This paper proposes a test of the single equilibrium in the data assumption commonly maintained when estimating static discrete games of incomplete information. By allowing for discrete common knowledge payoff-relevant unobserved heterogeneity, the test generalizes existing methods attributing all correlation between players’ decisions to multiple equilibria. It does not require the estimation of payoffs and is therefore useful in empirical applications leveraging multiple equilibria to identify the model’s primitives. The procedure boils down to testing the emptiness of the set of data generating processes that can rationalize the sample through a single equilibrium and a finite mixture over unobserved heterogeneity. Under verifiable conditions, this testable implication is generically sufficient for degenerate equilibrium selection. The main identifying assumption is the existence of an observable variable that plays the role of a proxy for the unobservable heterogeneity. Examples of such proxies are provided based on empirical applications from the existing literature.

This paper studies control function (CF) approaches in endogenous threshold regression where the threshold variable is allowed to be endogenous. We first use a simple example to show that the structural threshold regression (STR) estimator of the threshold point in Kourtellos, Stengos and Tan (2016, Econometric Theory 32, 827–860) is inconsistent unless the endogeneity level of the threshold variable is low compared to the threshold effect. We correct the CF in the STR estimator to generate our first CF estimator using a method that extends the two-stage least squares procedure in Caner and Hansen (2004, Econometric Theory 20, 813–843). We develop our second CF estimator which can be treated as an extension of the classical CF approach in endogenous linear regression. Both these approaches embody threshold effect information in the conditional variance beyond that in the conditional mean. Given the threshold point estimates, we propose new estimates for the slope parameters. The first is a by-product of the CF approach, and the second type employs generalized method of moment (GMM) procedures based on two new sets of moment conditions. Simulation studies, in conjunction with the limit theory, show that our second CF estimator and confidence interval for the threshold point together with the associated second GMM estimator and confidence interval for the slope parameter dominate the other methods. We further apply the new estimation methodology to an empirical application from international trade to illustrate its usefulness in practice.

This paper proposes a robust moment selection method aiming to pick the best model even if this is a moment condition model with mixed identification strength, that is, moment conditions including moment functions that are local to zero uniformly over the parameter set. We show that the relevant moment selection procedure of Hall et al. (2007, Journal of Econometrics 138, 488–512) is inconsistent in this setting as it does not explicitly account for the rate of convergence of parameter estimation of the candidate models which may vary. We introduce a new moment selection procedure based on a criterion that automatically accounts for both the convergence rate of the candidate model’s parameter estimate and the entropy of the estimator’s asymptotic distribution. The benchmark estimator that we consider is the two-step efficient generalized method of moments estimator, which is known to be efficient in this framework as well. A family of penalization functions is introduced that guarantees the consistency of the selection procedure. The finite-sample performance of the proposed method is assessed through Monte Carlo simulations.

We study large sample properties of likelihood ratio tests of the unit-root hypothesis in an autoregressive model of arbitrary order. Earlier research on this testing problem has developed likelihood ratio tests in the autoregressive model of order 1, but resorted to a plug-in approach when dealing with higher-order models. In contrast, we consider the full model and derive the relevant large sample properties of likelihood ratio tests under a local-to-unity asymptotic framework. As in the simpler model, we show that the full likelihood ratio tests are nearly efficient, in the sense that their asymptotic local power functions are virtually indistinguishable from the Gaussian power envelopes. Extensions to sieve-type approximations and different classes of alternatives are also considered.

Two adaptive bandwidth selection methods for minimizing the mean squared error of nonparametric estimators in locally stationary processes are proposed. We investigate a cross-validation approach and a method based on contrast minimization and derive asymptotic properties of both methods. The results are applicable for different statistics under a general setting of local stationarity including nonlinear processes. At the same time, we deepen the general framework for local stationarity based on stationary approximations. For example, a general Bernstein inequality is derived for such processes. The properties of the bandwidth selection methods are also investigated in several simulation studies.

This paper develops a test for homogeneity of the threshold parameter in threshold regression models. The test has a natural interpretation from time series perspectives and can also be applied to test for additional change points in the structural break models. The limiting distribution of the test statistic is derived, and the finite sample properties are studied in Monte Carlo simulations. We apply the new test to the tipping point problem studied by Card, Mas, and Rothstein (2008, Quarterly Journal of Economics 123, 177–218) and statistically justify that the location of the tipping point varies across tracts.

We propose a series-based nonparametric specification test for a regression function when data are spatially dependent, the “space” being of a general economic or social nature. Dependence can be parametric, parametric with increasing dimension, semiparametric or any combination thereof, thus covering a vast variety of settings. These include spatial error models of varying types and levels of complexity. Under a new smooth spatial dependence condition, our test statistic is asymptotically standard normal. To prove the latter property, we establish a central limit theorem for quadratic forms in linear processes in an increasing dimension setting. Finite sample performance is investigated in a simulation study, with a bootstrap method also justified and illustrated. Empirical examples illustrate the test with real-world data.

Combining cross-sectional and time-series data is a long and well-established practice in empirical economics. We develop a central limit theory that explicitly accounts for possible dependence between the two datasets. We focus on common factors as the mechanism behind this dependence. Using our central limit theorem (CLT), we establish the asymptotic properties of parameter estimates of a general class of models based on a combination of cross-sectional and time-series data, recognizing the interdependence between the two data sources in the presence of aggregate shocks. Despite the complicated nature of the analysis required to formulate the joint CLT, it is straightforward to implement the resulting parameter limiting distributions due to a formal similarity of our approximations with Murphy and Topel’s (1985, Journal of Business and Economic Statistics 3, 370–379) formula.

Controlling the bias is central to estimating semiparametric models. Many methods have been developed to control bias in estimating conditional expectations while maintaining a desirable variance order. However, these methods typically do not perform well at moderate sample sizes. Moreover, and perhaps related to their performance, nonoptimal windows are selected with undersmoothing needed to ensure the appropriate bias order. In this paper, we propose a recursive differencing estimator for conditional expectations. When this method is combined with a bias control targeting the derivative of the semiparametric expectation, we are able to obtain asymptotic normality under optimal windows. As suggested by the structure of the recursion, in a wide variety of triple index designs, the proposed bias control performs much better at moderate sample sizes than regular or higher-order kernels and local polynomials.

Conditional value-at-risk (CVaR) and conditional expected shortfall (CES) are widely adopted risk measures which help monitor potential tail risk while adapting to evolving market information. In this paper, we propose an approach to constructing simultaneous confidence bands (SCBs) for tail risk as measured by CVaR and CES, with the confidence bands uniformly valid for a set of tail levels. We consider one-sided tail risk (downside or upside tail risk) as well as relative tail risk (the ratio of upside to downside tail risk). A general class of location-scale models with heavy-tailed innovations is employed to filter out the return dynamics. Then, CVaR and CES are estimated with the aid of extreme value theory. In the asymptotic theory, we consider two scenarios: (i) the extreme scenario that allows for extrapolation beyond the range of the available data and (ii) the intermediate scenario that works exclusively in the case where the available data are adequate relative to the tail level. For finite-sample implementation, we propose a novel bootstrap procedure to circumvent the slow convergence rates of the SCBs as well as infeasibility of approximating the limiting distributions. A series of Monte Carlo simulations confirm that our approach works well in finite samples.

New methods are developed for identifying, estimating, and performing inference with nonstationary time series that have autoregressive roots near unity. The approach subsumes unit-root (UR), local unit-root (LUR), mildly integrated (MI), and mildly explosive (ME) specifications in the new model formulation. It is shown how a new parameterization involving a localizing rate sequence that characterizes departures from unity can be consistently estimated in all cases. Simple pivotal limit distributions that enable valid inference about the form and degree of nonstationarity apply for MI and ME specifications and new limit theory holds in UR and LUR cases. Normalizing and variance stabilizing properties of the new parameterization are explored. Simulations are reported that reveal some of the advantages of this alternative formulation of nonstationary time series. A housing market application of the methods is conducted that distinguishes the differing forms of house price behavior in Australian state capital cities over the past decade.

Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. The limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply in stationary and nonstationary cases. Implications for inference are discussed and a feasible pre-test inference procedure is proposed that takes unknown potential flatness into consideration and provides a practical approach to inference.

Diffusion over a network refers to the phenomenon of a change of state of a cross-sectional unit in one period leading to a change of state of its neighbors in the network in the next period. One may estimate or test for diffusion by estimating a cross-sectionally aggregated correlation between neighbors over time from data. However, the estimated diffusion can be misleading if the diffusion is confounded by omitted covariates. This paper focuses on the measure of diffusion proposed by He and Song (2022, Preprint, arXiv:1812.04195v4 [stat.ME]), provides a method of decomposition analysis to measure the role of the covariates on the estimated diffusion, and develops an asymptotic inference procedure for the decomposition analysis in such a situation. This paper also presents results from a Monte Carlo study on the small sample performance of the inference procedure.

This paper introduces a counting process for event arrivals in high-frequency trading, based on high-dimensional covariates. The novelty is that, under sparsity conditions on the true model, we do not need to impose any model penalty or parameters shrinkage, unlike Lasso. The procedure allows us to derive a central limit theorem to test restrictions in a two-stage estimator. We achieve this by the use of a sign constraint on the intensity which necessarily needs to be positive. In particular, we introduce an additive model to extract the nonlinear impact of order book variables on buy and sell trade arrivals. In the empirical application, we show that the shape and dynamics of the order book are fundamental in determining the arrival of buy and sell trades in the crude oil futures market. We establish our empirical results mapping the covariates into a higher-dimensional space. Consistently with the theoretical results, the estimated models are sparse in the number of parameters. Using this approach, we are also able to compare competing model hypotheses on the basis of an out-of-sample likelihood ratio type of test.

We describe a (nonparametric) prediction algorithm for spatial data, based on a canonical factorization of the spectral density function. We provide theoretical results showing that the predictor has desirable asymptotic properties. Finite sample performance is assessed in a Monte Carlo study that also compares our algorithm to a rival nonparametric method based on the infinite $AR$ representation of the dynamics of the data. Finally, we apply our methodology to predict house prices in Los Angeles.

We develop theoretical finite-sample results concerning the size of wild bootstrap-based heteroskedasticity robust tests in linear regression models. In particular, these results provide an efficient diagnostic check, which can be used to weed out tests that are unreliable for a given testing problem in the sense that they overreject substantially. This allows us to assess the reliability of a large variety of wild bootstrap-based tests in an extensive numerical study.

In many nonlinear panel data models with fixed effects maximum likelihood estimators suffer from the incidental parameters problem, which often entails that point estimates are markedly biased. While the recent literature has mostly generated methods that yield a first-order bias reduction relative to maximum likelihood, we derive a first- and second-order bias correction of the profile likelihood based on “expected quantities” which differs from the corresponding correction based on “sample averages” derived in Dhaene and Sun (2021, Journal of Econometrics 220, 227–252). While consistency and asymptotic normality of our estimator are derived in a setting where both the number of individuals and the number of time periods grow to infinity, we illustrate in a simulation study that our second-order bias reduction indeed yields an estimator with substantially improved small sample properties relative to its first-order unbiased counterpart, especially when less than 10 time periods are available.

We extend the notion of cointegration for time series taking values in a potentially infinite dimensional Banach space. Examples of such time series include stochastic processes in $C[0,1]$ equipped with the supremum distance and those in a finite dimensional vector space equipped with a non-Euclidean distance. We then develop versions of the Granger–Johansen representation theorems for I(1) and I(2) autoregressive (AR) processes taking values in such a space. To achieve this goal, we first note that an AR(p) law of motion can be characterized by a linear operator pencil (an operator-valued map with certain properties) via the companion form representation, and then study the spectral properties of a linear operator pencil to obtain a necessary and sufficient condition for a given AR(p) law of motion to admit I(1) or I(2) solutions. These operator-theoretic results form a fundamental basis for our representation theorems. Furthermore, it is shown that our operator-theoretic approach is in fact a closely related extension of the conventional approach taken in a Euclidean space setting. Our theoretical results may be especially relevant in a recently growing literature on functional time series analysis in Banach spaces.

This paper develops an estimation methodology for network data generated from a system of simultaneous equations, which allows for network interdependencies via spatial lags in the endogenous and exogenous variables, as well as in the disturbances. By allowing for higher-order spatial lags, our specification provides important flexibility in modeling network interactions. The estimation methodology builds, among others, on the two-step generalized method of moments estimation approach introduced in Kelejian and Prucha (1998, Journal of Real Estate Finance and Economics 17, 99–121; 1999, International Economic Review 40, 509–533; 2004, Journal of Econometrics 118, 27–50). The paper considers limited and full information estimators, and one- and two-step estimators, and establishes their asymptotic properties. In contrast to some of the earlier two-step estimation literature, our asymptotic results facilitate joint tests for the absence of all forms of network spillovers.

Three-dimensional panel models are widely used in empirical analysis. Researchers use various combinations of fixed effects for three-dimensional panels while the correct specification is unknown. When one imposes a parsimonious model and the true model is rich in complexity, the fitted model inevitably incurs the consequences of misspecification including potential bias. When a richly specified model is employed and the true model is parsimonious, then the consequences typically include a poor fit with larger standard errors than necessary. It is therefore useful for researchers to have good model selection techniques that assist in determining the “true” model or a satisfactory approximation. In this light, Lu, Miao, and Su (2021, Econometric Reviews 40, 867–898) propose methods of model selection. We advance this literature by proposing a method of post-selection inference for regression parameters. Despite our use of the lasso technique as the means of model selection, our assumptions allow for many and even all fixed effects to be nonzero. This property is important to avoid a degenerate distribution of fixed effects which often reflect economic sizes of countries in gravity analyses of trade. Using an international trade database, we document evidence that our key assumption of approximately sparse fixed effects is plausibly satisfied for gravity analyses of trade. We also establish the uniform size control over alternative data generating processes of fixed effects. Simulation studies demonstrate that the proposed method is less biased than under-fitting fixed effect estimators, is more efficient than over-fitting fixed effect estimators, and robustly allows for inference that is as accurate as the oracle estimator.