Shadow Interest Rate

Shadow rate Empirical evidence New Keynesian model International evidence

Jing Cynthia Wu Notre Dame and NBER

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

Cynthia Wu (Notre Dame & NBER) 1 / 34 Shadow rate Empirical evidence New Keynesian model International evidence ZLB: monetary policy

Before ZLB, policy rates are the tool for monetary policy and its research 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 facilities I forward guidance I negative interest rate policy

Cynthia Wu (Notre Dame & NBER) 2 / 34 New Keynesian models I Benchmark models: no unconventional monetary policy I My papers: Wu and Zhang (JEDC 2019), Wu and Zhang (JIE 2019) I incorporate unconventional monetary policy I Key feature: tractable

Shadow rate Empirical evidence New Keynesian model International evidence ZLB: economic models

Term structure models I Benchmark Gaussian ATSM I ZLB: Yields are unconstrained in the model, but constrained in the data I My papers: Wu and Xia (JMCB 2016), Wu and Xia (JAE forthcoming) I respect the ZLB

Cynthia Wu (Notre Dame & NBER) 3 / 34 Shadow rate Empirical evidence New Keynesian model International evidence ZLB: economic models

New Keynesian models I Benchmark models: no unconventional monetary policy I My papers: Wu and Zhang (JEDC 2019), Wu and Zhang (JIE 2019) I incorporate unconventional monetary policy I Key feature: tractable

Cynthia Wu (Notre Dame & NBER) 3 / 34 Shadow rate Empirical evidence New Keynesian model International evidence 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

-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 (Notre Dame & NBER) 4 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Wu and Xia (JMCB 2016) shadow rate

I Wu-Xia 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: then Governor Powell (2013), Altig (2014) of the Atlanta Fed, Hakkio and Kahn (2014) of the Kansas City Fed I Media:Wall Street Journal, Financial Times, The New York Times, Bloomberg news, Bloomberg Businessweek, Forbes, Business Insider, VOX

Cynthia Wu (Notre Dame & NBER) 5 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Outline

1. Wu-Xia shadow rate: Wu and Xia (JMCB 2016)

2. Empirical evidence: Wu and Xia (JMCB 2016), Wu and Zhang (JEDC 2019)

3. New Keynesian model: Wu and Zhang (JEDC 2019), Wu and Zhang (JIE 2019)

4. International evidence: Wu and Zhang (JIE 2019)

Cynthia Wu (Notre Dame & NBER) 6 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Shadow rate

Black (1995):

rt = max(st , r)

The shadow rate is aﬃne

0 st = δ0 + δ1Xt

I Xt : 3 factors

Cynthia Wu (Notre Dame & NBER) 7 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Bond pricing

Physical dynamics:

Xt+1 = µ + ρXt + Σεt+1, εt+1 ∼ N(0, I ). Risk-neutral Q dynamics:

Q Q Q Q Q Xt+1 = µ + ρ Xt + Σεt+1, εt+1 ∼ N(0, I ). Pricing equation Q Pnt = Et [exp(−rt )Pn−1,t+1] 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 (Notre Dame & NBER) 8 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Forward rates

Our approximation

0 Q an + bnXt − r fnt ≈ r +σ ˜n g Q σ˜n where g(z) = zΦ(z) + φ(z). Details

Forward rate in GATSM

0 fnt = an + bnXt .

Cynthia Wu (Notre Dame & NBER) 9 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Property of g(.)

3 y

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 (Notre Dame & NBER) 10 / 34 Shadow rate Empirical evidence New Keynesian model International evidence State space form

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

We apply extended Kalman ﬁlter for estimation

Cynthia Wu (Notre Dame & NBER) 11 / 34 Shadow rate Empirical evidence New Keynesian model International evidence 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

Cynthia Wu (Notre Dame & NBER) 12 / 34 Shadow rate Empirical evidence New Keynesian model International evidence 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 (Notre Dame & NBER) 13 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Outline

1. Wu-Xia shadow rate: Wu and Xia (JMCB 2016)

2. Empirical evidence: Wu and Xia (JMCB 2016), Wu and Zhang (JEDC 2019)

3. New Keynesian model: Wu and Zhang (JEDC 2019), Wu and Zhang (JIE 2019)

4. International evidence: Wu and Zhang (JIE 2019)

Cynthia Wu (Notre Dame & NBER) 14 / 34 Shadow rate Empirical evidence New Keynesian model International evidence 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 Shift in shadow rate summarizes this eﬀect

Cynthia Wu (Notre Dame & NBER) 15 / 34 Shadow rate Empirical evidence New Keynesian model International evidence 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 (Notre Dame & NBER) 16 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Evidence 3: structural break test in VAR

Wu and Xia (JMCB 2016): structural break test

I p = 0.29 for st I p = 0.0007 for EFFR model details robustness

Cynthia Wu (Notre Dame & NBER) 17 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Evidence 4: shadow rate Taylor rule n st = β0 + β1st−1 + β2(yt − yt ) + β3πt + εt Full sample 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 Post-85 sample

15 5 fed funds rate & shadow rate Taylor rule implied 10

5 0

−5 −5 1985 1991 1997 2003 2009 2015 1985 1991 1997 2003 2009 2015 No structural break I F statistics: 0.48 & 1.42 I Critical values: 2.64 & 2.68 Cynthia Wu (Notre Dame & NBER) 18 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Evidence 5: shadow rate and private rates

20 110 Wu-Xia shadow rate GSFCI 108 high yield effective yield 15 BBB effective yield GZ credit spread 106 fed funds rate 104 10

102

Interest rates 100 Goldman Sachs FCI 98 0 96

-5 94 2009 2011 2013 2015

I private rates are the relevant rates for agents and the economy I correlation with SR: 0.8 I private rate = st + rp

Cynthia Wu (Notre Dame & NBER) 19 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Summary

Shadow rate summarizes unconventional monetary policy I Taper tantrum I Fed’s balance sheet

There is no structural break in I VAR I shadow rate Taylor rule

Private rates I are the relevant interest rates for economic agents I respond to unconventional monetary policy I the shadow rate is a sensible summary

Cynthia Wu (Notre Dame & NBER) 20 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Outline

1. Wu-Xia shadow rate: Wu and Xia (JMCB 2016)

2. Empirical evidence: Wu and Xia (JMCB 2016), Wu and Zhang (JEDC 2019)

3. New Keynesian model: Wu and Zhang (JEDC 2019), Wu and Zhang (JIE 2019)

4. International evidence: Wu and Zhang (JIE 2019)

Cynthia Wu (Notre Dame & NBER) 21 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Shadow rate New Keynesian model

Deﬁnition 1 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 Phillips curve

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] .

Cynthia Wu (Notre Dame & NBER) 22 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Outline

1. Wu-Xia shadow rate: Wu and Xia (JMCB 2016)

2. Empirical evidence: Wu and Xia (JMCB 2016), Wu and Zhang (JEDC 2019)

3. New Keynesian model: Wu and Zhang (JEDC 2019), Wu and Zhang (JIE 2019)

4. International evidence: Wu and Zhang (JIE 2019)

Cynthia Wu (Notre Dame & NBER) 23 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Taylor rule

Deﬁnition

I rt : observed policy rate I st : Taylor rule implied

Cases

I Normal times: rt = st I ELB: rt = λst

I λ = 0: Standard model I λ = 1: UMP behaves the same as normal times; Wu and Zhang (2017) I 0 < λ < 1: partially active UMP I λ > 1: hyper active UMP

Cynthia Wu (Notre Dame & NBER) 24 / 34 Shadow rate Empirical evidence New Keynesian model International evidence VAR: how large is λ?

Model implication for a negative TFP shock y

ELB + UMP 0.4 ELB 0.3 normal times 0.2 0.1 % 0 −0.1 −0.2 −0.3

0 20 40

Cynthia Wu (Notre Dame & NBER) 25 / 34 Shadow rate Empirical evidence New Keynesian model International evidence VAR: how large is λ?

I 2 variables: growth rate of labor productivity and per-capita hours similar to Debortoli, Gal´ı,and Gambetti (2016) I Identiﬁcation: Cholesky decomposition I quarterly VAR(1)

Beneﬁts of the VAR I Simple and robust I Does not depend on any one shadow rate Samples

Cynthia Wu (Notre Dame & NBER) 26 / 34 Shadow rate Empirical evidence New Keynesian model International evidence IRF of output to a productivity shock: US

US 0 normal times ELB −0.2

−0.4

−0.6

−0.8 1 5 10 15

Conclusion: λ ≈ 1 Cynthia Wu (Notre Dame & NBER) 27 / 34 Shadow rate Empirical evidence New Keynesian model International evidence IRF of output to a productivity shock: Euro area

Euro area 0.4

0.2

−0.2

−0.4

−0.6 1 5 10 15

Conclusion: 0 < λ < 1 Cynthia Wu (Notre Dame & NBER) 28 / 34 Shadow rate Empirical evidence New Keynesian model International evidence IRF of output to a productivity shock: UK

UK 0

−0.2

−0.4

−0.6

−0.8

−1 1 5 10 15

Conclusion: λUS ≈ 1 > λEuro > λUK > 0 Cynthia Wu (Notre Dame & NBER) 29 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Taylor rule: quantify λ

Why Taylor rule? It gives us a quantitative value of λ

Basic idea: compare what has been done at the ELB with the interest rate implied by the historical Taylor rule

Cynthia Wu (Notre Dame & NBER) 30 / 34 Shadow rate Empirical evidence New Keynesian model International evidence United States

I pre-ELB: 1985Q2 - 2007Q4, ELB: 2009Q1 - 2015Q4 I Simple method: 1.02 I Iterative method: 1.12

Vary pre-ELB sample: t0 ∈ {1982Q1 : 1990Q1}, t1 ∈ {2003Q1 : 2008Q4}

simple iterative 400 300

350 250 300 200 250

200 150

150 100 100 50 50

0 0 0.9 1 1.1 1.2 1.3 1.4 0 0.5 1 1.5 2 2.5 3

Median (std): simple 1.03 (0.065), iterative 1.19 (0.45) Cynthia Wu (Notre Dame & NBER) 31 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Euro area and UK

I Euro area: simple 0.998(0.031), iterative 0.63(1.07) I UK: simple 0.98(0.10), iterative 0.39(4.10)

Again, we conclude λUS ≈ 1 > λEuro > λUK > 0

Cynthia Wu (Notre Dame & NBER) 32 / 34 Shadow rate Empirical evidence New Keynesian model International evidence Conclusion

Empirically, we ﬁnd

I λUS ≈ 1 > λEuro > λUK > 0

Cynthia Wu (Notre Dame & NBER) 33 / 34 Shadow rate Empirical evidence New Keynesian model International evidence

Cynthia Wu (Notre Dame & NBER) 34 / 34 Shadow rate 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 m x xt = µ + ρ Xt−1 + ut xs xs xs + 1(tJune 2009)ρ3 St−1

I monthly VAR(13)

Null hypothesis xs xs H0 : ρ1 = ρ3 back

Cynthia Wu (Notre Dame & NBER) 35 / 34 Shadow rate Robustness

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

back

Cynthia Wu (Notre Dame & NBER) 36 / 34 Shadow rate Samples

pre-ELB and ELB samples I US: 1985Q2 - 2007Q4 and 2009Q1 - 2015Q4 I Euro area: 1999Q1 - 2009Q1 and 2009Q2 - 2017Q4 I UK: 1993Q1 - 2009Q1 and 2009Q2 - 2017Q4 Back

Cynthia Wu (Notre Dame & NBER) 37 / 34