Constructing smooth tests without estimating the eigenpairs of the limiting process

Based on the well known Karhunen–Loève expansion, it can be shown that many omnibus tests lack power against “high frequency” alternatives. The smooth tests of  Neyman (1937) may be employed to circumvent this power deficiency problem. Yet, such tests may be difficult to compute in many applications. In this paper, we propose a more operational approach to constructing smooth tests. This approach hinges on a Fourier representation of the postulated empirical process with known Fourier coefficients, and the proposed test is based on the normalized principal components associated with the covariance matrix of finitely many Fourier coefficients. The proposed test thus needs only standard principal component analysis that can be carried out using most econometric packages. We establish the asymptotic properties of the proposed test and consider two data-driven methods for determining the number of Fourier coefficients in the test statistic. Our simulations show that the proposed tests compare favorably with the conventional smooth tests in finite samples.     

Efficient rank tests for semiparametric competing risk models

We consider a semiparametric competing risk model given by k independent survival times. The paper offers an asymptotic treatment of tests for the semiparametric null hypothesis of equality of the underlying risks. It turns out that modified rank tests are asymptotically efficient for certain semiparametric submodels, where the baseline hazard is a nuisance parameter. In addition, the asymptotic relative efficiency of the present tests is derived. A comparison of asymptotic power functions can then be used to classify various tests proposed earlier in the literature. For instance a chi-square type test is efficient for proportional hazards. Data driven tests of likelihood ratio type are proposed for cones of alternatives. We will consider certain stochastically increasing alternatives as a special example. The paper shows how the concept of local asymptotic normality of Le Cam works for hazard oriented models.     

Powerful tests for structural changes in volatility

Detecting structural changes in volatility is important for understanding volatility dynamics and stylized facts observed for financial returns such as volatility persistence. We propose modified CUSUM and LM tests that are built on a robust estimator of the long-run variance of squared series. We establish conditions under which the new tests have standard null distributions and diverge faster than standard tests under the alternative. The theory allows smooth and abrupt structural changes that can be small. The smoothing parameter is automatically selected such that the proposed test has good finite-sample size and meanwhile achieves decent power gain.     

Likelihood based testing for no fractional cointegration

We consider two likelihood ratio tests, the so-called maximum eigenvalue and trace tests, for the null of no cointegration when fractional cointegration is allowed under the alternative, which is a first step to generalize the so-called Johansen’s procedure to the fractional cointegration case. The standard cointegration analysis only considers the assumption that deviations from equilibrium can be integrated of order zero, which is very restrictive in many cases and may imply an important loss of power in the fractional case. We consider the alternative hypotheses with equilibrium deviations that can be mean reverting with order of integration possibly greater than zero. Moreover, the degree of fractional cointegration is not assumed to be known, and the asymptotic null distribution of both tests is found when considering an interval of possible values. The power of the proposed tests under fractional alternatives and size accuracy provided by the asymptotic distribution in finite samples are investigated.     

Nonparametric checks for varying coefficient models with missing response at random

In this paper, we propose a test on the parametric form of the coefficient functions in the varying coefficient model with missing response. Two groups of completed data sets are constructed by using imputation and inverse probability weighting methods respectively. By noting that the coefficient part can be a regression function for a specific model, we construct two empirical process-based tests. The asymptotical distributions of the proposed tests under null and local alternative hypothesis are investigated respectively. Simulation study is carried out to show the power performance of the test. We illustrate the proposed approaches with an environmental data set.     

Cross-sectional dependence robust block bootstrap panel unit root tests

In this paper we consider the issue of unit root testing in cross-sectionally dependent panels. We consider panels that may be characterized by various forms of cross-sectional dependence including (but not exclusive to) the popular common factor framework. We consider block bootstrap versions of the group-mean (Im et al., 2003) and the pooled (Levin et al., 2002) unit root coefficient DF tests for panel data, originally proposed for a setting of no cross-sectional dependence beyond a common time effect. The tests, suited for testing for unit roots in the observed data, can be easily implemented as no specification or estimation of the dependence structure is required. Asymptotic properties of the tests are derived for T going to infinity and N finite. Asymptotic validity of the bootstrap tests is established in very general settings, including the presence of common factors and cointegration across units. Properties under the alternative hypothesis are also considered. In a Monte Carlo simulation, the bootstrap tests are found to have rejection frequencies that are much closer to nominal size than the rejection frequencies for the corresponding asymptotic tests. The power properties of the bootstrap tests appear to be similar to those of the asymptotic tests.     

Heteroskedasticity and non-normality robust LM tests for spatial dependence

The standard LM tests for spatial dependence in linear and panel regressions are derived under the normality and homoskedasticity assumptions of the regression disturbances. Hence, they may not be robust against non-normality or heteroskedasticity of the disturbances. Following Born and Breitung (2011), we introduce general methods to modify the standard LM tests so that they become robust against heteroskedasticity and non-normality. The idea behind the robustification is to decompose the concentrated score function into a sum of uncorrelated terms so that the outer product of gradient (OPG) can be used to estimate its variance. We also provide methods for improving the finite sample performance of the proposed tests. These methods are then applied to several popular spatial models. Monte Carlo results show that they work well in finite sample.     

Testing forecast accuracy of expectiles and quantiles with the extremal consistent loss functions

Forecast evaluations aim to choose an accurate forecast for making decisions by using loss functions. However, different loss functions often generate different ranking results for forecasts, which complicates the task of comparisons. In this paper, we develop statistical tests for comparing performances of forecasting expectiles and quantiles of a random variable under consistent loss functions. The test statistics are constructed with the extremal consistent loss functions of Ehm et al. (2016). The null hypothesis of the tests is that a benchmark forecast at least performs equally well as a competing one under all extremal consistent loss functions. It can be shown that if such a null holds, the benchmark will also perform at least equally well as the competitor under all consistent loss functions. Thus under the null, when different consistent loss functions are used, the result that the competitor does not outperform the benchmark will not be altered. We establish asymptotic properties of the proposed test statistics and propose to use the re-centered bootstrap to construct their empirical distributions. Through simulations, we show that the proposed test statistics perform reasonably well. We then apply the proposed method to evaluations of several different forecast methods.     

Multivariate residual‐based finite‐sample tests for serial dependence and ARCH effects with applications to asset pricing models

In this paper, we propose several finite‐sample specification tests for multivariate linear regressions (MLR). We focus on tests for serial dependence and ARCH effects with possibly non‐Gaussian errors. The tests are based on properly standardized multivariate residuals to ensure invariance to error covariances. The procedures proposed provide: (i) exact variants of standard multivariate portmanteau tests for serial correlation as well as ARCH effects, and (ii) exact versions of the diagnostics presented by Shanken ( 1990 ) which are based on combining univariate specification tests. Specifically, we combine tests across equations using a Monte Carlo (MC) test method so that Bonferroni‐type bounds can be avoided. The procedures considered are evaluated in a simulation experiment: the latter shows that standard asymptotic procedures suffer from serious size problems, while the MC tests suggested display excellent size and power properties, even when the sample size is small relative to the number of equations, with normal or Student‐t errors. The tests proposed are applied to the Fama–French three‐factor model. Our findings suggest that the i.i.d. error assumption provides an acceptable working framework once we allow for non‐Gaussian errors within 5‐year sub‐periods, whereas temporal instabilities clearly plague the full‐sample dataset. Copyright © 2009 John Wiley & Sons, Ltd.     

Codependent cycles

This paper extends the work of Engle and Kozicki (1993) to test for co-movement in multiple time series when their cycles are not exactly synchronized. We call these codependent cycles and show that testing and estimation in this case will be a Generalized Method of Moments test and estimation procedure. We also show that the Tiao and Tsay (1985) proposed test for scalar components models of order (0, q) can be seen as a test for codependent cycles based on a consistent, but sub-optimal, estimate of the cofeature vector. We assess the small sample performance of the proposed tests through a series of simulations. Finally we apply this test to investigate comovement between durable and non-durable consumption expenditures.     

Generalized spectral testing for multivariate continuous-time models

We develop an omnibus specification test for multivariate continuous-time models using the conditional characteristic function, which often has a convenient closed-form or can be accurately approximated for many multivariate continuous-time models in finance and economics. The proposed test fully exploits the information in the joint conditional distribution of underlying economic processes and hence is expected to have good power in a multivariate context. A class of easy-to-interpret diagnostic procedures is supplemented to gauge possible sources of model misspecification. Our tests are also applicable to discrete-time distribution models. Simulation studies show that the tests provide reliable inference in finite samples.     

A General Approach to Testing Volatility Models in Time Series

Volatility models have been playing important roles in economics and finance. Using a generalized spectral second order derivative approach, we propose a new class of generally applicable omnibus tests for the adequacy of linear and nonlinear volatility models. Our tests have a convenient asymptotic null N(0,1) distribution, and can detect a wide range of misspecifications for volatility dynamics, including both neglected linear and nonlinear volatility dynamics. Distinct from the existing diagnostic tests for volatility models, our tests are robust to time-varying higher order moments of unknown form (e.g., time-varying skewness and kurtosis). They check a large number of lags and are therefore expected to be powerful against neglected volatility dynamics that occurs at higher order lags or display long memory properties. Despite using a large number of lags, our tests do not suffer much from the loss of a large number of degrees of freedom, because our approach naturally discounts higher order lags, which is consistent with the stylized fact that economic or financial markets are affected more by the recent past events than by the remote past events. No specific estimation method is required, and parameter estimation uncertainty has no impact on the convenient limit N(0,1) distribution of the test statistics. Moreover, there is no need to formulate an alternative volatility model, and only estimated standardized residuals are needed to implement our tests. We do not have to calculate tedious and model-specific score functions or derivatives of volatility models with respect to estimated parameters, which are required in some existing popular diagnostic tests for volatility models. We examine the finite sample performance of the proposed tests. It is documented that the new tests are rather powerful in detecting neglected nonlinear volatility dynamics which the existing tests can easily miss. They are useful diagnostic tools for practitioners when modelling volatility dynamics.     

On specification testing of ordered discrete choice models

We discuss how to test the specification of an ordered discrete choice model against a general alternative. Two main approaches can be followed: tests based on moment conditions and tests based on comparisons between parametric and nonparametric estimations. Following these approaches, various statistics are proposed and their asymptotic properties are discussed. The performance of the statistics is compared by means of simulations. An easy-to-compute variant of the standard moment-based statistic yields the best results in models with a single explanatory variable. In models with various explanatory variables the results are less conclusive, since the relative performance of the statistics depends on both the fit of the model and the type of misspecification that is considered.     

Evaluating quantile-bounded and expectile-bounded interval forecasts

In many different contexts, decision-making is improved by the availability of probabilistic predictions. The accuracy of probabilistic forecasting methods can be compared using scoring functions and insight provided by calibration tests. These tests evaluate the consistency of predictions with the observations. Our main agenda in this paper is interval forecasts and their evaluation. Such forecasts are usually bounded by two quantile forecasts. However, a limitation of quantiles is that they convey no information regarding the size of potential exceedances. By contrast, the location of an expectile is dictated by the whole distribution. This prompts us to propose expectile-bounded intervals. We provide interpretation, a consistent scoring function and a calibration test. Before doing this, we reflect on the evaluation of forecasts of quantile-bounded intervals and expectiles, and suggest extensions of previously proposed calibration tests in order to guard against strategic forecasting. We illustrate ideas using day-ahead electricity price forecasting.     

Locally Most Powerful and Other Rank Tests for Independence – with a Contaminated Weighted Alternative

Several authors in the literature have attempted the quantification of the concept of stochastic dependence for bivariate distribution. Two weighted rank tests for testing independence against a weighted contamination alternative is proposed and their distributional properties are studied. We also derived a locally most powerful rank test for the alternative setting. The rank tests proposed are shown to be asymptotic locally most powerful for specific distributions.     

Tests for Multiple Breaks in the Trend with Stationary or Integrated Shocks

In this paper, we propose new tests of the presence of multiple breaks in the trend of a univariate time‐series where the number and dates of the breaks are unknown and that are valid in the presence of stationary or unit root shocks. These tests can also be used to sequentially estimate the number of breaks. The behaviour of the proposed tests is studied through Monte Carlo experiments.     

Non-causality in bivariate binary time series

In this paper we develop a dynamic discrete-time bivariate probit model, in which the conditions for Granger non-causality can be represented and tested. The conditions for simultaneous independence are also worked out. The model is extended in order to allow for covariates, representing individual as well as time heterogeneity. The proposed model can be estimated by Maximum Likelihood. Granger non-causality and simultaneous independence can be tested by Likelihood Ratio or Wald tests. A specialized version of the model, aimed at testing Granger non-causality with bivariate discrete-time survival data is also discussed. The proposed tests are illustrated in two empirical applications.     

Is a co-jump in prices a sparse jump?

Systematic co-jumps in asset prices are generally thought to account for only a small proportion of overall jumps. In actual observations, however, jumps in asset prices are often persistent, and the time of persistence varies. In this context, we develop a new rule to identify co-jumps and improve traditional tests by considering different sampling frequencies and different sampling starting points to re-evaluate the occurrence rate of systematic co-jumps in financial assets. We conduct a simulation experiment to show that the current test procedures generally underestimate the number of co-jumps when considering persistence, but that the proposed procedure can identify co-jumps more accurately. We also perform an empirical analysis using price data from the Shanghai 50 Index and its 25 constituent stocks in China’s stock market. The average proportion of systematic co-jumps detected by the improved s-BNS is approximately 30%, which shows that the co-jump and even the systematic co-jump are not sparse jumps. The results also reveal the shortcomings of traditional jump tests in estimating persistent jumps and demonstrate that the proposed method can better detect the possible nondiversifiable risks between market indices and their constituent stocks, thereby contributing to financial risk management.     

Variables control chart limits and tests for special causes

Control charts are used to detect problems in control such as outliers, shifts in levels or excess variability in subgroup means that may have a special cause. This paper addresses itself to deriving control chart limits based on past data and based on initial samples in a current control situation. We present a general setting for control charts. Furthermore, an overview is given of tests for special causes. The tests are standardized so that the asymptotic type I error does not exceed a fixed level. The distributions of the run lengths of the tests and combinations of tests are also evaluated. We propose to use a low percen-tile of the run length distribution, instead of the average run length, to study the performance of the tests. These indicate that, in particular when tests are combined, the run length percentiles may be too small for practical purposes. It is shown that (nearly) exact control chart limits for observations from a normal distribution exist. The traditional limits differ considerably from the proposed ones and correspond to even smaller run length percentiles.     

A test for distributional assumptions for the stochastic frontier functions

Some asymptotic tests for testing distributional assumptions, namely, the half-normal and truncated normal distributions for the stochastic frontier functions have been proposed. The tests are Lagrangean multiplier tests based on the Pearson family of truncated distributions. The statistics can be easily computed. Simple interpretations of the statistics and two empirical examples are provided.