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Shadow rate SRNKM Microfoundations Economic implications

Shadow Interest Rate

Jing Cynthia WuChicago Booth and NBER

Coauthors: Dora Xia (BIS) and Ji Zhang (Tsinghua)

Cynthia Wu (Chicago Booth & NBER) 1 / 63


ZLB: monetary policy

Before ZLB

I Central banks lower policy rates to stimulate aggregate demand

I Economists rely on them to study monetary policy

Policy rates at ZLB

I Japan, US, Europe

I Unconventional policy tools

I large-scale asset purchases (QE)I lending facilitiesI forward guidance

I Negative interest rates



ZLB: economic models

Term structure modelsI Benchmark models

I Gaussian ATSM allows undesirably negative interest ratesI It is especially problematic when interest rates are low

I Our papers:I Wu and Xia (JMCB 2016): model US yield curve with ZLBI Wu and Xia (2017): model recent negative interest rates in Europe

New Keynesian modelsI Benchmark models: no unconventional monetary policy

I ZLB introduces structural breakI counterfactual economic implicationsI computationally demanding

I Wu and Zhang (2016): incorporate unconventional monetary policyI sensible economic implicationsI tractable



Common theme: shadow rate

Black (1995)rt = max(st , r)

Sources:BoardofGovernorsoftheFederalReserveSystemandWuandXia(2015)

Wu-XiaShadowFederalFundsRate

Effectivefederalfundsrate,end-of-monthWu-Xiashadowrate

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016-4%

-2%

0%

2%

4%

6%



Contributions - Wu and Xia (JMCB 2016)

Measuring the Macroeconomic Impact of Monetary Policy at the ZLB

I Develop an analytical approximation for SRTSM

I Shadow rate has similar dynamic correlations with macro variables as the fedfunds rate did previously

I Our shadow rates for US, Euro area, and UK are available at

I Atlanta FedI Haver AnalyticsI Thomson ReutersI Bloomberg

I Wu-Xia shadow rate has been discussed by

I Policy makers: Governor Powell (2013), Altig (2014) of the Atlanta Fed,Hakkio and Kahn (2014) of the Kansas City Fed

I Media:The Wall Street Journal, Bloomberg news, Bloomberg Businessweek,Forbes, Business Insider, VOX



Contributions - Wu and Zhang (2016)

A Shadow Rate New Keynesian Model

I Present new empirical evidence relating the shadow rate to

I private interest ratesI Fed’s balance sheetI Taylor rule

I Propose a New Keynesian model with the shadow rate

I accommodates both conventional and unconventional policies

I Microfoundations

I QEI lending facilities

I Economic implications

I a negative supply shock decreases outputI data-consistentI contradicts standard models

I government-spending multiplier is not larger than normal



Outline

1. Wu-Xia shadow rate

2. Motivating shadow rate New Keynesian model

3. Microfoundations

4. Economic implications



Shadow rate

Black (1995):

rt = max(st , r)

The shadow rate is affine

st = δ0 + δ′1Xt

I r = 0.25, interest on reserves

I Xt : 3 factors



Bond pricing

Factor dynamics:

Xt+1 = µ + ρXt + Σεt+1, εt+1 ∼ N(0, I ).

Pricing kernel

mt+1 = rt +1

2λ′tλt + λ′tεt+1

where λt = λ0 + λ1Xt

Pricing equation I

Pnt = Et [exp(−mt+1)Pn−1,t+1]

Pricing equation II

Pnt = EQt [exp(−rt)Pn−1,t+1]



Bond pricing

Factor dynamics under risk-neutral measure Q:

Xt+1 = µQ + ρQXt + ΣεQt+1, εQt+1Q∼ N(0, I ).

where µQ = µ− Σλ0, and ρQ = ρ− Σλ1

Yield

ynt = −1

nlog(Pnt)

Forward rate from t + n to t + n + 1

fnt = (n + 1)yn+1,t − nynt



Forward rates

Our approximation

fnt = r + σQn g

(an + b′nXt − r

σQn

)where g(z) = zΦ(z) + φ(z).

an = δ0 + δ′1

n−1∑j=0

(ρQ)jµQ −

1

2δ′1

n−1∑j=0

(ρQ)jΣΣ′

n−1∑j=0

(ρQ)j′ δ1

b′n = δ′1

(ρQ)n

(σQn )2 ≡ VQt (st+n) =

n−1∑j=0

δ′1(ρQ)jΣΣ′(ρQ′)jδ1



Comparison to GATSM

SRTSM

st = δ0 + δ′1Xt

rt = max(st , r)

Forward rate

fnt = r + σQn g

(an + b′nXt − r

σQn

)

GATSM

rt = δ0 + δ′1Xt

Forward rate

fnt = an + b′nXt



Property of g(.)

−5 −4 −3 −2 −1 0 1 2 3 4 50

1

2

3

4

5

z

y

y = g(z)y = z

f SRnt

≈ r , at the ZLB

≈ an + b′nXt = f Gnt , when interest rates are high



Extended Kalman filter

State equation

Xt+1 = µ+ ρXt + Σεt+1, εt+1 ∼ N(0, I )

Observation equation

f ont = r + σQn g

(an + b′nXt − r

σQn

)+ ηnt , ηnt ∼ N(0, ω)



Model fit

Figure: Average forward curve in 2012

1 2 5 7 100

0.5

1

1.5

2

2.5

3

3.5

4SRTSM

fittedobserved

1 2 5 7 100

0.5

1

1.5

2

2.5

3

3.5

4GATSM

fittedobserved

Log likelihood values

I SRTSM: 850; GATSM: 750data and normalization



Approximation error

Average absolute approximation error between 1990M1 and 2013M1 (in basis points)

3M 6M 1Y 2Y 5Y 7Y 10Y

forward rate error 0.01 0.02 0.04 0.13 0.69 1.14 2.29forward rate level 346 357 384 435 551 600 636yield error 0.00 0.01 0.01 0.04 0.24 0.42 0.78



Outline


2. Motivating shadow rate New Keynesian modelSR as summary of unconventional monetary policyLinear model

3. Microfoundations




Evidence 1: taper tantrum

3M 1Y 3Y 5Y 7Y 10Y0

1

2

Maturity

May 21, 2013May 22, 2013

−3

−2

−1

0

1

2

3

4

5Expected short rate

Apr 2013Apr 2014

Apr 2015Apr 2016

Apr 2017Apr 2018

Apr 2019Apr 2020

Apr 2021Apr 2022

Apr 2023

Apr 2013May 2013

−3

−2

−1

0

1

2

3

4

5Expected shadow rate

Apr 2013Apr 2014

Apr 2015Apr 2016

Apr 2017Apr 2018

Apr 2019Apr 2020

Apr 2021Apr 2022

Apr 2023

Apr 2013May 2013

I May 22, 2013: Bernanke told Congress Fed may decrease the size of QE

I shift in shadow rate summarizes this effect



Evidence 2: shadow rate and Fed’s balance sheet

2009 2010 2011 2012 2013 2014−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Per

cent

age

poin

ts

QE1

QE2

OT

QE3

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

Tril

lions

of D

olla

rs

Wu−Xia shadow rate− Fed balance sheet

Correlation

I QE1 - QE3: -0.94



Evidence 3: structural break test in FAVAR

Replace the fed funds rate with st in Bernanke, Boivin, and Eliasz (2005)

xmt = µx + ρxxXm

t−1 + uxt

+ 1(t<December 2007)ρxs1 St−1 + 1(December 2007≤t≤June 2009)ρ

xs2 St−1 + 1(t>June 2009)ρ

xs3 St−1

I monthly VAR(13)

I xmt : 3 underlying macro factors

Null hypothesisH0 : ρxs1 = ρxs3

Likelihood ratio test: χ2(39)

I p = 0.29 for st

I p = 0.0007 for EFFR

model details robustness



Evidence 4: shadow rate Taylor rule

st = β0 + β1st−1 + β2(yt − ynt ) + β3πt + εt

1955 1970 1985 2000 2015-5

0

5

10

15

20

fed funds rate & shadow rateTaylor rule implied

1955 1970 1985 2000 2015-10

-5

0

5

10

Test for structural break

I F statistic = 2

I Critical value: 2.37

I No structural break



Evidence 5: shadow rate and private rates

2009 2011 2013 2015

Inte

rest

rat

es

-5

0

5

10

15

20

Wu-Xia shadow ratefed funds ratehigh yield effective yieldBBB effective yieldAAA effective yieldGoldman Sachs Financial Conditions Index

Gol

dman

Sac

hs F

CI

95

100

105

110

I private rates are the relevant rates for agents and the economy

I correlations: 0.8

I rBt = st + rp



Summary

Shadow rate summarizes unconventional monetary policy

I Taper tantrum

I Fed’s balance sheet

There is no structural break in

I relationship between macro variables and the shadow rate

I shadow rate Taylor rule

Private rates

I are the relevant interest rates for economic agents

I respond to unconventional monetary policy

I reflect the overall effect of UMP on the economy

I the shadow rate is a sensible summary



Shadow rate New Keynesian model

Definition

The shadow rate New Keynesian model consists of the shadow rate IScurve

yt = − 1

σ(st − Etπt+1 − s) + Etyt+1,

New Keynesian Phillips curve

πt = βEtπt+1 + κ(yt − ynt ),

and shadow rate Taylor rule

st = φsst−1 + (1− φs) [φy (yt − ynt ) + φππt + s] .

QE lending facilities



Outline



3. MicrofoundationsMicrofoundations I: QEMicrofoundations II: lending facilities




Microfoundation for SRNKM I: QE

The risk premium channel

I government purchases outstanding loans

I decrease interest rates through reducing risk premiumI Gagnon et al. (2011) and Hamilton and Wu (2012)

I the same mechanism works for government bonds or corporate bonds



Households’ problem

Households’ utility function

E0

∞∑t=0

βt

(C 1−σt

1− σ− Lt

1+η

1 + η

)budget constraint

Ct +BHt

Pt=

RBt−1B

Ht−1

Pt+ WtLt + Tt

Euler equation

C−σt = βRBt Et

[C−σt+1

Πt+1

]The linear Euler equation

yt = − 1

σ

(rBt − Etπt+1 − rB

)+ Etyt+1,



Bond return and policy rate

Define

rpt ≡ rBt − rt

I interpreted as convenience yield by Krishnamurthy and Vissing-Jorgensen (2012)

I Gagnon et al. (2011), Krishnamurthy and Vissing-Jorgensen (2011), and Hamilton

and Wu (2012) suggest

rp′t(bGt ) < 0⇒ rpt(b

Gt ) = rp − ς(bGt − bG )

During normal times

I bGt = bG

I rpt(bG ) = rp ⇒ rBt = rt + rp

At the ZLB

I QE → bGt ↑→ rpt ↓→ rBt ↓



Shadow rate equivalence for QE

Euler equation

yt = − 1

σ

(rBt − Etπt+1 − rB

)+ Etyt+1,

I During normal times, bGt = bG , rt = st

rBt = rt + rp = st + rp

I At the ZLB, rt = 0

rBt = rpt = rp − ς(bGt − bG ) = st + rp

if st = −ς(bGt − bG )



Shadow rate equivalence for QE

Proposition

The shadow rate New Keynesian model represented by the shadow rate IS curve

yt = − 1

σ(st − Etπt+1 − s) + Etyt+1,

New Keynesian Phillips Curve, and shadow rate Taylor rule

st = φsst−1 + (1− φs) [φy (yt − ynt ) + φππt + s]

nests both conventional Taylor interest rate rule and QE operation that changesrisk premium if

rt = st , bGt = bG for st ≥ 0

rt = 0, bGt = bG − stς for st < 0.

Shadow rate NK model



Quantifying assumption in proposition

st = −ς(bGt − bG )

2009 2010 2011 2012 2013 2014-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Per

cent

age

poin

ts

QE1 QE2 QE3

Wu-Xia shadow rate-log(QE holdings)

-29.5

-29

-28.5

-28

-27.5

-27

I linear assumption: correlation = 0.92I ς = 1.83

I Fed increases its bond holdings by 1%, the shadow rate decreases by 0.0183%I QE1: 490 billion to 2 trillion ⇒ 2.5% decrease in the shadow rateI QE3: 2.6 trillion to 4.2 trillion ⇒ 0.9% decrease in the shadow rate



Outline



3. MicrofoundationsMicrofoundations I: QEMicrofoundations II: lending facilities




Microfoundation for SRNKM II: lending facilities

Lending facilities

I extend loans to the private sector ⇒ change the loan-to-value ratio

I Example: Term Asset-Backed Securities Loan Facility in the US

Combine this with a tax policy

I tax (subsidy) on the interest rate income (payment)



Model features

Entrepreneurs

I produce intermediate goods with labor and capital

I maximize utility

I discount factor γ < β

I borrow from households with a loan-to-value ratio M

I accumulate capital

I use capital as collateral

Government policy at the ZLBI lending facilities

I lend directly to entrepreneursI change the loan-to-value ratio from M to Mt

I tax (subsidy) on the interest rate income (payment)



Entrepreneurs’ problem

Utility function

max E0

∞∑t=0

γt logCEt

production function

Y Et = AKα

t−1(Lt)1−α

capital accumulation

Kt = It + (1− δ)Kt−1

budget constraint

Y Et

Xt+ Bt =

RBt−1Bt−1

Tt−1Πt+ WtLt + It + CE

t

borrowing constraint

Bt ≤ MtEt

(KtΠt+1/R

Bt

)Cynthia Wu (Chicago Booth & NBER) 35 / 63


Entrepreneurs’ FOCs

Labor demand

Wt =(1− α)AKα

t−1L−αt

Xt

Euler equation

1

CEt

(1− MtEtΠt+1

RBt

)= γEt

[1

CEt+1

(αY E

t+1

Xt+1Kt− Mt

Tt+ 1− δ

)]



Households’ problem

Households’ utility

E0

∞∑t=0

βt

(C 1−σt

1− σ− L1+η

t

1 + η

)

budget constraint

Ct + BHt =

RBt−1B

Ht−1

Tt−1Πt+ WtLt + Tt

Euler equation

C−σt = βEt

(RBt

C−σt+1

Πt+1Tt

)labor supply

Wt = Cσt Lηt



Sources of funding

Entrepreneurs’ borrowing constraint

Bt ≤ MtEt

(KtΠt+1

RBt

)Households lend

BHt ≤ MEt

(KtΠt+1

RBt

)I During normal times Bt = BH

t , and Mt = M

I At the ZLB Mt > M

Government lends the rest

BGt = (Mt −M)Et

(KtΠt+1

RBt

)



Conventional and unconventional policy

Suppose RBt = RtRP

Conventional and unconventional policy tools appear in the model in pairs:

I Rt/Tt – HH Euler equation, HH&E budget constraints

I Rt/Mt – E borrowing constraint, E Euler equation

I Mt/Tt – E Euler equation

model

Decreasing Rt is equivalent to increasing Tt and Mt .



Shadow rate equivalence for lending facilities

Proposition

If Rt = St , Tt = 1,Mt = M for St ≥ 1

Tt = Mt/M = 1/St for St < 1,

then Rt/Tt = St , Rt/Mt = St/M, Mt/Tt = M ∀St .

I St summarizes both conventional and unconventional policies

I Equivalence in the non-linear model



Shadow rate equivalence for lending facilities

Proposition

The shadow rate New Keynesian model represented by the Euler equation

ct = − 1

σ(st − Etπt+1 − s) + Etct+1

the shadow rate Taylor rule, Phillips curve, ..., nests both conventionalTaylor interest rate rule and lending facility – tax policy if

rt = st , τt = 0,mt = m for st ≥ 0

τt = mt −m = −st for st < 0.

Shadow rate NK model Detailed model



Outline



3. Microfoundations

4. Economic implicationsEconomic implication I: negative supply shockEconomic implication II: government spending multiplier



Economic implication I: negative supply shock

Technology shock

at ↓ = ρaat−1 + ea,t ↓

Phillips Curve

πt ↑ = βEtπt+1 + κ(yt − y)− κ(1 + η)

σ + ηat ↓




Standard model

Monetary policy

rt = maxφsst−1 + (1− φs) [φy (yt − ynt ) + φππt + s] , 0

Real interest raterrt = rt − Et [πt+1]

IS curve

yt = − 1

σ(rrt − r) + Etyt+1

normal times: π ↑ → r ↑↑ → rr ↑ → y ↓ZLB without UMP: π ↑ → r = 0 → rr ↓ → y ↑ Counterfactual




Shadow rate model

Shadow rate Taylor rule


Real interest raterrt = st − Et [πt+1]

Shadow rate IS curve

yt = − 1

σ(st − Etπt+1 − r) + Etyt+1

normal times: π ↑ → r ↑↑ → rr ↑ → y ↓ZLB without UMP: π ↑ → r = 0 → rr ↓ → y ↑ CounterfactualZLB with UMP: π ↑ → s ↑↑ → rr ↑ → y ↓ Data consistent



Extended model

Iacoviello (2005)I Patient households

I consume, work, hold house, lend

I Impatient householdsI consume, work, hold houseI borrow with collateral

I EntrepreneursI consume, invest, hold housesI borrow with collateralI produce intermediate goods using housing, capital, and labor

I RetailersI monopolistically competitiveI Calvo sticky

I GovernmentI Taylor rule

Calibration



Difference from Iacoviello (2005)

I Unconventional policyI QE: time-varying risk premiumI lending facilities: time-varying loan-to-value ratioI time-varying tax

I government spending

I preference shock: create ZLB



Models and methodology

Notation

I Standard model: without unconventional policy: rt = 0

I Shadow rate model: with unconventional policy: st < 0

Methodology for shadow rate model

I solve linear model with shadow rate

I then use propositions mapping into various UMP



Preference shock and the ZLB

5 10 15 20 25 30 35 400

0.05

0.1

0.15

0.2

0.25

Preference shock

% d

ev. f

rom

S.S

.

5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

rt policy rate

perc

enta

ge

more




10 20 30 40−1

−0.8

−0.6

−0.4

−0.2

0

At TFP

% d

ev. f

rom

S.S

.

10 20 30 40

0

0.5

1

1.5

Yt output

% d

ev. f

rom

S.S

.

10 20 30 40−0.5

0

0.5

1

1.5

2

Ct consumption

% d

ev. f

rom

S.S

.

10 20 30 400

1

2

3

It investment

% d

ev. f

rom

S.S

.

Note: blue: shadow rate NK model with UMP; red: standard NK model without UMP




10 20 30 400

0.5

1

1.5

2

πt inflation

Leve

l in

pect

.

10 20 30 400

0.2

0.4

0.6

0.8

rt policy rate

Leve

l in

pect

.

10 20 30 400

0.2

0.4

0.6

0.8

st shadow rate

Leve

l in

pect

.

10 20 30 400

0.2

0.4

0.6

0.8

rBt−τ

t private rate

Leve

l in

pect

.

10 20 30 40

−1.5

−1

−0.5

0

0.5

rrt real interest rate

Leve

l in

pect

. Note: blue: shadow rate NK model with UMP; red: standard NK model without UMP




10 20 30 400

0.2

0.4

0.6

0.8

rpt risk premium − QE

Leve

l in

pect

.

10 20 30 40

−0.2

−0.15

−0.1

−0.05

0

τt tax − Facilities

Leve

l in

pect

.

10 20 30 40

−0.2

−0.15

−0.1

−0.05

0

Mt loan−to−value ratio − Facilities

% d

ev. f

rom

S.S

.




Outline



3. Microfoundations

4. Economic implicationsEconomic implication I: negative supply shockEconomic implication II: government spending multiplier



Economic implication II: government spending multiplier

Government spending shock

gt ↑ = (1− ρg )g + ρggt−1 + eg ,t ↑

Market-clearing condition

yt ↑ = cyct + gygt ↑

Phillips Curve

πt ↑ = βEtπt+1 +κ

δ + η(σ(ct − c) + η(yt ↑ −y))




Standard model

Monetary policy

rt = maxφsst−1 + (1− φs) [φy (yt − ynt ) + φππt + s] , 0

Real interest raterrt = rt − Et [πt+1]

IS curve

ct = − 1

σ(rrt − r) + Etct+1

normal times: π ↑ y ↑ → r ↑↑ → rr ↑ → c ↓ → ∆y < ∆gZLB without UMP: π ↑ y ↑ → r = 0 → rr ↓ → c ↑ → ∆y > ∆g




Shadow rate model

Shadow rate Taylor rule


Real interest raterrt = st − Et [πt+1]

Shadow rate IS curve

ct = − 1

σ(st − Etπt+1 − r) + Etct+1

normal times: π ↑ y ↑ → r ↑↑ → rr ↑ → c ↓ → ∆y < ∆gZLB without UMP: π ↑ y ↑ → r = 0 → rr ↓ → c ↑ → ∆y > ∆gZLB with UMP: π ↑ y ↑ → s ↑↑ → rr ↑ → c ↓ → ∆y < ∆g




10 20 30 400

0.5

1

1.5

2

2.5

Gt government spending

% d

ev. f

rom

S.S

.

10 20 30 40

1

1.5

2

2.5

3

Government spending multiplier

leve

l





10 20 30 40

0

0.5

1

πt inflation

Leve

l in

pect

.

10 20 30 40

0

0.2

0.4

0.6

rt policy rate

Leve

l in

pect

.

10 20 30 400

0.1

0.2

0.3

0.4

st shadow rate

Leve

l in

pect

.

10 20 30 40

0

0.2

0.4

0.6

rBt−τ

t private rate

Leve

l in

pect

.

10 20 30 40

−0.5

0

0.5

rrt real interest rate

Leve

l in

pect

. Note: blue: shadow rate NK model with UMP; red: standard NK model without UMP




10 20 30 40

0

0.5

1

Ct consumption

% d

ev. f

rom

S.S

.

10 20 30 400

0.5

1

1.5

2

It investment

% d

ev. f

rom

S.S

.

10 20 30 400

0.5

1

1.5

Yt output

% d

ev. f

rom

S.S

.

10 20 30 40

1

1.5

2

2.5

3

Government spending multiplier

leve

l





10 20 30 400

0.1

0.2

0.3

0.4

rpt risk premium − QE

Leve

l in

pect

.

10 20 30 40−0.1

−0.08

−0.06

−0.04

−0.02

0

τt tax − Facilities

Leve

l in

pect

.

10 20 30 40−0.1

−0.08

−0.06

−0.04

−0.02

0

Mt loan−to−value ratio − Facilities

% d

ev. f

rom

S.S

.




Computational advantages

The ZLB imposes one of the biggest challenges for solving and estimatingthese models

Existing methods

I have undesirable economic implications

I computationally demanding

Shadow rate model

I no structural break

I can rely on traditional methods

I unique solution



Conclusion

Wu and Xia (JMCB 2016)

I provide an analytical approximation for SRTSM

I Wu-Xia shadow rate summarizes unconventional monetary policy

Wu and Zhang (2016)

I propose a New Keynesian model with the shadow rate

I map unconventional policy tools into the shadow rate framework

I make counterfactual results in standard NK model disappear

I computationally tractable



Related work: Wu and Xia (2017)

Time-Varying Lower Bound of Interest Rates in Europe

I A new shadow rate term structure model with rt = max(st , r t)I r t = rdt + spt

I Discrete policy lower bound rdtI Non-constant spread spt

I Agents are forward lookingI Derive bond prices

I AsymmetryI 10 bp drop in the ELB decreases the 10-year rate by 6.5-8.5 bpI 10 bp initial rise in the ELB increases the 10-year rate by 9-14 bp


Data and model specification

Gurkaynak, Sack, and Wright (2007) dataset

I monthly, January 1990 - December 2013

I maturities: 3 and 6 months, 1, 2, 5, 7 and 10 years

Normalization with repeated eigenvalues

ρQ =

ρQ1 0 0

0 ρQ2 1

0 0 ρQ2

.Back


FAVAR

Replace the fed funds rate with st in Bernanke, Boivin, and Eliasz (2005)

Ymt = am + bxx

mt + bsst + ηmt , ηmt ∼ N(0,Ω)

I Ymt : 97 economic variables from 1960 to 2013

I xmt : 3 underlying macro factors

Factor dynamics:[xmtst

]=

[µx

µs

]+

[ρxx ρxs

ρsx ρss

] [Xmt−1

St−1

]+ Σm

[εmtεMPt

],

[εmtεMPt

]∼ N(0, I )

I monthly VAR(13)

I Σm: Cholesky decomposition

back


Rubustness

p-value for ρxs1 = ρxs3 p-value for ρsx1 = ρsx3

Baseline 0.29 1.00A1 estimate r 0.18 1.00A2 2-factor SRTSM 0.13 0.97A3 Fama-Bliss 0.38 1.00A4 5-factor FAVAR 0.70 1.00A5 6-lag FAVAR 0.09 0.98

7-lag FAVAR 0.19 0.9712-lag FAVAR 0.22 1.00

back


Lending facilities

ct = −1

σ(rBt − τt − Etπt+1 − r − rp) + Etct+1

⇒ ct = −1

σ(st − Etπt+1 − r) + Etct+1

CE cEt = αY

X(yt − xt) + Bbt − RBB(rBt−1 + bt−1 − τt−1 − πt−1)− Iit + Λ1

⇒ CE cEt = αY

X(yt − xt) + Bbt − RBB(st−1 + rp + bt−1 − πt−1)− Iit + Λ1

bt = Et(kt + πt+1 + mt − rBt )

⇒ bt = Et(kt + πt+1 + m − st − rp)


Lending facilities

0 =

(1−

M

RB

)(cEt − Etc

Et+1) +

γαY

XKEt(yt+1 − xt+1 − kt)

+M

RBEt(πt+1 − rBt + mt) + γM(τt −mt) + Λ2

⇒ 0 =

(1−

M

RB

)(cEt − Etc

Et+1) +

γαY

XKEt(yt+1 − xt+1 − kt)

+M

RBEt(πt+1 − st − rp + m)− γMm + Λ2

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Calibration

para description source value

β discount factor of patient households Iacoviello (2005) 0.99βI discount factor of impatient households Iacoviello (2005) 0.95γ discount factor of entrepreneurs Iacoviello (2005) 0.98j steady-state weight on housing services Iacoviello (2005) 0.1η labor supply aversion Iacoviello (2005) 0.01µ capital share in production Iacoviello (2005) 0.3ν housing share in production Iacoviello (2005) 0.03δ capital depreciation rate Iacoviello (2005) 0.03X steady state gross markup Iacoviello (2005) 1.05θ probability that cannot re-optimize Iacoviello (2005) 0.75α patient households’ wage share Iacoviello (2005) 0.64ME loan-to-value ratio for entrepreneurs Iacoviello (2005) 0.89M I loan-to-value ratio for impatient households Iacoviello (2005) 0.55rR interest rate persistence Iacoviello (2005) 0.73rY interest rate response to output Iacoviello (2005) 0.27rΠ interest rate response to inflation Iacoviello (2005) 0.13GY steady-state government-spending-to-output ratio Fernandez-Villaverde et al. (2015) 0.20ρa autocorrelation of technology shock Fernandez-Villaverde et al. (2015) 0.90ρg autocorrelation of government-spending shock Fernandez-Villaverde et al. (2015) 0.80ρβ autocorrelation of discount rate shock Fernandez-Villaverde et al. (2015) 0.80σa standard deviation of technology shock Fernandez-Villaverde et al. (2015) 0.0025σg standard deviation of government-spending shock Fernandez-Villaverde et al. (2015) 0.0025σβ standard deviation of discount rate shock Fernandez-Villaverde et al. (2015) 0.0025ξp price indexation Smets and Wouters (2007) 0.24Π steady-state inflation 2% annual inflation 1.005BG steady-state government bond holdings no gov. intervention in private bond market 0T steady-state tax/subsidy on interest rate income/payment no tax in normal times 1rp steady-state risk premium 3.6% risk premium annually 1.009

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Preference shock and the ZLB

10 20 30 40

-1

0

1

2

perc

enta

ge

1. st shadow rate

10 20 30 400

0.5

1

1.5

2

2.5

perc

enta

ge

2. rt policy rate

10 20 30 40

3

4

5

6

perc

enta

ge

3. rBt-τ

t private rate

10 20 30 40

-2

-1

0

1

2

perc

enta

ge

4. πt inflation

10 20 30 40

-1.5

-1

-0.5

0

0.5

1

perc

enta

ge

5. rrt real interest rate

10 20 30 40

2.5

3

3.5

perc

enta

ge

6. rpt risk premium - QE

10 20 30 400

0.1

0.2

0.3

perc

enta

ge

7. τt tax - Facilities

10 20 30 400

0.1

0.2

0.3

% d

ev. f

rom

S.S

.

8. Mt loan-to-value ratio - Facilities

10 20 30 40

-3

-2

-1

% d

ev. f

rom

S.S

.

9. Yt output

10 20 30 40

-4

-3

-2

-1

0

% d

ev. f

rom

S.S

.

10. Ct consumption

10 20 30 40

-4

-3

-2

-1

0

% d

ev. f

rom

S.S

.11. I

t investment

10 20 30 400

0.05

0.1

0.15

0.2

0.25

% d

ev. f

rom

S.S

.

12. Preference shock

ZLB periodwithout UMPwith UMP

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