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Shadow Interest Rates, Macroeconomic Trends, and Time

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Shadow Interest Rates, Macroeconomic Trends, and Time SHADOW INTEREST RATES, MACROECONOMIC TRENDS, & TIME-VARYING UNCERTAINTY Benjamin K. Johannsen Elmar Mertens Federal Reserve Board The results presented here do not necessarily represent the views of the Federal Reserve System or the Federal Open Market Committee June 2015 NOMINAL INTEREST RATE U.S. DATA Three-month T-Bill (APR) 15 10 5 0 −5 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 RESEARCH AGENDA Nominal short rate near zero since late 2008 • Typical time-series tools unusable (VARs, unobserved component models, time-varying parameter macro) • How to measure variations in the stance of policy? Shadow rate approach • Hypothetical nominal rate, unconstrained by lower bound • Our focus: Pure time series approach Trend real rate depends only weakly on growth, stands currently at just about one percent \Not much" as decline in trends started earlier Expected future short rates get less sensitive to news SOME QUESTIONS 1 What is the long-run level of the real rate, how does it relate to growth? 2 How did the recent recession affect trends in real and nominal variables? 3 How does the term structure change with short rates near the lower bound? RELATED LITERATURE Macro-Time Series at the ZLB Iwata & Wu (2006), Nakajima (2011), Chan & Strachan (2014) Dynamic Term-Structure Models Kim & Wright (2005), Wright (2011), Kim & Singleton (2011), Krippner (2013), Wu & Xia (2014), Bauer & Rudebusch (2014) Unobserved Component Models of the Macroeconomy Gordon (1997), Gerlach & Smets (1997), Staiger, Stock & Watson (1997), Laubach & Williams (2003), Clark & Kozicki (2005), Stock & Watson (2007), Stella & Stock (2013), Watson (2014), Mertens (2014) AGENDA 1 Shadow-Rate Concept 2 Shadow-Rate Sampling 3 Estimates from an Empirical Macro Model 4 Term-Premium Estimates Observed Rate ∗ it = max (0; it) Key idea Model it with conventional tools and handle max operator THE SHADOW RATE OF INTEREST FISCHER BLACK (1995) Shadow Rate it Nominal interest rate that would prevail in the absence of lower bound constraint Key idea Model it with conventional tools and handle max operator THE SHADOW RATE OF INTEREST FISCHER BLACK (1995) Shadow Rate it Nominal interest rate that would prevail in the absence of lower bound constraint Observed Rate ∗ it = max (0; it) THE SHADOW RATE OF INTEREST FISCHER BLACK (1995) Shadow Rate it Nominal interest rate that would prevail in the absence of lower bound constraint Observed Rate ∗ it = max (0; it) Key idea Model it with conventional tools and handle max operator AGENDA 1 Shadow-Rate Concept 2 Shadow-Rate Sampling 3 Estimates from an Empirical Macro Model 4 Term-Premium Estimates ∗ ∗ ¯∗ t ρ (it−1 − i ) t AR(1) EXAMPLE Observer ∗ it = max (0; it) Shadow Rate as Latent State ¯ ¯ (it − i) = ρ (it−1 − i) + σ et t ¯ ρ (it−1 − i) t AR(1) EXAMPLE Observer ∗ it = max (0; it) Shadow Rate as Latent State ¯ ∗ ∗ ¯∗ (it − i) = ρ (it−1 − i ) + σ et t t AR(1) EXAMPLE Observer ∗ it = max (0; it) Shadow Rate as Latent State ¯ ¯ ∗ ∗ ¯∗ (it − i) = ρ (it−1 − i) + ρ (it−1 − i ) + σ et ∗ ∗ ¯∗ t ρ (it−1 − i ) t AR(1) EXAMPLE Observer ∗ it = max (0; it) Shadow Rate as Latent State ¯ ¯ (it − i) = ρ (it−1 − i) + σ et ∗ ∗ ¯∗ ρ (it−1 − i ) AR(1) EXAMPLE Observer ∗ it = max (0; it) Shadow Rate as Latent State ¯ ¯ (it − it) = ρt(it−1 − it−1) + σtet SHADOW RATE SAMPLING 2 ∗ 3 i1 6i∗ 7 ∗ 6 2 7 • Let Y = 6 . 7 = max (0;Y ) 4 . 5 ∗ iT • Denote non-zero values of Y ∗ by X • Kalman smoother implies Y jX ∼ N(µ, V ) • Thus Y ∗jX ∼ truncN(µ, V ) • Model only needs to be conditionally linear NOMINAL INTEREST RATE U.S. DATA Three-month T-Bill (APR), 90 percent quantile range 15 10 5 0 −5 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 AGENDA 1 Shadow-Rate Concept 2 Shadow-Rate Sampling 3 Estimates from an Empirical Macro Model 4 Term-Premium Estimates Taylor-type reaction function for the gap ~ ~ i it = ρ it−1 + dπ(πt − π¯t) + dy(yt − y¯t) + "t ¯ Real shadow-rate trend: r¯t = it − π¯t r¯t = r¯t−1 + et ? = β · µt + rt TREND AND CYCLE IN NOMINAL INTEREST RATE Trend-cycle decomposition for the shadow rate ¯ ~ ¯ ~ it = it + it it = Etit+1 it ∼ I(0) ¯ Real shadow-rate trend: r¯t = it − π¯t r¯t = r¯t−1 + et ? = β · µt + rt TREND AND CYCLE IN NOMINAL INTEREST RATE Trend-cycle decomposition for the shadow rate ¯ ~ ¯ ~ it = it + it it = Etit+1 it ∼ I(0) Taylor-type reaction function for the gap ~ ~ i it = ρ it−1 + dπ(πt − π¯t) + dy(yt − y¯t) + "t Trends y µ y¯t = µt−1 +y ¯t−1 +" ¯t µt = µt−1 + ηt u u¯t = u¯t−1 +" ¯t π π¯t = π¯t−1 +" ¯t Dynamic factor model for the gaps u π a(L)~yt = "~t u~t = bu(L)~yt + "t π~t = bπ(L)~yt + "t TREND-CYCLE MODEL FOR MACRO VARIABLES Observables Real GDP: yt =y ¯t +y ~t Unemployment rate: ut =u ¯t +u ~t Inflation: πt =π ¯t +π ~t ∗ ~ Nominal Rate: it = max (0; it) it =r ¯t +π ¯t + it TREND-CYCLE MODEL FOR MACRO VARIABLES Observables Real GDP: yt =y ¯t +y ~t Unemployment rate: ut =u ¯t +u ~t Inflation: πt =π ¯t +π ~t ∗ ~ Nominal Rate: it = max (0; it) it =r ¯t +π ¯t + it Trends y µ y¯t = µt−1 +y ¯t−1 +" ¯t µt = µt−1 + ηt u u¯t = u¯t−1 +" ¯t π π¯t = π¯t−1 +" ¯t Dynamic factor model for the gaps u π a(L)~yt = "~t u~t = bu(L)~yt + "t π~t = bπ(L)~yt + "t TREND UNEMPLOYMENT 11 Unemployment Rate Trend Unemployment Rate 10 9 8 7 6 5 4 3 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 OUTPUT GAP 8 Posterior distribution from macro model CBO estimate 6 4 2 0 −2 −4 −6 −8 −10 −12 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 SHADOW RATE ESTIMATES ∗ it = 0 treated as missing data 6 Nominal Short Rate Shadow Rate Distribution 4 2 0 −2 −4 −6 −8 −10 2007 2008 2009 2010 2011 2012 2013 2014 SHADOW RATE ESTIMATES ∗ Censored sampling: it < 0 when it = 0 6 Nominal Short Rate Shadow Rate Distribution 4 2 0 −2 −4 −6 −8 −10 2007 2008 2009 2010 2011 2012 2013 2014 SHADOW RATE ESTIMATES BIVARIATE MODEL ∗ it = 0 treated as missing data 6 Nominal Short Rate Shadow Rate Distribution 4 2 0 −2 −4 −6 −8 −10 2007 2008 2009 2010 2011 2012 2013 2014 SHADOW RATE ESTIMATES IN PSEUDO-REAL TIME 4 2 0 −2 −4 −6 −8 2008 2009 2010 2011 2012 2013 2014 SHADOW RATE ESTIMATES IN PSEUDO-REAL TIME 4 2 0 −2 −4 −6 −8 2008 2009 2010 2011 2012 2013 2014 REAL (SHADOW) RATE TREND Real shadow-rate trend r¯t ¯ it = r¯t +π ¯t r¯t = r¯t−1 + et ? = β · µt + rt Trend growth µt yt = y¯t +y ~t y y¯t = µt−1 +y ¯t−1 +" ¯t µ µt = µt−1 + ηt TREND INFLATION BACKUP Level 12 Trend Headline PCE Inflation 10 8 6 4 2 0 −2 −4 −6 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 REAL RATE TREND 8 Real Rate Trend Nominal Rate less Inflation Trend 6 4 2 0 −2 −4 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 TREND GROWTH AND REAL RATE TREND ? r¯t = β · µt + rt 5 Expected Growth in Trend GDP Real Rate Trend 4 3 2 1 0 −1 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 AGENDA 1 Shadow-Rate Concept 2 Shadow-Rate Sampling 3 Estimates from an Empirical Macro Model 4 Term-Premium Estimates TERM PREMIUM A decomposition for h-period bond yield h−1 ∗ 1 X ∗ i = Eti + pt;t+h t;t+h h t+j j=0 Expected future short rates ∗ Etit+j: mean of truncated predictive density for it+j ∗ @Etit+j < 1 the more mass on it+j < 0 @Etit+j LONG RATES AND EXPECTED FUTURE SHORT RATES Expected future shadow rates (10-year avg) 16 14 12 10 8 6 4 2 0 −2 −4 −6 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 LONG RATES AND EXPECTED FUTURE SHORT RATES Expected future rates (actual vs. shadow, 10-year avg) 16 14 12 10 8 6 4 2 0 −2 −4 Avg of Exp. Future Short Rates Avg of Exp. Future Shadow Rates −6 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 LONG RATES AND EXPECTED FUTURE SHORT RATES Expected future rates (actual vs. shadow, 10-year avg) 5 4 3 2 1 0 −1 −2 Avg of Exp. Future Short Rates Avg of Exp. Future Shadow Rates −3 2000 2002 2005 2007 2010 2012 LONG RATES AND EXPECTED FUTURE SHORT RATES Expected future rates (actual vs. shadow, 10-year avg) 16 14 12 10 8 6 4 2 0 −2 −4 Avg of Exp. Future Short Rates Avg of Exp. Future Shadow Rates 10−year Rate −6 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 TERM PREMIUM ESTIMATES 10-year rate 6 4 2 0 −2 −4 −6 −8 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 TERM PREMIUM ESTIMATES 10-year rate 6 Macro model Kim−Wright 5 4 3 2 1 0 −1 −2 1992 1995 1997 2000 2002 2005 2007 2010 2012 SOME QUESTIONS and answers 1 What is the long-run level of the real rate, how does it relate to growth? Trend real rate depends only weakly on growth, stands currently at just about one percent 2 How did the recent recession affect trends in real and nominal variables? \Not much" as decline in trends started earlier 3 How does the term structure change with short rates near the lower bound? Expected future short rates get less sensitive to news.
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