Publications

Forecasting with Shadow-Rate VARs (2025, forth. QE) 

with Andrea Carriero (Queen Mary University of London, U Bologna), Todd Clark (Federal Reserve Bank of Cleveland), Massimiliano Marcellino (Bocconi, IGIER and CEPR)

• draft: pdf 

• supplementary appendix: pdf

• draft and supplement at QE: https://www.econometricsociety.org/publications/quantitative-economics/forthcoming-papers

• slides: pdf updated version of our presentation at NBER SI 2023)

• earlier Bundesbank DP at IDEAS

• earlier Cleveland Fed WP version: https://doi.org/10.26509/frbc-wp-202109

• replication code (github) and https://doi.org/10.5281/zenodo.14814807 

Abstract: Vector autoregressions (VARs) are popular for forecasting, but ill-suited to handle occasionally binding constraints, like the effective lower bound on nominal interest rates. We examine reduced-form ``shadow-rate VARs'' that model interest rates as censored observations of a latent shadow-rate process and develop an efficient Bayesian estimation algorithm that accommodates large models.  When compared to a standard VAR, our better-performing shadow-rate VARs generate superior predictions for interest rates and broadly similar predictions for macroeconomic variables.  We obtain this result for shadow-rate VARs in which the federal funds rate is the only interest rate and in models including additional interest rates. Our shadow-rate VARs also deliver notable gains in forecast accuracy relative to a VAR that omits shorter-term interest rate data in order to avoid modeling the lower bound.

Earlier versions of this paper were also earlier circulated under the title “Shadow-Rate VARs.”

Constructing Fan Charts from the Ragged Edge of SPF Forecasts (2025,  forth. REStat)

with Todd E. Clark (FRB Cleveland), and Gergely Ganics (Banco de España)

• https://doi.org/10.1162/rest_a_01565 

• latest draft: pdf and online appendix: pdf

• Slides: pdf

• replication code: https://github.com/elmarmertens/ClarkGanicsMertensSPFfancharts and REStat Dataverse

Abstract: We develop models that take point forecasts from the Survey of Professional Forecasters (SPF) as inputs and produce estimates of survey-consistent term structures of expectations and uncertainty at arbitrary forecast horizons. Our models combine fixed-horizon and fixed-event forecasts, accommodating time-varying horizons and availability of survey data, as well as potential inefficiencies in survey forecasts. The estimated term structures of SPF-consistent expectations are comparable in quality to the published, widely used short-horizon forecasts. Our estimates of time-varying forecast uncertainty reflect historical variations in realized errors of SPF point forecasts, and generate fan charts with reliable coverage rates.

Parts of this paper were earlier circulated under the title “Constructing the Term Structure of Uncertainty from the Ragged Edge of SPF Forecasts.”

Survey Expectations and Forecast Uncertainty (2024, Edward Elgar Handbook chapter)

with Todd E. Clark (Federal Reserve Bank of Cleveland)

• Chapter for the  Handbook of Research Methods and Applications on Macroeconomic Forecasting  edited by Mike Clements and Ana Galvao, 2024, Edward Elgar Publishing Ltd. 

• Edward Elgar weblink 

Abstract: In recent decades, the collection of survey expectations for macroeconomic variables has gained considerable attention. A bourgeoning literature has developed that studies not only the predictive content of survey data, but also its usefulness in testing economic theories about the behavior of forward-looking decision makers under uncertainty. Leading examples of economic survey data include surveys of experts (including professional forecasters and financial market participants), households and firms.  Point forecasts of professional forecasters have, on balance, emerged as competitive (albeit not fully optimal) predictors of future outcomes. Meanwhile, ex-ante measures of uncertainty derived from probabilistic surveys have been found to be systematically at variance with the distribution of ex-post outcomes. Nevertheless, puzzles remain and the analysis and design of expectation surveys is an active field of research.

Addressing COVID-19 Outliers in BVARs with Stochastic Volatility (2024, Review of Economics and Statistics)

with Andrea Carriero (Queen Mary University of London, U Bologna), Todd Clark (Federal Reserve Bank of Cleveland), Massimiliano Marcellino (Bocconi, IGIER and CEPR)

• Review of Economics and Statistics. Vol 106 (5): 1403–1417

• supplementary results: pdf

• slides:  pdf

• also presented at NBER SI and the 11th ECB forecasting conference (youtube,  VoxEU)

• replication code: github

Abstract: The COVID-19 pandemic has led to enormous movements in economic data that strongly affect parameters and forecasts obtained from standard VARs. One way to address these issues is to model extreme observations as random shifts in the stochastic volatility (SV) of BVAR residuals. Specifically, we propose VAR models with outlier-augmented SV that combine transitory and persistent changes in volatility. The resulting density forecasts for the COVID-19 period are much less sensitive to outliers in the data than standard VARs.  Evaluating forecast performance over the last few decades, we find that outlier-augmented SV schemes do at least as well as a conventional SV model. Predictive Bayes factors indicate that our outlier-augmented SV model provides the best fit for the period since the pandemic's outbreak, as well as for earlier subsamples of high volatility. 

Precision-Based Sampling for State Space Models that Have No Measurement Error (2023, Journal of Economic Dynamics and Control)

•  Journal of Economic Dynamics and Control. Vol. 154, 104720.

• earlier draft: pdf (revised June 2023)

• Bundesbank DP at IDEAS

• online appendix: pdf

• replication code: https://github.com/elmarmertens/ABCprecisionsampler

Abstract: This article presents a computationally efficient approach to sample from Gaussian state space models. The method is an instance of precision-based sampling methods that operate on the inverse variance-covariance matrix of the states (also known as precision). The novelty is to handle cases where the observables are modeled as a linear combination of the states without measurement error. In this case, the posterior variance of the states is singular and precision is ill-defined. As in other instances of precision-based sampling, computational gains are considerable. Relevant applications include trend-cycle decompositions, (mixed-frequency) VARs with missing variables and DSGE models.

Stochastic Volatility in Bayesian Vector Autoregressions (2023, Oxford Research Encyclopdia)

with Todd E. Clark (Federal Reserve Bank of Cleveland)

• Oxford Research Encyclopedia of Economics and Finance, 2023 

Abstract: Vector autoregressions with stochastic volatility are widely used in macroeconomic forecasting and structural inference.   The stochastic volatility component of the model conveniently allows for time variation in the variance-covariance matrix of the model's forecast errors.  In turn, that feature of the model generates time variation in predictive densities.  The models are most commonly estimated with Bayesian methods, most typically Markov Chain Monte Carlo methods such as Gibbs sampling. Recently developed equation-by-equation methods enable the estimation of models with large variable sets at much lower computational cost than the standard approach of estimating the model as a system of equations.  The Bayesian framework also facilitates the accommodation of mixed frequency data, non-Gaussian error distributions, and non-parametric specifications. With recent advances, researchers are also addressing some of the framework's outstanding challenges, particularly the dependence of estimates on the ordering of variables in the model and reliable estimation of the marginal likelihood, which is the fundamental measure of model fit in Bayesian methods.

Indeterminacy and Imperfect Information (2023, Review of Economic Dynamics)

with Thomas A. Lubik (FRB Richmond) and Christian Matthes (U Indiana)

• Review of Economic Dynamics. Vol. 49, 37 - 57.

• appendix: pdf 

• replication codes: https://github.com/elmarmertens/LMMREDcode

Abstract: We study equilibrium determination in an environment where two types of agents have different information sets: Fully informed agents observe histories of all exogenous and endogenous variables. Less informed agents observe only a strict subset of the full information set and need to solve a dynamic signal extraction problem to gather information about the variables they do not directly observe. Both types of agents know the structure of the model and form expectations rationally. In this environment, we identify a new channel that generates equilibrium indeterminacy: Optimal information processing of the less informed agent introduces stable dynamics into the equation system that lead to self-fulling expectations. For parameter values that imply a unique equilibrium under full information, the limited information rational expectations equilibrium is indeterminate. We illustrate our framework with monetary policy models where an imperfectly informed central bank follows an interest rate rule.

A Time Series Model of Interest Rates With the Effective Lower Bound (2021, Journal of Money, Credit, and Banking)

with Benjamin K. Johannsen

• Journal of Money, Credit, and Banking. Vol. 53, 1005 - 1046. 

• Working paper: pdf (revised 2019; older FEDS)

• Supplementary appendix: pdf

• slides: pdf, poster: pdf

• replication files: GitHub

• Updated estimates through 2024:Q3 (using FRED data available on Oct 29,  2024) : Real-rate trend, shadow rate. (The usual disclaimer applies; none of the estimates or related material necessarily represents the views of the Deutsche Bundesbank, the Eurosystem, the Board of Governors of the Federal Reserve System or the Federal Open Market Committee.)

Abstract:  Modeling nominal interest rates requires taking account of their effective lower bound (ELB). We propose a flexible time-series approach that includes a ``shadow rate'' -- a notional rate identical to the actual nominal rate except when the ELB binds. We apply this approach to a trend-cycle decomposition of interest rates and macroeconomic variables that generates competitive interest-rate forecasts. Our estimates of the real-rate trend edged down somewhat in recent decades, but not significantly so. We identify monetary policy shocks from shadow-rate surprises and find they were particularly effective at stimulating economic activity during the ELB period.

Inflation and Professional Forecast Dynamics: An Evaluation of Stickiness, Persistence and Volatility (2020, Quantitative Economics)

with James M. Nason 

• Quantitative Economics. Vol, 11(4), 1485 – 1520.

• online results appendix

• slides (pdf), poster (pdf)

• replication files: GitHub 

• Estimates of trend and gap inflation: Filtered trend, smoothed trend, filtered gap, smoothed gap (all in csv format).  

Abstract: 

This paper studies the joint dynamics of U.S. inflation and a term structure of average inflation predictions taken from the Survey of Professional Forecasters (SPF).  We estimate these joint dynamics by combining an unobserved components (UC) model of inflation and a sticky-information forecast mechanism.  The UC model decomposes inflation into trend and gap components, and innovations to trend and gap inflation are affected by stochastic volatility.  A novelty of our model is to allow for time-variation in inflation-gap persistence as well as in the frequency of forecast updating under sticky information. The model is estimated with sequential Monte Carlo methods that include a particle learning filter and a Rao-Blackwellized particle smoother.  Based on data from 1968Q4 to 2018Q3, estimates show that (i) longer horizon average SPF inflation predictions inform estimates of trend inflation; (ii) inflation gap  persistence is countercyclical before the Volcker disinflation and acyclical afterwards; (iii) by 1990 sticky-information inflation forecast updating is less frequent than it was earlier in the sample; and (iv) the drop in the frequency of the sticky-information forecast updating occurs at the same time persistent shocks become less important for explaining movements in inflation. Our findings support the view that stickiness in survey forecasts is not invariant to the inflation process.

Modeling Time-Varying Uncertainty of Multiple-Horizon Forecast Errors (2020, Review of Economics and Statistics)

with Todd E. Clark, Michael W. McCracken

• Review of Economics and Statistics Vol. 102(1), 17-33. 

• supplementary appendix: pdf

• slides: pdf (This is an updated version of our talk at the NBER SI 2017) 

• replication code: Harvard Dataverse (html) Github.

Abstract:

We estimate uncertainty measures for point forecasts obtained from survey data, pooling information embedded in observed forecast errors for different forecast horizons. To track time-varying uncertainty in the associated forecast errors, we derive a multiple-horizon specification of stochastic volatility. We apply our method to forecasts for various macroeconomic variables from the Survey of Professional Forecasters. Compared to simple variance approaches, our stochastic volatility model improves the accuracy of uncertainty measures for survey forecasts.

Measuring the Level and Uncertainty of Trend Inflation (2016, Review of Economics and Statistics)

• Review of Economics and Statistics Vol. 98 (5), 950 – 967.

• FEDS working paper 

• web-appendix

• replication data: Harvard Dataverse (html)

• Data and code for updating trend estimates of the paper (based on public data sources): FRED data set, https://github.com/elmarmertens/trendInflation

Abstract: 

Firmly-anchored inflation expectations are widely viewed as playing a central role for the conduct of monetary policy. This paper presents estimates of trend inflation, based on information contained in monthly data on realized inflation, survey expectations, and the term structure of interest rates. In order to assess whether inflation expectations are anchored, a time-varying volatility of trend shocks is estimated as well. There is quite some commonality in inflation- and survey-based estimates of trend inflation, but yield-based trend estimates embed a highly persistent component orthogonal to trend inflation. Trimmed-mean inflation rates and survey forecasts are most indicative of trend inflation.

Managing Beliefs about Monetary Policy under Discretion (2016, Journal of Money, Credit, and Banking)

• Journal of Money, Credit, and Banking, Vol 48 (4).

• web-appendix: pdf

• replication code: zip

• Note on Westelius, 2009, JEDC: pdf

Abstract: 

In models of monetary policy, discretionary policymaking is typically constrained in its ability to manage public beliefs. However, when a policymaker possesses private information, policy actions serve as signals to the public about unobserved economic conditions and belief management becomes an integral part of optimal discretion policies. My paper derives the optimal time-consistent policy for a general linear-quadratic setting.

The optimal policy is illustrated in a simple New Keynesian model, where analytical so- lutions can be derived as well. In this model, imperfect information about the policymaker’s output target leads to lower policy losses.

Trend Inflation in Advanced Economies (2015,  International Journal of Central Banking)

with Christine Garnier and Edward Nelson

• International Journal of Central Banking, Vol 11, Supplement 1, September 2015

• Code https://github.com/elmarmertens/trendInflation 

Abstract: 

We derive estimates of trend inflation for fourteen advanced economies from a framework in which trend shocks exhibit stochastic volatility. The estimated specification allows for time variation in the degree to which longer-term inflation expectations are well anchored in each economy. Our results bring out the effect of changes in monetary regime (such as the adoption of inflation targeting in several countries) on the behavior of trend inflation. Our estimates represent an expansion of those in the previous literature along several dimensions. For each country, we employ a multivariate approach that pools different inflation series in order to identify their common trend. In addition, our estimates of the inflation gap (that is, the difference between trend and observed inflation) are allowed to exhibit considerable persistence—a treatment that affects the trend estimates to some extent. A forecast evaluation based on quasi-real-time estimates registers sizable improvements in inflation forecasts at different horizons for almost all countries considered. It remains the case, however, that simple random-walk forecasts of inflation are difficult to outperform by a statistically significant amount.

Stock Prices, News, and Economic Fluctuations: Comment (2014,  American Economic Review)

with Andre Kurmann

• American Economic Review, Vol 104(4), 1439–45.

• FEDS working paper

• web appendix (pdf)

• Replication Files (zip)

• Discussion by Kilian and Luetkepohl in their monograph on SVARs, see Chapter 10

Abstract: 

Beaudry and Portier (2006) propose an identification scheme to study the effects of news shocks about future productivity in Vector Error Correction Models (VECM). This comment shows that their methodology does not have a unique solution, when applied to their VECMs with more than two variables. The problem arises from the interplay of cointegration assumptions and long-run restrictions imposed by Beaudry and Portier (2006).

Are Spectral Estimators Useful for Long-Run Restrictions in SVARs? (2012, Journal of Economic Dynamics and Control)

• Journal of Economic Dynamics and Control, Vol 36 (12), 1831–1844.

• preprint: pdf, online appendix: pdf

• FEDS Working paper

Abstract: 

No, not really. In response to concerns about the reliability of SVARs, one proposal has been to combine OLS estimates of a VAR with non-parametric estimates of the spectral density. But as shown here, spectral estimators are no panacea for implementing long-run restrictions. They can suffer from small sample and misspecification biases just as VARs do. As a novelty, this paper uses a spectral factorization to ensure a correct representation of the data's variance. But this cannot overcome the basic small sample issues, which arise when trying to estimate long-run properties from relatively short samples of time-series data.

Structural Shocks and the Comovements between Output and Interest Rates (2010, Journal of Economic Dynamics and Control)

• Journal of Economic Dynamics and Control 34 (2010), pp. 1171-1186.

• WP version incl. appendix: pdf

Abstract: 

Stylized facts on U.S. output and interest rates have so far proved hard to match with DSGE models. But model predictions hinge on the joint specification of economic structure and a set of driving processes. In a model, different shocks often induce different comovements, such that the overall pattern depends as much on the specified transmission mechanisms from shocks to outcomes, as well as on the composition of these driving processes. I estimate covariances between output, nominal and real interest rate conditional on several shocks, since such evidence has largely been lacking in previous discussions of the output-interest rate puzzle.

Conditional on shocks to neutral technology and monetary policy, the results square with simple models, like the standard RBC model or a textbook version of the New Keynesian model. In addition, news about future productivity help to explain the overall counter-cyclical behavior of the real rate.

A sub-sample analysis documents also interesting changes in these pattern. During the Great Inflation (1959--1979), permanent shocks to inflation accounted for the counter-cyclical behavior of the real rate and its inverted leading indicator property. Over the Great Moderation (1982--2006), neutral technology shocks were more dominant in explaining comovements between output and interest rates, and the real rate has been pro-cyclical.

Predictability in Financial Markets: What do Survey Expectations Tell Us? (2009, Journal of International Money and Finance)

with Philippe Bacchetta and Eric van Wincoop.

• Journal of International Money and Finance, 2009, Vol 28 (3), 406 - 426

• Appendix

• Slides

Abstract: 

There is widespread evidence of excess return predictability in financial markets. For the foreign exchange market a number of studies have documented that the predictability of excess returns is closely related to the predictability of expectational errors of excess returns. In this paper we investigate the link between the predictability of excess returns and expectational errors in a much broader set of financial markets, using data on survey expectations of market participants in the stock market, the foreign exchange market, the bond market and money markets in various countries. The results are striking. First, in markets where there is significant excess return predictability, expectational errors of excess returns are predictable as well, with the same sign and often even with similar magnitude. This is the case for foreign exchange, stock and bond markets. Second, in the only market where excess returns are generally not predictable, the money market, expectational errors are not predictable either. These findings suggest that an explanation for the predictability of excess returns must be closely linked to an explanation for the predictability of expectational errors.