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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
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 2 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 3 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 3 / 63
Shadow rate SRNKM Microfoundations Economic implications
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%
Cynthia Wu (Chicago Booth & NBER) 4 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 5 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 6 / 63
Shadow rate SRNKM Microfoundations Economic implications
Outline
1. Wu-Xia shadow rate
2. Motivating shadow rate New Keynesian model
3. Microfoundations
4. Economic implications
Cynthia Wu (Chicago Booth & NBER) 7 / 63
Shadow rate SRNKM Microfoundations 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
Cynthia Wu (Chicago Booth & NBER) 8 / 63
Shadow rate SRNKM Microfoundations Economic implications
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]
Cynthia Wu (Chicago Booth & NBER) 9 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 10 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 11 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 12 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 13 / 63
Shadow rate SRNKM Microfoundations Economic implications
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)
Cynthia Wu (Chicago Booth & NBER) 14 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 15 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 16 / 63
Shadow rate SRNKM Microfoundations Economic implications
Outline
1. Wu-Xia shadow rate
2. Motivating shadow rate New Keynesian modelSR as summary of unconventional monetary policyLinear model
3. Microfoundations
4. Economic implications
Cynthia Wu (Chicago Booth & NBER) 17 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 18 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 19 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 20 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 21 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 22 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 23 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 24 / 63
Shadow rate SRNKM Microfoundations Economic implications
Outline
1. Wu-Xia shadow rate
2. Motivating shadow rate New Keynesian model
3. MicrofoundationsMicrofoundations I: QEMicrofoundations II: lending facilities
4. Economic implications
Cynthia Wu (Chicago Booth & NBER) 25 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 26 / 63
Shadow rate SRNKM Microfoundations Economic implications
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,
Cynthia Wu (Chicago Booth & NBER) 27 / 63
Shadow rate SRNKM Microfoundations Economic implications
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 ↓
Cynthia Wu (Chicago Booth & NBER) 28 / 63
Shadow rate SRNKM Microfoundations Economic implications
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 )
Cynthia Wu (Chicago Booth & NBER) 29 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 30 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 31 / 63
Shadow rate SRNKM Microfoundations Economic implications
Outline
1. Wu-Xia shadow rate
2. Motivating shadow rate New Keynesian model
3. MicrofoundationsMicrofoundations I: QEMicrofoundations II: lending facilities
4. Economic implications
Cynthia Wu (Chicago Booth & NBER) 32 / 63
Shadow rate SRNKM Microfoundations Economic implications
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)
Cynthia Wu (Chicago Booth & NBER) 33 / 63
Shadow rate SRNKM Microfoundations Economic implications
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)
Cynthia Wu (Chicago Booth & NBER) 34 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Shadow rate SRNKM Microfoundations Economic implications
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
)]
Cynthia Wu (Chicago Booth & NBER) 36 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 37 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
)
Cynthia Wu (Chicago Booth & NBER) 38 / 63
Shadow rate SRNKM Microfoundations Economic implications
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 .
Cynthia Wu (Chicago Booth & NBER) 39 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 40 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 41 / 63
Shadow rate SRNKM Microfoundations Economic implications
Outline
1. Wu-Xia shadow rate
2. Motivating shadow rate New Keynesian model
3. Microfoundations
4. Economic implicationsEconomic implication I: negative supply shockEconomic implication II: government spending multiplier
Cynthia Wu (Chicago Booth & NBER) 42 / 63
Shadow rate SRNKM Microfoundations Economic implications
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 ↓
Cynthia Wu (Chicago Booth & NBER) 43 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication I: negative supply shock
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
Cynthia Wu (Chicago Booth & NBER) 44 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication I: negative supply shock
Shadow rate model
Shadow rate Taylor rule
st = φsst−1 + (1− φs) [φy (yt − ynt ) + φππt + s]
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
Cynthia Wu (Chicago Booth & NBER) 45 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 46 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 47 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 48 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 49 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication I: negative supply shock
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
Cynthia Wu (Chicago Booth & NBER) 50 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication I: negative supply shock
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
Cynthia Wu (Chicago Booth & NBER) 51 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication I: negative supply shock
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
.
Note: blue: shadow rate NK model with UMP; red: standard NK model without UMP
Cynthia Wu (Chicago Booth & NBER) 52 / 63
Shadow rate SRNKM Microfoundations Economic implications
Outline
1. Wu-Xia shadow rate
2. Motivating shadow rate New Keynesian model
3. Microfoundations
4. Economic implicationsEconomic implication I: negative supply shockEconomic implication II: government spending multiplier
Cynthia Wu (Chicago Booth & NBER) 53 / 63
Shadow rate SRNKM Microfoundations Economic implications
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))
Cynthia Wu (Chicago Booth & NBER) 54 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication II: government spending multiplier
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
Cynthia Wu (Chicago Booth & NBER) 55 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication II: government spending multiplier
Shadow rate model
Shadow rate Taylor rule
st = φsst−1 + (1− φs) [φy (yt − ynt ) + φππt + s]
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
Cynthia Wu (Chicago Booth & NBER) 56 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication II: government spending multiplier
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
Note: blue: shadow rate NK model with UMP; red: standard NK model without UMP
Cynthia Wu (Chicago Booth & NBER) 57 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication II: government spending multiplier
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
Cynthia Wu (Chicago Booth & NBER) 58 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication II: government spending multiplier
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
Note: blue: shadow rate NK model with UMP; red: standard NK model without UMP
Cynthia Wu (Chicago Booth & NBER) 59 / 63
Shadow rate SRNKM Microfoundations Economic implications
Economic implication II: government spending multiplier
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
.
Note: blue: shadow rate NK model with UMP; red: standard NK model without UMP
Cynthia Wu (Chicago Booth & NBER) 60 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 61 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 62 / 63
Shadow rate SRNKM Microfoundations Economic implications
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
Cynthia Wu (Chicago Booth & NBER) 63 / 63
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
Cynthia Wu (Chicago Booth & NBER) 64 / 63
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
Cynthia Wu (Chicago Booth & NBER) 65 / 63
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
Cynthia Wu (Chicago Booth & NBER) 66 / 63
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)
Cynthia Wu (Chicago Booth & NBER) 67 / 63
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
Back
Cynthia Wu (Chicago Booth & NBER) 68 / 63
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
Back
Cynthia Wu (Chicago Booth & NBER) 69 / 63
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
Back
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