In high-dimensional (HD) sparse linear regression, parameter selection and estimation are addressed using a constraint $l_0$ on the direction of the parameter vector. We begin by establishing a general result that identifies this direction through the leading generalized eigenspace of specific measurable matrices. Using this result, we propose a novel approach to the selection of the best subsets by solving an empirical generalized eigenvalue problem to estimate the direction of the HD parameter. We then introduce a new estimator based on the RIFLE algorithm, providing a non-asymptotic bound for the estimation risk, minimax convergence, and a central limit theorem. Simulations demonstrate the superiority of our method over existing $l_0$-constrained estimators.

This article proposes and studies two Huber-type estimation approaches, namely, the Huber instrumental variable (IV) estimation and the Huber generalized method of moments (GMM) estimation, for a spatial autoregressive model. We establish the consistency, asymptotic distributions, finite sample breakdown points, and influence functions of these estimators. Simulation studies show that compared to the corresponding traditional estimators (the two-stage least squares estimator, the best IV estimator, and the GMM estimator), our estimators are more robust when the unknown disturbances are long-tailed, and our estimators only lose a little efficiency when the disturbances are short-tailed. Moreover, the Huber GMM estimator also outperforms several robust estimators in the literature. Finally, we apply our estimation method to investigate the impact of the urban heat island effect on housing prices. A package is published on GitHub for practitioners to use in their empirical studies.

Panel data often contain stayers and slow movers. The literature proposes an estimator for the average partial effects (APEs) for this setting without a formal theory. The literature is also silent about inference in the presence of stayers and many slow movers. We contribute to this state of the art. First, we develop an asymptotic theory to guarantee that such an estimator is consistent in the presence of stayers and slow movers. Second, we propose its standard error. Third, we relax the existing assumption to allow for “many” slow movers. Fourth, we generalize the existing estimator. Fifth, we establish that this generalized estimator can achieve larger extents of bias reduction and hence faster convergence rates. Simulation studies demonstrate that the conventional 95% confidence interval covers the true value of the APE with 37%–93% frequencies whereas our proposed one achieves 93%–96% coverage frequencies. Using the U.S. Panel Study of Income Dynamics, we find that estimates of the marginal propensity to consume based on our generalized estimator remarkably differ in values from those of the existing estimators. Moreover, the generalized estimator achieves more than three times as small standard errors as those of the existing robust estimator.

A first-order Gaussian autoregressive model is considered. The exact finite-sample joint density of the minimal sufficient statistic is derived, for any value of the autoregressive parameter. This allows us to derive explicitly the exact density of the autocorrelation coefficient and its Studentized t-ratio, whose densities were available only in the asymptotic case and not for all values of the parameter and the statistic. This article also demonstrates how to solve a general problem in statistical distribution theory (well beyond the specific case of autoregressive models), that of inverting confluent characteristic functions in multiple variables.

This article presents novel methods and theories for estimation and inference about parameters in statistical models using machine learning for nuisance parameter estimation when data are dyadic. We propose a dyadic cross-fitting method to remove over-fitting biases under arbitrary dyadic dependence. Together with the use of Neyman orthogonal scores, this novel cross-fitting method enables root-n consistent estimation and inference robustly against dyadic dependence. We demonstrate its versatility by applying it to high-dimensional network formation models and reexamine the determinants of free trade agreements.

This article provides a general asymptotic theory for mildly explosive autoregression. We confirm that Cauchy limit theory remains invariant across a broad range of error processes, including general linear processes with martingale difference innovations, stationary causal processes, and nonlinear autoregressive time series, such as threshold autoregressive and bilinear models. Our results unify the Cauchy limit theory for long memory, short memory, and anti-persistent innovations within a single framework. Notably, we demonstrate that in the presence of anti-persistent innovations, the Cauchy limit theory may be violated when the regression coefficient approaches the local-to-unity range. Additionally, we explore extensions to models with varying drift, which is of significant interest in its own right.

This article develops a new test to detect changes in generalized autoregressive conditionally heteroscedastic (GARCH(1,1)) processes without imposing a stationary assumption. Specifically, the procedure tests the null hypothesis of a GARCH process with constant parameters, either in (strictly) stationary or explosive regimes, against the alternative hypothesis of parameter changes. We derive the limiting distribution of the test statistics and establish their asymptotic consistency. Monte Carlo simulations show that the proposed test has good size control and high power. We demonstrate a prototype application on a small group of stocks and report a further extensive application to more than ten thousand U.S. stocks.

This article proposes a local projection (LP) residual bootstrap method to construct confidence intervals for impulse response coefficients of AR(1) models. Our bootstrap method is based on the LP approach and involves a residual bootstrap procedure applied to AR(1) models. We present theoretical results for our bootstrap method and proposed confidence intervals. First, we prove the uniform consistency of the LP-residual bootstrap over a large class of AR(1) models that allow for a unit root, conditional heteroskedasticity of unknown form, and martingale difference shocks. Then, we prove the asymptotic validity of our confidence intervals over the same class of AR(1) models. Finally, we show that the LP-residual bootstrap provides asymptotic refinements for confidence intervals on a restricted class of AR(1) models relative to those required for the uniform consistency of our bootstrap.

We propose a one-to-many matching estimator of the average treatment effect based on propensity scores estimated by isotonic regression. This approach is predicated on the assumption of monotonicity in the propensity score function, a condition that can be justified in many economic applications. We show that the nature of the isotonic estimator can help us to fix many problems of existing matching methods, including efficiency, choice of the number of matches, choice of tuning parameters, robustness to propensity score misspecification, and bootstrap validity. As a by-product, a uniformly consistent isotonic estimator is developed for our proposed matching method.

Detecting multiple structural breaks at unknown dates is a central challenge in time-series econometrics. Step-indicator saturation (SIS) addresses this challenge during model selection, and we develop its asymptotic theory for tuning parameter choice. We study its frequency gauge—the false detection rate—and show it is consistent and asymptotically normal. Simulations suggest that a smaller gauge minimizes bias in post-selection regression estimates. For the small gauge situation, we develop a complementary Poisson theory. We compare the local power of SIS to detect shifts with that of Andrews’ break test. We find that SIS excels when breaks are near the sample end or closely spaced. An application to U.K. labor productivity reveals a growth slowdown after the 2008 financial crisis.

We investigate the properties of the linear two-way fixed effects (FE) estimator for panel data when the underlying data generating process (DGP) does not have a linear parametric structure. The FE estimator is consistent for some pseudo-true value and we characterize the corresponding asymptotic distribution. We show that the rate of convergence is determined by the degree of model misspecification, and that the asymptotic distribution can be non-normal. We propose a novel autoregressive double adaptive wild (AdaWild) bootstrap procedure applicable for a large class of DGPs. Monte Carlo simulations show that it performs well for panels of small and moderate dimensions. We use data from U.S. manufacturing industries to illustrate the benefits of our procedure.

Time series of counts often display complex dynamic and distributional characteristics. For this reason, we develop a flexible framework combining the integer-valued autoregressive (INAR) model with a latent Markov structure, leading to the hidden Markov model-INAR (HMM-INAR). First, we illustrate conditions for the existence of an ergodic and stationary solution and derive closed-form expressions for the autocorrelation function and its components. Second, we show consistency and asymptotic normality of the conditional maximum likelihood estimator. Third, we derive an efficient expectation–maximization algorithm with steps available in closed form which allows for fast computation of the estimator. Fourth, we provide an empirical illustration and estimate the HMM-INAR on the number of trades of the Standard & Poor’s Depositary Receipts S&P 500 Exchange-Traded Fund Trust. The combination of the latent HMM structure with a simple INAR$(1)$ formulation not only provides better fit compared to alternative specifications for count data, but it also preserves the economic interpretation of the results.

In this article, we develop a novel high-dimensional coefficient estimation procedure based on high-frequency data. Unlike usual high-dimensional regression procedures such as LASSO, we additionally handle the heavy-tailedness of high-frequency observations as well as time variations of coefficient processes. Specifically, we employ the Huber loss and a truncation scheme to handle heavy-tailed observations, while $\ell _{1}$-regularization is adopted to overcome the curse of dimensionality. To account for the time-varying coefficient, we estimate local coefficients which are biased due to the $\ell _{1}$-regularization. Thus, when estimating integrated coefficients, we propose a debiasing scheme to enjoy the law of large numbers property and employ a thresholding scheme to further accommodate the sparsity of the coefficients. We call this robust thresholding debiased LASSO (RED-LASSO) estimator. We show that the RED-LASSO estimator can achieve a near-optimal convergence rate. In the empirical study, we apply the RED-LASSO procedure to the high-dimensional integrated coefficient estimation using high-frequency trading data.

This article proposes a novel method for estimating quantile regression models that account for sample selection. Unlike the approach by Arellano and Bonhomme (2017, Econometrica 85(1), 1–28; hereafter referred to as AB17), which employs a parametric selection equation, our method utilizes a standard binary quantile regression model to handle the selection issue, thereby accommodating general heterogeneity in both the selection and outcome equations. We adopt a semiparametric estimation technique for the outcome quantile regression by integrating local moment conditions, resulting in $\sqrt {n}$-consistent estimators for the quantile coefficients and copula parameter. Monte Carlo simulation results demonstrate that our estimator performs well in finite samples. Additionally, we apply our method to examine the wage distribution among women using a randomly simulated sample from the US General Social Survey. Our key finding is the presence of significant positive selection among women in the US, which is notably more pronounced than the estimates produced by the AB17’s model.

This article studies the identification of complete economic models with testable assumptions. We start with a local average treatment effect ($LATE$) model where the “No Defiers,” the independent IV assumption, and the exclusion restrictions can be jointly refuted by some data distributions. We propose two relaxed assumptions that are not refutable, with one assumption focusing on relaxing the “No Defiers” assumption while the other relaxes the independent IV assumption. The identified set of $LATE$ under either of the two relaxed assumptions coincides with the classical $LATE$ Wald ratio expression whenever the original assumption is not refuted by the observed data distribution. We propose an estimator for the identified $LATE$ and derive the estimator’s limit distribution. We then develop a general method to relax a refutable assumption A. This relaxation method requires finding a function that measures the deviation of an econometric structure from the original assumption A, and a relaxed assumption $\tilde {A}$ is constructed using this measure of deviation. We characterize a condition to ensure the identified sets under $\tilde {A}$ and A coincide whenever A is not refuted by the observed data distribution and discuss the criteria to choose among different relaxed assumptions.

For time series with high temporal correlation, the empirical process converges rather slowly to its limiting distribution. Many statistics in change-point analysis, goodness-of-fit testing, and uncertainty quantification admit a representation as functionals of the empirical process and therefore inherit its slow convergence. As a result, inference based on the asymptotic distribution of those quantities is significantly affected by relatively small sample sizes. We assess the quality of higher-order approximations (HOAs) of the empirical process by deriving the asymptotic distribution of the corresponding error terms. Based on the limiting distribution of the higher-order terms, we propose a novel approach to calculate confidence intervals for statistical quantities such as the median. In a simulation study, we compare coverage rates and lengths of these confidence intervals with those based on the asymptotic distribution of the empirical process and highlight some benefits of HOAs of the empirical process.

Detecting structural changes in economic relationships has been a longstanding challenge in econometrics. Most of the literature on structural breaks has considered abrupt structural breaks. Existing tests for detecting smooth structural change typically rely on kernel estimation. In this article, we introduce a novel tuning-parameter-free test that minimizes a criterion function over all possible nondecreasing or nonincreasing structural change functions. This test is pivotal (after appropriate scaling) in the scalar case and remains computationally simple even in multivariate settings. Compared to existing nonparametric tests, our method offers superior power against local monotonic structural changes and does not involve the choice of a bandwidth parameter. A simulation study and two empirical examples highlight the merits of the proposed test relative to some popular tests for structural changes in the literature.

This article considers a general class of varying coefficient models defined by a set of moment equalities and/or inequalities, where unknown functional parameters are not necessarily point-identified. We propose an inferential procedure for a subvector of the varying parameters and establish the asymptotic validity of the resulting confidence sets uniformly over a broad family of data-generating processes. We also propose a practical specification test for a set of necessary conditions of our model. Monte Carlo studies show that the proposed methods have good finite sample properties. We apply our method to estimate the return to education in China using its 1%-population census data from 2005.

This article studies the robustness of quasi-maximum-likelihood estimation in hidden Markov models when the regime-switching structure is misspecified. Specifically, we examine the case where the data-generating process features a hidden Markov regime sequence with covariate-dependent transition probabilities, but estimation proceeds under a simplified mixture model that assumes regimes are independent and identically distributed. We show that the parameters governing the conditional distribution of the observables can still be consistently estimated under this misspecification, provided certain regularity conditions hold. Our results highlight a practical benefit of using computationally simpler mixture models in settings where regime dependence is complex or difficult to model directly.

We study the uniform convergence rates of nonparametric estimators for a probability density function and its derivatives when the density has a known pole. Such situations arise in some structural microeconometric models, for example, in auction, labor, and consumer search, where uniform convergence rates of density functions are important for nonparametric and semiparametric estimation. Existing uniform convergence rates based on Rosenblatt’s kernel estimator are derived under the assumption that the density is bounded. They are not applicable when there is a pole in the density. We treat the pole nonparametrically and show various kernel-based estimators can attain any convergence rate that is slower than the optimal rate when the density is bounded uniformly over an appropriately expanding support under mild conditions.