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Learning and Escape Dynamics

Dynamic Regime-Switching Macroeconomics

Econometric Theory and Application

Other Macroeconomic Papers



Learning and Escape Dynamics



LEARNING, ADAPTIVE EXPECTATIONS, AND TECHNOLOGY SHOCKS (Forthcoming in Economic Journal)

BY KEVIN HUANG, ZHENG LIU, AND TAO ZHAmodel. When rational expectations are replaced by adaptive expectations, we prove that the self-confirming equilibrium is the same as the steady state rational expectations equilibrium, but that dynamics around the steady state are substantially different between the two equilibria. We show that, in contrast to Williams(2003), the differences are driven mainly by the lack of the wealth effect and the strengthening of the intertemporal substitution effect, not by escapes. As a result, adaptive expectations substantially alter the amplification and propagation mechanisms and allow technology shocks to exert much more impact on macroeconomic variables than do rational expectations.


THE CONQUEST OF SOUTH AMERICAN INFLATION (Atlanta Fed Working Paper 2006-20 (November, 2006))
BY THOMAS SARGENT, NOAH WILLIAMS, AND TAO ZHA

Abstract We infer determinants of Latin American hyperinflations and stabilizations by using the method of maximum likelihood to estimate a hidden Markov model that potentially assigns roles both to fundamentals in the form of government deficits that are financed by money creation and to destabilizing expectations dynamics that can occasionally divorce inflation from fundamentals. Our maximum likelihood estimates allow us to interpret observed inflation rates in terms of variations in the deficits, sequences of shocks that trigger temporary episodes of expectations driven hyperinflations, and occasional superficial reforms that cut inflation without reforming deficits. Our estimates also allow us to infer the deficit adjustments that seem to have permanently stabilized inflation processes.


BAYESIAN ECONOMETRICS OF LEARNING MODELS
BY TAO ZHA

Abstract  Lecture notes for the October 2005 Dynare workshop on learning and monetary policy.


SHOCKS AND GOVERNMENT BELIEFS: THE RISE AND FALL OF AMERICAN INFLATION (American Economic Review, September 2006, vol 96, no 4, pp. 1193-1224)
BY THOMAS SARGENT, NOAH WILLIAMS, AND TAO ZHA
For the program C source code, click on here

Abstract We use a Bayesian Markov Chain Monte Carlo algorithm jointly to estimate the parameters of a `true' data generating mechanism and those of a sequence of approximating models that a monetary authority uses to guide its decisions. Gaps between a true expectational Phillips curve and the monetary authority's approximating non-expectational Phillips curve models unleash inflation that a monetary authority that knows the true model would avoid. A sequence of dynamic programming problems implies that the monetary authority's inflation target evolves as its estimated Phillips curve moves. Our estimates attribute the rise and fall of post WWII inflation in the US to an intricate interaction between the monetary authority's beliefs and economic shocks. Shocks in the 1970s made the monetary authority perceive a tradeoff between inflation and unemployment that ignited big inflation. The monetary authority's beliefs about the Phillips curve changed in ways that account for Volcker's conquest of US inflation.


Dynamic Regime-Switching Macroeconomics


GENERALIZING THE TAYLOR PRINCIPLE: A COMMENT (Forthcoming in American Economic Review)
BY ROGER E. A. FARMER, DANIEL F. WAGGONER, AND TAO ZHA

Abstract  Davig and Leeper (2007) have proposed a condition they call the \emph{generalized Taylor principle} to rule out indeterminate equilibria in a version of the new-Keynesian model where the parameters of the policy rule follow a Markov-switching process. We show that although their condition rules out a subset of indeterminate equilibria, it does not establish uniqueness of the fundamental equilibrium. We discuss the differences between indeterminate fundamental equilibria included by Davig and Leeper's condition and fundamental equilibria that their condition misses. JEL E40, E52, Taylor principle, indeterminacy, Markov switching.


ASYMMETRIC EXPECTATION EFFECTS OF REGIME SWITCHES IN MONETARY POLICY (Forthcoming in Review of Economic Dynamics)
BY ZHENG LIU, DAN WAGGONER, TAO ZHA

Abstract  This paper addresses two substantive issues: (1) Does the magnitude of the expectation effect of regime switching in monetary policy depend on a particular policy regime? (2) Under which regime is the expectation effect quantitatively important? Using two canonical DSGE models, we show that there exists asymmetry in the expectation effect across regimes. The expectation effect under the dovish policy regime is quantitatively more important than that under the hawkish regime. These results suggest that the possibility of regime shifts in monetary policy can have important effects on rational agents' expectation formation and on equilibrium dynamics. They offer a theoretical explanation for the empirical possibility that a policy shift from the dovish regime to the hawkish regime may not be the main source of substantial reductions in the volatilities of inflation and output.


HAS THE FEDERAL RESERVE'S INFLATION TARGET CHANGED?
BY ZHENG LIU, DAN WAGGONER, TAO ZHA

Abstract  We confront a variety of medium-scale regime-switching DSGE models against U.S. macroeconomic time series data. Our goal is to employ a unified Bayesian framework for these models to test empirical evidence of regime changes in the Federal Reserve's inflation target in the post-war period when heteroscedastic shock disturbances are properly taken into account.


UNDERSTANDING THE NEW-KEYNESIAN MODEL WHEN MONETARY POLICY SWITCHES REGIMES (Atlanta Fed Working Paper 2007-12 (July, 2007); NBER Working Paper No. 12965 (March 2007))
BY ROGER E. A. FARMER, DANIEL F. WAGGONER, AND TAO ZHA

Abstract  This paper studies a New-Keynesian model in which monetary policy may switch between regimes. The study is of substantive importance because the extent to which there is indeterminacy or determinacy in this kind of model is largely unknown. We derive a set of sufficient conditions that enable one to construct a wide range of indeterminate solutions. We show that the necessary and sufficient condition for determinacy, provided by \citet{DL07AER}, is necessary, but not sufficient. A number of numerical examples are used to illustrate our general point that indeterminacy is much more prevalent than previously thought not only in theory but in practice.


INDETERMINACY IN A FORWARD LOOKING REGIME SWITCHING MODEL (Forthcoming in a special issue of International Journal of Economic Theory in honor of Jess Benhabib)
BY ROGER E. A. FARMER, DANIEL F. WAGGONER, AND TAO ZHA

Abstract  This paper is about the properties of Markov switching rational expectations (MSRE) models. We present a simple monetary policy model that switches between two regimes with known transition probabilities. The first regime, treated in isolation, has a unique determinate rational expectations equilibrium and the second contains a set of indeterminate sunspot equilibria. We show that the Markov switching model, which randomizes between these two regimes, may contain a continuum of indeterminate equilibria. We provide examples of stationary sunspot equilibria and bounded sunspot equilibria which exist even when the MSRE model satisfies a 'generalized Taylor principle'. Our result suggests that it may be more difficult to rule out non-fundamental equilibria in MRSE models than in the single regime case where the Taylor principle is known to guarantee local uniqueness.


MINIMAL STATE VARIABLE SOLUTIONS TO MARKOV-SWITCHING RATIONAL EXPECTATIONS MODELS
BY ROGER E. A. FARMER, DANIEL F. WAGGONER AND TAO ZHA

Abstract We develop a new method for computing minimal state variable solutions (MSV) to Markov switching rational expectations models. We provide an algorithm to compute an MSV solution and we show how to test a given solution for uniqueness and boundedness. We construct an example, calibrated to US data, and we show that the MSV solution in our example is unique. This solution can potentially explain the observed reduction in the variance of inflation and the interest rate after 1980, in three different ways. The policy rule might have changed, the variance of the fundamental shocks might have fallen or the private sector equations might have been different across regimes. We compare these three explanations for the change in variance and we show that any one of them can potentially account for the facts. Our paper provides the necessary tools for a future empirical study of this issue.


MARKOV-SWITCHING STRUCTURAL VECTOR AUTOREGRESSIONS: THEORY AND APPLICATION
BY JUAN F. RUBIO-RAMIREZ, DANIEL WAGGONER, AND TAO ZHA

Abstract This paper develops a new and easily implementable necessary and sufficient condition for the exact identification of a Markov-switching SVAR model. The theorem applies to models with both linear and some nonlinear restrictions on the structural parameters. We also derive efficient MCMC algorithms to implement sign and long-run restrictions in Markov-switching SVARs. Using our methods, four well-known identification schemes are used to study whether monetary policy has changed in the euro area since the introduction of the European Monetary Union. We find that models restricted to only time-varying shock variances dominate the other models. We find a persistent post-1993 regime that is associated with low volatility of shocks to output, prices, and interest rates. Finally, the output effects of monetary policy shocks are small and uncertain across regimes and models. These results are robust to the four identification schemes studied in this paper.


WERE THERE REGIME SWITCHES IN US MONETARY POLICY?
BY CHRISTOPHER A. SIMS AND TAO ZHA

Abstract A multivariate model, identifying monetary policy and allowing for simultaneity and regime switching in coefficients and variances, is confronted with US data since 1959. The best fit is with a version that allows time variation in structural disturbance variances only. Among versions that allow for changes in equation coefficients also, the best fit is for a one that allows coefficients to change only in the monetary policy rule. That version allows switching among three main regimes and one rarely and briefly occurring regime. The three main regimes correspond roughly to periods when most observers believe that monetary policy actually differed, but the differences among regimes are not large enough to account for the rise, then decline, in inflation of the 70's and 80's. In versions that insist on changes in the policy rule, the estimates imply monetary targeting was central in the early 80's, but also important sporadically in the 70's.


MODEST POLICY INTERVENTIONS (Journal of Monetary Economics, 2003, vol 50, no 8, pp. 1673-1700)
BY ERIC M. LEEPER AND TAO ZHA

Abstract We present a theoretical and empirical framework for computing and evaluating linear projections conditional on hypothetical paths of monetary policy. A modest policy intervention does not significantly shift agents’ beliefs about policy regime and does not induce the changes in behavior that Lucas (1976) emphasizes. Applied to an econometric model of U.S. monetary policy, we find that a rich class of interventions routinely considered by the Federal Reserve is modest and their impacts can be reliably forecasted by an identified linear model. Modest interventions can shift projected paths and probability distributions of macro variables in economically meaningful ways.


Econometric Theory and Application


METHODS FOR INFERENCE IN LARGE MULTIPLE-EQUATION MARKOV-SWITCHING MODELS (Forthcoming in Journal of Econometrics)
BY CHRISTOPHER A. SIMS, DANIEL F. WAGGONER, AND TAO ZHATAO ZHA

Abstract Inference for hidden Markov chain models in which the structure is a multiple-equation macroeconomic model raises a number of difficulties that are not as likely to appear in smaller models. One is likely to want to allow for many regimes in the Markov chain, without allowing the number of free parameters in the transition matrix to grow as the square as the number of regimes, but also without losing a convenient form for the posterior distribution of the transition matrix. Calculation of marginal data densities for assessing model fit is often difficult in high-dimensional models, and seems particularly difficult in these models. This paper gives a detailed explanation of methods for maximizing posterior density and initiating MCMC simulations that provide some robustness against the complex shape of the likelihood in these models. These difficulties and remedies are likely to be useful generally for Bayesian inference in large time series models.


LOCAL AND GLOBAL IDENTIFICATION OF DSGE MODELS: A SIMULTANEOUS-EQUATION APPROACH
 By Martin Fukac, Dan Waggoner, and Tao Zha

Abstract We address some issues about local and global identification of DSGE models and link these issues to identification in the simultaneous-equation VAR framework.


STRUCTURAL VECTOR AUTOREGRESSIONS: THEORY OF IDENTIFICATION AND ALGORITHMS FOR INFERENCE
 By Juan Rubio-Ramirez, Dan Waggoner, and Tao Zha

Abstract SVARs are widely used for policy analysis and to provide stylized facts for dynamic general equilibrium models. Yet there have been no workable rank conditions to ascertain whether an SVAR is globally identified. When identifying restrictions, such as long-run restrictions, are imposed on impulse responses, there have been no efficient algorithms for small-sample estimation and inference. To fill these important gaps in the literature, this paper makes four contributions. First, we establish general rank conditions for global identification of both overidentified and exactly identified models. Second, we show that these conditions can be checked as a simple matrix-filling exercise and that they apply to a wide class of identifying restrictions, including linear and certain nonlinear restrictions. Third, we establish a very simple rank condition for exactly identified models that amounts to a straightforward counting exercise. Fourth, we develop a number of efficient algorithms for small-sample estimation and inference.


MCMC METHOD FOR MARKOV MIXTURE SIMULTANEOUS-EQUATION MODELS: A NOTE
BY CHRISTOPHER A. SIMS AND TAO ZHA
(If you are unable to download this paper, click here)

Abstract This paper extends the existing MCMC simulation methods to a system of simultaneous equations with hidden Markov chains. It overcomes analytical and computational difficulties that arise when one restricts the degree of time variation on the system. We derive the probability density functions of conditional posterior distributions used for the MCMC simulations and develope software that enables one to obtain the solution on a standard PC desktop. Sims and Zha 2004 have applied this method to addressing various questions regarding monetary policy. Despite intensive computation needed to get reliable results, we hope that further innovations in numerical methods and computer technology will make our method easier for applied researchers to use.


NORMALIZATION IN ECONOMETRICS (Econometric Reviews, 2007, vol 26, no 2-4, pp. 221-252)
BY JAMES D. HAMILTON, DANIEL F. WAGGONER, AND TAO ZHA

Abstract The issue of normalization arises whenever two different values for a vector of unknown parameters imply the identical economic model. A normalization implies not just a rule for selecting which among equivalent points to call the MLE, but also governs the topography of the set of points that go into a small-sample confidence interval associated with that MLE. A poor normalization can lead to multimodal distributions, disjoint confidence intervals, and very misleading characterizations of the true statistical uncertainty. This paper introduces the identification principle as a framework upon which a normalization should be imposed, according to which the boundaries of the allowable parameter space should correspond to loci along which the model is locally unidentified. We illustrate these issues with examples taken from mixture models, structural VARs, and cointegration.


BLOCK RECURSION AND STRUCTURAL VECTOR AUTOREGRESSIONS (Journal of Econometrics, 1999, vol 90, pp. 291-316)
BY TAO ZHA

Abstract In applications of structural VAR modeling, finite-sample properties may be difficult to obtain when certain identifying restrictions are imposed on lagged relationships. As a result, even though imposing some lagged restrictions makes economic sense, lagged relationships are often left unrestricted to make statistical inference more convenient. This paper develops block Monte Carlo methods to obtain both maximum likelihood estimates and exact Bayesian inference when certain types of restrictions are imposed on the lag structure. These methods are applied to two examples to illustrate the importance of imposing restrictions on lagged relationships.


VECTOR AUTOREGRESSIONS (The New Palgrave Dictionary of Economics, 2nd Edition, (eds) Blume and Durlauf, eds.)
 BY TAO ZHA

Abstract Vector autoregressions are a class of dynamic multivariate models introduced by Sims (1980) to macroeconomics. These models have been primarily used to bring empirical regularities out of the time series data, to provide forecasting and policy analysis, and to serve as a benchmark for model comparison. Economic applications often impose more restrictions on vector autoregressions than originally thought necessary. Recent econometric developments have made it feasible to handle vector autoregressions with a wide class of restrictions and have narrowed the gap between these models and dynamic stochastic general equilibrium models.


Other Macroeconomic Papers


DOES MONETARY POLICY GENERATE RECESSIONS? (Macroeconomic Dynamics, April 2006, vol 10, no 2, pp. 231-272)
BY CHRISTOPHER A. SIMS AND TAO ZHA

Abstract We consider two kinds of answers to the title question: Do random shifts in monetary policy account for historical recessions, and would changes in the systematic component of monetary policy have allowed reductions in inflation or output variance without substantial costs. The answer to both questions is no. We use weak identifying assumptions and include extensive discussion of these assumptions, including a completely specified dynamic stochastic equilibrium model in which our identifying assumptions can be shown to be approximately satisfied.


BANKRUPTCY LAW, CAPITAL ALLOCATION, AND AGGREGATE EFFECT: A DYNAMIC HETEROGENEOUS AGENT MODEL WITH INCOMPLETE MARKETS (Annals of Economics and Finance 2, 2001, pp. 379–400)
 BY TAO ZHA

Abstract  Under the assumption that asset markets are incomplete, this paper introduces bankruptcy in a stochastic general equilibrium model with capital accumulation and heterogeneous agents. It explores the role of regulatory intervention and argues that intervention in the form of a level of bankruptcy exemption can enhance not only social welfare but also distributive equity. The bankruptcy law is carefully specified in the model. The model generates distributional changes in consumption, capital, and bankruptcy risk in response to changes in bankruptcy law and highlights the macroeconomic effects of these redistributions.

 

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