A time-domain test for some types of nonlinearity

Hypothesis tests

Testing time series linearity via goodness-of-fit methods

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.

Semiparametric Innovation-Based Tests of Orthogonality and Causality between Two Infinite-Order Cointegrated Ceries with Application to Canada/US Monetary Interactions

Sieve IV estimation of cross-sectional interaction models with nonparametric endogenous effect

We propose a test for the equality of the autocovariance functions of two independent and stationary time series. The test statistic is a quadratic form in the vector of differences of the first J + 1 autocovariances. Its asymptotic distribution is derived under the null hypothesis, and the finite‐sample properties of the test, namely the bias and the power, are investigated by Monte Carlo methods. A by‐product of this study is a new estimator of the covariance between two sample autocovariances which provides a positive definite covariance matrix. We establish the convergence of this estimator in the L1 norm.

A bootstrap test for the comparison of nonlinear time series

Meta-analysis plays an important role in the analysis and interpretation of clinical trials in medicine and of trials in the social sciences but is of importance in other fields (e.g., particle physics 1) as well. In 2001, Hartung and Knapp 2, 3 introduced a new approach to test for a nonzero treatment effect in a meta-analysis of k studies. Hartung and Knapp 2, 3 suggest to use the random effects estimate according to DerSimonian and Laird 4 and propose a variance estimator q so that the test statistics for the treatment effect is t distributed with k − 1 degrees of freedom. In their paper on dichotomous endpoints, results of a simulation study with 6 and 12 studies illustrate for risk differences, log relative risks and log odds ratios, the excellent properties regarding control of the type I error, and the achieved power 2. They investigate different sample sizes in each study, and different amounts of heterogeneity between studies and compare their new approach (Hartung and Knapp approach (HK)) with the fixed effects approach (FE) and the classical random effects approach by DerSimonian and Laird (DL). It can be clearly seen that, with increasing heterogeneity, the FE as well as the DL does not control the type I error rate, while the HK keeps the type I error rate in nearly every situation and in every scale. Advantages and disadvantages of the two standard approaches and respective test statistics have been extensively discussed (e.g., 5-7). While it is well known that the FE is too liberal in the presence of heterogeneity, the DL is often thought to be rather conservative because heterogeneity is incorporated into the standard error of the estimate for the treatment effect and this should lead to larger confidence intervals and smaller test statistics for the treatment effect (8 chapter 9.4.4.3). This was disproved among others by Ziegler and Victor 7, who observed in situations with increasing heterogeneity severe inflation of the type I error for the DerSimonian and Laird test statistic. Notably, the asymptotic properties of this approach will be valid, if both the number of studies and the number of patients per study are large enough (8 chapter 9.54, 9, 10). Although power issues of meta-analysis tests have received some interest, comparisons between the approaches and the situation with two studies were not the main interest 11, 12. Borenstein et al. (10, pp. 363/364) recommend the random effects approach in general for meta-analysis and do not recommend meta-analyses of small numbers of studies. However, meta-analyses of few and of even only two trials are of importance. In drug licensing in many instances, two successful phase III clinical trials have to be submitted as pivotal evidence for drug licensing 13, and summarizing the findings of these studies is required according to the International Conference on Harmonisation guidelines E9 and M4E (14, 15). It is stated that ‘An overall summary and synthesis of the evidence on safety and efficacy from all the reported clinical trials is required for a marketing application [...]. This may be accompanied, when appropriate, by a statistical combination of results’ (14, p. 31). For the summary, ‘The use of meta-analytic techniques to combine these estimates is often a useful addition, because it allows a more precise overall estimate of the size of the treatment effects to be generated, and provides a complete and concise summary of the results of the trials’ (14, p. 32). While in standard drug development, this summary will include usually more than two studies; in rare diseases for the same intervention, barely ever more than two studies are available because of the limited number of patients. Likewise, decision making in the context of health technology assessment is based on systematic reviews and meta-analyses. Often in practice, only two studies are considered homogeneous enough from clinical grounds to be included into a meta-analysis and then form the basis for decision making about reimbursement 16. Despite the fact that meta-analysis is non-experimental observational (secondary) research 17 and p-values should be interpreted with caution, meta-analyses of randomized clinical trials are termed highest-level information in evidence-based medicine and are the recommended basis for decision making 18. As statistical significance plays an important role in the assessment of the meta-analysis, it is mandatory to understand the statistical properties of the relevant methodology also in a situation, where only two clinical trials are included into a meta-analysis. We found Cochrane reviews including meta-analyses with two studies only, which are considered for evidence-based decision making even in the presence of a large amount of heterogeneity (I2≈75%) 19-21 We repeated the simulation study for dichotomous endpoints of Hartung and Knapp 2 with programs written in R 3.1.0 22 to compare the statistical properties of the FE, the DL, and the HK for testing the overall treatment effect θ (H0: θ = 0) in a situation with two to six clinical trials. We considered scenarios under the null and alternative hypothesis for the treatment effect with and without underlying heterogeneity. We present the findings for the odds ratio with pC=0.2 and did vary probability of success in the treatment group pT to investigate the type I error and the power characteristics. The total sample size per meta-analysis was kept constant in the different scenarios (n = 480) and n/k number of patients per study to clearly demonstrate the effect of the number of included studies on power and type I error of the various approaches. Likewise, we attempted to avoid problems with zero cell counts or extremely low event rates that may impact on type I error and power as well. I2 was used to describe heterogeneity because thresholds have been published (low: I2=25%, moderate: I2=50%, and high: I2=75%) 23 for the quantification of the degree of heterogeneity with this measure. We termed I2≤15% negligible, and this refers to simulations assuming no heterogeneity (i.e., the fixed effects model). Table 1 summarizes the results of our simulation study. The well-known anticonservative behavior of the FE and the DL in the presence of even low heterogeneity is visible for small numbers of studies in the meta-analysis. Particularly for the FE, the increase in the type I error is pronounced. With more than four studies even in situations with substantial heterogeneity, the HK perfectly controls the type I error. There is almost no impact on the power of the test in situations with no or low heterogeneity, and overall, it seems as if the only price to be paid for an increased heterogeneity is a reduced power of the test. This is in strong contrast to the situation with only two studies. Again, the HK perfectly controls the prespecified type I error. However, even in a homogeneous situation, the power of the meta-analysis test was lower than 15% in situations where the power of the FE and the DL approximates 70% and 60%, respectively. In the presence of even low heterogeneity with the HK, there is not much chance to arrive at a positive conclusion even with substantial treatment effects. Figure 1 summarizes the main finding of our simulation study with k = 2 and 6 studies impressively. In the homogeneous situation with two studies, the DL and even better the FE can be used to efficiently base conclusions on a meta-analysis. In contrast, already with mild to moderate heterogeneity, both standard tests severely violate the prespecified type I error, and there is a high risk of false positive conclusion with the classical approaches. This has major implications for decision making in drug licensing as well. We have noted previously that a meta-analysis can be confirmatory if a drug development program was designed to include a preplanned meta-analysis of the two pivotal trials 24. As an example, thrombosis prophylaxis was discussed in the paper by Koch and Röhmel 24, where venous thromboembolism is accepted as primary endpoint in the pivotal trials. In case when both pivotal trials are successful, they can be combined to demonstrate a positive impact on, for example, mortality. This can be preplanned as a hierarchical testing procedure: first, both pivotal trials will be assessed individually before confirmatory conclusions will be based on the meta-analysis. As explained, neither the FE, nor the DL, nor the HK can be the methodology to be recommended for a priori planning in this sensitive area unless any indication for heterogeneity is taken as a trigger not to combine studies in a meta-analysis at all. It is our belief that not enough emphasis has been given to this finding in the original paper and the important role of heterogeneity is not acknowledged enough in the discussion of findings from meta-analyses, in general. This work has been funded partly by the FP7-HEALTH-2013-INNOVATION-1 project Advances in Small Trials Design for Regulatory Innovation and Excellence (ASTERIX) Grant Agreement No. 603160. We thank the editors and reviewers for useful comments to tailor the main messages.

ABSTRACTWe consider multiple comparison test procedures among treatment effects in a randomized block design. We propose closed testing procedures based on maximum values of some two-sample t test statistics and based on F test statistics. It is shown that the proposed procedures are more powerful than single-step procedures and the REGW (Ryan/Einot–Gabriel/Welsch)-type tests. Next, we consider the randomized block design under simple ordered restrictions of treatment effects. We propose closed testing procedures based on maximum values of two-sample one-sided t test statistics and based on Batholomew’s statistics for all pairwise comparisons of treatment effects. Although single-step multiple comparison procedures are utilized in general, the power of these procedures is low for a large number of groups. The closed testing procedures stated in the present article are more powerful than the single-step procedures. Simulation studies are performed under the null hypothesis and some alternative hypotheses. In this studies, the proposed procedures show a good performance.

Approximating a class of goodness-of-fit test statistics

A consistent nonparametric test for linearity of AR( p) models

A recent article in this journal by Tabak (2006) highlighted a potentially serious source of bias that can arise when multiple statistical tests of a damages theory are performed and even one test rejecting the null hypothesis is regarded as supporting the damages theory. Repeated testing will eventually produce a “false discovery,” that is, rejection of the null hypothesis in favor of the alternative hypothesis when the null hypothesis is true, which statisticians refer to as type I error. Consequently, performing multiple tests without adjusting the critical value can be problematical because it can lead to improperly accepting statistical evidence that apparently supports rejection of the null hypothesis as reliable when it is not. Tabak (2006) recommends making the Sidak (1968, 1971) multiple-comparison adjustment to the standard statistical t-test to correct for the false-discovery bias inherent in multiple-comparison testing. In particular, he recommends making this adjustment when performing 10b-5 securities fraud event studies when more than one corrective disclosure date is involved. This article clarifies the circumstances in which a multiple-comparison adjustment is appropriate and explains why the correction is normally not needed in securities fraud event-study testing. More generally, I explain why it is not required when each of several tests is performed and its results are reported separately, as for example, where the objective is simply to test the statistical significance of the abnormal stock return on each day on which a new and distinct curative disclosure occurred. I show that the Sidak multiple-comparison adjustment is nearly as stringent as the classical Bonferroni procedure (Simes, 1986), which can increase the risk of type II error. This article discusses a more powerful alternative to the Sidak adjustment due to Benjamini and Hochberg (1993), which directly corrects for the false-discovery bias in multiple-comparison testing and reduces the risk of type II error. II. Application of Multiple-Comparison Adjustments to Securities Fraud Event Studies Multiple-comparison-false-discovery bias can arise when (a) several statistical tests are performed on subsets of the same larger data set in an effort to

We propose generalized portmanteau-type test statistics in the frequency domain to test independence between two stationary time series. The test statistics are formed analogous to the one in the paper by Chen and Deo (2004,Econometric Theory20, 382–416), who extended the applicability of the portmanteau goodness-of-fit test to the long memory case. Under the null hypothesis of independence, the asymptotic standard normal distributions of the proposed statistics are derived under fairly mild conditions. In particular, each time series is allowed to possess short memory, long memory, or antipersistence. A simulation study shows that the tests have reasonable size and power properties.

In this article, we introduce two new test statistics based on pairwise distance to check whether a parametric model with missing response is correctly specified or not. The proposed test statistics are easy to implement and no non parametric function estimation is involved. The asymptotic properties of the newly introduced test statistics are investigated. Under the null hypothesis, it is shown that they converge to a non degenerate distribution. And it can also detect any local alternatives that are different from the null at the parametric rate. In addition, the convergence rates of the proposed test are unaffected by the dimension of covariates. Some simulation studies and a real data example are conducted to show the good performance of the proposed test.

In this study, we consider the testing problem about the null hypothesis that the nonlinear part in the partial linear regression model with missing response at random is a parametric function or not against the alternative that it is nonparametric. By imputation and inverse probability weighting methods, we then construct two completed data sets. Two empirical process-based tests, from these completed data sets, are introduced. Under the null hypothesis and local alterative hypotheses, the limiting null distributions and power study of the test statistics are, respectively, investigated. A nonparametric Monte Carlo test procedure, which can automatically make the test procedure scale-invariant even when the test statistics are not scale-invariant, is applied to approximate the limiting null distributions of the test statistics. Simulation study is carried out to examine the performance of the tests. We illustrate the proposed method with a real data set on monozygotic twins.