Stationarity Testing Under Endogenous Smooth Deterministic Components: Some Asymptotic Results

Stationarity testing for nonlinear time series models which include several smooth trend components with (possibly) unknown parameters is considered. A pseudo-Lagrange multiplier stationarity test is proposed and its asymptotic behaviour is derived. The limiting null distribution generally depends on the unknown parameters of the model. A bootstrap approach permits this problem to be circumvented and consistency of the bootstrapped test is obtained. The theoretical analysis is complemented with a simulation study which allows us to check the performance of the test in finite samples. The article ends with an empirical application.

In some survival studies part of the population may be no longer subject to the event of interest. The called cure rate models take this fact into account. They have been extensively studied for several authors who have proposed extensions and applications in real lifetime data. Classic large sample tests are usually considered in these applications, especially the likelihood ratio. Recently a new test called \textit{gradient test} has been proposed. The gradient statistic shares the same asymptotic properties with the classic likelihood ratio and does not involve knowledge of the information matrix, which can be an advantage in survival models. Some simulation studies have been carried out to explore the behavior of the gradient test in finite samples and compare it with the classic tests in different models. However little is known about the properties of these large sample tests in finite sample for cure rate models. In this work we performed a simulation study based on the promotion time model with Weibull distribution, to assess the performance of likelihood ratio and gradient tests in finite samples. An application is presented to illustrate the results.

Copula-based dynamic models for multivariate time series

This paper addresses the problem of testing the hypothesis that an observed series is difference stationary. The alternative hypothesis is that the series is another nonstationary process; in particular, an autoregressive model with a random parameter is used. A locally best invariant test is developed assuming Gaussianity, and a representation of its asymptotic distribution as a mixture of Brownian motions is found. The performance of the test in finite samples is investigated by simulation. An example is given where the difference stationary assumption for a well-known data series is rejected.

Robustifying Multivariate Trend Tests to Nonstationary Volatility

This paper proposes some new tests for detecting the presence of a unit root in quite general time series models. Our approach is nonparametric with respect to nuisance parameters and thereby allows for a very wide class of weakly dependent and possibly heterogeneously distributed data. The tests accommodate models with a fitted drift and a time trend so that they may be used to discriminate between unit root nonstationarity and stationarity about a deterministic trend. The limiting distributions of the statistics are obtained under both the unit root null and a sequence of local alternatives. The latter noncentral distribution theory yields local asymptotic power functions for the tests and facilitates comparisons with alternative procedures due to Dickey and Fuller. Some simulations are reported which provide evidence on the performance of the new tests in finite samples.

A simple specification test based on fully modified residuals and the cumulative sum (CUSUM) test for cointegration of Xiao and Phillips (2002, Journal of Econometrics, 108, 43–61) are considered as means of testing for functional form in long-run cointegrating relations. It is shown that both tests are consistent under functional form misspecification and lack of cointegration. A simulation experiment is carried out to assess the properties of the tests in finite samples. The Dickey–Fuller test is also considered. The simulation results reveal that the first two tests perform reasonably well. However, the Dickey–Fuller test performs poorly under functional form misspecification.

On a family of test statistics for discretely observed diffusion processes

Constructing smooth tests without estimating the eigenpairs of the limiting process

The first chapter proposes a nonparametric test of significant variables in the partial derivative of a regression mean function. The test is constructed through a variation based measure of the derivative in the directions of the significant variables, with the derivative estimation through a local polynomial estimator. The test is shown to have the asymptotic null distribution and demonstrated to be consistent. The chapter further proposes a wild-bootstrap test, which exhibits the same null distribution regardless of whether the null is valid or not. Through Monte Carlo studies, the test shows encouraging finite sample performances. Through an empirical application, the test is applied to infer certain aspects of regression structures on labor's earning function. The second chapter investigates the role of debt in the firm's production frontier and technical efficiency by employing a firm-level dataset over 1998-2007 and 1998-2013. The impact of debt on frontier is decomposed into a stand-alone neutral effect and indirect non-neutral effects, which alter the output elasticity of production inputs. The effects are estimated through a semiparametric smooth coefficient stochastic frontier model. A nonzero probability for the firms to be fully efficient is allowed, modeled as a function of debt and technical progress. The study shows that an increase in debt significantly shifts firms' frontier downward across different ownerships, regions, and industries. Foreign and private firms are more efficient, with their full efficiency probability increased by debt and technical progress. By contrast, state-owned enterprises (SOEs) and collective firms are much less efficient and their probability of being fully efficient does not increase with more debt. Furthermore, lower efficiency levels are concentrated in the central and western regions and in the mining and public utility industries. The third chapter proposes a semiparametric additive stochastic frontier model for panel data, where inputs and environment variables can enter the frontier individually and interactively through unknown smooth functions. The inefficiency has its mean function known up to certain parameters, and influenced by its determinants that may or may not appear on the frontier. The model disentangles time invariant unobserved heterogeneities from inefficiency, which can be helpful to avoid overestimating the inefficiency level. Different from conventional stochastic frontier models, the proposed model can be identified without the distribution assumption on the composite error, and consistently estimated without suffering from the curse of dimensionality. Thus, a large number of interested variables for frontier or inefficiency determinants can be included, a potentially attractive

Bootstrapping divergence statistics for testing homogeneity in multinomial populations

Response to Zang et al. and Han et al.

Distance correlation and Chatterjee's correlation are two highly representative measures of dependence. Their simplicity and desirable properties have attracted much attention over the past years, leading to numerous extensions and in-depth discussions. The theoretical connections between these two measures, however, have remained largely underexplored, despite some empirical comparisons of their statistical power. This paper investigates some connections between Chatterjee's correlation and the rank-based distance correlation. We show that both belong to a class of dependence measures based on non-negative weighted rank differences, but with distinct weight allocations resulting in different asymptotic null distributions. Motivated by this finding, we construct a new rank distance correlation using Gaussian kernel that, quite interestingly, exhibits Chatterjee-like behaviour under finite sample and specific bandwidth conditions.

In this paper, we extend the heterogeneous panel data stationarity test of Hadri [Econometrics Journal, Vol. 3 (2000) pp. 148–161] to the cases where breaks are taken into account. Four models with different patterns of breaks under the null hypothesis are specified. Two of the models have been already proposed by Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175]. The moments of the statistics corresponding to the four models are derived in closed form via characteristic functions. We also provide the exact moments of a modified statistic that do not asymptotically depend on the location of the break point under the null hypothesis. The cases where the break point is unknown are also considered. For the model with breaks in the level and no time trend and for the model with breaks in the level and in the time trend, Carrion‐i‐Silvestre et al. [Econometrics Journal, Vol. 8 (2005) pp. 159–175] showed that the number of breaks and their positions may be allowed to differ across individuals for cases with known and unknown breaks. Their results can easily be extended to the proposed modified statistic. The asymptotic distributions of all the statistics proposed are derived under the null hypothesis and are shown to be normally distributed. We show by simulations that our suggested tests have in general good performance in finite samples except the modified test. In an empirical application to the consumer prices of 22 OECD countries during the period from 1953 to 2003, we found evidence of stationarity once a structural break and cross‐sectional dependence are accommodated.

SummaryWe consider the problem of testing functional constraints in a class of functional concurrent linear models where both the predictors and the response are functional data measured at discrete time points. We propose test procedures based on the empirical likelihood with bias-corrected estimating equations to conduct both pointwise and simultaneous inferences. The asymptotic distributions of the test statistics are derived under the null and local alternative hypotheses, where sparse and dense functional data are considered in a unified framework. We find a phase transition in the asymptotic null distributions and the orders of detectable alternatives from sparse to dense functional data. Specifically, the tests proposed can detect alternatives of √n-order when the number of repeated measurements per curve is of an order larger than nη0 with n being the number of curves. The transition points η0 for pointwise and simultaneous tests are different and both are smaller than the transition point in the estimation problem. Simulation studies and real data analyses are conducted to demonstrate the methods proposed.