Shadow rate SRNKM Economic implications

Shadow

Jing Cynthia Wu Chicago 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:

Before ZLB

I Central lower policy rates to stimulate

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 facilities I forward guidance

I Negative interest rates

Cynthia Wu (Chicago Booth & NBER) 2 / 63 New Keynesian models I Benchmark models: no unconventional monetary policy

I ZLB introduces structural break I counterfactual economic implications I computationally demanding I Wu and Zhang (2016): incorporate unconventional monetary policy

I sensible economic implications I tractable

Shadow rate SRNKM Microfoundations Economic implications ZLB: economic models

Term structure models I Benchmark models

I Gaussian ATSM allows undesirably negative interest rates I It is especially problematic when interest rates are low I Our papers:

I Wu and Xia (JMCB 2016): model US with ZLB I Wu and Xia (2017): model recent negative interest rates in Europe

Cynthia Wu (Chicago Booth & NBER) 3 / 63 Shadow rate SRNKM Microfoundations Economic implications ZLB: economic models

Term structure models I Benchmark models

I Gaussian ATSM allows undesirably negative interest rates I It is especially problematic when interest rates are low I Our papers:

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

New Keynesian models I Benchmark models: no unconventional monetary policy

I ZLB introduces structural break I counterfactual economic implications I computationally demanding I Wu and Zhang (2016): incorporate unconventional monetary policy

I sensible economic implications I tractable

Cynthia Wu (Chicago Booth & NBER) 3 / 63 Shadow rate SRNKM Microfoundations Economic implications Common theme: shadow rate

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

Wu-Xia Shadow Federal Funds Rate

Effective federal funds rate, end-of-month Wu-Xia shadow rate

6%

4%

2%

0%

-2%

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

Sources: Board of Governors of the Federal Reserve System and Wu and Xia (2015)

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 fed funds rate did previously

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

I Atlanta Fed I Haver Analytics I Thomson Reuters I 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 rates I Fed’s balance sheet I Taylor rule

I Propose a New Keynesian model with the shadow rate

I accommodates both conventional and unconventional policies

I Microfoundations

I QE I lending facilities

I Economic implications

I a negative decreases output I data-consistent I contradicts standard models

I government-spending 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

0 st = δ0 + δ1Xt

I r = 0.25, interest on reserves

I Xt : 3 factors

Cynthia Wu (Chicago Booth & NBER) 8 / 63 Shadow rate SRNKM Microfoundations Economic implications pricing

Factor dynamics:

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

Pricing kernel 1 m = r + λ0 λ + λ0 ε t+1 t 2 t t t t+1 where λt = λ0 + λ1Xt

Pricing equation I

Pnt = Et [exp(−mt+1)Pn−1,t+1]

Pricing equation II

EQ Pnt = t [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:

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

Q Q where µ = µ − Σλ0, and ρ = ρ − Σλ1

Yield 1 y = − log(P ) nt n nt 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

 0  Q an + bnXt − r fnt = r + σn g Q σn where g(z) = zΦ(z) + φ(z).

0 n−1  n−1  n−1  j j j 0 X  Q Q 1 0 X  Q 0 X  Q an = δ0 + δ  ρ  µ − δ  ρ  ΣΣ  ρ  δ1 1 2 1 j=0 j=0 j=0 n 0 0  Q bn = δ1 ρ

n−1 Q 2 VQ X 0 Q j 0 Q0 j (σn ) ≡ t (st+n) = δ1(ρ ) ΣΣ (ρ ) δ1 j=0

Cynthia Wu (Chicago Booth & NBER) 11 / 63 Shadow rate SRNKM Microfoundations Economic implications Comparison to GATSM

SRTSM GATSM

0 0 st = δ0 + δ1Xt rt = δ0 + δ1Xt rt = max(st , r)

Forward rate Forward rate  0  Q an + b Xt − r 0 n fnt = an + b Xt fnt = r + σn g Q n σn

Cynthia Wu (Chicago Booth & NBER) 12 / 63 Shadow rate SRNKM Microfoundations Economic implications Property of g(.)

5

4

3 y

2

1 y = g(z) y = z 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 z

( ≈ r, at the ZLB f SR nt 0 G ≈ an + bnXt = fnt , 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

 0  o Q an + bnXt − r fnt = r + σn g Q + ηnt , ηnt ∼ N(0, ω) σn

Cynthia Wu (Chicago Booth & NBER) 14 / 63 Shadow rate SRNKM Microfoundations Economic implications Model fit

Figure: Average forward curve in 2012 SRTSM GATSM 4 4 fitted fitted 3.5 observed 3.5 observed

3 3

2.5 2.5

2 2

1.5 1.5

1 1

0.5 0.5

0 0 1 2 5 7 10 1 2 5 7 10

Log likelihood values I SRTSM: 850; GATSM: 750 data 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 2.29 forward rate level 346 357 384 435 551 600 636 yield 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 model SR as summary of unconventional monetary policy Linear model

3. Microfoundations

4. Economic implications

Cynthia Wu (Chicago Booth & NBER) 17 / 63 Shadow rate SRNKM Microfoundations Economic implications Evidence 1: taper tantrum

Expected short rate Expected shadow rate 5 5 2 4 4

3 3

2 2

1 1 1 0 0

−1 −1

−2 −2 May 21, 2013 Apr 2013 Apr 2013 May 22, 2013 May 2013 May 2013 0 −3 −3 3M 1Y 3Y 5Y 7Y 10Y Apr 2013 Apr 2014 Apr 2015 Apr 2016 Apr 2017 Apr 2018 Apr 2019 Apr 2020 Apr 2021 Apr 2022 Apr 2023 Apr 2013 Apr 2014 Apr 2015 Apr 2016 Apr 2017 Apr 2018 Apr 2019 Apr 2020 Apr 2021 Apr 2022 Apr 2023 Maturity

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

1 Wu−Xia shadow rate −1 QE1 − Fed balance sheet 0.5 QE2 OT −1.5 0 QE3 −2 −0.5 −2.5 −1

−3 −1.5 Trillions of Dollars Percentage points

−3.5 −2

−2.5 −4

−3 −4.5 2009 2010 2011 2012 2013 2014

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)

m x xx m x xt = µ + ρ Xt−1 + ut xs xs xs + 1(tJune 2009)ρ3 St−1

I monthly VAR(13) m I xt : 3 underlying macro factors

Null hypothesis xs xs H0 : ρ1 = ρ3 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

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

20 10

15 5 10 0 5 -5 0 fed funds rate & shadow rate Taylor rule implied -5 -10 1955 1970 1985 2000 2015 1955 1970 1985 2000 2015

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

20 110 Wu-Xia shadow rate fed funds rate 15 high yield effective yield BBB effective yield AAA effective yield 105 Goldman Sachs Financial Conditions Index 10

5 Interest rates 100 Goldman Sachs FCI 0

-5 95 2009 2011 2013 2015

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

I correlations: 0.8 B I rt = 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 IS curve 1 y = − (s − π − s) + y , t σ t Et t+1 Et t+1 New Keynesian

n πt = βEt πt+1 + κ(yt − yt ),

and shadow rate Taylor rule

n st = φs st−1 + (1 − φs )[φy (yt − yt ) + 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. Microfoundations Microfoundations I: QE Microfoundations 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

I decrease interest rates through reducing risk premium

I 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

∞  1−σ 1+η  X C Lt E βt t − 0 1 − σ 1 + η t=0 budget constraint H B H Bt Rt−1Bt−1 Ct + = + Wt Lt + Tt Pt Pt Euler equation " −σ # −σ B Ct+1 Ct = βRt Et Πt+1 The linear Euler equation 1 y = − r B − π − r B  + y , t σ t Et t+1 Et t+1

Cynthia Wu (Chicago Booth & NBER) 27 / 63 Shadow rate SRNKM Microfoundations Economic implications Bond return and policy rate

Define

B rpt ≡ rt − 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

0 G G G G rpt (bt ) < 0 ⇒ rpt (bt ) = rp − ς(bt − b )

During normal times G G I bt = b G B I rpt (b ) = rp ⇒ rt = rt + rp At the ZLB G B I QE → bt ↑→ rpt ↓→ rt ↓

Cynthia Wu (Chicago Booth & NBER) 28 / 63 Shadow rate SRNKM Microfoundations Economic implications Shadow rate equivalence for QE

Euler equation 1   y = − r B − π − r B + y , t σ t Et t+1 Et t+1

G G I During normal times, bt = b , rt = st

B rt = rt + rp = st + rp

I At the ZLB, rt = 0

B G G rt = rpt = rp − ς(bt − b ) = st + rp

G G if st = −ς(bt − b )

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 1 y = − (s − π − s)+ y , t σ t Et t+1 Et t+1 New Keynesian Phillips Curve, and shadow rate Taylor rule

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

nests both conventional Taylor interest rate rule and QE operation that changes risk premium if ( G G rt = st , bt = b for st ≥ 0 G G st rt = 0, bt = b − ς for st < 0.

Shadow rate NK model

Cynthia Wu (Chicago Booth & NBER) 30 / 63 Shadow rate SRNKM Microfoundations Economic implications Quantifying assumption in proposition

G G st = −ς(bt − b )

1.5 Wu-Xia shadow rate -27 -log(QE holdings) 1

0.5 QE1 QE2 QE3 -27.5

0

-28 -0.5

-1 -28.5

Percentage points -1.5

-2 -29

-2.5

-3 -29.5 2009 2010 2011 2012 2013 2014

I linear assumption: correlation = 0.92 I 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 rate I 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. Microfoundations Microfoundations I: QE Microfoundations 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 -to-value ratio

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

Combine this with a 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 ZLB I lending facilities

I lend directly to entrepreneurs I 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 ∞ X t E max E0 γ log Ct t=0 production function

E α 1−α Yt = AKt−1(Lt ) capital accumulation

Kt = It + (1 − δ)Kt−1 budget constraint E RB B˜ Yt ˜ t−1 t−1 E + Bt = + Wt Lt + It + Ct Xt Tt−1Πt borrowing constraint ˜ B  Bt ≤ Mt Et Kt Πt+1/Rt

Cynthia Wu (Chicago Booth & NBER) 35 / 63 Shadow rate SRNKM Microfoundations Economic implications Entrepreneurs’ FOCs

Labor demand

α −α (1 − α)AKt−1Lt Wt = Xt Euler equation

  " E !# 1 Mt Et Πt+1 1 αYt+1 Mt E 1 − B = γEt E − + 1 − δ Ct Rt Ct+1 Xt+1Kt Tt

Cynthia Wu (Chicago Booth & NBER) 36 / 63 Shadow rate SRNKM Microfoundations Economic implications Households’ problem

Households’ utility

∞ 1+η ! X C 1−σ L βt t − t E0 1 − σ 1 + η t=0 budget constraint

RB B˜H ˜H t−1 t−1 Ct + Bt = + Wt Lt + Tt Tt−1Πt Euler equation −σ ! −σ B Ct+1 Ct = βEt Rt Πt+1Tt labor supply σ η Wt = Ct Lt

Cynthia Wu (Chicago Booth & NBER) 37 / 63 Shadow rate SRNKM Microfoundations Economic implications Sources of funding

Entrepreneurs’ borrowing constraint   ˜ Kt Πt+1 Bt ≤ Mt Et B Rt Households lend   ˜H Kt Πt+1 Bt ≤ MEt B Rt

˜ ˜H I During normal times Bt = Bt , and Mt = M I At the ZLB Mt > M Government lends the rest   ˜G Kt Πt+1 Bt =( Mt − M)Et B Rt

Cynthia Wu (Chicago Booth & NBER) 38 / 63 Shadow rate SRNKM Microfoundations Economic implications Conventional and unconventional policy

B Suppose Rt = Rt RP 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 1 c = − (s − π − s) + c t σ t Et t+1 Et t+1 the shadow rate Taylor rule, Phillips curve, ..., nests both conventional Taylor 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 implications Economic implication I: negative supply shock Economic implication II: 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

κ(1 + η) π ↑ = β π + κ(y − y) − a ↓ t Et t+1 t σ + η t

Cynthia Wu (Chicago Booth & NBER) 43 / 63 Shadow rate SRNKM Microfoundations Economic implications Economic implication I: negative supply shock

Standard model

Monetary policy

n rt = max{φs st−1 + (1 − φs )[φy (yt − yt ) + φππt + s] , 0}

Real interest rate rrt = rt − Et [πt+1] IS curve 1 y = − (rr − r) + y t σ t Et t+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

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

Real interest rate rrt = st − Et [πt+1] Shadow rate IS curve 1 y = − (s − π − r) + y t σ t Et t+1 Et t+1

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

I consume, work, hold house I borrow with collateral I Entrepreneurs

I consume, invest, hold houses I borrow with collateral I produce intermediate goods using housing, capital, and labor I Retailers

I monopolistically competitive I Calvo sticky I Government

I Taylor rule

Calibration

Cynthia Wu (Chicago Booth & NBER) 46 / 63 Shadow rate SRNKM Microfoundations Economic implications Difference from Iacoviello (2005)

I Unconventional policy

I QE: time-varying risk premium I lending facilities: time-varying loan-to-value ratio I 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

Preference shock r policy rate t

0.25 2.5

0.2 2

0.15 1.5

0.1 1 percentage % dev. from S.S.

0.05 0.5

0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40

more

Cynthia Wu (Chicago Booth & NBER) 49 / 63 Shadow rate SRNKM Microfoundations Economic implications Economic implication I: negative supply shock

A TFP Y output t t 0

−0.2 1.5

−0.4 1

−0.6 0.5

% dev. from S.S. −0.8 % dev. from S.S. 0 −1 10 20 30 40 10 20 30 40 C consumption I investment t t 2 3 1.5

1 2

0.5 1 0 % dev. from S.S. % dev. from S.S.

−0.5 0 10 20 30 40 10 20 30 40 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

π r policy rate t t 2 0.8 1.5 0.6

1 0.4 Level in pect. Level in pect. 0.5 0.2

0 0 10 20 30 40 10 20 30 40

s shadow rate rB−τ private rate rr real interest rate t t t t 0.8 0.8 0.5 0.6 0.6 0

0.4 0.4 −0.5 Level in pect. Level in pect. 0.2 Level in pect. 0.2 −1

0 −1.5 0 10 20 30 40 10 20 30 40 10 20 30 40 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

rp risk premium − QE t

0.8

0.6

0.4

Level in pect. 0.2

0 10 20 30 40

τ tax − Facilities M loan−to−value ratio − Facilities t t 0 0

−0.05 −0.05

−0.1 −0.1

−0.15 −0.15 Level in pect. % dev. from S.S. −0.2 −0.2 10 20 30 40 10 20 30 40 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 implications Economic implication I: negative supply shock Economic 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 + ρg gt−1 + eg,t ↑

Market-clearing condition

yt ↑ = cy ct + gy gt ↑

Phillips Curve κ π ↑ = β π + (σ(c − c) + η(y ↑ −y)) t Et t+1 δ + η t t

Cynthia Wu (Chicago Booth & NBER) 54 / 63 Shadow rate SRNKM Microfoundations Economic implications Economic implication II: government spending multiplier

Standard model

Monetary policy

n rt = max{φs st−1 + (1 − φs )[φy (yt − yt ) + φππt + s] , 0}

Real interest rate rrt = rt − Et [πt+1] IS curve 1 c = − (rr − r) + c t σ t Et t+1

normal times: π ↑ y ↑ → r ↑↑ → rr ↑ → c ↓ → ∆y < ∆g ZLB 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

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

Real interest rate rrt = st − Et [πt+1] Shadow rate IS curve 1 c = − (s − π − r) + c t σ t Et t+1 Et t+1

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

G government spending Government spending multiplier t

2.5 3

2 2.5

1.5 2 1 level 1.5 % dev. from S.S. 0.5 1 0 10 20 30 40 10 20 30 40 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

π inflation r policy rate t t

0.6 1 0.4

0.5 0.2 Level in pect. Level in pect.

0 0 10 20 30 40 10 20 30 40

s shadow rate rB−τ private rate rr real interest rate t t t t 0.4 0.6 0.5 0.3 0.4 0.2 0 0.2 Level in pect. Level in pect. 0.1 Level in pect. −0.5 0 0 10 20 30 40 10 20 30 40 10 20 30 40 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

C consumption I investment t t 2

1 1.5

1 0.5 0.5 % dev. from S.S. % dev. from S.S. 0 0 10 20 30 40 10 20 30 40

Y output Government spending multiplier t

1.5 3 2.5 1 2

level 1.5 0.5 % dev. from S.S. 1 0 10 20 30 40 10 20 30 40 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

rp risk premium − QE t 0.4

0.3

0.2

Level in pect. 0.1

0 10 20 30 40

τ tax − Facilities M loan−to−value ratio − Facilities t t 0 0

−0.02 −0.02

−0.04 −0.04

−0.06 −0.06 Level in pect.

−0.08 % dev. from S.S. −0.08

−0.1 −0.1 10 20 30 40 10 20 30 40 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 estimating these 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 ) d I r t = r t + spt d I Discrete policy lower bound r t I Non-constant spread spt

I Agents are forward looking I Derive bond prices I Asymmetry

I 10 bp drop in the ELB decreases the 10-year rate by 6.5-8.5 bp I 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

G¨urkaynak, 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

 Q  ρ1 0 0 Q  Q  ρ = 0 ρ2 1 .  Q 0 0 ρ2

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Cynthia Wu (Chicago Booth & NBER) 64 / 63 FAVAR

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

m m m m Yt = am + bx xt + bs st + ηt , ηt ∼ N(0, Ω)

m I Yt : 97 economic variables from 1960 to 2013 m I xt : 3 underlying macro factors

Factor dynamics:

 m  x   xx xs   m   m   m  xt µ ρ ρ Xt−1 m εt εt = s + sx ss + Σ MP , MP ∼ N(0, I ) st µ ρ ρ St−1 εt εt

I monthly VAR(13) m I Σ : Cholesky decomposition

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Cynthia Wu (Chicago Booth & NBER) 65 / 63 Rubustness

xs xs sx sx p-value for ρ1 = ρ3 p-value for ρ1 = ρ3 Baseline 0.29 1.00 A1 estimate r 0.18 1.00 A2 2-factor SRTSM 0.13 0.97 A3 Fama-Bliss 0.38 1.00 A4 5-factor FAVAR 0.70 1.00 A5 6-lag FAVAR 0.09 0.98 7-lag FAVAR 0.19 0.97 12-lag FAVAR 0.22 1.00

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Cynthia Wu (Chicago Booth & NBER) 66 / 63 Lending facilities

1 B ct = − (rt − τt − Et πt+1 − r − rp) + Et ct+1 σ 1 ⇒ ct = − (st − Et πt+1 − r) + Et ct+1 σ

E E Y B B C c = α (yt − xt ) + Bbt − R B(r + bt−1 − τt−1 − πt−1) − Iit + Λ1 t X t−1 E E Y B ⇒ C c = α (yt − xt ) + Bbt − R B(st−1 + rp + bt−1 − πt−1) − Iit + Λ1 t X

B bt = Et (kt + πt+1 + mt − rt ) ⇒ bt = Et (kt + πt+1 + m − st − rp)

Cynthia Wu (Chicago Booth & NBER) 67 / 63 Lending facilities

  M E E γαY 0 = 1 − (ct − Et ct+1) + Et (yt+1 − xt+1 − kt ) RB XK

M B + Et (πt+1 − rt + mt ) + γM(τt − mt ) + Λ2 RB   M E E γαY ⇒ 0 = 1 − (ct − Et ct+1) + Et (yt+1 − xt+1 − kt ) RB XK M + Et (πt+1 − st − rp + m) − γMm + Λ2 RB

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Cynthia Wu (Chicago Booth & NBER) 68 / 63 Calibration

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

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Cynthia Wu (Chicago Booth & NBER) 69 / 63 Preference shock and the ZLB

1. s shadow rate 2. r policy rate 3. rB-τ private rate 4. π inflation t t t t t 2.5 2 6 2 2 1 5 1 1.5 0 1 4 0 -1 percentage percentage percentage percentage 0.5 3 -2 -1 0 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

5. rr real interest rate 6. rp risk premium - QE 7. τ tax - Facilities 8. M loan-to-value ratio - Facilities t t t t 1 0.3 0.3 0.5 3.5 0 0.2 0.2 -0.5 3 -1 0.1 0.1 percentage percentage percentage % dev. from S.S. -1.5 2.5 0 0 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

9. Y output 10. C consumption 11. I investment t t t 12. Preference shock 0 0 0.25 ZLB period -1 without UMP -1 -1 0.2 with UMP -2 -2 0.15 -2 -3 0.1 -3 % dev. from S.S. -3 % dev. from S.S. % dev. from S.S. % dev. from S.S. 0.05 -4 -4 0 10 20 30 40 10 20 30 40 10 20 30 40 10 20 30 40

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Cynthia Wu (Chicago Booth & NBER) 70 / 63