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.     

Quantile and Probability Curves Without Crossing

This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem (Bassett and Koenker (1982)). The method consists in sorting or monotone rearranging the original estimated non‐monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples, establish a functional delta method for rearrangement‐related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural distribution and quantile functions using data on Vietnam veteran status and earnings.     

On Confidence Intervals for Autoregressive Roots and Predictive Regression

Local to unity limit theory is used in applications to construct confidence intervals (CIs) for autoregressive roots through inversion of a unit root test (Stock (1991)). Such CIs are asymptotically valid when the true model has an autoregressive root that is local to unity (ρ = 1 + c/n), but are shown here to be invalid at the limits of the domain of definition of the localizing coefficient c because of a failure in tightness and the escape of probability mass. Failure at the boundary implies that these CIs have zero asymptotic coverage probability in the stationary case and vicinities of unity that are wider than O(n^−1/3). The inversion methods of Hansen (1999) and Mikusheva (2007) are asymptotically valid in such cases. Implications of these results for predictive regression tests are explored. When the predictive regressor is stationary, the popular Campbell and Yogo (2006) CIs for the regression coefficient have zero coverage probability asymptotically, and their predictive test statistic Q erroneously indicates predictability with probability approaching unity when the null of no predictability holds. These results have obvious cautionary implications for the use of the procedures in empirical practice.     

Generalized Method of Integrated Moments for High‐Frequency Data

We propose a semiparametric two‐step inference procedure for a finite‐dimensional parameter based on moment conditions constructed from high‐frequency data. The population moment conditions take the form of temporally integrated functionals of state‐variable processes that include the latent stochastic volatility process of an asset. In the first step, we nonparametrically recover the volatility path from high‐frequency asset returns. The nonparametric volatility estimator is then used to form sample moment functions in the second‐step GMM estimation, which requires the correction of a high‐order nonlinearity bias from the first step. We show that the proposed estimator is consistent and asymptotically mixed Gaussian and propose a consistent estimator for the conditional asymptotic variance. We also construct a Bierens‐type consistent specification test. These infill asymptotic results are based on a novel empirical‐process‐type theory for general integrated functionals of noisy semimartingale processes.     

Hybrid and Size‐Corrected Subsampling Methods

This paper considers inference in a broad class of nonregular models. The models considered are nonregular in the sense that standard test statistics have asymptotic distributions that are discontinuous in some parameters. It is shown in Andrews and Guggenberger (2009a) that standard fixed critical value, subsampling, and m out of n bootstrap methods often have incorrect asymptotic size in such models. This paper introduces general methods of constructing tests and confidence intervals that have correct asymptotic size. In particular, we consider a hybrid subsampling/fixed‐critical‐value method and size‐correction methods. The paper discusses two examples in detail. They are (i) confidence intervals in an autoregressive model with a root that may be close to unity and conditional heteroskedasticity of unknown form and (ii) tests and confidence intervals based on a post‐conservative model selection estimator.     

Risk of Bayesian Inference in Misspecified Models,and the Sandwich Covariance Matrix

It is well known that, in misspecified parametric models, the maximum likelihood estimator (MLE) is consistent for the pseudo‐true value and has an asymptotically normal sampling distribution with “sandwich” covariance matrix. Also, posteriors are asymptotically centered at the MLE, normal, and of asymptotic variance that is, in general, different than the sandwich matrix. It is shown that due to this discrepancy, Bayesian inference about the pseudo‐true parameter value is, in general, of lower asymptotic frequentist risk when the original posterior is substituted by an artificial normal posterior centered at the MLE with sandwich covariance matrix. An algorithm is suggested that allows the implementation of this artificial posterior also in models with high dimensional nuisance parameters which cannot reasonably be estimated by maximizing the likelihood.     

Subsampling Intervals in Autoregressive Models with Linear Time Trend

A new method is proposed for constructing confidence intervals in autoregressive models with linear time trend. Interest focuses on the sum of the autoregressive coefficients because this parameter provides a useful scalar measure of the long‐run persistence properties of an economic time series. Since the type of the limiting distribution of the corresponding OLS estimator, as well as the rate of its convergence, depend in a discontinuous fashion upon whether the true parameter is less than one or equal to one (that is, trend‐stationary case or unit root case), the construction of confidence intervals is notoriously difficult. The crux of our method is to recompute the OLS estimator on smaller blocks of the observed data, according to the general subsampling idea of Politis and Romano (1994a), although some extensions of the standard theory are needed. The method is more general than previous approaches in that it works for arbitrary parameter values, but also because it allows the innovations to be a martingale difference sequence rather than i.i.d. Some simulation studies examine the finite sample performance.     

Impossibility Results for Nondifferentiable Functionals

We examine challenges to estimation and inference when the objects of interest are nondifferentiable functionals of the underlying data distribution. This situation arises in a number of applications of bounds analysis and moment inequality models, and in recent work on estimating optimal dynamic treatment regimes. Drawing on earlier work relating differentiability to the existence of unbiased and regular estimators, we show that if the target object is not differentiable in the parameters of the data distribution, there exist no estimator sequences that are locally asymptotically unbiased or α‐quantile unbiased. This places strong limits on estimators, bias correction methods, and inference procedures, and provides motivation for considering other criteria for evaluating estimators and inference procedures, such as local asymptotic minimaxity and one‐sided quantile unbiasedness.     

Intersection Bounds: Estimation and Inference

We develop a practical and novel method for inference on intersection bounds, namely bounds defined by either the infimum or supremum of a parametric or nonparametric function, or, equivalently, the value of a linear programming problem with a potentially infinite constraint set. We show that many bounds characterizations in econometrics, for instance bounds on parameters under conditional moment inequalities, can be formulated as intersection bounds. Our approach is especially convenient for models comprised of a continuum of inequalities that are separable in parameters, and also applies to models with inequalities that are nonseparable in parameters. Since analog estimators for intersection bounds can be severely biased in finite samples, routinely underestimating the size of the identified set, we also offer a median‐bias‐corrected estimator of such bounds as a by‐product of our inferential procedures. We develop theory for large sample inference based on the strong approximation of a sequence of series or kernel‐based empirical processes by a sequence of “penultimate” Gaussian processes. These penultimate processes are generally not weakly convergent, and thus are non‐Donsker. Our theoretical results establish that we can nonetheless perform asymptotically valid inference based on these processes. Our construction also provides new adaptive inequality/moment selection methods. We provide conditions for the use of nonparametric kernel and series estimators, including a novel result that establishes strong approximation for any general series estimator admitting linearization, which may be of independent interest.     

Estimating the Probability of Rare Events: Addressing Zero Failure Data

Traditional statistical procedures for estimating the probability of an event result in an estimate of zero when no events are realized. Alternative inferential procedures have been proposed for the situation where zero events have been realized but often these are ad hoc, relying on selecting methods dependent on the data that have been realized. Such data‐dependent inference decisions violate fundamental statistical principles, resulting in estimation procedures whose benefits are difficult to assess. In this article, we propose estimating the probability of an event occurring through minimax inference on the probability that future samples of equal size realize no more events than that in the data on which the inference is based. Although motivated by inference on rare events, the method is not restricted to zero event data and closely approximates the maximum likelihood estimate (MLE) for nonzero data. The use of the minimax procedure provides a risk adverse inferential procedure where there are no events realized. A comparison is made with the MLE and regions of the underlying probability are identified where this approach is superior. Moreover, a comparison is made with three standard approaches to supporting inference where no event data are realized, which we argue are unduly pessimistic. We show that for situations of zero events the estimator can be simply approximated with , where n is the number of trials.     

Structural Nonparametric Cointegrating Regression

Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel estimates, and frequently leads to ill‐posed inverse problems. In functional cointegrating regressions where the regressor is an integrated or near‐integrated time series, it is shown here that inverse and ill‐posed inverse problems do not arise. Instead, simple nonparametric kernel estimation of a structural nonparametric cointegrating regression is consistent and the limit distribution theory is mixed normal, giving straightforward asymptotics that are useable in practical work. It is further shown that use of augmented regression, as is common in linear cointegration modeling to address endogeneity, does not lead to bias reduction in nonparametric regression, but there is an asymptotic gain in variance reduction. The results provide a convenient basis for inference in structural nonparametric regression with nonstationary time series when there is a single integrated or near‐integrated regressor. The methods may be applied to a range of empirical models where functional estimation of cointegrating relations is required.     

GMM Estimation of Autoregressive Roots Near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data and incidental deterministic trends. Such models arise in empirical econometric studies of firm size and in dynamic panel data modeling with weak instruments. The two moment conditions in the GMM approach are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay (1998)). Assuming that the localizing parameter takes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate , slower than , when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data.     

Strict Stationarity Testing and Estimation of Explosive and Stationary Generalized Autoregressive Conditional Heteroscedasticity Models

This paper studies the asymptotic properties of the quasi‐maximum likelihood estimator of (generalized autoregressive conditional heteroscedasticity) GARCH(1, 1) models without strict stationarity constraints and considers applications to testing problems. The estimator is unrestricted in the sense that the value of the intercept, which cannot be consistently estimated in the explosive case, is not fixed. A specific behavior of the estimator of the GARCH coefficients is obtained at the boundary of the stationarity region, but, except for the intercept, this estimator remains consistent and asymptotically normal in every situation. The asymptotic variance is different in the stationary and nonstationary situations, but is consistently estimated with the same estimator in both cases. Tests of strict stationarity and nonstationarity are proposed. The tests developed for the classical GARCH(1, 1) model are able to detect nonstationarity in more general GARCH models. A numerical illustration based on stock indices and individual stock returns is proposed.     

Cointegration in Fractional Systems with Unknown Integration Orders

Cointegrated bivariate nonstationary time series are considered in a fractional context, without allowance for deterministic trends. Both the observable series and the cointegrating error can be fractional processes. The familiar situation in which the respective integration orders are 1 and 0 is nested, but these values have typically been assumed known. We allow one or more of them to be unknown real values, in which case Robinson and Marinucci (2001, 2003) have justified least squares estimates of the cointegrating vector, as well as narrow‐band frequency‐domain estimates, which may be less biased. While consistent, these estimates do not always have optimal convergence rates, and they have nonstandard limit distributional behavior. We consider estimates formulated in the frequency domain, that consequently allow for a wide variety of (parametric) autocorrelation in the short memory input series, as well as time‐domain estimates based on autoregressive transformation. Both can be interpreted as approximating generalized least squares and Gaussian maximum likelihood estimates. The estimates share the same limiting distribution, having mixed normal asymptotics (yielding Wald test statistics with χ² null limit distributions), irrespective of whether the integration orders are known or unknown, subject in the latter case to their estimation with adequate rates of convergence. The parameters describing the short memory stationary input series are √n‐consistently estimable, but the assumptions imposed on these series are much more general than ones of autoregressive moving average type. A Monte Carlo study of finite‐sample performance is included.     

Program Evaluation and Causal Inference With High‐Dimensional Data

In this paper, we provide efficient estimators and honest confidence bands for a variety of treatment effects including local average (LATE) and local quantile treatment effects (LQTE) in data‐rich environments. We can handle very many control variables, endogenous receipt of treatment, heterogeneous treatment effects, and function‐valued outcomes. Our framework covers the special case of exogenous receipt of treatment, either conditional on controls or unconditionally as in randomized control trials. In the latter case, our approach produces efficient estimators and honest bands for (functional) average treatment effects (ATE) and quantile treatment effects (QTE). To make informative inference possible, we assume that key reduced‐form predictive relationships are approximately sparse. This assumption allows the use of regularization and selection methods to estimate those relations, and we provide methods for post‐regularization and post‐selection inference that are uniformly valid (honest) across a wide range of models. We show that a key ingredient enabling honest inference is the use of orthogonal or doubly robust moment conditions in estimating certain reduced‐form functional parameters. We illustrate the use of the proposed methods with an application to estimating the effect of 401(k) eligibility and participation on accumulated assets. The results on program evaluation are obtained as a consequence of more general results on honest inference in a general moment‐condition framework, which arises from structural equation models in econometrics. Here, too, the crucial ingredient is the use of orthogonal moment conditions, which can be constructed from the initial moment conditions. We provide results on honest inference for (function‐valued) parameters within this general framework where any high‐quality, machine learning methods (e.g., boosted trees, deep neural networks, random forest, and their aggregated and hybrid versions) can be used to learn the nonparametric/high‐dimensional components of the model. These include a number of supporting auxiliary results that are of major independent interest: namely, we (1) prove uniform validity of a multiplier bootstrap, (2) offer a uniformly valid functional delta method, and (3) provide results for sparsity‐based estimation of regression functions for function‐valued outcomes.     

On the Failure of the Bootstrap for Matching Estimators

Matching estimators are widely used in empirical economics for the evaluation of programs or treatments. Researchers using matching methods often apply the bootstrap to calculate the standard errors. However, no formal justification has been provided for the use of the bootstrap in this setting. In this article, we show that the standard bootstrap is, in general, not valid for matching estimators, even in the simple case with a single continuous covariate where the estimator is root‐N consistent and asymptotically normally distributed with zero asymptotic bias. Valid inferential methods in this setting are the analytic asymptotic variance estimator of Abadie and Imbens (2006a) as well as certain modifications of the standard bootstrap, like the subsampling methods in Politis and Romano (1994).     

GMM,GEL, Serial Correlation,and Asymptotic Bias

For stationary time series models with serial correlation, we consider generalized method of moments (GMM) estimators that use heteroskedasticity and autocorrelation consistent (HAC) positive definite weight matrices and generalized empirical likelihood (GEL) estimators based on smoothed moment conditions. Following the analysis of Newey and Smith (2004) for independent observations, we derive second order asymptotic biases of these estimators. The inspection of bias expressions reveals that the use of smoothed GEL, in contrast to GMM, removes the bias component associated with the correlation between the moment function and its derivative, while the bias component associated with third moments depends on the employed kernel function. We also analyze the case of no serial correlation, and find that the seemingly unnecessary smoothing and HAC estimation can reduce the bias for some of the estimators.     

Robust Estimation and Inference for Jumps in Noisy High Frequency Data: A Local‐to‐Continuity Theory for the Pre‐Averaging Method

We develop an asymptotic theory for the pre‐averaging estimator when asset price jumps are weakly identified, here modeled as local to zero. The theory unifies the conventional asymptotic theory for continuous and discontinuous semimartingales as two polar cases with a continuum of local asymptotics, and explains the breakdown of the conventional procedures under weak identification. We propose simple bias‐corrected estimators for jump power variations, and construct robust confidence sets with valid asymptotic size in a uniform sense. The method is also robust to certain forms of microstructure noise.     

Robustness,Infinitesimal Neighborhoods,and Moment Restrictions

This paper is concerned with robust estimation under moment restrictions. A moment restriction model is semiparametric and distribution‐free; therefore it imposes mild assumptions. Yet it is reasonable to expect that the probability law of observations may have some deviations from the ideal distribution being modeled, due to various factors such as measurement errors. It is then sensible to seek an estimation procedure that is robust against slight perturbation in the probability measure that generates observations. This paper considers local deviations within shrinking topological neighborhoods to develop its large sample theory, so that both bias and variance matter asymptotically. The main result shows that there exists a computationally convenient estimator that achieves optimal minimax robust properties. It is semiparametrically efficient when the model assumption holds, and, at the same time, it enjoys desirable robust properties when it does not.     

Applied Nonparametric Instrumental Variables Estimation

Instrumental variables are widely used in applied econometrics to achieve identification and carry out estimation and inference in models that contain endogenous explanatory variables. In most applications, the function of interest (e.g., an Engel curve or demand function) is assumed to be known up to finitely many parameters (e.g., a linear model), and instrumental variables are used to identify and estimate these parameters. However, linear and other finite‐dimensional parametric models make strong assumptions about the population being modeled that are rarely if ever justified by economic theory or other a priori reasoning and can lead to seriously erroneous conclusions if they are incorrect. This paper explores what can be learned when the function of interest is identified through an instrumental variable but is not assumed to be known up to finitely many parameters. The paper explains the differences between parametric and nonparametric estimators that are important for applied research, describes an easily implemented nonparametric instrumental variables estimator, and presents empirical examples in which nonparametric methods lead to substantive conclusions that are quite different from those obtained using standard, parametric estimators.