Mehmet Caner

THRESHOLD AUTOREGRESSION WITH A UNIT ROOT

This paper develops an asymptotic theory of inference for an unrestricted two-regime threshold autoregressive (TAR) model with an autoregressive unit root. We find that the asymptotic null distribution of Wald tests for a threshold are nonstandard and different from the stationary case, and suggest basing inference on a bootstrap approximation. We also study the asymptotic null distributions of tests for an autoregressive unit root, and find that they are nonstandard and dependent on the presence of a threshold effect. We propose both asymptotic and bootstrap-based tests. These tests and distribution theory allow for the joint consideration of nonlinearity (thresholds) and nonstationary (unit roots). Our limit theory is based on a new set of tools that combine unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical process that converges weakly to a two-parameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this two-parameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemployment rate. We find strong evidence of a threshold effect. The point estimates suggest that the threshold effect is in the short-run dynamics, rather than in the dominate root. While the conventional ADF test for a unit root is insignificant, our TAR unit root tests are arguably significant. The evidence is quite strong that the unemployment rate is not a unit root process, and there is considerable evidence that the series is a stationary TAR process. Copyright The Econometric Society.

INSTRUMENTAL VARIABLE ESTIMATION OF A THRESHOLD MODEL

Threshold models (sample splitting models) have wide application in economics. Existing estimation methods are confined to regression models, which require that all right-hand-side variables are exogenous. This paper considers a model with endogenous variables but an exogenous threshold variable. We develop a two-stage least squares estimator of the threshold parameter and a generalized method of moments estimator of the slope parameters. We show that these estimators are consistent, and we derive the asymptotic distribution of the estimators. The threshold estimate has the same distribution as for the regression case (Hansen, 2000, Econometrica 68, 575–603), with a different scale. The slope parameter estimates are asymptotically normal with conventional covariance matrices. We investigate our distribution theory with a Monte Carlo simulation that indicates the applicability of the methods.We thank the two referees and co-editor for constructive comments. Hansen thanks the National Science Foundation for financial support. Caner thanks University of Pittsburgh Central Research Development Fund for financial support.

Size distortions of tests of the null hypothesis of stationarity: evidence and implications for the PPP debate

Tests of the null hypothesis of stationarity against the unit root alternative play an increasingly important role in empirical work in macroeconomics and in international finance. We show that the use of conventional asymptotic critical values for stationarity tests may cause extreme size distortions, if the model under the null hypothesis is highly persistent. This fact calls into question the use of these tests in empirical work. We illustrate the practical importance of this point for tests of long-run purchasing power parity under the recent float. We show that the common practice of viewing tests of stationarity as complementary to tests of the unit root null will tend to result in contradictions and in spurious rejections of long-run PPP. While the size distortions may be overcome by the use of finite-sample critical values, the resulting tests tend to have low power under economically plausible assumptions about the half-life of deviations from PPP. Thus, the fact that stationarity is not rejected cannot be interpreted as convincing evidence in favor of mean reversion. Only in the rare case that stationarity is rejected do size-corrected tests shed light on the question of long-run PPP.

LASSO-TYPE GMM ESTIMATOR

This paper proposes the least absolute shrinkage and selection operator–type (Lasso-type) generalized method of moments (GMM) estimator. This Lasso-type estimator is formed by the GMM objective function with the addition of a penalty term. The exponent of the penalty term in the regular Lasso estimator is equal to one. However, the exponent of the penalty term in the Lasso-type estimator is less than one in the analysis here. The magnitude of the exponent is reduced to avoid the asymptotic bias. This estimator selects the correct model and estimates it simultaneously. In other words, this method estimates the redundant parameters as zero in the large samples and provides the standard GMM limit distribution for the estimates of the nonzero parameters in the model. The asymptotic theory for our estimator is nonstandard. We conduct a simulation study that shows that the Lasso-type GMM correctly selects the true model much more often than the Bayesian information Criterion (BIC) and another model selection procedure based on the GMM objective function.

Finding the Tipping Point ­ When Sovereign Debt Turns Bad

Public debt has surged during the current global economic crisis and is expected to increase further. This development has raised concerns whether public debt is starting to hit levels where it might negatively affect economic growth. Does such a tipping point in public debt exist? How severe would the impact of public debt be on growth beyond this threshold? What happens if debt stays above this threshold for an extended period of time? The present study addresses these questions with the help of threshold estimations based on a yearly dataset of 101 developing and developed economies spanning a time period from 1980 to 2008. The estimations establish a threshold of 77 percent public debt-to-GDP ratio. If debt is above this threshold, each additional percentage point of debt costs 0.017 percentage points of annual real growth. The effect is even more pronounced in emerging markets where the threshold is 64 percent debt-to-GDP ratio. In these countries, the loss in annual real growth with each additional percentage point in public debt amounts to 0.02 percentage points. The cumulative effect on real GDP could be substantial. Importantly, the estimations control for other variables that might impact growth, such as the initial level of per-capita-GDP.

The Validity of Instruments Revisited

This paper shows how valid inferences can be made when an instrumental variable does not perfectly satisfy the orthogonality condition. When there is a mild violation of the orthogonality condition, the Anderson and Rubin (1949) test is oversized. In order to correct this problem, the fractionally resampled Anderson–Rubin test is derived by modifying Wu’s (1990) resampling technique. We select half of the sample when resampling and obtain valid but conservative critical values. Simulations show that our technique performs well even with moderate to large violation of exogeneity when there is a finite sample correction for the block size choice.

Are “Nearly Exogenous Instruments” reliable?

We show that when instruments are nearly exogenous, the two stage least squares t-statistic unpredictably over-rejects or under-rejects the null hypothesis that the endogenous regressor is insignificant and Anderson-Rubin test over-rejects the null. We prove that in the limit these tests are no longer nuisance parameter free.

Time-Varying Betas Help in Asset Pricing: The Threshold CAPM

Although there is a consensus about time variation in market betas, it is not clear how this variation should be captured. Several researchers continue to analyze different versions of the conditional CAPM. However, Ghysels (1998) shows that these conditional CAPM models fail to capture the dynamics of beta risk. In this study, we introduce a new model, threshold CAPM, which outperforms both the conditional and unconditional CAPMs by generating smaller pricing errors. We also show that the beta risk changes through time with the changes in the economic environment and the dynamics of time variation of beta differ across industries. These findings have important implications for asset allocation, portfolio selection, and hedging decisions.

A locally optimal seasonal unit-root test

This article proposes a locally best invariant test of the null hypothesis of seasonal stationarity against the alternative of seasonal unit roots at all or individual seasonal frequencies. An asymptotic distribution theory is derived and the finite-sample properties of the test are examined in a Monte Carlo simulation. The authors test is also compared with the Canova and Hansen test. The proposed test is superior to the Canova and Hansen test in terms of both size and power.

Adaptive Elastic Net for Generalized Methods of Moments

Model selection and estimation are crucial parts of econometrics. This article introduces a new technique that can simultaneously estimate and select the model in generalized method of moments (GMM) context. The GMM is particularly powerful for analyzing complex datasets such as longitudinal and panel data, and it has wide applications in econometrics. This article extends the least squares based adaptive elastic net estimator by Zou and Zhang to nonlinear equation systems with endogenous variables. The extension is not trivial and involves a new proof technique due to estimators’ lack of closed-form solutions. Compared to Bridge-GMM by Caner, we allow for the number of parameters to diverge to infinity as well as collinearity among a large number of variables; also, the redundant parameters are set to zero via a data-dependent technique. This method has the oracle property, meaning that we can estimate nonzero parameters with their standard limit and the redundant parameters are dropped from the equations simultaneously. Numerical examples are used to illustrate the performance of the new method.