We consider the adaptive Lasso estimator with componentwise tuning in the framework of a low-dimensional linear regression model. In our setting, at least one of the components is penalized at the rate of consistent model selection and certain components may not be penalized at all. We perform a detailed study of the consistency properties and the asymptotic distribution which includes the effects of componentwise tuning within a so-called moving-parameter framework. These results enable us to explicitly provide a set $\mathcal {M}$ such that every open superset acts as a confidence set with uniform asymptotic coverage equal to 1, whereas removing an arbitrarily small open set along the boundary yields a confidence set with uniform asymptotic coverage equal to 0. The shape of the set $\mathcal {M}$ depends on the regressor matrix as well as the deviations within the componentwise tuning parameters. Our findings can be viewed as a broad generalization of Pötscher and Schneider (2009, Journal of Statistical Planning and Inference 139, 2775–2790; 2010, Electronic Journal of Statistics 4, 334–360), who considered distributional properties and confidence intervals based on components of the adaptive Lasso estimator for the case of orthogonal regressors.

Deviations of asset prices from the random walk dynamic imply the predictability of asset returns and thus have important implications for portfolio construction and risk management. This paper proposes a real-time monitoring device for such deviations using intraday high-frequency data. The proposed procedures are based on unit root tests with in-fill asymptotics but extended to take the empirical features of high-frequency financial data (particularly jumps) into consideration. We derive the limiting distributions of the tests under both the null hypothesis of a random walk with jumps and the alternative of mean reversion/explosiveness with jumps. The limiting results show that ignoring the presence of jumps could potentially lead to severe size distortions of both the standard left-sided (against mean reversion) and right-sided (against explosiveness) unit root tests. The simulation results reveal satisfactory performance of the proposed tests even with data from a relatively short time span. As an illustration, we apply the procedure to the Nasdaq composite index at the 10-minute frequency over two periods: around the peak of the dot-com bubble and during the 2015–2106 stock market sell-off. We find strong evidence of explosiveness in asset prices in late 1999 and mean reversion in late 2015. We also show that accounting for jumps when testing the random walk hypothesis on intraday data is empirically relevant and that ignoring jumps can lead to different conclusions.

The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge–Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge–Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincaré inequality. Under a normalization of the potential, the result extends to convergence in the $L^2$ norm, while the Poincaré inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge–Kantorovich problem, which is of independent interest.

In predictive regressions with variables of unknown persistence, the use of extended IV (IVX) instruments leads to asymptotically valid inference. Under highly persistent regressors, the standard normal or chi-squared limiting distributions for the usual t and Wald statistics may, however, differ markedly from the actual finite-sample distributions which exhibit in particular noncentrality. Convergence to the limiting distributions is shown to occur at a rate depending on the choice of the IVX tuning parameters and can be very slow in practice. A characterization of the leading higher-order terms of the t statistic is provided for the simple regression case, which motivates finite-sample corrections. Monte Carlo simulations confirm the usefulness of the proposed methods.

In recent decades, in the research community of macroeconometric time series analysis, we have observed growing interest in the smoothing method known as the Hodrick–Prescott (HP) filter. This article examines the properties of an alternative smoothing method that looks like the HP filter, but is much less well known. We show that this is actually more like the exponential smoothing filter than the HP filter although it is obtainable through a slight modification of the HP filter. In addition, we also show that it is also like the low-frequency projection of Müller and Watson (2018, Econometrica 86, 775–804). We point out that these results derive from the fact that all three similar smoothing methods can be regarded as a type of graph spectral filter whose graph Fourier transform is discrete cosine transform. We then theoretically reveal the relationship between the similar smoothing methods and provide a way of specifying the smoothing parameter that is necessary for its application. An empirical examination illustrates the results.

This article investigates the instrumental variable quantile regression model (Chernozhukov and Hansen, 2005, Econometrica 73, 245–261; 2013, Annual Review of Economics, 5, 57–81) with a binary endogenous treatment. It offers two identification results when the treatment status is not directly observed. The first result is that, remarkably, the reduced-form quantile regression of the outcome variable on the instrumental variable provides a lower bound on the structural quantile treatment effect under the stochastic monotonicity condition. This result is relevant, not only when the treatment variable is subject to misclassification, but also when any measurement of the treatment variable is not available. The second result is for the structural quantile function when the treatment status is measured with error; the sharp identified set is characterized by a set of moment conditions under widely used assumptions on the measurement error. Furthermore, an inference method is provided in the presence of other covariates.

There are many environments in econometrics which require nonseparable modeling of a structural disturbance. In a nonseparable model with endogenous regressors, key conditions are validity of instrumental variables and monotonicity of the model in a scalar unobservable variable. Under these conditions the nonseparable model is equivalent to an instrumental quantile regression model. A failure of the key conditions, however, makes instrumental quantile regression potentially inconsistent. This article develops a methodology for testing the hypothesis whether the instrumental quantile regression model is correctly specified. Our test statistic is asymptotically normally distributed under correct specification and consistent against any alternative model. In addition, test statistics to justify the model simplification are established. Finite sample properties are examined in a Monte Carlo study and an empirical illustration is provided.

Long-run restrictions are a very popular method for identifying structural vector autoregressions, but they suffer from weak identification when the data is very persistent, i.e., when the highest autoregressive roots are near unity. Near unit roots introduce additional nuisance parameters and make standard weak-instrument-robust methods of inference inapplicable. We develop a method of inference that is robust to both weak identification and strong persistence. The method is based on a combination of the Anderson-Rubin test with instruments derived by filtering potentially nonstationary variables to make them near stationary using the IVX instrumentation method of Magdalinos and Phillips (2009). We apply our method to obtain robust confidence bands on impulse responses in two leading applications in the literature.

In this article, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction models, game theoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of Lp functionals of kernel-type estimators (1 ≤ p < ∞) and is easy to implement in general, mainly due to its recourse to the bootstrap method. The bootstrap procedure is based on the nonparametric bootstrap applied to kernel-based test statistics, with an option of estimating “contact sets.” We provide regularity conditions under which the bootstrap test is asymptotically valid uniformly over a large class of distributions, including cases where the limiting distribution of the test statistic is degenerate. Our bootstrap test is shown to exhibit good power properties in Monte Carlo experiments, and we provide a general form of the local power function. As an illustration, we consider testing implications from auction theory, provide primitive conditions for our test, and demonstrate its usefulness by applying our test to real data. We supplement this example with the second empirical illustration in the context of wage inequality.

This paper is concerned with the problem of identifiability of the parameters of a high frequency multivariate autoregressive model from mixed frequency time series data. We demonstrate identifiability for generic parameter values using the population second moments of the observations. In addition we display a constructive algorithm for the parameter values and establish the continuity of the mapping attaching the high frequency parameters to these population second moments. These structural results are obtained using two alternative tools viz. extended Yule Walker equations and blocking of the output process. The cases of stock and flow variables, as well as of general linear transformations of high frequency data, are treated. Finally, we briefly discuss how our constructive identifiability results can be used for parameter estimation based on the sample second moments.