Shadow rate Empirical evidence New Keynesian model International evidence Shadow Interest Rate 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 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 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 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 (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 + Σ"t+1;"t+1 ∼ N(0; I ): Risk-neutral Q dynamics: Q Q Q Q Q Xt+1 = µ + ρ Xt + Σ"t+1;"t+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(:) 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 (Notre Dame & NBER) 10 / 34 Shadow rate Empirical evidence New Keynesian model International evidence State space form State equation Xt+1 = µ + ρXt + Σ"t+1;"t+1 ∼ N(0; I ) Observation equation 0 o Q an + bnXt − r fnt = r + σn g Q + ηnt ; ηnt ∼ N(0;!) 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 + "t 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 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 5 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.