Central Banks Balance Sheet Policies Without Rational Expectations · 2019. 6. 5. · Beine-Benassy-Quere-Lecourt (2002) 3. Theory The irrelevance result if 1. people can freely trade

Central Banks Balance Sheet PoliciesWithout Rational Expectations

Luigi Iovino Dmitriy Sergeyev

Bocconi University, IGIER, CEPR

ESSIM 2019 - Tarragona

May 7, 2019

1

Central Banks Balance Sheet Policies

Examples

I QE (long-term public and private assets purchases)

I FX interventions

“The problem with QE is that it works in practice,but it does not work in theory”

Ben Bernanke (2014)

2

Central Banks Balance Sheet Policies

Examples

I QE (long-term public and private assets purchases)

I FX interventions

“The problem with QE is that it works in practice,but it does not work in theory”

Ben Bernanke (2014)

2

Empirics

QE

I Gagnon-Raskin-Remache-Sack (2011),Krishnamurthy-Vissing-Jorgensen (2011),Hancock-Passmore (2011), Di Maggio-Kermani-Palmer(2016), Chakraborty-Goldstein-MacKinlay (2016),Fieldhouse-Mertens-Ravn (2018)

, Stroebel-Taylor(2012),Greenlaw-Hamilton-Harris-West (2018)

FX interventions

I Dominguez-Frankel (1990, 1993), Dominguez (1990, 2006),Catte-Galli-Rebecchini (1994), Kearns-Rigobon (2005),Blanchard-Adler-de Carvalho (2014),Fratzscher-Gloede-Menkhoff-Sarno-Stohr (2015)

Beine-Benassy-Quere-Lecourt (2002)

3

Empirics

QE

I Gagnon-Raskin-Remache-Sack (2011),Krishnamurthy-Vissing-Jorgensen (2011),Hancock-Passmore (2011), Di Maggio-Kermani-Palmer(2016), Chakraborty-Goldstein-MacKinlay (2016),Fieldhouse-Mertens-Ravn (2018), Stroebel-Taylor(2012),Greenlaw-Hamilton-Harris-West (2018)

FX interventions

I Dominguez-Frankel (1990, 1993), Dominguez (1990, 2006),Catte-Galli-Rebecchini (1994), Kearns-Rigobon (2005),Blanchard-Adler-de Carvalho (2014),Fratzscher-Gloede-Menkhoff-Sarno-Stohr (2015)Beine-Benassy-Quere-Lecourt (2002)

3

Theory

The irrelevance result

if

1. people can freely trade targeted assets

2. symmetric info between policy maker and markets

3. people correctly predict future effects of policies

Prominent channels

1. Portfolio balance channel (segmented markets)

2. Signaling channel (asymmetric info or limited commitment)

This paper: bounded rationality channel

I Beliefs about future deviate from rational expectations

I Agents do not fully understand future effects of the policies

4

Theory

The irrelevance result if




Prominent channels






4

Theory





Prominent channels






4

Theory





Prominent channels






4

Roadmap

1. A simple real closed-economy model

2. Full model with endogenous output

3. Empirical evidence

5

Deviations from Rational Expectations

Induction

I Idea: beliefs about future are refined over time through aprocess of induction from observed outcomes.

I Econometric learning : Evans-Honkapohja; Shleifer; etc.

Eduction

I Idea: agents understand the model and form expectationsfrom it about future outcomes through the process ofreflection

I Level-k thinking

6


Induction



Eduction


I Level-k thinking

6


Induction



Eduction


I Level-k thinking

6

Level-k Thinking

Game Theory

I Stahl-Wilson (1994,1995); Nagel (1995); Crawford (2013)

I Idea: Agents know the game; rationally respond to beliefs;form beliefs about opponents actions recursively

I Result: Level-k thinking better approximates experimentalresults in strategic games (more so in new games)

Macro

I Evans-Ramey; GarciaSchmidt-Woodford; Farhi-Werning

I Idea: Agents know the model; optimize; form expectationsabout future endogenous variables recursively

I Result: Level-k thinking dampens changes in expectationsabout future endogenous variables after new policies

7

Level-k Thinking

Game Theory

I Stahl-Wilson (1994,1995); Nagel (1995); Crawford (2013)

I Idea: Agents know the game; rationally respond to beliefs;form beliefs about opponents actions recursively

I Result: Level-k thinking better approximates experimentalresults in strategic games (more so in new games)

Macro

I Evans-Ramey; GarciaSchmidt-Woodford; Farhi-Werning

I Idea: Agents know the model; optimize; form expectationsabout future endogenous variables recursively

I Result: Level-k thinking dampens changes in expectationsabout future endogenous variables after new policies

7

Simple Model

Infinitely-lived households solve

max{xt+1,bt+1,ct}

E0

[−1

γ

∞∑t=0

e−ρt−γct

],

subject to

ct + bt+1 + qtxt+1 ≤Wt − Tt + (1 + r)bt + (Dt + qt)xt

need to forecast

(i) Asset returns (asset prices)

(ii) Human capital (taxes)

8

Government

1. Central bank

I announces path {Xt+1, Rt+1} s.t.

Rt+1 = qtXt+1

I transfers profits/losses to the treasury

Trt = (Dt + qt − (1 + r)qt−1)Xt

2. Treasury

I chooses path {Tt, Bt+1} so as to satisfy

(1 + r)Bt = Bt+1 + Trt + Tt

9

Government

1. Central bank

I announces path {Xt+1, Rt+1} s.t.

Rt+1 = qtXt+1

I transfers profits/losses to the treasury

Trt = (Dt + qt − (1 + r)qt−1)Xt

2. Treasury

I chooses path {Tt, Bt+1} so as to satisfy

(1 + r)Bt = Bt+1 + Trt + Tt

9

Asset demand

Assumptions

I Households understand treasury’s BC,

(but may fail to forecast transfers)

I Beliefs about (relevant) future endogenous variables:

ys+1 = αy,s + βy,sεxs+1, y ∈ {q, Tr}

Risky-asset demand

x(qt; {qt+s, T rt+s}) =Et (Dt+1 + qt+1)− (1 + r)qt

γ r1+rσ

2x

− βTr,t

10

Asset demand

Assumptions

I Households understand treasury’s BC,

(but may fail to forecast transfers)

I Beliefs about (relevant) future endogenous variables:

ys+1 = αy,s + βy,sεxs+1, y ∈ {q, Tr}

Risky-asset demand

x(qt; {qt+s, T rt+s}) =Et (Dt+1 + qt+1)− (1 + r)qt

γ r1+rσ

2x

− βTr,t

10

Temporary Equilibrium (TE)

Idea: TE takes as given a sequence of beliefs and imposes thatmarkets clear in every period (Hicks; Lindahl; Grandmont)

Definition Given agents’ beliefs, a TE is{Xt+1, Rt+1, T rt, , Bt+1, Tt; qt; bt+1, xt+1, ct} s.t. {xt+1, bt+1, ct}are optimal, Rt+1 = qtXt+1, the treasury’s BC is satisfied,

D + Etqt+1 − (1 + r)qtγ r1+rσ

2x

− βTr,t = X −Xt+1,

andTrt = (Dt + qt − (1 + r)qt−1)Xt.

REE

11

Temporary Equilibrium (TE)

Idea: TE takes as given a sequence of beliefs and imposes thatmarkets clear in every period (Hicks; Lindahl; Grandmont)

Definition Given agents’ beliefs, a TE is{Xt+1, Rt+1, T rt, , Bt+1, Tt; qt; bt+1, xt+1, ct} s.t. {xt+1, bt+1, ct}are optimal, Rt+1 = qtXt+1, the treasury’s BC is satisfied,


2x

− βTr,t = X −Xt+1,

andTrt = (Dt + qt − (1 + r)qt−1)Xt.

REE

11

Level-k thinking Belief Formation

Status quo {qt+s, T rt+s} = {q∗, 0} (REE before intervention)

Level-1Thinking

x(q1t ; {q∗, 0}) = X −Xt+1

Tr1t =(Dt + q1t − (1 + r)q1t−1

)Xt

}⇒ {q1t , T r1t }

Level-2Thinking

x(q2t ; {q1t+s, T r1t+s}) = X −Xt+1

Tr2t =(Dt + q2t − (1 + r)q2t−1

)Xt

}⇒ {q2t , T r2t }

Level-kThinking

{qkt , T rkt } = Ψ({qk−1t , T rk−1t }; {Xt+1})

REE {qt, T rt} = Ψ({qt, T rt}; {Xt+1})

12



Level-1Thinking

x(q1t ; {q∗, 0}) = X −Xt+1

Tr1t =(Dt + q1t − (1 + r)q1t−1

)Xt

}⇒ {q1t , T r1t }

Level-2Thinking

x(q2t ; {q1t+s, T r1t+s}) = X −Xt+1

Tr2t =(Dt + q2t − (1 + r)q2t−1

)Xt

}⇒ {q2t , T r2t }

Level-kThinking



12



Level-1Thinking

x(q1t ; {q∗, 0}) = X −Xt+1

Tr1t =(Dt + q1t − (1 + r)q1t−1

)Xt

}⇒ {q1t , T r1t }

Level-2Thinking

x(q2t ; {q1t+s, T r1t+s}) = X −Xt+1

Tr2t =(Dt + q2t − (1 + r)q2t−1

)Xt

}⇒ {q2t , T r2t }

Level-kThinking



12



Level-1Thinking

x(q1t ; {q∗, 0}) = X −Xt+1

Tr1t =(Dt + q1t − (1 + r)q1t−1

)Xt

}⇒ {q1t , T r1t }

Level-2Thinking

x(q2t ; {q1t+s, T r1t+s}) = X −Xt+1

Tr2t =(Dt + q2t − (1 + r)q2t−1

)Xt

}⇒ {q2t , T r2t }

Level-kThinking



12



Level-1Thinking

x(q1t ; {q∗, 0}) = X −Xt+1

Tr1t =(Dt + q1t − (1 + r)q1t−1

)Xt

}⇒ {q1t , T r1t }

Level-2Thinking

x(q2t ; {q1t+s, T r1t+s}) = X −Xt+1

Tr2t =(Dt + q2t − (1 + r)q2t−1

)Xt

}⇒ {q2t , T r2t }

Level-kThinking



12


qkt =

D + q∗ − γ r

1+rσ2x

(X −Xt+1

)1 + r

, k = 1

D + qk−1t+1 − γ r1+rσ

2xX

1 + r, k > 1

13

Forward Iteration

Pricing equation (REE)

qt =D + Etqt+1 − γ r

1+rσ2xX

1 + r

Forward Iteration

t4 50 1 2 3

Constant solution

qt =D − γ r

1+rσ2xX

r≡ q∗

14

Forward Iteration

Pricing equation (REE)

qt =D + Etqt+1 − γ r

1+rσ2xX

1 + r

Forward Iteration

t4 50 1 2 3

Constant solution

qt =D − γ r

1+rσ2xX

r≡ q∗

14

Diagonal Iteration

qkt =D + qk−1t+1 − γ r

1+rσ2xX

1 + r,

q1t+k−1 =D + q∗ − γ r

1+rσ2x

(X −Xt+k

)1 + r

t

k

4 50 1 2 31

2

3

4

5

6

Endogenous discounting

15

Diagonal Iteration

qkt =D + qk−1t+1 − γ r

1+rσ2xX

1 + r,

q1t+k−1 =D + q∗ − γ r

1+rσ2x

(X −Xt+k

)1 + r

t

k

4 50 1 2 31

2

3

4

5

6

Endogenous discounting15

Reflective Equilibrium

Idea: agents form beliefs according to level-k thinking, theeconomy is populated by agents with different k with pdf f(k)(Woodford;GarciaSchmidt-Woodford)

In case of exponential f(k) with average k

qt = q∗ + γσ2xr

1 + r·

∑∞k=1

(k−1k

)k−1 Xt+k

(1+r)k

k

Higher k

1. reduces the direct effect of interventions

2. makes the price react more to expected future interventions

Numerical Illustration

16

Reflective Equilibrium

Idea: agents form beliefs according to level-k thinking, theeconomy is populated by agents with different k with pdf f(k)(Woodford;GarciaSchmidt-Woodford)

In case of exponential f(k) with average k

qt = q∗ + γσ2xr

1 + r·

∑∞k=1

(k−1k

)k−1 Xt+k

(1+r)k

k

Higher k

1. reduces the direct effect of interventions

2. makes the price react more to expected future interventions


16

Does QE Affect Output?

Simple Model

I Endowment economy

I Balance sheet policies affect prices and taxes only

An extension

I Output is endogenous (“demand determined”)

I Prices are perfectly rigid (avoids Phillips curve)

I Inelastic supply of safe bonds (but r is still fixed)

I Risky assets are claims on traded part of output

17


Simple Model

I Endowment economy

I Balance sheet policies affect prices and taxes only

An extension

I Output is endogenous (“demand determined”)

I Prices are perfectly rigid (avoids Phillips curve)

I Inelastic supply of safe bonds (but r is still fixed)

I Risky assets are claims on traded part of output

17

A Model with Endogenous Output

Households

max{xt+1,bt+1,ct}

E0

[−1

γ

∞∑t=0

eεxt−ρt−γct

]s.t.: ct + bt+1 + qtxt+1 ≤Wt − Tt + (1 + r)bt + (Dt + qt)xt

Total Income/Output Yt distributed as

I Wt = (1− δ)Yt – labor (non-traded) income

I DtX = δYt – dividends

What determines output? goods market clearing (in TE)

Yt = C[Wt(Yt)− Tt(Yt), Dt(Yt), qt, {W e

t+s − T et+s, Det+s, q

et+s}

]

18


Households

max{xt+1,bt+1,ct}

E0

[−1

γ

∞∑t=0

eεxt−ρt−γct








et+s}

]

18


Households

max{xt+1,bt+1,ct}

E0

[−1

γ

∞∑t=0

eεxt−ρt−γct








et+s}

]18


Rational expectations equilibrium (no effect of interventions)

q∗ =1

r· δX

(Y − σ2x

γ

)Y ∗t = Y − 1

γεxt

Level-1 equilibrium

q1t − q∗ = ΓqXt+1

X; Y 1

t − Y ∗t = ΓY

(Xt+1

X

)2

Level-k equilibrium

qkt − q∗ =1

1 + r

δ

X(Y k−1t+1 − Y

∗t+1) +

1

1 + r(qk−1t+1 − q

∗)

Y kt − Y ∗t =

∞∑j=1

1

(1 + r)j

[Ψ1(Y

k−1t+j − Y

∗t+j) + Ψ2(q

k−1t+j − q

kt+j)

2]

19


Rational expectations equilibrium (no effect of interventions)

q∗ =1

r· δX

(Y − σ2x

γ

)Y ∗t = Y − 1

γεxt

Level-1 equilibrium

q1t − q∗ = ΓqXt+1

X; Y 1

t − Y ∗t = ΓY

(Xt+1

X

)2

Level-k equilibrium

qkt − q∗ =1

1 + r

δ

X(Y k−1t+1 − Y

∗t+1) +

1

1 + r(qk−1t+1 − q

∗)

Y kt − Y ∗t =

∞∑j=1

1

(1 + r)j

[Ψ1(Y

k−1t+j − Y

∗t+j) + Ψ2(q

k−1t+j − q

kt+j)

2]

19

Does QE affect output?

Result

I First-order effect on asset prices

I Second-order effect on output

Intuition

I Higher asset prices lead to first-order wealth effect

I Offset by higher borrowing by the CB

I Only effect is to reduce perceived risk

I Less precautionary saving, more consumption/output(second order)

20

Extensions

1. Foreign exchange interventions (+nominal variables) Details

2. Long-term public bonds purchases (+nominal variables)

I Model with level-1 thinkers resembles Vayanos-Vila (2009)

3. Learning to play equilibrium Details

I Existing policy effect disappears over timeI New policies are less effective

4. Limited participation + level-k thinking

I Negative interaction

5. Presence of rational expectations agents Details

I Does not change qualitative resultsI Can amplify effects due to level-k thinking

21

Testable Predictions

Forecast errors

I Agents make predictable forecast errors

I The errors can differentiate the model from other theories(segmented markets, signaling channel)

Forecast errors in the data

I Forecasts of future taxes?

I Forecasts of asset prices

Forecast errors in the model

Individual: ukt+s ≡ qt+s − Ekt qt+s

Average: ut+s ≡∞∑k=1

f(k)ukt+s = µsγσ2x

r1+rXt+1

k[(1 + r − µ)k + µ]

22


Forecast errors










r1+rXt+1

k[(1 + r − µ)k + µ]

22


Forecast errors










r1+rXt+1

k[(1 + r − µ)k + µ]

22

Empirics

Fieldhouse-Mertens-Ravn (2018, QJE)

I Monthly data on GSEs mortgage purchases: 1967-2006

I “Unexpected exogenous” purchases narrative identification

I Result: mortgage yield reacts significantly to interventions

Forecast errors

I Blue Chip conventional mortgage rate forecasts: 1982-2006

I Project median forecast errors on “exogenous” purchases

23

Fieldhouse-Mertens-Ravn (2018)

1st stage ∑hj=0 pt+j

Xt= αh + γh

mt

Xt+ ϕh(L)Zt−1 + ut+h

I pt – policy action (agency commitments)

I mt – a noncyclical narrative policy indicator

I Xt – a deterministic trend in real personal income

2nd stage

yt+h − yt−1 = αh + γh

(12

8×∑7

j=0 pt+j

Xt

)︸︷︷︸

≡QEt

+ ϕh (L)Zt−1 + ut+h

24

Forecast Errors

Forecasts Etrt+h

m m+1 m+2

Q Q+1 Q+2

t

Forecast errorut,t+h = rt+h − Etrt+h

H0: ut,t+h is uncorrelated with QEt−1

25

Forecast Errors

Forecasts Etrt+h

m m+1 m+2

Q Q+1 Q+2

t



25

Forecast Errors

Forecasts Etrt+h

m m+1 m+2

Q Q+1 Q+2

t



25

Forecast Errors

Blue Chip Financial Forecasts (BCFF)

m m+1 m+2

Q Q+1 Q+2

t

BCFF next calendar quarter forecast error

ut,t+“1:3” =

∑2i=0 rt+3+i−mod(t+2,3)

3− f“1:3”t

H0: ut,t+“3(k−1)+1:3k” is uncorrelated with QEt−1

26

Forecast Errors

Blue Chip Financial Forecasts (BCFF)

m m+1 m+2

Q Q+1 Q+2

t

BCFF next calendar quarter forecast error

ut,t+“1:3” =

∑2i=0 rt+3+i−mod(t+2,3)

3− f“1:3”t

H0: ut,t+“3(k−1)+1:3k” is uncorrelated with QEt−126

0 6 12 18 24h (months)

-20

-15

-10

-5

0

5

basi

s po

ints

27

1-3 4-6 7-9 9-12h (months)

-15

-10

-5

0

basi

s po

ints

28

Conclusion

1. Bounded rationality channel of balance sheet policies

I Level-k thinking belief formation

2. Testable predictions

I Forecast errors respond to interventions

I Evidence from mortgage rate forecasts errors

29

Rational Expectations Equilibrium

Definition: REE is a TE such that

qt = qt, T rt = Trt

Specifically

αTr,t + βTr,tεxt+1︸︷︷︸

beliefs T rt+1

REE=(D + qt+1 − (1 + r)qt

)Xt+1 +Xt+1ε

xt+1︸︷︷︸

realized transfers Trt+1

Risky assets market in t


2x

− βTr,t = X −Xt+1

⇒ Balance sheet policy does not affect price qt in REE!Back

30



qt = qt, T rt = Trt

Specifically


beliefs T rt+1

REE=(D + qt+1 − (1 + r)qt

)Xt+1 +Xt+1ε

xt+1︸︷︷︸




2x



30



qt = qt, T rt = Trt

Specifically


beliefs T rt+1

REE=(D + qt+1 − (1 + r)qt

)Xt+1 +Xt+1ε

xt+1︸︷︷︸




2x



30



qt = qt, T rt = Trt

Specifically


beliefs T rt+1

REE=(D + qt+1 − (1 + r)qt

)Xt+1 +Xt+1ε

xt+1︸︷︷︸




2x



30


0 5 10 15 20 25 300

0.005

0.01

0.015

Back

31

Open Economy Model

New elements

I 2 countries: Home (size ω), Foreign (size 1− ω)

I Additional assets: riskless nominal bonds (in eachcountry), money (in each country)

I Goods: single traded good (LOOP holds)

I Risk: money supply (=inflation risk)

I Monetary policy: logMt+1 = logM + εht (home)

I FX intervention: {−Bht+1, B

ft+1} (home)

Back

32

Solution

Money markets

mt − pt = −vitm∗t − p∗t = −vi∗t

Bond markets

Home: ωbH,t+1 + (1− ω)b∗H,t+1 = −Bht+1

Foreign: ωbF,t+1 + (1− ω)b∗F,t+1 = B∗ −Bft+1

Back

33

Nominal Exchange Rate

Rational expectations equilibrium

et =v

1 + vEtet+1 +

1

1 + v(mt −m∗t ) ⇒ eREEt

Reflective equilibrium (summing over different level-k’s)

et = eREEt +γ

(1 + v)2

∞∑k=1

f (k)

(v

1 + v

)k+2 [σ2fB

ft+k + σ2h(−Bh

t+k)]

Back

34

Unraveling

Assumption: agents become more sophisticated over time

ft (k) =

{f(k − ht), k ≥ 1 + ht,

0, k < 1 + ht,

When asset purchases follow Xt+1 = µtX1

qt − qREE =γσ2xω· 11+r−µ1−λ + µ

X1

(µ1+h

(1 + r)h

)tBack

35

REE Agents

Assumption

I φ agents form expectations rationally

I (1− φ) agents form expectations using level-k thinking

Risky asset price

qt = qREE + (1− φ) γσ2x

∞∑k=1

f(k)

(1 + r)k

∞∑s=0

(φ

1 + r

)sXt+s+k

Risky asset price without REE agents

qt = qREE + γσ2x

∞∑k=1

f(k)

(1 + r)kXt+k

Back

36

Central Banks Balance Sheet Policies Without Rational Expectations · 2019. 6. 5. · Beine-Benassy-Quere-Lecourt (2002) 3. Theory The irrelevance result if 1. people can freely trade

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