Data-Driven Smooth Tests for the Martingale Difference Hypothesis

A general method for testing the martingale difference hypothesis is proposed. The new tests are data-driven smooth tests based on the principal components of certain marked empirical processes that are asymptotically distribution-free, with critical values that are already tabulated. The data-driven smooth tests are optimal in a semiparametric sense discussed in the paper, and they are robust to conditional heteroskedasticity of unknown form. A simulation study shows that the smooth tests perform very well for a wide range of realistic alternatives and have more power than the omnibus and other competing tests. Finally, an application to the S&P 500 stock index and some of its components highlights the merits of our approach.

On the Asymptotic Power Properties of Specification Tests for Dynamic Parametric Regressions

Economic theories in dynamic contexts usually impose certain restrictions on the conditional mean of the underlying economic variables. Omnibus specification tests are the primary tools to test such restrictions when there is no information on the possible alternative. In this paper we study in detail the power properties of a large class of omnibus specification tests for parametric conditional means under time series processes. We show that all omnibus specification tests have a preference for a finite-dimensional space of alternatives (usually unknown to the practitioner) and we characterize such space for Cramér-von Mises (CvM) tests. This fact motivates the use of optimal tests against such preferred spaces instead of the omnibus tests. We proposed new asymptotically optimal directional and smooth tests that are optimally designed for cases in which a finite-dimensional space of alternatives is in mind. The new proposed optimal procedures are asymptotically distribution-free and are valid under weak assumptions on the underlying data generating process. In particular, they are valid under possibly time varying higher conditional moments of unknown form, e.g., conditional heteroskedasticity. A Monte Carlo experiment shows that previous asymptotic results provide good approximations in small sample sizes. Finally, an application of our theory to test the martingale difference hypothesis of some exchange rates provides new information on the rejection of omnibus tests and illustrates the relevance of our results for practitioners.

Nonparametric Tests for Conditional Symmetry in Dynamic Models

This article proposes omnibus tests for conditional symmetry around a parametric function in a dynamic context. Conditional moments may not exist or may depend on the explanatory variables. Test statistics are suitable functionals of the empirical process of residuals and explanatory variables, whose limiting distribution under the null is nonpivotal. The tests are implemented with the assistance of a bootstrap method, which is justified assuming very mild regularity conditions on the specification of the center of symmetry and the underlying serial dependence structure. Finite sample properties are examined by means of a Monte Carlo experiment.Publicad

A Simple Test for the Absence of Covariate Dependence in Hazard Regression Models

This paper extends commonly used tests for equality of hazard rates in a two-sample or k-sample setup to a situation where the covariate under study is continuous. In other words, we test the hypothesis that the conditional hazard rate is the same for all covariate values, against the omnibus alternative as well as more specific alternatives, when the covariate is continuous. The tests developed are particularly useful for detecting trend in the underlying conditional hazard rates or changepoint trend alternatives. Asymptotic distribution of the test statistics are established and small sample properties of the tests are studied. An application to the e¤ect of aggregate Q on corporate failure in the UK shows evidence of trend in the covariate e¤ect, whereas a Cox regression model failed to detect evidence of any covariate effect. Finally, we discuss an important extension to testing for proportionality of hazards in the presence of individual level frailty with arbitrary distribution

How People Judge What Is Reasonable

A classic debate concerns whether reasonableness should be understood statistically (e.g., reasonableness is what is common) or prescriptively (e.g., reasonableness is what is good). This Article elaborates and defends a third possibility. Reasonableness is a partly statistical and partly prescriptive “hybrid,” reflecting both statistical and prescriptive considerations. Experiments reveal that people apply reasonableness as a hybrid concept, and the Article argues that a hybrid account offers the best general theory of reasonableness. First, the Article investigates how ordinary people judge what is reasonable. Reasonableness sits at the core of countless legal standards, yet little work has investigated how ordinary people (i.e., potential jurors) actually make reasonableness judgments. Experiments reveal that judgments of reasonableness are systematically intermediate between judgments of the relevant average and ideal across numerous legal domains. For example, participants’ mean judgment of the legally reasonable number of weeks’ delay before a criminal trial (ten weeks) falls between the judged average (seventeen weeks) and ideal (seven weeks). So too for the reasonable num- ber of days to accept a contract offer, the reasonable rate of attorneys’ fees, the reasonable loan interest rate, and the reasonable annual number of loud events on a football field in a residential neighborhood. Judgment of reasonableness is better predicted by both statistical and prescriptive factors than by either factor alone. This Article uses this experimental discovery to develop a normative view of reasonableness. It elaborates an account of reasonableness as a hybrid standard, arguing that this view offers the best general theory of reasonableness, one that applies correctly across multiple legal domains. Moreover, this hybrid feature is the historical essence of legal reasonableness: the original use of the “reasonable person” and the “man on the Clapham omnibus” aimed to reflect both statistical and prescriptive considerations. Empirically, reasonableness is a hybrid judgment. And normatively, reasonableness should be applied as a hybrid standard

A simple test for normality for time series

This paper considers testing for normality for correlated data. The proposed test procedure employs the skewness-kurtosis test statistic, but studentized by standard error estimators that are consistent under serial dependence of the observations. The standard error estimators are sample versions of the asymptotic quantities that do not incorporate any downweighting, and, hence, no smoothing parameter is needed. Therefore, the main feature of our proposed test is its simplicity, because it does not require the selection of any user-chosen parameter such as a smoothing number or the order of an approximating model.Publicad