Adaptive testing in arch models

Specification tests for conditional heteroskedasticity that are derived under the assumption that the density of the innovation is Gaussian may not be powerful in light of the recent empirical results that the density is not Gaussian. We obtain specification tests for conditional heteroskedasticity under the assumption that the innovation density is a member of a general family of densities. Our test statistics maximize asymptotic local power and weighted average power criteria for the general family of densities. We establish both first-order and second-order theory for our procedures. Simulations indicate that asymptotic power gains are achievable in finite samples.     

NONPARAMETRIC TWO SAMPLE TESTS BASED ON AREA STATISTICS

ABSTRACT

Area statistics are sample versions of areas occurring in a probability plot of two distribution functions F and G. This paper presents a unified basis for five statistics of this type. They can be used for various testing problems in the framework of the two sample problem for independent observations, such as testing equality of distributions against inequality or testing stochastic dominance of distributions in one or either direction against nondominance. Though three of the statistics considered have already been suggested in literature, two of them are new and deserve our interest. The finite sample distributions of the statistics (under F=G) can be calculated via recursion formulae. Two tables with critical values of the new statistics are included. The asymptotic distribution of the properly normalized versions of the area statistics are functionals of the Brownian bridge. The distribution functions and quantiles thereof are obtained by Monte Carlo simulation. Finally, the power functions of the two new tests based on area statistics are compared to the power functions of the tests based on the corresponding supremum statistics, i.e., statistics of the Kolmogorov–Smirnov type.     

Accurately sized test statistics with misspecified conditional homoskedasticity

We study the finite-sample performance of test statistics in linear regression models where the error dependence is of unknown form. With an unknown dependence structure, there is traditionally a trade-off between the maximum lag over which the correlation is estimated (the bandwidth) and the amount of heterogeneity in the process. When allowing for heterogeneity, through conditional heteroskedasticity, the correlation at far lags is generally omitted and the resultant inflation of the empirical size of test statistics has long been recognized. To allow for correlation at far lags, we study the test statistics constructed under the possibly misspecified assumption of conditional homoskedasticity. To improve the accuracy of the test statistics, we employ the second-order asymptotic refinement in Rothenberg [Approximate power functions for some robust tests of regression coefficients, Econometrica 56 (1988), pp. 997–1019] to determine the critical values. The simulation results of this paper suggest that when sample sizes are small, modelling the heterogeneity of a process is secondary to accounting for dependence. We find that a conditionally homoskedastic covariance matrix estimator (when used in conjunction with Rothenberg's second-order critical value adjustment) improves test size with only a minimal loss in test power, even when the data manifest significant amounts of heteroskedasticity. In some specifications, the size inflation was cut by nearly 40% over the traditional heteroskedasticity and autocorrelation consistent (HAC) test. Finally, we note that the proposed test statistics do not require that the researcher specify the bandwidth or the kernel.     

A simulation study on power comparisons for group sequential tests of non-parametric statistics

In this article, a group sequential test (GST) of non-parametric statistics for survival data is briefly reviewed. An asymptotic joint distribution of the test statistics, obtained after each interim analysis, is given to illustrate the applicability of the critical values of the GST procedures. It should be noted that censored observations are generally seen in survival data. Therefore, if one makes power calculations irrespective of censoring, reliable results may not be achieved, due to the lack of information about the censoring structure. A wide simulation study, covering different censoring rates and tied observations, is conducted to make the power comparisons under various scenarios. The simulation results are interpreted and compared with the results obtained by using power analysis and sample size (PASS) software.     

The null hypothesis of (common) jumps in case of irregular and asynchronous observations

This paper proposes novel tests for the absence of jumps in a univariate semimartingale and for the absence of common jumps in a bivariate semimartingale. Our methods rely on ratio statistics of power variations based on irregular observations, sampled at different frequencies. We develop central limit theorems for the statistics under the respective null hypotheses and apply bootstrap procedures to assess the limiting distributions. Furthermore, we define corrected statistics to improve the finite sample performance. Simulations show that the test based on our corrected statistic yields good results and even outperforms existing tests in the case of regular observations.     

A Lagrange multiplier test for testing the adequacy of constant conditional correlation GARCH model

ABSTRACT

A Lagrange multiplier test for testing the parametric structure of a constant conditional correlation-generalized autoregressive conditional heteroskedasticity (CCC-GARCH) model is proposed. The test is based on decomposing the CCC-GARCH model multiplicatively into two components, one of which represents the null model, whereas the other one describes the misspecification. A simulation study shows that the test has good finite sample properties. We compare the test with other tests for misspecification of multivariate GARCH models. The test has high power against alternatives where the misspecification is in the GARCH parameters and is superior to other tests. The test is not greatly affected by misspecification in the conditional correlations and is therefore well suited for considering misspecification of GARCH equations.     

The robustness, reliabiligy and power of heteroskedasticity tests

Several tests for heteroskedasticity in linear regression models are examined. Asymptoticrobustness to heterokurticity, nonnormality and skewness is discussed. The finite sample eliability of asymptotically valid tests is investigated using Monte Carlo experiments. It is found that asymptotic critical values cannot, in general. be relied upon to give good agreement between nominal and actual finite sample significance levels. The use of the bootstrap overcomes this problem for general approaches that lead to asymptotically pivotal test statistics. Power comparisons are made for bootstrap tests and modified Glejser and Koenker tests are recommended.     

More powerful sign test using median ranked set sample: Finite sample power comparison

Sign test using median ranked set samples (MRSS) is introduced and investigated. We show that, this test is more powerful than the sign tests based on simple random sample (SRS) and ranked set sample (RSS) for finite sample size. It is found that, when the set size of MRSS is odd, the null distribution of the MRSS sign test is the same as the sign test obtained by using SRS. The exact null distributions and the power functions, in case of finite sample sizes, of these tests are derived. Also, the asymptotic distribution of the MRSS sign tests are derived. Numerical comparison of the MRSS sign test power with the power of the SRS sign test and the RSS sign test is given. Illustration of the procedure, using real data set of bilirubin level in Jaundice babies who stay in neonatal intensive care is introduced.     

The effect of conditional heteroskedasticity on common statistical procedures for means and variances

Summary: Commonly used standard statistical procedures for means and variances (such as the t–test for means or the F–test for variances and related confidence procedures) require observations from independent and identically normally distributed variables. These procedures are often routinely applied to financial data, such as asset or currency returns, which do not share these properties. Instead, they are nonnormal and show conditional heteroskedasticity, hence they are dependent. We investigate the effect of conditional heteroskedasticity (as modelled by GARCH(1,1)) on the level of these tests and the coverage probability of the related confidence procedures. It can be seen that conditional heteroskedasticity has no effect on procedures for means (at least in large samples). There is, however, a strong effect of conditional heteroskedasticity on procedures for variances. These procedures should therefore not be used if conditional heteroskedasticity is prevalent in the data.*We are grateful to the referees for their useful and constructive comments.     

THE BEHAVIOR OF HEGY TESTS FOR QUARTERLY TIME SERIES WITH SEASONAL MEAN SHIFTS

ABSTRACT

This paper studies the behavior of the HEGY statistics for quarterly data, for seasonal autoregressive unit roots, when the analyzed time series is deterministic seasonal stationary but exhibits a change in the seasonal pattern. We analyze also the HEGY test for the nonseasonal unit root. the data generation process being trend stationary too. Our results show that when the break magnitudes are finite, the HEGY test statistics are not asymptotically biased toward the nonrejection of the seasonal and nonseasonal unit root hypotheses. However, the finite sample power properties may be substantially affected, the behavior of the tests depending on the type of the break.     

Lag Length Selection for Unit Root Tests in the Presence of Nonstationary Volatility

A number of recent papers have focused on the problem of testing for a unit root in the case where the driving shocks may be unconditionally heteroskedastic. These papers have, however, taken the lag length in the unit root test regression to be a deterministic function of the sample size, rather than data-determined, the latter being standard empirical practice. We investigate the finite sample impact of unconditional heteroskedasticity on conventional data-dependent lag selection methods in augmented Dickey–Fuller type regressions and propose new lag selection criteria which allow for unconditional heteroskedasticity. Standard lag selection methods are shown to have a tendency to over-fit the lag order under heteroskedasticity, resulting in significant power losses in the (wild bootstrap implementation of the) augmented Dickey–Fuller tests under the alternative. The proposed new lag selection criteria are shown to avoid this problem yet deliver unit root tests with almost identical finite sample properties as the corresponding tests based on conventional lag selection when the shocks are homoskedastic.     

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear Models

We propose two test statistics for testing serial correlation in semiparametric varying-coefficient partially linear models. The proposed test statistics are not only for testing zero first-order serial correlation, but also for testing higher-order serial correlations. Under the null hypothesis of no serial correlation, the test statistics are shown to have asymptotic normal or chi-square distributions. By using R, some Monte Carlo experiments are conducted to examine the finite sample performances of the proposed tests. Simulation results show that the estimated size and power of the proposed tests behave well.     

A non-parametric statistic for testing conditional heteroscedasticity for unobserved component models

When prediction intervals are constructed using unobserved component models (UCM), problems can arise due to the possible existence of components that may or may not be conditionally heteroscedastic. Accurate coverage depends on correctly identifying the source of the heteroscedasticity. Different proposals for testing heteroscedasticity have been applied to UCM; however, in most cases, these procedures are unable to identify the heteroscedastic component correctly. The main issue is that test statistics are affected by the presence of serial correlation, causing the distribution of the statistic under conditional homoscedasticity to remain unknown. We propose a nonparametric statistic for testing heteroscedasticity based on the well-known Wilcoxon''s rank statistic. We study the asymptotic validation of the statistic and examine bootstrap procedures for approximating its finite sample distribution. Simulation results show an improvement in the size of the homoscedasticity tests and a power that is clearly comparable with the best alternative in the literature. We also apply the test on real inflation data. Looking for the presence of a conditionally heteroscedastic effect on the error terms, we arrive at conclusions that almost all cases are different than those given by the alternative test statistics presented in the literature.     

Tests of fit for the Gumbel distribution: EDF-based tests against entropy-based tests

In this article, we propose some tests of fit based on sample entropy for the composite Gumbel (Extreme Value) hypothesis. The proposed test statistics are constructed using different entropy estimates. Through a Monte Carlo simulation, critical values of the test statistics for various sample sizes are obtained. Since the tests based on the empirical distribution function (EDF) are commonly used in practice, the power values of the entropy-based tests with those of the EDF tests are compared against various alternatives and different sample sizes. Finally, two real data sets are modeled by the Gumbel distribution.KEYWORDS: Entropy estimator, Gumbel distribution, Monte Carlo simulation, test power     

Positive quadrant dependence testing and constrained copula estimation

Positive quadrant dependence is a specific dependence structure that is of practical importance in for example modelling dependencies in insurance and actuarial sciences. This dependence structure imposes a constraint on the copula function. The interest in this paper is to test for positive quadrant dependence. One way to assess the distribution of the test statistics under the null hypothesis of positive quadrant dependence is to resample from a constrained copula. This requires constrained estimation of a copula function. We show that this use of resampling under a constrained copula improves considerably the power performance of existing testing procedures. We propose two resampling procedures, one based on a parametric constrained copula estimation and one relying on nonparametric estimation of a positive quadrant dependence copula, and discuss their properties. The finite‐sample performances of the resulting testing procedures are evaluated via a simulation study that also includes comparisons with existing tests. Finally, a data set of Danish fire insurance claims is tested for positive quadrant dependence. The Canadian Journal of Statistics 41: 36–64; 2013 © 2012 Statistical Society of Canada     

Divergences and duality for estimation and test under moment condition models

We introduce estimation and test procedures through divergence minimization for models satisfying linear constraints with unknown parameter. These procedures extend the empirical likelihood (EL) method and share common features with generalized empirical likelihood approach. We treat the problems of existence and characterization of the divergence projections of probability distributions on sets of signed finite measures. We give a precise characterization of duality, for the proposed class of estimates and test statistics, which is used to derive their limiting distributions (including the EL estimate and the EL ratio statistic) both under the null hypotheses and under alternatives or misspecification. An approximation to the power function is deduced as well as the sample size which ensures a desired power for a given alternative.     

Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

Size and power of some cointegration tests under structural breaks and heteroskedastic noise

In this paper we examine the power and size distortions of a number of representative cointegration tests in some large-scale Monte Carlo simulations, when the underlying system is subjected to regime shifts and conditional (ARCH-type) heteroskedasticity. Following the suggestion by Gregory and Hansen, we select the minimum (maximum for the Johansen test) value for the statistics evaluated over a set of tentative break points for the regime shifts. The performance of these statistics is compared to the corresponding ordinary statistics in the presence of regime shifts. We demonstrate that the size of the cointegration tests is severely distorted by conditional heteroskedasticity.     

Exact unconditional testing procedures for comparing two independent Poisson rates

We consider seven exact unconditional testing procedures for comparing adjusted incidence rates between two groups from a Poisson process. Exact tests are always preferable due to the guarantee of test size in small to medium sample settings. Han [Comparing two independent incidence rates using conditional and unconditional exact tests. Pharm Stat. 2008;7(3):195–201] compared the performance of partial maximization p-values based on the Wald test statistic, the likelihood ratio test statistic, the score test statistic, and the conditional p-value. These four testing procedures do not perform consistently, as the results depend on the choice of test statistics for general alternatives. We consider the approach based on estimation and partial maximization, and compare these to the ones studied by Han (2008) for testing superiority. The procedures are compared with regard to the actual type I error rate and power under various conditions. An example from a biomedical research study is provided to illustrate the testing procedures. The approach based on partial maximization using the score test is recommended due to the comparable performance and computational advantage in large sample settings. Additionally, the approach based on estimation and partial maximization performs consistently for all the three test statistics, and is also recommended for use in practice.     

Conditional Score Tests for Heteroscedasticity in the Two-Way Error Components Model

This paper extends the one-way heteroskedasticity score test of Holly and Gardiol (2000, In: Krishnakumar, J, Ronchetti, E (Eds.), Panel Data Econometrics: Future Directions, North-Holland, Amsterdam, pp. 199–211) to two conditional Lagrange Multiplier (LM) tests of heteroskedasticity under contiguous alternatives within the two-way error components model framework. In each case, the derivation of Rao's efficient score statistics for testing heteroskedasticity is first obtained. Then, based on a specific set of assumptions, the asymptotic distribution of the score under contiguous alternatives is established. Finally, the expression for the score test statistic in the presence of heteroskedasticity and related asymptotic local powers of these score test statistics are derived and discussed.