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.

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.

This paper proposes a consistent nonparametric test with good sampling properties to detect instantaneous causality between vector autoregressive (VAR) variables with time-varying variances. The new test takes the form of the U-statistic, and has a limiting standard normal distribution under the null. We further show that the test is consistent against any fixed alternatives, and has nontrivial asymptotic power against a class of local alternatives with a rate slower than $T^{-1/2}$. We also propose a wild bootstrap procedure to better approximate the finite sample null distribution of the test statistic. Monte Carlo experiments are conducted to highlight the merits of the proposed test relative to other popular tests in finite samples. Finally, we apply the new test to investigate the instantaneous causality relationship between money supply and inflation rates in the USA.

Since the implementation of the Basel III Accord, expected shortfall (ES) has gained increasing attention from regulators as a complement to value-at-risk (VaR). The problem of elicitability for ES makes jointly modeling VaR and ES a popular method to study ES. In this article, we develop model averaging for joint VaR and ES regression models that selects the two weight vectors by minimizing a jackknife criterion. We show the large sample properties of the estimators under potential model misspecification with increasing dimension of parameters and the asymptotic optimality of the selected weights in the sense of minimizing the out-of-sample excess final prediction error. Simulation studies and three empirical analyses reveal good finite sample performance.

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.

This paper considers linear rational expectations models in the frequency domain. The paper characterizes existence and uniqueness of solutions to particular as well as generic systems. The set of all solutions to a given system is shown to be a finite-dimensional affine space in the frequency domain. It is demonstrated that solutions can be discontinuous with respect to the parameters of the models in the context of nonuniqueness, invalidating mainstream frequentist and Bayesian methods. The ill-posedness of the problem motivates regularized solutions with theoretically guaranteed uniqueness, continuity, and even differentiability properties.

This study proposes a test for coefficient randomness in autoregressive models where the autoregressive coefficient is local to unity, which is empirically relevant given earlier work. Under this specification, we analyze the effect of the correlation between the random coefficient and disturbance on the properties of tests, a matter that remains largely unexplored in the literature. Our analysis reveals that tests proposed in earlier studies can have poor power when the correlation is moderate to large. The test proposed here is designed to have power functions robust to the correlation. A modified version of the test is suggested that can be applied when the disturbance is serially correlated and conditionally heteroskedastic. The test is shown to have better power properties than existing ones in large and finite samples.

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 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.

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.

We consider bootstrap inference in predictive (or Granger-causality) regressions when the parameter of interest may lie on the boundary of the parameter space, here defined by means of a smooth inequality constraint. For instance, this situation occurs when the definition of the parameter space allows for the cases of either no predictability or sign-restricted predictability. We show that in this context constrained estimation gives rise to bootstrap statistics whose limit distribution is, in general, random, and thus distinct from the limit null distribution of the original statistics of interest. This is due to both (i) the possible location of the true parameter vector on the boundary of the parameter space and (ii) the possible non-stationarity of the posited predicting (resp. Granger-causing) variable. We discuss a modification of the standard fixed-regressor wild bootstrap scheme where the bootstrap parameter space is shifted by a data-dependent function in order to eliminate the portion of limiting bootstrap randomness attributable to the boundary and prove validity of the associated bootstrap inference under non-stationarity of the predicting variable as the only remaining source of limiting bootstrap randomness. Our approach, which is initially presented in a simple location model, has bearing on inference in parameter-on-the-boundary situations beyond the predictive regression problem.

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.

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.

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.

A new explicit solution representation is provided for ARMA recursions with drift and either deterministically or stochastically varying coefficients. It is expressed in terms of the determinants of banded Hessenberg matrices and, as such, is an explicit function of the coefficients. In addition to computational efficiency, the proposed solution provides a more explicit analysis of the fundamental properties of such processes, including their Wold–Cramér decomposition, their covariance structure, and their asymptotic stability and efficiency. Explicit formulae for optimal linear forecasts based either on finite or infinite sequences of past observations are provided. The practical significance of the theoretical results in this work is illustrated with an application to U.S. inflation data. The main finding is that inflation persistence increased after 1976, whereas from 1986 onward, the persistence declines and stabilizes to even lower levels than the pre-1976 period.

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 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.

We study minimax regret treatment rules under matched treatment assignment in a setup where a policymaker, informed by a sample of size N, needs to decide between T different treatments for a $T\geq 2$. Randomized rules are allowed for. We show that the generalization of the minimax regret rule derived in Schlag (2006, ELEVEN—Tests needed for a recommendation, EUI working paper) and Stoye (2009, Journal of Econometrics 151, 70–81) for the case $T=2$ is minimax regret for general finite $T>2$ and also that the proof structure via the Nash equilibrium and the “coarsening” approaches generalizes as well. We also show by example, that in the case of random assignment the generalization of the minimax rule in Stoye (2009, Journal of Econometrics 151, 70–81) to the case $T>2$ is not necessarily minimax regret and derive minimax regret rules for a few small sample cases, e.g., for $N=2$ when $T=3.$

In the case where a covariate x is included, it is shown that a minimax regret rule is obtained by using minimax regret rules in the “conditional-on-x” problem if the latter are obtained as Nash equilibria.

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.

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.