We propose a statistical procedure to determine the dimension of the nonstationary subspace of cointegrated functional time series taking values in the Hilbert space of square-integrable functions defined on a compact interval. The procedure is based on sequential application of a proposed test for the dimension of the nonstationary subspace. To avoid estimation of the long-run covariance operator, our test is based on a variance ratio-type statistic. We derive the asymptotic null distribution and prove consistency of the test. Monte Carlo simulations show good performance of our test and provide evidence that it outperforms the existing testing procedure. We apply our methodology to three empirical examples: age-specific U.S. employment rates, Australian temperature curves, and Ontario electricity demand.

We develop a nonparametric Bayesian analysis of regression discontinuity (RD) designs, allowing for covariates, in which we model and estimate the unknown functions of the forcing variable by basis expansion methods. In a departure from current methods, we use the entire data on the forcing variable, but we emphasize the data near the threshold by placing some knots at and near the threshold, a technique we refer to as soft-windowing. To handle the nonequally spaced knots that emerge from soft-windowing, we construct a prior on the spline coefficients, from a second-order Ornstein–Uhlenbeck process, which is hyperparameter light, and satisfies the Kullback–Leibler support property. In the fuzzy RD design, we explain the divergence between the treatment implied by the forcing variable, and the actual intake, by a discrete confounder variable, taking three values, complier, never-taker, and always-taker, and a model with four potential outcomes. Choice of the soft-window, and the number of knots, is determined by marginal likelihoods, computed by the method of Chib [Journal of the American Statistical Association, 1995, 90, 1313–1321] as a by-product of the Markov chain Monte Carlo (MCMC)-based estimation. Importantly, in each case, we allow for covariates, incorporated nonparametrically by additive natural cubic splines. The potential outcome error distributions are modeled as student-t, with an extension to Dirichlet process mixtures. We derive the large sample posterior consistency, and posterior contraction rate, of the RD average treatment effect (ATE) (in the sharp case) and RD ATE for compliers (in the fuzzy case), as the number of basis parameters increases with sample size. The excellent performance of the methods is documented in simulation experiments, and in an application to educational attainment of women from Meyersson [Econometrica, 2014, 82, 229–269].

We estimate and test for multiple structural breaks in distribution via an empirical characteristic function approach. By minimizing the sum of squared generalized residuals, we can consistently estimate the break fractions. We propose a sup-F type test for structural breaks in distribution as well as an information criterion and a sequential testing procedure to determine the number of breaks. We further construct a class of derivative tests to gauge possible sources of structural breaks, which is asymptotically more powerful than the smoothed nonparametric tests for structural breaks. Simulation studies show that our method performs well in determining the appropriate number of breaks and estimating the unknown breaks. Furthermore, the proposed tests have reasonable size and excellent power in finite samples. In an application to exchange rate returns, our tests are able to detect structural breaks in distribution and locate the break dates. Our tests also indicate that the documented breaks appear to occur in variance and higher-order moments, but not so often in mean.

Unobserved heterogeneous treatment effects have been emphasized in the recent policy evaluation literature (see, e.g., Heckman and Vytlacil (2005, Econometrica 73, 669–738)). This paper proposes a nonparametric test for unobserved heterogeneous treatment effects in a treatment effect model with a binary treatment assignment, allowing for individuals’ self-selection to the treatment. Under the standard local average treatment effects assumptions, i.e., the no defiers condition, we derive testable model restrictions for the hypothesis of unobserved heterogeneous treatment effects. Furthermore, we show that if the treatment outcomes satisfy a monotonicity assumption, these model restrictions are also sufficient. Then, we propose a modified Kolmogorov–Smirnov-type test which is consistent and simple to implement. Monte Carlo simulations show that our test performs well in finite samples. For illustration, we apply our test to study heterogeneous treatment effects of the Job Training Partnership Act on earnings and the impacts of fertility on family income, where the null hypothesis of homogeneous treatment effects gets rejected in the second case but fails to be rejected in the first application.

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.