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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 Feds 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 = 0 + 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 = + Xt + t+1, t+1 N(0, I ).

Pricing kernel

mt+1 = rt +1

2tt +

tt+1

where t = 0 + 1Xt

Pricing equation I

Pnt = Et [exp(mt+1)Pn1,t+1]

Pricing equation II

Pnt = EQt [exp(rt)Pn1,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 = Q + QXt +

Qt+1,

Qt+1

Q N(0, I ).

where Q = 0, and Q = 1

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 + Qn g

(an + b

nXt rQn

)where g(z) = z(z) + (z).

an = 0 + 1

n1j=0

(Q)jQ 1

21

n1j=0

(Q)j

n1j=0

(Q)j 1

bn = 1

(Q)n

(Qn )2 VQt (st+n) =

n1j=0

1(Q)j(Q)j1

Cynthia Wu (Chicago Booth & NBER) 11 / 63

Shadow rate SRNKM Microfoundations Economic implications

Comparison to GATSM

SRTSM

st = 0 + 1Xt

rt = max(st , r)

Forward rate

fnt = r + Qn g

(an + b

nXt rQn

)

GATSM

rt = 0 + 1Xt

Forward rate

fnt = an + bnXt

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 + bnXt = 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 = + Xt + t+1, t+1 N(0, I )

Observation equation

f ont = r + Qn g

(an + b

nXt rQn

)+ nt , nt N(0, )

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 2013

Apr 2014

Apr 2015

Apr 2016

Apr 2017

Apr 2018

Apr 2019

Apr 2020

Apr 2021

Apr 2022

Apr 2023

Apr 2013May 2013

3

2

1

0

1

2

3

4

5Expected shadow rate

Apr 2013

Apr 2014

Apr 2015

Apr 2016

Apr 2017

Apr 2018

Apr 2019

Apr 2020

Apr 2021

Apr 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 Feds balance sheet

2009 2010 2011 2012 2013 20143

2.5

2

1.5

1

0.5

0

0.5

1

Per

cent

age

poin

ts

QE1

QE2

OT

QE3