Macro-finance theory implies that trend inflation and the equilibrium real interest rate are fundamental determinants of the yield curve. However, empirical models of the terms structure of interest rates generally assume that these fundamentals are constant. We show that accounting for time variation in these underlying long-run trends is crucial for understanding the dynamics of Treasury yields and predicting excess bond returns. We introduce a new arbitrage-free model that captures the key role that long-run trends play for interest rates. The model also provides new, more plausible estimates of the term premium and accurate out-of-sample yield forecasts.
The ability of the usual factors from empirical arbitrage-free representations of the term structure — that is, spanned factors — to account for interest rate volatility dynamics has been much debated. We examine this issue with a comprehensive set of new arbitrage-free term structure specifications that allow for spanned stochastic volatility to be linked to one or more of the yield curve factors. Using U.S. Treasury yields, we find that much realized stochastic volatility cannot be associated with spanned term structure factors. However, a simulation study reveals that the usual realized volatility metric is misleading when yields contain plausible measurement noise. We argue that other metrics should be used to validate stochastic volatility models
Published Articles (Refereed Journals and Volumes)
Researchers have debated the extent of the decline in the steady-state short-term real interest rate—that is, in the so-called equilibrium or natural rate of interest. We examine this issue using a dynamic term structure finance model estimated directly on the prices of individual inflation-indexed bonds with adjustments for real term and liquidity risk premiums. Our methodology avoids two pitfalls of previous macroeconomic analyses: structural breaks at the zero lower bound and potential misspecification of output and inflation dynamics. We estimate that the equilibrium real rate has fallen about 2 percentage points and appears unlikely to rise quickly.
Term Structure Analysis with Big Data: One-Step Estimation Using Bond Prices
Journal of Econometrics 212(1), September 2018, 26-46 | With Andreasen and Christensen
Analysis of the term structure of interest rates almost always takes a two-step approach.
First, actual bond prices are summarized by interpolated synthetic zero-coupon yields, and second, a small set of these yields are used as the source data for further empirical examination. In contrast, we consider the advantages of a one-step approach that directly analyzes the universe of bond prices. To illustrate the feasibility and desirability of the onestep approach, we compare arbitrage-free dynamic term structure models estimated using both approaches. We also provide a simulation study showing that a one-step approach can extract the information in large panels of bond prices and avoid any arbitrary noise introduced from a first-stage interpolation of yields.
Most existing macro-finance term structure models (MTSMs) appear incompatible with regression evidence of unspanned macro risk. This “spanning puzzle” appears to invalidate those models in favor of new unspanned MTSMs. However, our empirical analysis supports the previous spanned models. Using simulations to investigate the spanning implications of MTSMs, we show that a canonical spanned model is consistent with the regression evidence; thus, we resolve the spanning puzzle. In addition, direct likelihood-ratio tests find that the knife-edge restrictions of unspanned models are rejected with high statistical significance, though these restrictions have only small effects on cross-sectional fit and estimated term premia.
We show that conventional dynamic term structure models (DTSMs) estimated on recent U.S. data severely violate the zero lower bound (ZLB) on nominal interest rates and deliver poor forecasts of future short rates. In contrast, shadow-rate DTSMs account for the ZLB by construction, capture the resulting distributional asymmetry of future short rates, and achieve good forecast performance. These models provide more accurate estimates of the most likely path for future monetary policy—including the timing of policy liftoff from the ZLB and the pace of subsequent policy tightening. We also demonstrate the benefits of including macroeconomic factors in a shadow-rate DTSM when yields are constrained near the ZLB.
In standard macroeconomic models, the two objectives in the Federal Reserve’s dual mandate—full employment and price stability—are closely intertwined. We motivate and estimate an alternative model in which long-term unemployment varies endogenously over the business cycle but does not affect price inflation. In this new model, an increase in long-term unemployment as a share of total unemployment creates short-term tradeoffs for optimal monetary policy and a wedge in the dual mandate. In particular, faced with high long-term unemployment following the Great Recession, optimal monetary policy would allow inflation to overshoot its target more than in standard models.
We use an arbitrage-free term structure model with spanned stochastic volatility to determine the value of the deflation protection option embedded in Treasury inflation protected securities (TIPS). The model accurately prices the deflation protection option prior to the financial crisis when its value was near zero; at the peak of the crisis in late 2008 when deflationary concerns spiked sharply; and in the post-crisis period. During 2009, the average value of this option at the five-year maturity was 41 basis points on a par-yield basis. The option value is shown to be closely linked to overall market uncertainty as measured by the VIX, especially during and after the 2008 financial crisis.
Recent U.S. Treasury yields have been constrained to some extent by the zero lower bound (ZLB) on nominal interest rates. Therefore, we compare the performance of a standard affine Gaussian dynamic term structure model (DTSM), which ignores the ZLB, to a shadow-rate DTSM, which respects the ZLB. Near the ZLB, we find notable declines in the forecast accuracy of the standard model, while the shadow-rate model forecasts well. However, 10-year yield term premiums are broadly similar across the two models. Finally, in applying the shadow-rate model, we find no gain from estimating a slightly positive lower bound on U.S. yields
Standard Gaussian affine dynamic term structure models do not rule out negative nominal interest rates—a conspicuous defect with yields near zero in many countries. Alternative shadow-rate models, which respect the nonlinearity at the zero lower bound, have been rarely used because of the extreme computational burden of their estimation. However, by valuing the call option on negative shadow yields, we provide the first estimates of a three-factor shadow-rate model. We validate our option-based results by closely matching them using a simulation-based approach. We also show that the shadow short rate is sensitive to model fit and specification.
To support the economy, the Federal Reserve amassed a large portfolio of long-term bonds. We assess the Fed’s associated interest rate risk — including potential losses to its Treasury securities holdings and declines in remittances to the Treasury. Unlike past examinations of this interest rate risk, we attach probabilities to alternative interest rate scenarios. These probabilities are obtained from a dynamic term structure model that respects the zero lower bound on yields. The resulting probability-based stress test finds that the Fed’s losses are unlikely to be large and remittances are unlikely to exhibit more than a brief cessation.
Previous research has emphasized the portfolio balance effects of Federal Reserve bond purchases, in which a reduced bond supply lowers term premia. In contrast, we find that such purchases have important signaling effects that lower expected future short-term interest rates. Our evidence comes from a model-free analysis and from dynamic term structure models
that decompose declines in yields following Federal Reserve announcements into changes in risk premia and expected short
rates. To overcome problems in measuring term premia, we consider bias-corrected model estimation and restricted risk price estimation. In comparison with other studies, our estimates of signaling effects are larger in magnitude and statistical significance.
Term premia implied by maximum likelihood estimates of affine term structure models are misleading because of small-sample bias. We show that accounting for this bias alters the conclusions about the trend, cycle, and macroeconomic determinants of the term premia estimated in Wright (2011). His term premium estimates are essentially acyclical, and often just parallel the secular trend in long-term interest rates. In contrast, bias-corrected term premia show pronounced countercyclical behavior, consistent with theoretical and empirical arguments about movements in risk premia.
In response to the global financial crisis that started in August 2007, central banks provided extraordinary amounts of liquidity to the financial system. To investigate the effect of central bank liquidity facilities on term interbank lending rates, we estimate a six-factor
arbitrage-free model of U.S. Treasury yields, financial corporate bond yields, and term interbank rates. This model can account for fluctuations in the term structure of credit risk and liquidity risk. A significant shift in model estimates after the announcement of
the liquidity facilities suggests that these central bank actions did help lower the liquidity premium in term interbank rates.
We construct probability forecasts for episodes of price deflation (i.e., a falling price level) using yields on nominal and real U.S. Treasury bonds. The deflation probability forecasts identify two “deflation scares” during the past decade: a mild one following the 2001 recession and a more serious one starting in late 2008 with the deepening of the financial crisis. The estimated deflation probabilities are generally consistent with those from macroeconomic models and surveys of professional forecasters, but they also provide high-frequency insight into the views of financial market participants. The probabilities can also be used to price the deflation protection option embedded in real Treasury bonds.
We analyze declines in government bond yields following announcements by the Federal Reserve and the Bank of England of plans to buy longer term debt. Using dynamic term structure models, we decompose US and UK yields into expectations about future short-term interest rates and term premiums. We find that declines in US yields mainly reflected lower expectations of future short-term interest rates, while declines in UK yields appeared to reflect reduced term premiums. Thus, the relative importance of the signalling and portfolio balance channels of quantitative easing may depend on market institutional structures and central bank communication policies.
The affine dynamic term structure model (DTSM) is the canonical empirical finance representation of the yield curve. However, the possibility that DTSM estimates may be distorted by small-sample bias has been largely ignored. We show that conventional estimates of DTSM coefficients are indeed severely biased, and this bias results in misleading estimates of expected future short-term interest rates and of long-maturity term premia. We provide a variety of bias-corrected estimates of affine DTSMs, both for maximally-flexible and over-identified specifications. Our estimates imply short rate expectations and term premia that are more plausible from a macro-finance perspective.
The term premium on nominal long-term bonds in the standard dynamic stochastic general equilibrium (DSGE) model used in macroeconomics is far too small and stable relative to empirical measures obtained from the data–an example of the “bond premium puzzle.” However, in models of endowment economies, researchers have been able to generate reasonable term premiums by assuming that investors have recursive Epstein-Zin preferences and face long-run economic risks. We show that introducing Epstein-Zin preferences into a canonical DSGE model can also produce a large and variable term premium without compromising the model’s ability to fi t key macroeconomic variables. Long-run real and nominal risks further improve the model’s ability to fit the data with a lower level of household risk aversion.
The Affine Arbitrage-Free Class of Nelson-Siegel Term Structure Models
Journal of Econometrics 164, September 2011, 4-20 | With Christensen and Diebold
We derive the class of affine arbitrage-free dynamic term structure models that approximate the widely-used Nelson-Siegel yield curve specification. These arbitrage-free Nelson-Siegel (AFNS) models can be expressed as slightly restricted versions of the canonical
representation of the three-factor affine arbitrage-free model. Imposing the Nelson-Siegel structure on the canonical model greatly facilitates estimation and can improve predictive performance. In the future, AFNS models appear likely to be a useful workhorse
representation for term structure research.
Inflation Expectations and Risk Premiums in an Arbitrage-Free Model of Nominal and Real Bond Yields
Journal of Money, Credit, and Banking 42, September 2010, 143-178 | With Christensen and Lopez
Differences between yields on comparable-maturity U.S. Treasury nominal and real debt, the so-called breakeven inflation (BEI) rates, are widely used indicators of inflation expectations. However, better measures of inflation expectations could be obtained by subtracting inflation risk premiums from the BEI rates. We provide such decompositions using an estimated affine arbitrage-free model of the term structure that captures the pricing of both nominal and real Treasury securities. Our empirical results suggest that long-term inflation expectations have been well anchored over the past few years, and inflation risk premiums, although volatile, have been close to zero on average.
Macro-Finance Models of Interest Rates and the Economy
During the past decade, much new research has combined elements of finance, monetary economics and macroeconomics in order to study the relationship between the term structure of interest rates and the economy. In this survey, I describe three different strands of such interdisciplinary macro-finance term structure research. The first adds macroeconomic variables and structure to a canonical arbitrage-free finance representation of the yield curve. The second examines bond pricing and bond risk premiums in a canonical macroeconomic dynamic stochastic general equilibrium model. The third develops a new class of arbitrage-free term structure models that are empirically tractable and well suited to macro-finance investigations.
The Svensson generalization of the popular Nelson-Siegel term structure model is widely used by practitioners and central banks. Unfortunately, like the original Nelson-Siegel specification, this generalization, in its dynamic form, does not enforce arbitrage-free consistency over time. Indeed, we show that the factor loadings of the Svensson generalization cannot be obtained in a standard finance arbitrage-free affine term structure representation. Therefore, we introduce a closely related generalized Nelson-Siegel model on which the no-arbitrage condition can be imposed. We estimate this new arbitrage-free generalized Nelson-Siegel model and demonstrate its tractability and good in-sample fit.
Forecasting Recessions: The Puzzle of the Enduring Power of the Yield Curve
Journal of Business and Economic Statistics 27(4), 2009, 492-503 | With Williams
We show that professional forecasters have essentially no ability to predict future recessions a few quarters ahead. This is particularly puzzling because, for at least the past two decades, researchers have provided much evidence that the yield curve, specifically the spread between long- and short-term interest rates, does contain useful information at that forecast horizon for predicting aggregate economic activity and, especially, for signalling future recessions. We document this puzzle and suggest that forecasters have generally placed too little weight on yield curve information when projecting declines in the aggregate economy.
Examining the Bond Premium Puzzle with a DSGE Model
Journal of Monetary Economics 55, October 2008, S111-S126 | With Swanson
The basic inability of standard theoretical models to generate a sufficiently large and variable nominal bond risk premium has been termed the “bond premium puzzle.” We show that the term premium on long-term bonds in the canonical dynamic stochastic general equilibrium (DSGE) model used in macroeconomics is far too small and stable relative to the data. We find that introducing long-memory habits in consumption as well as labor market frictions can help fit the term premium, but only by seriously distorting the DSGE model’s ability to fit other macroeconomic variables, such as the real wage; therefore, the bond premium puzzle remains.
This article develops and estimates a macro-finance model that combines a canonical affine no-arbitrage finance specification of the term structure of interest rates with standard macroeconomic aggregate relationships for output and inflation. Based on this combination of yield curve and macroeconomic structure and data, we obtain several interesting results: (1) the latent term structure factors from no-arbitrage finance models appear to have important macroeconomic and monetary policy underpinnings, (2) there is no evidence of a slow partial adjustment of the policy interest rate by the central bank, and (3) both forward-looking and backward-looking elements play roles in macroeconomic dynamics.
The modern view of monetary policy stresses its role in shaping the entire yield curve of interest rates in order to achieve various macroeconomic objectives. A crucial element of this process involves guiding financial market expectations of future central bank actions. Recently, a few central banks have started to explicitly signal their future policy intentions to the public, and two of these banks have even begun publishing their internal interest rate projections. We examine the macroeconomic effects of direct revelation of a central bank’s expectations about the future path of the policy rate. We show that, in an economy where private agents have imperfect information about the determination of monetary policy, central bank communication of interest rate projections can help shape financial market expectations and may improve macroeconomic performance.
Linearized New Keynesian models and empirical no-arbitrage macro-finance models offer little insight regarding the implications of changes in bond term premiums for economic activity. This paper investigates these implications using both a structural model and a reduced-form framework. The authors show that there is no structural relationship running from the term premium to economic activity, but a reduced-form empirical analysis does suggest that a decline in the term premium has typically been associated with stimulus to real economic activity, which contradicts earlier results in the literature.
Accounting for a Shift in Term Structure Behavior with No-Arbitrage and Macro-Finance Models
Journal of Money, Credit, and Banking 39 (2-3), March 2007, 395-422 | With Wu
This paper examines a shift in the dynamics of the term structure of interest rates in the U.S. during the mid-1980s. We document this shift using standard interest rate regressions and using dynamic, affine, no-arbitrage models estimated for the pre- and post-shift subsamples. The term structure shift largely appears to be the result of changes in the pricing of risk associated with a “level” factor. Using a macro-finance model, we suggest a link between this shift in term structure behavior and changes in the dynamics and risk pricing of the Federal Reserve’s inflation target as perceived by investors.
In 2004 and 2005, long-term interest rates remained remarkably low
despite improving economic conditions and rising short-term interest rates,
a situation that then-Federal Reserve Board Chairman Alan Greenspan
dubbed a “conundrum.” We document the extent and timing of this conundrum using two empirical no-arbitrage macro-finance models of the term structure of interest rates. These models confirm that the recent behavior of long-term yields has been unusual–that is, it cannot be explained within the framework of the models. Therefore, we consider other macroeconomic factors omitted from the models and find that some of these variables, particularly declines in long-term bond volatility, may explain a portion of the conundrum. Foreign official purchases of U.S. Treasuries appear to have played little or no role.
Many interpret estimated monetary policy rules as suggesting that central banks conduct very sluggish partial adjustment of short-term policy interest rates. In contrast, others argue that this appearance of policy inertia is an illusion and simply reflects the spurious omission of important persistent influences on the actual setting of policy. Similarly, the real-world implications of the theoretical arguments for policy inertia are debatable. However, empirical evidence on policy gradualism obtained by examining expectations of future monetary policy embedded in the term structure of interest rates is definitive and indicates that the actual amount of policy inertia is quite low.
The Macroeconomy and the Yield Curve: A Dynamic Latent Factor Approach
Journal of Econometrics 131(1-2), March 2006, 309-338 | With Diebold and Aruoba
We estimate a model that summarizes the yield curve using latent factors (specifically, level, slope, and curvature) and also includes observable macroeconomic variables (specifically, real activity, inflation, and the monetary policy instrument). Our goal is to provide a characterization of the dynamic interactions between the macroeconomy and the yield curve. We find strong evidence of the effects of macro variables on future movements in the yield curve and evidence for a reverse influence as well. We also relate our results to the expectations hypothesis.
Using a Long-Term Interest Rate as the Monetary Policy Instrument
Journal of Monetary Economics 52(5), July 2005, 855-879 | With Williams and McGough
Using a short-term interest rate as the monetary policy instrument can be
problematic near its zero-bound constraint. An alternative strategy is to use a long-term
interest rate as the policy instrument. We find when Taylor-type policy rules are used to
set the long rate in a standard New Keynesian model, indeterminacy–that is, multiple
rational expectations equilibria–may often result. However, a policy rule with a long rate
policy instrument that responds in a “forward-looking” fashion to inflation expectations
can avoid the problem of indeterminacy.
From a macroeconomic perspective, the short-term interest rate is a policy instrument under the direct control of the central bank. From a finance perspective, long rates are risk-adjusted averages of expected future short rates. Thus, as illustrated by much recent research, a joint macro-finance modeling strategy will provide the most comprehensive understanding of the term structure of interest rates. We discuss various questions that arise in this research, and we also present a new examination of the relationship between two prominent dynamic, latent factor models in this literature: the Nelson-Siegel and affine no-arbitrage term structure models.
Assessing the Lucas Critique in Monetary Policy Models
Journal of Money, Credit, and Banking 37(2), April 2005, 245-272
Empirical estimates of monetary policy rules suggest that the behavior of U.S. monetary policymakers changed during the past few decades. However, for that same time period, statistical analyses of lagged representations of the economy, such as VARs, often have not rejected the null of structural stability. These two sets of empirical results appear to contradict the Lucas critique. This paper reconciles these results with the Lucas critique by showing that the apparent policy invariance of reduced forms is consistent with the magnitude of historical policy shifts and the relative insensitivity of the reduced forms of plausible forward-looking macroeconomic specifications to policy shifts.
Estimating the Euler Equation for Output
Journal of Monetary Economics 51(6), September 2004, 1133-1153 | With Fuhrer
New Keynesian macroeconomic models have generally emphasized that
expectations of future output are a key factor in determining current output.
The theoretical motivation for such forward-looking behavior relies on a
straightforward generalization of the well-known Euler equation for consumption. In this paper, we use maximum likelihood and generalized method of moments (GMM) methods to explore the empirical importance of output expectations. We find little evidence that rational expectations of future output help determine current output, especially after taking into account the small-sample bias in GMM.
Term Structure Evidence on Interest Rate Smoothing and Monetary Policy Inertia
Journal of Monetary Economics 49(6), September 2002, 1161-1187
Numerous studies have used quarterly data to estimate monetary policy rules or reaction functions that appear to exhibit a very slow partial adjustment of the policy interest rate. The conventional wisdom asserts that this gradual adjustment reflects a policy inertia or interest rate smoothing behavior by central banks. However, such quarterly monetary policy inertia would imply a large amount of forecastable variation in interest rates at horizons of more than three months, which is contradicted by evidence from the term structure of interest rates. The illusion of monetary policy inertia evident in the estimated policy rules likely reflects the persistent shocks that central banks face.
Nominal income rules for monetary policy have long been debated, but two issues are of particular recent interest. First, there are questions about the performance of such rules over a range of plausible empirical models–especially models with and without explicit rational inflation expectations. Second, there are questions about the performance of these rules in real time using the type of data that is actually available contemporaneously to policymakers rather than final revised data. This paper determines optimal monetary policy rules in the presence of such model uncertainty and real-time data uncertainty and finds only a limited role for nominal output growth.
Eurosystem Monetary Targeting: Lessons from U.S. Data
European Economic Review 46, March 2002, 417-442 | With Svensson
Using a small empirical model of inflation, output, and money estimated on U.S. data, we compare the relative performance of monetary targeting and inflation targeting. The results show monetary targeting to be quite inefficient, yielding both higher inflation and output variability. This is true, even with a nonstochastic money demand formulation. Our results are also robust to using a P* model of inflation. Therefore, in these popular frameworks, there is no support for the prominent role given to money growth in the Eurosystem’s monetary policy strategy.
Estimates of the Taylor rule using historical data from the past decade or two suggest that monetary policy in the U.S. can be characterized as having reacted in a moderate fashion to output and inflation gaps. In contrast, the parameters of optimal Taylor rules derived using empirical models of the economy often recommend much more vigorous policy responses. This paper attempts to match the historical policy rule with an optimal policy rule by incorporating uncertainty into the derivation of the optimal rule and by examining plausible variations in the policymaker’s model and preferences.
Opportunistic and Deliberate Disinflation under Imperfect Credibility
Journal of Money, Credit, and Banking 32, November 2000, 707-721 | With Bomfim