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Model Shadow rate Macroeconomic Implications Conclusion
Measuring the Macroeconomic Impact of MonetaryPolicy at the Zero Lower Bound
Jing Cynthia WuChicago Booth and NBER
Fan Dora XiaMerrill Lynch
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 1 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Key question
What is the macroeconomic impact of monetary policy at the ZLB?
Conventional approach before ZLB
I VAR with the fed funds rate
But since December 2008, the fed funds rate has been near zero
1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014−2
0
2
4
6
8
10
shadow rateeffective federal funds rater
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 2 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Challenges of zero lower bound
Challenges
I Conventional monetary policy doesn’t work. Fed has implementedunconventional policy tools
I large-scale asset purchasesI forward guidance
I What framework to study unconventional monetary policy?
I Gaussian ATSM allows negative interest rates
Shadow rate term structure model: Black (1995)
I Non-negative short rate: rt = max(r , st)
I Analytical solution does not exist in general
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 3 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Contributions
This paper
I an analytical approximation for SRTSM
I shadow rate has similar dynamic correlations with macro variables asthe fed funds rate did previously
I our shadow rate updated monthly by Atlanta Fedwww.frbatlanta.org/cqer/researchcq/shadow_rate.cfm
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 4 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Contributions
This paper
I an analytical approximation for SRTSM
I shadow rate has similar dynamic correlations with macro variables asthe fed funds rate did previously
I our shadow rate updated monthly by Atlanta Fedwww.frbatlanta.org/cqer/researchcq/shadow_rate.cfm
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 4 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Contributions
This paper
I an analytical approximation for SRTSM
I shadow rate has similar dynamic correlations with macro variables asthe fed funds rate did previously
I our shadow rate updated monthly by Atlanta Fedwww.frbatlanta.org/cqer/researchcq/shadow_rate.cfm
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 4 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Outline
1 Model
2 Shadow rate
3 Macroeconomic Implications
4 Conclusion
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 5 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Bond pricing
Risk-neutral factor dynamics:
Xt+1 = µQ + ρQXt + ΣεQt+1, εQt+1Q∼ N(0, I ).
Pricing kernel
Pricing equation
Pnt = E
Qt [exp(−rt − rt+1 − ...− rt+n−1)]
Yield
ynt = −1
nlog(Pn
t )
Forward rate
fn,n+1,t = (n + 1)yn+1,t − nynt
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 6 / 39
Model Shadow rate Macroeconomic Implications Conclusion
SRTSM and GATSM
SRTSM
rt = max(r , st)
st = δ0 + δ′1Xt
Forward rate
f SRTSMn,n+1,t = r + σQn g
(an + b′nXt − r
σQn
)where g(z) = zΦ(z) + φ(z)
an, bn
GATSM
rt = δ0 + δ′1Xt
Forward rate
f GATSMn,n+1,t = an + b′nXt .
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 7 / 39
Model Shadow rate Macroeconomic Implications Conclusion
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 SRTSMn,n+1,t
≈ r , at the ZLB
≈ an + b′nXt = f GATSMn,n+1,t , when interest rates are high
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 8 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Model fit
GSW Data: monthly 1990-2013; maturities: 3m, 6m, 1y, 2y, 5y, 7y, 10yEstimation: Kalman filters details
Log likelihood values specification
I SRTSM: 850; GATSM: 750
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
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 9 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Approximation error
Average absolute approximation error between 1990M1 and 2013M1
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
ZLB
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 10 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Shadow rate
1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 2014−2
0
2
4
6
8
10
shadow rateeffective federal funds rater
Summary for unconventional monetary policy?
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 11 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Yield curve on May 21, 2013
3M 1Y 3Y 5Y 7Y 10Y0
1
2
Maturity
May 21, 2013
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 12 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Hint of tapering (yield)
3M 1Y 3Y 5Y 7Y 10Y0
1
2
Maturity
May 21, 2013May 22, 2013
May 22: Bernanke tells Congress Fed may decrease the size of monthly large-scale assetpurchases
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 13 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Hint of tapering (forward rate)
−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
May 22: Bernanke tells Congress Fed may decrease the size of monthly large-scale assetpurchases
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 14 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Shift in shadow rate might summarize the effect
−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
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 15 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Monetary policy
1960 1970 1980 1990 2000 2010
0
5
10
15
20
Wu−Xia policy rate: sto
effective fed funds rate
sot =
effective federal funds rate before 2009
shadow rate since 2009
Can we use shadow rate as similar summary of Fed actions as fed fundsrate provided historically?
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 16 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Factor augmented vector autoregression
Replace the fed funds rate with sot in Bernanke, Boivin, and Eliasz (2005)
Ymt = am + bxx
mt + bss
ot + ηmt , ηmt ∼ N(0,Ω)
I Ymt : 97 economic variables from 1960 to 2013
I xmt : 3 underlying macro factors
Factor dynamics:[xmtsot
]=
[µx
µs
]+
[ρxx ρxs
ρsx ρss
] [Xmt−1
Sot−1
]+ Σm
[εmtεMPt
],
[εmtεMPt
]∼ N(0, I )
I monthly VAR(13)
I Σm: Cholesky decomposition
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 17 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Measures of monetary policy
Can we use shadow rate as similar summary of Fed actions as fed fundsrate provided historically?
Hypothesis I
H0 : ρxs(t < Great Recession) = ρxs(t > Great Recession)
I p = 0.29 for sot
I p = 0.0007 for EFFR
Hypothesis II
H0 : ρsx(t < Great Recession) = ρsx(t > Great Recession)
I p = 1 for sot
I p = 1 for EFFR
Implication: researchers can use shadow rate to update earlier studiesthat had been based on the historical fed funds rate. Robustness
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 18 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Measures of monetary policy
Can we use shadow rate as similar summary of Fed actions as fed fundsrate provided historically?
Hypothesis I
H0 : ρxs(t < Great Recession) = ρxs(t > Great Recession)
I p = 0.29 for sot
I p = 0.0007 for EFFR
Hypothesis II
H0 : ρsx(t < Great Recession) = ρsx(t > Great Recession)
I p = 1 for sot
I p = 1 for EFFR
Implication: researchers can use shadow rate to update earlier studiesthat had been based on the historical fed funds rate. Robustness
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 18 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Measures of monetary policy
Can we use shadow rate as similar summary of Fed actions as fed fundsrate provided historically?
Hypothesis I
H0 : ρxs(t < Great Recession) = ρxs(t > Great Recession)
I p = 0.29 for sot
I p = 0.0007 for EFFR
Hypothesis II
H0 : ρsx(t < Great Recession) = ρsx(t > Great Recession)
I p = 1 for sot
I p = 1 for EFFR
Implication: researchers can use shadow rate to update earlier studiesthat had been based on the historical fed funds rate. Robustness
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 18 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Measures of monetary policy
Can we use shadow rate as similar summary of Fed actions as fed fundsrate provided historically?
Hypothesis I
H0 : ρxs(t < Great Recession) = ρxs(t > Great Recession)
I p = 0.29 for sot
I p = 0.0007 for EFFR
Hypothesis II
H0 : ρsx(t < Great Recession) = ρsx(t > Great Recession)
I p = 1 for sot
I p = 1 for EFFR
Implication: researchers can use shadow rate to update earlier studiesthat had been based on the historical fed funds rate. Robustness
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 18 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Measures of monetary policy
Can we use shadow rate as similar summary of Fed actions as fed fundsrate provided historically?
Hypothesis I
H0 : ρxs(t < Great Recession) = ρxs(t > Great Recession)
I p = 0.29 for sot
I p = 0.0007 for EFFR
Hypothesis II
H0 : ρsx(t < Great Recession) = ρsx(t > Great Recession)
I p = 1 for sot
I p = 1 for EFFR
Implication: researchers can use shadow rate to update earlier studiesthat had been based on the historical fed funds rate. Robustness
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 18 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Historical decomposition
What if there had been no monetary policy shocks?
I realized: εMPt = εMP
t
I counterfactual: εMPt = 0 for ZLB
Unconventional monetary policy
I reduced the shadow rate by 0.4% between 2011 and 2013.
2010 2011 2012 2013−2.5
−2
−1.5
−1
−0.5
0Policy rate
realizedcounterfactural I
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 19 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Historical decomposition
What if there had been no monetary policy shocks?
I realized: εMPt = εMP
t
I counterfactual: εMPt = 0 for ZLB
Unconventional monetary policy
I reduced unemployment by 0.13% in Dec 2013. More
2010 2011 2012 20136.5
7
7.5
8
8.5
9
9.5
10Unemployment
realizedcounterfactural I
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 20 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Counterfactual II
What if the shadow rate had been kept at r?
I counterfactual: εMPt is such that sot = r at ZLB
Unconventional monetary policy
I reduced unemployment by 1% in December 2013 More
2010 2011 2012 2013−2.5
−2
−1.5
−1
−0.5
0
0.5Policy rate
realizedcounterfactural II
2010 2011 2012 20136.5
7
7.5
8
8.5
9
9.5
10
10.5Unemployment
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 21 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Impulse resposne: full sample
A -25bps monetary policy shock
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 22 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Full sample FAVAR(13) vs. ZLB FAVAR(1)
ZLB with effective federal funds rate
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 23 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Full sample FAVAR(13) vs. ZLB FAVAR(1)
ZLB with shadow rate
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 24 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Forward guidance
ZLB durationτt = infτt ≥ 0|st+τ ≥ r.
2009 2010 2011 2012 2013 2014
2010
2011
2012
2013
2014
2015
2016
market anticipation
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 25 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Forward guidance
ZLB durationτt = infτt ≥ 0|st+τ ≥ r.
2009 2010 2011 2012 2013 2014
2010
2011
2012
2013
2014
2015
2016
market anticipationFed announcement
at least throughmid−2013
at least throughlate 2014
at least throughmid−2015
2015
Impulse responses
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 26 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Conclusion
Method
I Develop an approximation for bond prices in the SRTSM
Economics
I The shadow rate exhibits similar dynamic correlations with economicvariables after the Great Recession as the fed funds rate did earlier in data.
I Unconventional monetary policy lowered the unemployment rate by 0.13% inDecember 2013.
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 27 / 39
Model Shadow rate Macroeconomic Implications Conclusion
Sources:BoardofGovernorsoftheFederalReserveSystemandWuandXia(2014)
Wu-XiaShadowFederalFundsRatethroughFebruary2015
Effectivefederalfundsrate,end-of-monthWu-Xiashadowrate
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015-4%
-2%
0%
2%
4%
6%
Source: www.frbatlanta.org/cqer/researchcq/shadow_rate.cfm
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 28 / 39
Model Shadow rate Macroeconomic Implications Conclusion
ECB shadow rate
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 29 / 39
Pricing kernel
Factor dynamics:
Xt+1 = µ + ρXt + Σεt+1, εt+1 ∼ N(0, I ).
Pricing kernel
mt+1 = rt +1
2λ′tλt + λ′tεt+1
λt = λ0 + λ1Xt
where µQ = µ− Σλ0, and ρQ = ρ− Σλ1
Pricing equation
Pnt = Et [exp(−mt+1)Pn−1
t+1 ]
Back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 30 / 39
Bond recursions
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.
Back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 31 / 39
Model specification
r = 0.25, interest rate on reserves
three factors
Normalization: restrict Q parameters
Repeated eigenvalues
ρQ =
ρQ1 0 0
0 ρQ2 1
0 0 ρQ2
.Back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 32 / 39
Kalman filters
State equation
Xt+1 = µ+ ρXt + Σεt+1, εt+1 ∼ N(0, I )
observation equation for SRTSM ⇒ extended Kalman filter
f on,n+1,t = r + σQn g
(an + b′nXt − r
σQn
)︸ ︷︷ ︸
f SRTSMn,n+1,t
+ηnt , ηnt ∼ N(0, ω)
observation equation for GATSM ⇒ Kalman filter
f on,n+1,t = an + b′nXt︸ ︷︷ ︸f GATSMn,n+1,t
+ηnt , ηnt ∼ N(0, ω)
Back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 33 / 39
Approximation error for ZLB
Average absolute approximation error between 2009M1 and 2013M1
3M 6M 1Y 2Y 5Y 7Y 10Y
forward rate error 0.00 0.01 0.06 0.43 2.50 3.51 5.41forward rate level 23 26 46 111 326 418 481yield error 0.00 0.00 0.01 0.10 0.91 1.50 2.37
back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 34 / 39
Robustness
p-value for ρxs1 = ρxs3 p-value for ρsx1 = ρsx3
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 0.987-lag FAVAR 0.19 0.9712-lag FAVAR 0.22 1.00
Back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 35 / 39
Historical decomposition
2010 2011 2012 2013−2.5
−2
−1.5
−1
−0.5
0Policy rate
2010 2011 2012 201380
85
90
95
100
105Industrial production index
2010 2011 2012 2013210
220
230
240Consumer price index
2010 2011 2012 201360
65
70
75
80Capacity utilization
2010 2011 2012 20136
7
8
9
10Unemployment
2010 2011 2012 2013400
600
800
1000
1200Housing starts
realizedcounterfactural I
Back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 36 / 39
Counterfactual II
2010 2011 2012 2013−3
−2
−1
0
1Policy rate
2010 2011 2012 201380
85
90
95
100
105Industrial production index
2010 2011 2012 2013210
220
230
240
250Consumer price index
2010 2011 2012 201360
65
70
75
80Capacity utilization
2010 2011 2012 20136
7
8
9
10
11Unemployment
realizedcounterfactual II
2010 2011 2012 2013400
600
800
1000
1200Housing starts
Back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 37 / 39
Impulse responses: forward guidance
A monetary policy shock to increase the ZLB by 1 year
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 38 / 39
Forward guidance vs. shadow rate
Unemployment rate decreases by 0.25% with
I a one year increase in the expected ZLB duration
I 35 basis-point decrease in the policy rate Back
Cynthia Wu (Chicago Booth & NBER) and Dora Xia (Merrill Lynch) 39 / 39