

• 1 di

1

Ch = p (j)

1 1

Ph b1 (Ph)

• 1

Ch = p (j)

1 1

Ph b1 (Ph)

• 1

a

1 1

1

P Ph

!

1 1

C ! c (j) = a

1 1

1

b1 Ph p (j)

! 1

1

P Ph

!

1 1

C

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Reparameterization for Benigno’s model

In Benigno’s JME paper, home consumer’s preferences for h 2 [0; n] are Uh

t = Et 1

X

j=t

(jt)

2 4U

• Ch

t+j

• + N

@Mh

t+j

Pt+j

1 A V

• Lh

t+j

• 3

5

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where for elasticity of substitution between foreign and home and elasticity

• f substitution across within country goods ;

C =

• n1=C(1)=

H

(1 n)1= C(1)=

F

=(1)

; > 0 CH =

"1

n

1= Z n

0 c (z)(1)= dz

#=(1)

CF =

"

1 1 n

1= Z 1

n c (z)(1)= dz

#=(1)

P =

h

nP 1

H

+ (1 n) P 1

F

i1=(1)

PH =

1

n

Z n

0 p (z)1 dz

1=(1)

PF =

"

1 1 n

Z 1

n p (z)1 dz

#1=(1)

Parameter correspondence:

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• =

1

• !

= 1 1 !

• 1 = 1 ! 1

=

• 1

a1 = n

1

• For the general indices,

C = (a1x + a2y)

1 ! C =

h

n1=x(1)= + (1 n)1= y(1)=i=(1) P =

"

a

1 1

1

p

• 1

x

+ a

1 1

2

p

• 1

y

#1

• ! P =

h

np1

1

+ (1 n) p1

y

i1=(1)

For sub-price indices,

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= ( 1)

• ! =

1 (1 ); ! 1 = (1 :) ! 1 ( 1) = (1 )

• Ch

= b

(1)

• 1

Z n

0 c (i) di

1

! CH =

"1

n

1= Z n

0 c (z)(1)= dz

#=(1)

Ph = b

(1)

• 1

Z n

0 (p (i))

• 1 di

11

! PH =

1

n

Z n

0 p (z)1 dz

1=(1)

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3.1 Benigno’s model

Households – Notation Home: [0; n]: Foreign: (n; 1]: Continuum of goods and agents. Each agent owns one …rm and supplies labor to that …rm. h denotes home agent. f denotes foreign agent. st an event in period t: st = (st; st1; :::; s0) is the history of events. Bh (st+1) : home agent’s holdings of one-period nominal state- contingent bonds.

SLIDE 43

Q

• st+1jst

is the home-currency nominal price of the state-contingent bond. – Preferences: Home consumer: For h 2 [0; n] ; Uh

t = Et 1

X

j=t

(jt)

2 4U

• Ch

t+j

• + N

@Mh

t+j

Pt+j

1 A V

• Lh

t+j

• 3

5

where for elasticity of substitution between foreign and home and

SLIDE 44

elasticity of substitution across within country goods ; C =

• n1=C(1)=

H

(1 n)1= C(1)=

F

=(1)

; > 0 CH =

"1

n

1= Z n

0 c (z)(1)= dz

#=(1)

CF =

"

1 1 n

1= Z 1

n c (z)(1)= dz

#=(1)

P =

h

nP 1

H

+ (1 n) P 1

F

i1=(1)

PH =

1

n

Z n

0 p (z)1 dz

1=(1)

PF =

"

1 1 n

Z 1

n p (z)1 dz

#1=(1)

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– Budget constraint: Income comes from pro…ts of …rms owned by households, payo¤s from state-contingent bonds, wages, and trans- fers from the government. Ch

t +

X

st+1

Q

• st+1jst

Bh (st+1) Pt +Mh

t

Pt = Bh (st) Pt +Mh

t1

Pt +W h

t Lh t

Pt +h

t

Pt +Trh

t

Pt – Government budget constraint: Transfers are helicopter drops of cash.

Z n

• Mh

t Mh t1

• dh =

Z n

0 Trh t dh

SLIDE 46

– For the intertemporal allocation problem, write down two terms from the Lagrangian t X

st

• st; st1

2 4U

• Ch

t

• st

+ N

@Mh

t

• st

Pt (st)

1 A V

• Lh

t

• st

3 5

• X

st

t

• st

B B B @

Ch

t

• st

+ P

st+1 Q(st+1jst)Bh(st+1) Pt(pt)

+ Mh

t (st)

Pt(st)

• "

Bh(st)+Mh

t1(st1)+W h t (st)Lh t (st)+h t (st)+Trh t (st)

Pt(st)

# 1 C C C A

t+1 X

st+1

• st+1

2 4U

• Ch

t+1

• st+1

+ N

@Mh

t+1

• st+1

Pt+1

• st+1
• 1

A V

• Lh

t+1

• st+1

3 5

• X

st+1

• st+1

t+1

• st+1

B B B @

Ch

t+1

• st+1

+ P

st+2 Q(st+2jst+1 Pt+1

+

Mh

t+1(st+1)

Pt+1(st+1)

"

Bh(st+1)+Mh

t (st)+W h t+1(st+1)Lh t+1(

Pt+1(st+1

SLIDE 47

Obtain the …rst-order conditions for consumption, Ch

t

: t

• st; st1

U0 Ch

t

• st

= t

• st

Bh (st+1) : t

• st Q
• st+1jst

Pt (st) = t+1 (st+1) 1 Pt+1 (st+1) Eliminating the multiplier gives the price of state contingent bonds Q

• st+1jst

=

• st+1jst U0

Ch

t+1

• st+1

U0

• Ch

t

• Pt

Pt+1 from which it follows that the price of one-period riskless nominal bonds are 1 1 + it =

X

st+1

Q

• st+1jst

= Et

8 < :

U0 Ch

t+1

• U0
• Ch

t

• Pt

Pt+1

9 = ;

SLIDE 48

Euler equations for labor and real money holdings: L : ! VL

• st

= U0 Ch

t

• st W h

t

• st

Pt (st) M : ! NM Mt Pt

!

= U0 Ch

t

• st

it 1 + it In equilibrium, we can drop the h superscript on consumption. Due to complete markets, everyone’s consumption will be equal. Home consumer demand for the home good and the foreign good ch (h) = p (h) PH

! PH

P

• C

ch (f) = p (f) PF

! PF

P

• C

with similar demand functions for the foreign guy.

SLIDE 49

Complete international asset markets. One-period state contingent nom- inal bonds denominated in the home currency are traded internationally. Home guy’s Euler equation has already been found. For foreign guy, Cf

t

: t

• st

U0 Cf

t

• st

= f

t

• st

Bh (st+1) : f

t

• st Q
• st+1jst

StP

t (st)

= t+1 (st+1) 1 StP

t+1 (st+1)

Q

• st+1jst

=

• st+1jst U0

Cf

t+1

• st+1

U0

• Cf

t (st)

• StP

t

• st

St+1P

t+1 (st+1)

=

• st+1jst U0

Ch

t+1

• st+1

U0

• Ch

t

• Pt

Pt+1

SLIDE 50

By the law of one price, StP

t

Pt Pt+1 St+1P

t+1

= U0 Ch

t+1

• st+1

U0

• Cf

t+1

• st+1
• U0

Cf

t

• U0
• Ch

t

• and after backwards recursive substitution,

U0 Cf

t

• U0
• Ch

t

= StP

t

Pt = RSt where depends on initial conditions. This is the Backus–Smith con-

• dition. Benigno achieves real exchange rate dynamics through relative

movements in consumption.

SLIDE 51

Monetary policy is conducted through interest-rate feedback rules (Taylor rules). Let "H

t , "F t be monetary policy shocks, F and F be a set of

target variables. The general speci…cation of the rules are 1 + it 1 + { =

• F; st; "H

t

• 1 + i

t

1 + { = F ; st; "F

t

• 1 = (1 + it) Et

8 < :

U0 Ch

t+1

• U0
• Ch

t

• Pt

Pt+1

9 = ; = (1 + i

t) Et

8 > < > :

U0 Cf

t+1

• U0
• Cf

t

• P

t

P

t+1

9 > = > ;

SLIDE 52

Production with local currency pricing. All produced goods are trad-

• able. Deviations from PPP are due to international market segmentation.

Prices are sticky in terms of the local currency. – Sticky prices through the Calvo mechanism. Say the Calvo-lottery chooses home …rm h to set a new price at time t: The …rm does so to maximize expected present value of net pro…ts. Let s be the probability that the price will remain in e¤ect from t + 1 through t + s: The owner’s discount factor is t;t+s = sU0 (Ct+s) U0 (Ct) Pt Pt+s and the production technology ch

t (h) + cf t (h) = yt (h) = AtLh t

where At is a country-speci…c technology (productivity) shock.

SLIDE 53

– The objective is to maximize Et

1

X

j=0

jt;t+j

2 6 6 4pt+j (h) yh

t+j (h)

| {z }

home sales

+ St+jp

t+j (h) yf t+j (h)

| {z }

foreign sales

W h

t+jLh t+j

| {z }

wage bill

3 7 7 5

subject to yh

t (h)

= pt (h) PHt

! "PHt

Pt

#

nCt = h

t pt (h)

yf

t (h)

=

@p

t (h)

P

Ht

1 A

"P H;t

P

t

#

(1 n) C

t = tp t (h)

where h

t

= 1 PHt

! "PHt

Pt

#

nCt

• t

=

@ 1

P

Ht

1 A

"P H;t

P

t

#

(1 n) C

t

SLIDE 54

The …rm must set pt (h) and p

t (h).

t;t

h

pt (h) h

t pt (h) + Stp t (h) tp t (h) W h t Lh t

i

+t

• AtLh

t h t pt (h) tp t (h)

+Ett;t+1

h

pt+1 (h) h

t+1pt+1 (h) + St+1p t+1 (h) t+1p t+1 (h) W h t+1L

+Ett+1

• At+1Lh

t+1 h t+1pt+1 (h) t+1p t+1 (h)

+ First, choose pt (h) ; with the expectation that it will stay in e¤ect. pt (h) : (1 ) pt (h) Et

1

X

j=0

jt;t+jh

t+jpt (h)(1+) Et 1

X

j=0

h

t+j

and rearranging gives pt (h) =

• (1 )

Et

P1

j=0 h t+j

Et

P1

j=0 jt;t+jh t+j

SLIDE 55

Similarly, the optimal way to set the foreign price is, p

t (h) =

• (1 )

Et

P1

j=0 St+j t+j

Et

P1

j=0 jt;t+j t+j

Evolution of price sub-indices. A measure of the …rms have to keep the price as before. (1 ) set to the new price pt (h) ; therefore, PH;t = hPH;t1 + (1 h) pt(h) P

H;t

=

• hP

H;t1 + (1 h) p t (h)

with similar equations holding for the foreign …rm P

F;t

=

• fP

F;t1 +

• 1

f

• p

t (f)

PF;t = fPF;t1 +

• 1 f
• pt (f)

Benigno allows for the possibility that the degree of price stickiness may di¤er across countries.

SLIDE 56

Relative price indices of the imported good in terms of the domes- tically produced good in terms of the local currency T = PF PH T = P

H

P

F

SLIDE 57

Aggregate ‡uctuations. – Expand around a steady state with zero in‡ation and exchange rate depreciation, nominal interest rates equal to utility-based discount rate, and a real exchange rate of 1 and equal consumption across countries. – Highlights of the log-linearization.

• 1. = ln (1 + )
• 2. d

RSt = ln (RSt)

• 3. ^

T = ln

PF

PH

• ; ^

T = ln

P

H

P

F

• 4. ~

C; ~ T are ‡exible-price equilibrium values.

• 5. = LVLL

VL ; = CUCC Uc

SLIDE 58

– Relative prices and the real exchange rate have unit roots. Obtain an IS-like curve, uncovered interest parity, and an expenditure switching e¤ect. ^ Tt = ^ Tt1 + F;t H;t ^ T

t

= ^ T

t F;t + H;t

d

RSt =

d

RSt1 +

t t + St

Et ^ Ct+1 = ^ Ct + 1 [^ {t Ett+1] EtSt+1 = ^ {t ^ {

t

yH;t yF;t =

• n

^

Tt ~ Tt

• (1 n)

^

T

t ~

T

t

SLIDE 59

– Dynamics under sticky prices h =

h = f = f: Chari-Kehoe-McGrattan case.

h = f;

h = f : Location of consumption determines degree

• f price stickiness.

h =

h;

f = :

f :

Price stickiness is …rm location speci…c and faces the same stickiness in the home and foreign market. Monetary policy rules re‡ect interest rate smoothing and exchange rate feedback. ^ {t = H^ {t1 + Ht + HyH

t + HSt + H d

RSt + "H

t

^ {

t

= F^ {

t1 + F t + FyF t FSt F d

RSt + "F

t

"j

t = j"j t1 + vj;t

SLIDE 60

Alternative policy rules – Fixed exchange rates. Foreign country equates ito i with a reaction to deviations of the exchange rate from target. Let ^ S = ln

• S=

S

• ,

where S is the target. home country follows Taylor rule ^ {

t

= ^ {t ^ St ^ {t = t + yH;t – Taylor rules ^ {t = t + yH;t + ^ "H;t ^ {

t

=

• t + yF;t + "F;t

– Managed ‡oat ^ {t = t + yH;t ^ {

t

=

• t + yF;t St

SLIDE 61

Tradeo¤ between persistence and volatility as response to exchange rate increases.

SLIDE 62

Interest rate smoothing induces additional inertia. is the smootheing coe¢cient. For symmetric Taylor rules

SLIDE 63

Preliminary results – Suppose H =

H = F = F:

Then price dynamics are synchronized and the relative prices T = PF=PH and T = P

H=P F are uncorrelated with monetary policy.

Relative prices will be a¤ected only by productivity shocks. The real exchange rate displays no persistence following a monetary shock under in‡ation targeting or under the Taylor rule. After a monetary shock, the nominal and real exchange rates return to equilibrium after one period, as in the Redux model. There will be persistence in the exchange rates if monetary shocks are serially correlated. Under in‡ation targeting, the real exchange rate is isolated from productivity shocks.

SLIDE 64

When there is no weight on output stabilization, productivity shocks wil have no e¤ect on the real exchange rate. Also, the in‡ation di¤erential is independent of relative price changes. The link between productivity shocks and the real exchange rate is broken.

SLIDE 65

Plausible examples of persistence generation in the real exchange rate: Calibration and simulations. = 0:99; = 2; n = 0:5; = 6; = 10; = 1:5: The only shock is from domestic monetary policy with no autoregressive component.

SLIDE 66

– Interest rate smoothing

SLIDE 67

– Firm-speci…c price rigidity H =

H 6= F = F

SLIDE 68

– H = 0:8;

H = 0:66; F = 0:67; F = 0:4:, = 0:85; =

0:225

SLIDE 69

SLIDE 70

Takeaway: Previously thought: Persistence of real exchange rate increas- ing in degree of price stickiness, and unrealistically long period of price stickiness to match the persistence in real exchange rate. Now thought: Price rigidity is not su¢cient by itself to generate persistence following a monetary shock.

SLIDE 71

4 In‡ation News and the Exchange rate: Clar- ida and Waldman

Theoretical model for how bad news about in‡ation (higher than expected) is good news for the exchange rate (home currency strengthens). Interesting and nicely constructed empirical work. Questions:

• 1. What is the correlation between in‡ation surprises and changes in the

nominal exchange rate?

• 2. Is the sign di¤erent for in‡ation targeters and non-in‡ation targeters?

SLIDE 72

Data: 5-minute nominal spot exchange rate data for USD–JPY,CAD,NOK,SEK,CHF,EUR,GB and NZD, July 2001 through December 2005. Construct returns 10-minute percentage changes to capture behavior in a plus/minus 5 minute window of the announcement. Positive in‡ation surprise means higher than expected. Expectations is the median survey response from Bloomberg News Service, which surveys commercial and investment banks on macroeconomic announcements. Use consumer price in‡ation. Month-over-month (MoM) and year-over- year (YoY) in‡ation for headline and core in‡ation.

• Regression. Rt is 10 minute return around the announcement. R > 0

means home currency appreciates. St is the in‡ation surprise. Rit = + Sit + uit

SLIDE 73

– Normalize coe¢cients to interpret as an elasticity. – Pool the data and run as stacked OLS regression.

SLIDE 74

– In‡ation targeters versus non-in‡ation targeters.

SLIDE 75

– Regime changes: Bank of England independence (1997), Norway’s formal adoption of in‡ation targeting (2001). Before the regime change, coe¢cients are negative (not signi…cant).

SLIDE 76

5 Calibrated partial equilibrium models of the exchange rate and Taylor rules

Engel and West JMCB. Rational expectations model. Mark (mimeo). Adaptive learning

SLIDE 77

Partial equilibrium setup. We assume – The public believes Fed and Bundesbank use some form of the Taylor rule. – Public views in‡ation and output gap are exogenously generated by a VAR(4). – Economic model is real interest parity, a stochastic di¤erence equa- tion, whose solution gives the real exchange rate. – Public employs least-squares learning rules to form beliefs about model’s coe¢cients since their true values are unknown and may change over time. Feed historically observed data on in‡ation and the output gap into the model. Observe the implied learning equilibrium path of the real ex- change rate and compare with historical real exchange rate path.

SLIDE 78

5.1 Can we assume that the Fed and Bundesbank (ECB) reacts the same to in‡ation and the output gap?

German variables subscripted by ‘G’ U.S. variables subscripted by ‘U.’ German–U.S. di¤erentials have no special notation. t =

• G;t U;t
• it

=

• iG;t iU;t
• xt

=

• xG;t xU;t
• are German-U.S. di¤erentials in in‡ation, short-term nominal interest

rates and activity gaps, respectively. Activity gap de…ned such that the economy operates in excess of its potential when xtj > 0; j = G; U: The log real exchange rate is qt.

SLIDE 79

Fed rule for target rate iT

U;t = iU +

• EtU;t+1 U
• + xxU;t:

Actual interest rate subject to exogenous and i.i.d. policy shock U;t and interest rate smoothing iU;t = (1 )iT

U;t + iU;t1 + U;t:

Bundesbank target rule iT

G;t =

{G +

• EtG;t+1 G
• + xxG:t + qqt1;

SLIDE 80

Impose homogeneity of the coe¢cients (; x) across countries and write interest di¤erential as it = + it1 + (1 )

• Ett+1 + xxt + qqt1;
• + t;

– (1 )

• iG iU
• (G U)
• ; and t

iid

(0; 2

):

– Add and subtract (1 ) e t+1 on the right side of and rearrange it = + (1 ) [t+1 + xxt + sqt1] + it1 + 0

t;

where

t = t (1 ) [t+1 Ett+1] :

is uncorrelated with date t information. Estimate by GMM. Instru- ments are a constant, the current value and three lags of the in‡ation di¤erential, the current value and three lags of the output (alterna- tively unemployment) gap di¤erential, four lags of the nominal inter- est di¤erential, and four lags of the real exchange rate.

SLIDE 81

Table 2: GMM Estimates of Bundesbank–Fed Relative Interest-Rate Reaction Function with Lagged Real Exchange Rate Feedback. Bold indicates signi…cance at the 5% level Source output gap Sample

• x

q J-statistic (t-ratio) (t-ratio) (t-ratio) (t-ratio) (t-ratio) (p-valu 61.2-79.2

• 0.006

0.666 0.034 0.134 0.019 9.684 (-6.477) (10.545) (0.376) (5.092) (4.069) (0.644) 79.3-05.4

• 0.001

0.895 1.318 0.376 0.011 4.330 (-1.194) (22.450) (3.063) (2.318) (0.880) (0.977) Structural change test All coe¤s. Test statistic 40.037 0.000 In‡ation coe¤. Test statistic 8.525 0.004

SLIDE 82

HP output gap

• x

q J-statistic (t-ratio) (t-ratio) (t-ratio) (t-ratio) (t-ratio) (p-valu 61.2-79.2

• 0.003

0.745 0.121 0.045 0.011 9.370

• 3.589

12.551 1.357 1.166 2.708 0.671 79.3-05.4 0.000 0.873 1.544 0.428 0.004 5.314

• 0.202

24.480 4.406 3.239 0.558 0.947 Structural change test All coe¤s. Test statistic 36.747 0.000 In‡ation coe¤. Test statistic 15.501 0.000

SLIDE 83

HP unemployment gap

• x

q J-statistic (t-ratio) (t-ratio) (t-ratio) (t-ratio) (t-ratio) (p-valu 61.2-79.2

• 0.003

0.579 0.034

• 0.644

0.005 7.746

• 3.077

8.394 0.554

• 4.847

1.873 0.805 79.3-05.4 0.000 0.860 1.154

• 0.828

0.004 5.083

• 0.163

22.028 2.812

• 3.171

0.443 0.955 Structural change test All coe¤s. Test statistic 25.042 0.015 In‡ation coe¤. Test statistic 7.280 0.007

SLIDE 84

6 Modeling real exchange rate dynamics with learning

Economic model is uncovered interest parity. For the log nominal ex- change rate st (DM/$) st = Etst+1 it: where i is the German–US interest di¤erential. Add and subtract Ett+1 from the right side, rearrange qt = Etqt+1 it + Ett+1 it = + it1 + (1 )

• Ett+1 + xxt + qqt1;
• + t

SLIDE 85

Let and x be exogenously given by VAR(4). Let Y 0

t = (t; : : : ; t3; xt; : : : ; xt3) ;

and Z0

1;t = Z0 2;t =

Y 0

t ; 1

: Regression form of the VAR is

t = B0

1Z0 1;t1 + v1t;

xt = B0

2Z0 2;t1 + v2t;

B1 and B2 are 9 x 1 least-squares coe¢cient vectors. Companion representation Yt = + AYt1 + vt; e1 = (1; 0; 0; 0; 0; 0; 0; 0) t = e1Yt e2 = (0; 0; 0; 0; 1; 0; 0; 0) xt = e2Yt Ett+1 = e1 ( + AYt) :

SLIDE 86

The above gives a second-order stochastic di¤erence equation in qt, qt = ((1 ) 1) e1 + (1 ) qqt1 + it1 + ((1 ) xe2 + ((1 ) 1) e1A) Yt + t + Etqt+1 Rational expectations solution qt = a0 + a1it1 + a2qt1 + a3t + bYt a2 = 1 2 (1 )

q

(1 )2 (1 ) 4 q 2 ; a1 = a2 (1 ) q ; a0 = ((1 + (( 1) a1 + (1 )) ) e1 b) a2

!

(a1 1 ) a2 ; a3 = 1 a1 1 a2 ; b = (((1 + (a1 1) (1 ) ) e1) A + (a1 1) (1 ) xe2) ((1 a2) I A)1 :

SLIDE 87

– Non-uniqueness. Choose solution with positive a2 – b depends on in‡ation response coe¢cient . < 1 in pre-Volker sample: Decline in Ett+1 leadds public to expect increase in the r and real dollar depreciation. > 1 in the post-1979 sample: Decline in Ett+1 leads to real dollar appreciation

SLIDE 88

Learning the rational expectations equilibrium. Solve real interest parity condition using expectations formed from perceived law of motion. – At t, coe¢cient vectors B1;t1; B2;t1 are given (estimated in t1): From regression form, t = B0

1;t1Z1;t + u1;t;

xt = B0

2;t1Z2;t + u2;t;

form companion form Yt = t1 + At1Yt1 + vt; Construct expected in‡ation Ett+1 = e1 (t1 + At1Yt) : – Believe the rational expectation solution and use it for perceived law

• f motion

qt = a0;t1+a1;t1it1+a2;t1qt1+a3;t1t+bt1Yt B0

3;t1Z3;t

SLIDE 89

– Observe the policy shock t from perceived law of motion for the interest di¤erential it = t1 + t1it1 + ;t1e1 (t1 + At1Yt) + x;t1e2Yt + q;t1qt1 +

• B0

4;t1Z4;t + t:

– The expected exchange rate Etqt+1 = a0;t1 + a1;t1it + a2;t1qt + bt1 (t1 + At1Yt) : – Plug in‡ation forecast and q forecast into real UIP gives the actual law of motion, qt = 1

• 1 a3;t1
• h

a0;t1 + (e1 + bt1) (t1 + At1Yt) +

• a2;t1 1
• it

i

: – For next period, least-squares update the coe¢cients Let y1;t = t; y2;t = xt; y3;t = qt; and y4;t = it: For given Rj;t1 (j = 1; :::; 4) ;

SLIDE 90

and a …xed gain g; the updating formulae are Rj;t = Rj;t1 + g

• Zj;t1Z0

j;t1 Rj;t1

• ;

(1) Bj;t = Bj;t1 + gR1

j;t Zj;t1(yj;t B0 j;t1Zj;t1):

(2) The learning path and coe¢cient updating is generated using ob- servations of t; xt; and it from the data, but not with exchange rate data. The learning values of qt generated by the actual law of motion and employed in coe¢cient updating are generated solely as functions of ; x; and i.

SLIDE 91

Calibrated Learning and Rational Exchange Rate Paths The observations are standardized to highlight comovements between the series.

SLIDE 92

Figure 5. Learning path with source output gap.

SLIDE 93

The learning path shown in Figure 5 generally captures closer comovements with the data and does a good job of capturing the dollar cycle of the 1980s. The learning path exhibits the dollar appreciation and subsequent depreciation in the latter part of the sample although timing of the turning points are o¤ a bit with the phase of the learning cycle leading the data.

SLIDE 94

Figure 7. Learning path with HP output gap.

SLIDE 95

The learning path in Figure 7 shows the dollar prematurely appreciating in 1979.2 and it does not generate quite the strength attained in the data by 1984.4. The learning path also leads the turning points in the dollar appre- ciation and depreciation beginning in 1995.2 but otherwise comoves with the data.

SLIDE 96

Figure 9. Learning path with HP unemployment gap.

SLIDE 97

Figure 9, it is seen that the comovmements of the learning path with the data are also quite good. The learning path matches the timing of the 1980s dollar cycle very well. Except for a phase shift that leads the data, it also captures the cycle from 1995 to 2005

SLIDE 98

Figure 4: Rational path with source output gap.

SLIDE 99

Using the source output gap, the rational path shown in Figure 4 misses a good deal of the real dollar appreciation from 1980.4 to 1984.4 and falsely predicts a dollar appreciation from 1989.3 to 1991.2. It also erroneously predicts a large dollar depreciation in 1981.1 Apart from these episodes, there is a close connection between the implied rational expectations path and the data. The rational expectations path does a good job of explaining the real dollar appreciation from 1996.2 to 2002.2 and the subsequent real dollar depreciation.

SLIDE 100

Figure 6: Rational path with HP output gap.

SLIDE 101

In Figure 6, the rational path generated with the HP output gap is quite similar to the path generated with the source output gap. Here, the rational path also misses the real dollar appreciation of the 1980s and signals a false appreciation in the early 1990s.

SLIDE 102

Figure 8: Rational path with HP unemployment gap.

SLIDE 103

Figure 8 plots the rational path using HP detrended unemployment. The comovements between the rational path and the data are generally quite close.

SLIDE 104

Figure 11. Learning path with contemporaneous exchange rate in Taylor rule, source output gap.

SLIDE 105

Figure 10 Rational path with contemporaneous exchange rate in Taylor rule, source output gap.

SLIDE 106

Table 3: Correlations and Relative Volatility Activity Rational Learning Form variable Corr T-ratio Volatility Corr T-ratio Volatility Source gap 0.308 2.461 0.965 0.346 2.170 1.054 Level HP output

• 0.029
• 0.201

3.678 0.298 2.094 0.336 HP unemployment 0.484 3.426 0.644 0.340 1.834 1.130 Source gap 0.030 0.700 2.085

• 0.039
• 0.596

1.254 1-qtr return HP output 0.019 0.536 7.918 0.033 0.601 0.458 HP unemployment 0.031 0.499 1.561

• 0.008
• 0.135

1.870 Source gap 0.235 3.142 1.525 0.044 0.393 1.164 4-qtr return HP output 0.026 0.323 5.662 0.054 0.593 0.402 HP unemployment 0.459 5.575 1.050

• 0.026
• 0.265

1.486 Source gap 0.308 2.699 1.264 0.157 1.023 1.014 8-qtr return HP output 0.066 0.639 4.374 0.037 0.338 0.357 HP unemployment 0.624 5.418 0.848 0.109 0.706 1.142 Source gap 0.424 4.106 1.055 0.335 2.393 1.086 16-qtr return HP output 0.019 0.113 3.753 0.093 0.799 0.345 HP unemployment 0.691 4.381 0.746 0.093 0.799 1.033

SLIDE 107

To sum up, there were six major swings in the real DM-dollar rate in the

• sample. The real dollar depreciation from 1973.1 to 1979.4, the sharp ap-

preciation (1980.1–1984.4) and subsequent depreciation (1985.1–1987.4), a more tempered dollar depreciation (1988.1 to 1995.1), a dollar appreciation (1995.2–2001.2), and the dollar decline (2001.3–2005.4). Each of the ratio- nal paths falsely predicted a strong real dollar appreciation in 1991 and none

• f them adequately matched the volatility in the data. The learning model

provides a plausible description for the data. Regardless of the de…nition used for the activity gap, each of the learning paths captured the major swings in the real exchange rate. Two obvious extensions. (1) relax homogeneity restrictions on coe¢cients across countries. (2) Have an economic model for the output gap and in‡a- tion.

SLIDE 108

7 Taylor-rule fundamentals and exchange rate forecasts

Molodtsova, Nikolsko-Rzhevskyy, and Papell, ‘Taylor Rules with Real-Time Data: A Tale of Two Countries and One Exchange Rate.’ Real-time versus (revised) historical data. – Use the information (data) available to monetary authorities and

• ther economic agents when they made their decisions.

– Revision in output and output gap is more substantial than revision to price indices. – Alternative source of data: Greenbook forecasts of in‡ation and out- put gap observed by Fed but not the public until …ve years later.

SLIDE 109

– German real time data: Gerberding, Worms, and Seitz. – US real time data. Crouschore and Stark, Philadelphia Fed. Estimate Taylor rules with real-time data for US and Germany from 1979 to 1998. – U.S. potential GDP constructed by Orphanides. – German output gap is deviation of GDP to quadratic time trend.

SLIDE 110

SLIDE 111

– Find that Taylor rule coe¢cients are robust to choice of real-time or historical data, and choice of forward looking forecasts, current and lagged in‡ation/output gap data.

SLIDE 112

SLIDE 113

Reduced form exchange rate forecasting equation. U.S. is home country st+1 = 0+1t2

t +3yt4y t +1it1+2i t1+4qt+t

Symmetric means homogeneity restrictions imposed on coe¢cients.

SLIDE 114

7.0.1 Clark-West test of equal forecast accuracy (in mean square sense).

• 1. Out-of-sample forecasting. Let the exchange rate return be

yt st st1 (a) RMSE, Theil’s U-statistic. H0 : yt = et; Et1et = 0 Ha : yt = Z0

t + et;

Et1et = 0 Under Ha; yt = ^ ft + ^ et; ^ ft = Z0

t^

SLIDE 115

Make P one-step ahead predictions. MSPE(1) = ^ 2

1 = 1

P

T

X

t=TP+1

e2

t+1

MSPE(2) = ^ 2

2 = 1

P

T

X

t=TP+1

^ e2

t+1 = 1

P

T

X

t=TP+1

• yt+1 ^

ft+1

2

U = ^ 2 ^ 1 Can bootstrap Theil’s U.

SLIDE 116

(b) Diebold-Mariano (1995)–West (1996) statistic. For non-nested hy- potheses, Ft+1 = e2

t+1 ^

e2

t+1

• F

= 1 P

T

X

t=TP+1

Ft+1 ^ V = 1 P

T

X

t=TP+1

• Ft+1

F

2

DMW =

• F

^ V N (0; 1) To compute DMW: Regress Ft+1 on a constant. DMW is the t- statistic for the constant. Asymptotics hold for non nested hypothe- ses.

• i. DMW turns out not to be asymptotically normal for nested hy-
• potheses. The random walk model is nested, causing DMR to be

badly undersized.

SLIDE 117

• ii. Under H0; sampling variability in estimated coe¢cients under al-

ternative cause E

• ^

2

1

• < E
• ^

2

2

• Therefore, one accepts the random walk too often.

SLIDE 118

(c) Clark-West (2005) test. A simple adjustment.Works reasonably well for rolling or recursive regression. Both are slightly undersized but

• ¤er large improvements over DMW. Recall ^

ft+1 = x0

t+1^

: F a

t+1

= e2

t+1 ^

e2

t+1 + ^

f2

t+1

(3) CW =

• F

^ V

a

N (0; 1) where a superscript stands for ‘adjusted.’

• i. CW statistic: Regress F a

t+1 on a constant. CW is the t-ratio from

regression output.

• ii. For long-horizon forecasts.

yt;k = yt+1 + y2+2 + + yt+k H0 : yt;k = et+1;k Ha : yt;k = x0

t+1 + et+1;k = ^

ft;k + ^ et+1;k

SLIDE 119

Compute F a

t ;

F a as above. F a

t+1;k = et+1;k ^

e2

t+1;k ^

f2

t;k

Due to serial correlation induced by overlapping forecasts, need to adjust ^ V : Ken West suggests the Hodrick (1992) or West (1997) covariance matrix ^ gt = 2yt

^

ft;k + + ^ ftk+1

• ^

V = 1 P 2k + 2

Tk+1

X

t=tP+k

(^ gt+k g)2 7.0.2 Evaluate choice of real-time or historical data for exchange rate forecasts. Typical results

SLIDE 120

Signi…cant predictability observed in asymmetric model using Greenbook in- ‡ation forecasts and Survey of Professional Forecaster’s in‡ation forecasts.

SLIDE 121

SLIDE 122

In‡ation coe¢cients say bad news about in‡ation is good news for the ex- change rate. Increase in U.S. output gap predicts depreciation. German output gap coef- …cients never signi…cant.

SLIDE 123

8 Conclusion

• 1. Explicit treatment of endogenous monetary policy and interest rate rules

in exchange rate economics is relatively new.

• 2. The Taylor-rule approach might provide a resolution to the PPP puzzle.
• 3. The Taylor-rule approach identi…es a very di¤erent set of macroeconomic

fundamentals in discussions about exchange rate determination.

• 4. Calibrated partial equilibrium Taylor-rule models of the exchange rate

seem to …t the data reasonably well, and evidently much better than PPP or traditional monetary approaches.

SLIDE 124

• 5. Taylor-rule fundamentals have statistically signi…cant predictive power

for the future exchange rate.

• 6. There is scope for more research in the area.