Estimation of Nonlinear Error CorrectionModels

Asymptotic inference in nonlinear vector error correction models (VECM) thatexhibit regime-specific short-run dynamics is nonstandard and complicated. Thispaper contributes the literature in several important ways. First, we establish theconsistency of the least squares estimator of the cointegrating vector allowing forboth smooth and discontinuous transition between regimes. This is a nonregularproblem due to the presence of cointegration and nonlinearity. Second, we obtainthe convergence rates of the cointegrating vector estimates. They differ dependingon whether the transition is smooth or discontinuous. In particular, we find that therate in the discontinuous threshold VECM is extremely fast, which is n^{3/2},compared to the standard rate of n: This finding is very useful for inference onshort-run parameters. Third, we provide an alternative inference method for thethreshold VECM based on the smoothed least squares (SLS). The SLS estimatorof the cointegrating vector and threshold parameter converges to a functional of avector Brownian motion and it is asymptotically independent of that of the slopeparameters, which is asymptotically normal.Threshold Cointegration, Smooth Transition Error Correction,Least Squares, Smoothed Least Squares, Consistency,Convergence Rate.

The Lasso for High-Dimensional Regression with a Possible Change-Point

We consider a high-dimensional regression model with a possible change-point due to a covariate threshold and develop the Lasso estimator of regression coefficients as well as the threshold parameter. Our Lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non-asymptotic oracle inequalities for both the prediction risk and the $\ell_1$ estimation loss for regression coefficients. Since the Lasso estimator selects variables simultaneously, we show that oracle inequalities can be established without pretesting the existence of the threshold effect. Furthermore, we establish conditions under which the estimation error of the unknown threshold parameter can be bounded by a nearly $n^{-1}$ factor even when the number of regressors can be much larger than the sample size ($n$). We illustrate the usefulness of our proposed estimation method via Monte Carlo simulations and an application to real data

Specification tests for lattice processes

We consider an omnibus test for the correct specification of the dynamics of a sequence S0266466614000310_inline1 in a lattice. As it happens with causal models and d = 1, its asymptotic distribution is not pivotal and depends on the estimator of the unknown parameters of the model under the null hypothesis. One first main goal of the paper is to provide a transformation to obtain an asymptotic distribution that is free of nuisance parameters. Secondly, we propose a bootstrap analog of the transformation and show its validity. Thirdly, we discuss the results when S0266466614000310_inline2 are the errors of a parametric regression model. As a by product, we also discuss the asymptotic normality of the least squares estimator of the parameters of the regression model under very mild conditions. Finally, we present a small Monte Carlo experiment to shed some light on the finite sample behavior of our test

Local M-estimation with discontinuous criterion for dependent and limited observations

This paper examines asymptotic properties of local M-estimators under three sets of high-level conditions. These conditions are sufficiently general to cover the minimum volume predictive region, conditional maximum score estimator for a panel data discrete choice model, and many other widely used estimators in statistics and econometrics. Specifically, they allow for discontinuous criterion functions of weakly dependent observations, which may be localized by kernel smoothing and contain nuisance parameters whose dimension may grow to infinity. Furthermore, the localization can occur around parameter values rather than around a fixed point and the observation may take limited values, which leads to set estimators. Our theory produces three different nonparametric cube root rates and enables valid inference for the local M-estimators, building on novel maximal inequalities for weakly dependent data. Our results include the standard cube root asymptotics as a special case. To illustrate the usefulness of our results, we verify our conditions for various examples such as the Hough transform estimator with diminishing bandwidth, maximum score-type set estimator, and many others

Testing for Non-Nested Conditional Moment Restrictions Using Unconditional Empirical Likelihood

We propose non-nested hypotheses tests for conditional moment restriction models based on the method of generalized empirical likelihood (GEL). By utilizing the implied GEL probabilities from a sequence of unconditional moment restrictions that contains equivalent information of the conditional moment restrictions, we construct Kolmogorov-Smirnov and Cramer-von Mises type moment encompassing tests. Advantages of our tests over Otsu and Whang's (2007) tests are: (i) they are free from smoothing parameters, (ii) they can be applied to weakly dependent data, and (iii) they allow non-smooth moment functions. We derive the null distributions, validity of a bootstrap procedure, and local and global power properties of our tests. The simulation results show that our tests have reasonable size and power performance in finite samples.Empirical likelihood, Non-nested tests, Conditional moment restrictions

Robust Inference for Dynamic Panel Threshold Models

This paper develops robust bootstrap inference for a dynamic panel threshold model to improve the finite sample coverage and to be applicable irrespective of the regression's continuity. When the true model becomes continuous and kinked but this restriction is not imposed in the estimation, we find that the usual rank condition for the GMM identification fails, since the Jacobian of the moment function for the GMM loses the full-column rank property. Instead, we establish the identification in a higher-order expansion and derive a slower $n^{1/4}$ convergence rate for the GMM threshold estimator. Furthermore, we show that it destroys asymptotic normality for both coefficients and threshold estimators and invalidates the standard nonparametric bootstrap. We propose two alternative bootstrap schemes that are robust to the continuity and improve the finite sample coverage of the unknown threshold. One is a grid bootstrap that imposes null values of the threshold location. The other is a robust bootstrap where a resampling scheme is adjusted by a data-driven criterion. We show that both bootstraps are consistent. Finite sample performances of proposed methods are checked through Monte Carlo experiments, and an empirical application is shown

Testing for threshold effects in regression models

In this article, we develop a general method for testing threshold effects in regression models, using sup-likelihood-ratio (LR)-type statistics. Although the sup-LR-type test statistic has been considered in the literature, our method for establishing the asymptotic null distribution is new and nonstandard. The standard approach in the literature for obtaining the asymptotic null distribution requires that there exist a certain quadratic approximation to the objective function. The article provides an alternative, novel method that can be used to establish the asymptotic null distribution, even when the usual quadratic approximation is intractable. We illustrate the usefulness of our approach in the examples of the maximum score estimation, maximum likelihood estimation, quantile regression, and maximum rank correlation estimation. We establish consistency and local power properties of the test. We provide some simulation results and also an empirical application to tipping in racial segregation. This article has supplementary materials online.

Factor-Driven Two-Regime Regression

We propose a novel two-regime regression model where regime switching is driven by a vector of possibly unobservable factors. When the factors are latent, we estimate them by the principal component analysis of a panel data set. We show that the optimization problem can be reformulated as mixed integer optimization, and we present two alternative computational algorithms. We derive the asymptotic distribution of the resulting estimator under the scheme that the threshold effect shrinks to zero. In particular, we establish a phase transition that describes the effect of first-stage factor estimation as the cross-sectional dimension of panel data increases relative to the time-series dimension. Moreover, we develop bootstrap inference and illustrate our methods via numerical studies

Powerful Inference

We develop an inference method for a (sub)vector of parameters identified by conditional moment restrictions, which are implied by economic models such as rational behavior and Euler equations. Building on Bierens (1990), we propose penalized maximum statistics and combine bootstrap inference with model selection. Our method is optimized to be powerful against a set of local alternatives of interest by solving a data-dependent max-min problem for tuning parameter selection. We demonstrate the efficacy of our method by a proof of concept using two empirical examples: rational unbiased reporting of ability status and the elasticity of intertemporal substitution.Comment: 29 pages, 4 figures, 3 table