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.

We contribute to the recent debate on the instability of the slope of the Phillips curve by offering insights from a flexible time-varying instrumental variable (IV) approach robust to weak instruments. Our robust approach focuses directly on the Phillips curve and allows general forms of instability, in contrast to current approaches based either on structural models with time-varying parameters or IV estimates in ad-hoc sub-samples. We find evidence of a weakening of the slope of the Phillips curve starting around 1980. We also offer novel insights on the Phillips curve during the recent pandemic: The flattening has reverted and the Phillips curve is back.

We consider spline-based additive models for estimation of conditional treatment effects. To handle the uncertainty due to variable selection, we propose a method of model averaging with weights obtained by minimizing a J-fold cross-validation criterion, in which a nearest neighbor matching is used to approximate the unobserved potential outcomes. We show that the proposed method is asymptotically optimal in the sense of achieving the lowest possible squared loss in some settings and assigning all weight to the correctly specified models if such models exist in the candidate set. Moreover, consistency properties of the optimal weights and model averaging estimators are established. A simulation study and an empirical example demonstrate the superiority of the proposed estimator over other methods.

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.

For a multidimensional Itô semimartingale, we consider the problem of estimating integrated volatility functionals. Jacod and Rosenbaum (2013, The Annals of Statistics 41(3), 1462–1484) studied a plug-in type of estimator based on a Riemann sum approximation of the integrated functional and a spot volatility estimator with a forward uniform kernel. Motivated by recent results that show that spot volatility estimators with general two-sided kernels of unbounded support are more accurate, in this article, an estimator using a general kernel spot volatility estimator as the plug-in is considered. A biased central limit theorem for estimating the integrated functional is established with an optimal convergence rate. Central limit theorems for properly de-biased estimators are also obtained both at the optimal convergence regime for the bandwidth and when applying undersmoothing. Our results show that one can significantly reduce the estimator’s bias by adopting a general kernel instead of the standard uniform kernel. Our proposed bias-corrected estimators are found to maintain remarkable robustness against bandwidth selection in a variety of sampling frequencies and functions.

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.

This article considers a three-dimensional latent factor model in the presence of one set of global factors and two sets of local factors. We show that the numbers of global and local factors can be estimated uniformly and consistently. Given the number of global and local factors, we propose a two-step estimation procedure based on principal component analysis (PCA) and establish the asymptotic properties of the PCA estimators. Monte Carlo simulations demonstrate that they perform well in finite samples. An application to the dataset of international trade reveals the relative importance of different types of factors.

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 studies estimation and inference in the autoregressive (AR) models with unspecified and heavy-tailed heteroskedastic noises. A piece-wise locally stationary structure of the noise is constructed to capture various forms of heterogeneity, without imposing any restrictions on the tail index. The new nonstationary AR model allows for not only time-varying conditional features but also unconditional variance and tail index. This makes it appealing in practice, with wide applications in economics and finance. To obtain a feasible inference, we investigate the self-weighted least absolute deviation estimator and derive its asymptotic normality. Since the asymptotic variance relies on an unobserved density, a bootstrap method is proposed to approximate the limiting distribution. Based on the conditional moment condition, a portmanteau test from residuals is further proposed to detect misspecifications in the proposed model. A simulation study and two applications to time series illustrate our inference procedures.

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.

The primary focus of this article is to capture heterogeneous treatment effects measured by the conditional average treatment effect. A model averaging estimation scheme is proposed with multiple candidate linear regression models under heteroskedastic errors, and the properties of this scheme are explored analytically. First, it is shown that our proposal is asymptotically optimal in the sense of achieving the lowest possible squared error. Second, the convergence of the weights determined by our proposal is provided when at least one of the candidate models is correctly specified. Simulation results in comparison with several related existing methods favor our proposed method. The method is applied to a dataset from a labor skills training program.

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.