In a linear-regression model with heteroscedastic errors, we consider two tests: a Hausman test comparing the ordinary least squares (OLS) and least absolute error (LAE) estimators and a test based on the signs of the errors from OLS. It turns out that these are related by the well-known equivalence between Hausman and the generalized method of moments tests. Particular cases, including homoscedasticity and asymmetry in the errors, are discussed.

The limit distribution of conventional test statistics for predictability may depend on the degree of persistence of the predictors. Therefore, diverging results and conclusions may arise because of the different asymptotic theories adopted. Using differencing transformations, we introduce a new class of estimators and test statistics for predictive regression models with Gaussian limit distribution that is instead insensitive to the degree of persistence of the predictors. This desirable feature allows to construct Gaussian confidence intervals for the parameter of interest in stationary, nonstationary, and even locally explosive settings. Besides the limit distribution, we also study the efficiency and the rate of convergence of our new class of estimators. We show that the rate of convergence is $\sqrt n $ in stationary cases, while it can be arbitrarily close to n in nonstationary settings, still preserving the Gaussian limit distribution. Monte Carlo simulations confirm the high reliability and accuracy of our test statistics.

We consider pure-jump transaction-level models for asset prices in continuous time, driven by point processes. In a bivariate model that admits cointegration, we allow for time deformations to account for such effects as intraday seasonal patterns in volatility and nontrading periods that may be different for the two assets. We also allow for asymmetries (leverage effects). We obtain the asymptotic distribution of the log-price process. For the weak fractional cointegration case, we obtain the asymptotic distribution of the ordinary least squares estimator of the cointegrating parameter based on data sampled from an equally spaced discretization of calendar time, and we justify a feasible method of hypothesis testing for the cointegrating parameter based on the corresponding t-statistic. In the strong fractional cointegration case, we obtain the limiting distribution of a continuously averaged tapered estimator as well as other estimators of the cointegrating parameter, and we find that the rate of convergence can be affected by properties of intertrade durations. In particular, the persistence of durations (hence of volatility) can affect the degree of cointegration. We also obtain the rate of convergence of several estimators of the cointegrating parameter in the standard cointegration case. Finally, we consider the properties of the ordinary least squares estimator of the regression parameter in a spurious regression, i.e., in the absence of cointegration.

In this paper, we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications, we also obtain stationarity and $\beta $ -mixing with subgeometrically decaying mixing coefficients.

The paper examines the effect of conditional heteroskedasticity on least squares inference in stochastic regression models of unknown integration order and proposes an inference procedure that is robust to models within the (near) I(0)–(near) I(1) range with GARCH innovations. We show that a regressor signal of exact order $O_{p}\left ( n\kappa _{n}\right ) $ for arbitrary $\,\kappa _{n}\rightarrow \infty $ is sufficient to eliminate stationary GARCH effects from the limit distributions of least squares based estimators and self-normalized test statistics. The above order dominates the $O_{p}\left ( n\right ) $ signal of stationary regressors but may be dominated by the $O_{p}\left ( n^{2}\right ) $ signal of I(1) regressors, thereby showing that least squares invariance to GARCH effects is not an exclusively I(1) phenomenon but extends to processes with persistence degree arbitrarily close to stationarity. The theory validates standard inference for self normalized test statistics based on the ordinary least squares estimator when $\kappa _{n}\rightarrow \infty $ and $\kappa _{n}/n\rightarrow 0$ and the IVX estimator (Phillips and Magdalinos (2009a), Econometric Inference in the Vicinity of Unity. Working paper, Singapore Management University; Kostakis, Magdalinos, and Stamatogiannis, 2015a, Review of Financial Studies 28(5), 1506–1553.) when $\kappa _{n}\rightarrow \infty $ and the innovation sequence of the system is a covariance stationary vec-GARCH process. An adjusted version of the IVX–Wald test is shown to also accommodate GARCH effects in purely stationary regressors, thereby extending the procedure’s validity over the entire (near) I(0)–(near) I(1) range of regressors under conditional heteroskedasticity in the innovations. It is hoped that the wide range of applicability of this adjusted IVX–Wald test, established in Theorem 4.4, presents an advantage for the procedure’s suitability as a tool for applied research.

Lai and Wei (1983, Annals of Statistics 10, 154–166) state in their Theorem 1 that the estimators of the regression coefficients in the regression , t ∈ ℕ are almost surely (a.s.) consistent under the assumption that the minimum eigenvalue λmin(T) of tends to infinity (a.s.) and log(λmax(T))/λmin(T) → 0 (a.s.) where λmax(T) denotes the maximal eigenvalue. Moreover the rate of convergence in this case equals . In this note xt is taken to be a particular multivariate multifrequency I(1) processes, and almost sure rates of convergence for least squares estimators are established.

Hahn and Ridder (2013, Econometrica 81, 315–340) formulated influence functions of semiparametric three-step estimators where generated regressors are computed in the first step. This class of estimators covers several important examples for empirical analysis, such as production function estimators by Olley and Pakes (1996, Econometrica 64, 1263–1297) and propensity score matching estimators for treatment effects by Heckman, Ichimura, and Todd (1998, Review of Economic Studies 65, 261–294). The present article studies a nonparametric likelihood-based inference method for the parameters in such three-step estimation problems. In particular, we apply the general empirical likelihood theory of Bravo, Escanciano, and van Keilegom (2018, Annals of Statistics, forthcoming) to modify semiparametric moment functions to account for influences from plug-in estimates into the above important setup, and show that the resulting likelihood ratio statistic becomes asymptotically pivotal without undersmoothing in the first and second step nonparametric estimates.

02.6.1. Oblique projectorsTwo solutions are published, proposed independently by Götz Trinkler (the poser of the problem) and by Hans Joachim Werner. Excellent solutions have been independently proposed by G. Dhaene, L. Lauwers, S. Lawford, and H. Neudecker.—solution.

Professor Olav Reiersøl was born on 28 June 1908 in the countryside in Norway. He enrolled at Oslo University in 1928, where he studied chemistry, mathematics, physics, and botany. Choosing mathematics as his main subject, he received the cand. real. degree in 1935. As a graduate in mathematics and mathematical statistics he was employed as an assistant by Professor Ragnar Frisch.