Multiple hypothesis test for parameter constancy based on recursive residuals

This article presents a multiple hypothesis test procedure that combines two well known tests for structural change in the linear regression model, the CUSUM test and the recursive t test. The CUSUM test is run through the sequence of recursive residuals as usual; if the CUSUM plot does not violate the critical lines, one more step is taken to perform the t test for hypothesis of zero mean based on all recursive residuals. The asymptotic size of this multiple hypothesis test is derived; power simulation results suggest that it outperforms the traditional CUSUM test and complements other tests that are currently stressed in econometrics.     

Multiple hypothesis test for parameter constancy based on recursive residuals

This article presents a multiple hypothesis test procedure that combines two well known tests for structural change in the linear regression model, the CUSUM test and the recursive t test. The CUSUM test is run through the sequence of recursive residuals as usual; if the CUSUM plot does not violate the critical lines, one more step is taken to perform the t test for hypothesis of zero mean based on all recursive residuals. The asymptotic size of this multiple hypothesis test is derived; power simulation results suggest that it outperforms the traditional CUSUM test and complements other tests that are currently stressed in econometrics.     

Testing for random individual effects using recursive residuals

This paper derives a simple ANOVA-F-statistic which tests for random individual effects in a one-way error component model, using recursive residuals. Power comparisons are performed for this F-test when it is computed using true disturbances and recursive residuals from a panel data regression. Under the null, both statistics have an exact F distribution. The standardized version of the Breusch and Pagan (1980) Lagrange Multiplier test (SLM) as well as a fixed effects F-statistic (FE) recommended by Moulton and Randolph (1989), are also included in this comparison. The exact power function can be computed in all cases using Imhof's (1961) procedure. Our results suggest that the F-test based on recursive residuals is inferior to the popular SLM and FE tests based on computational simplicity, power comparisons and its sensitivity to the K observations starting the recursion.     

Structural change tests for GEL criteria

This article examines structural change tests based on generalized empirical likelihood methods in the time series context, allowing for dependent data. Standard structural change tests for the Generalized method of moments (GMM) are adapted to the generalized empirical likelihood (GEL) context. We show that when moment conditions are properly smoothed, these test statistics converge to the same asymptotic distribution as in the GMM, in cases with known and unknown breakpoints. New test statistics specific to GEL methods, and that are robust to weak identification, are also introduced. A simulation study examines the small sample properties of the tests and reveals that GEL-based robust tests performed well, both in terms of the presence and location of a structural change and in terms of the nature of identification.     

Testing for monotonic trend in time series based on resampling methods

In this paper, we propose several tests for monotonic trend based on the Brillinger's test statistic (1989, Biometrika, 76, 23–30). When there are highly correlated residuals or short record lengths, Brillinger's test procedure tends to have significance level much higher than the nominal level. It is found that this could be related to the discrepancy between the empirical distribution of the test statistic and the asymptotic normal distribution. Hence, in this paper, we propose three bootstrap-based procedures based on the Brillinger's test statistic to test for monotonic trend. The performance of the proposed test procedures is evaluated through an extensive Monte Carlo simulation study, and is compared to other trend test procedures in the literature. It is shown that the proposed bootstrap-based Brillinger test procedures can well control the significance levels and provide satisfactory power performance in testing the monotonic trend under different scenarios.     

Testing Monotonicity of Regression Functions – An Empirical Process Approach

We propose several new tests for monotonicity of regression functions based on different empirical processes of residuals and pseudo‐residuals. The residuals are obtained from an unconstrained kernel regression estimator whereas the pseudo‐residuals are obtained from an increasing regression estimator. Here, in particular, we consider a recently developed simple kernel‐based estimator for increasing regression functions based on increasing rearrangements of unconstrained non‐parametric estimators. The test statistics are estimated distance measures between the regression function and its increasing rearrangement. We discuss the asymptotic distributions, consistency and small sample performances of the tests.     

A comparison of algorithms for exact goodness-of-fit tests for multinomial data

Multinomial goodness-of-fit tests arise in a diversity of milieu. The long history of the problem has spawned a multitude of asymptotic tests. If the sample size relative to the number of categories is small, the accuracy of these tests is compromised. In that case, an exact test is a prudent option. But such tests are computationally intensive and need efficient algorithms. This paper gives a conceptual overview, and empirical comparisons of two avenues, namely the network and fast Fourier transform (FFT) algorithms, for an exact goodness-of-fit test on a multinomial. We show that a recursive execution of a polynomial product forms the basis of both these approaches. Specific details to implement the network method, and techniques to enhance the efficiency of the FFT algorithm are given. Our empirical comparisons show that for exact analysis with the chi-square and likelihood ratio statistics, the network-cum-polynomial multiplication algorithm is the more efficient and accurate of the two.     

Change-point detection in a shape-restricted regression model

A regression model with a possible structural change and with a small number of measurements is considered. A priori information about the shape of the regression function is used to formulate the model as a linear regression model with inequality constraints and a likelihood ratio test for the presence of a change-point is constructed. The exact null distribution of the test statistic is given. Consistency of the test is proved when the noise level goes to zero. Numerical approximations to the powers against various alternatives are given and compared with the powers of the k-linear-r-ahead recursive residuals tests and CUSUM tests. Performance of four different estimators of the change-point is studied in a Monte Carlo experiment. An application of the procedures to some real data is also presented.     

Bootstrap misspecification tests for ARCH based on the empirical process of squared residuals

We propose and study by means of simulations and graphical tools a class of goodness-of-fit tests for ARCH models. The tests are based on the empirical distribution function of squared residuals and smooth (parametric) bootstrap. We examine empirical size and power by means of a simulation study. While the tests have overall correct size, their power strongly depends on the type of alternative and is particularly high when the assumption of Gaussian innovations is violated. As an example, the tests are applied to returns on Foreign Exchange rates.     

Goodness-of-fit tests for one-shot device accelerated life testing data

In this article, we develop a formal goodness-of-fit testing procedure for one-shot device testing data, in which each observation in the sample is either left censored or right censored. Such data are also called current status data. We provide an algorithm for calculating the nonparametric maximum likelihood estimate (NPMLE) of the unknown lifetime distribution based on such data. Then, we consider four different test statistics that can be used for testing the goodness-of-fit of accelerated failure time (AFT) model by the use of samples of residuals: a chi-square-type statistic based on the difference between the empirical and expected numbers of failures at each inspection time; two other statistics based on the difference between the NPMLE of the lifetime distribution obtained from one-shot device testing data and the distribution specified under the null hypothesis; as a final statistic, we use White's idea of comparing two estimators of the Fisher Information (FI) to propose a test statistic. We then compare these tests in terms of power, and draw some conclusions. Finally, we present an example to illustrate the proposed tests.     

Model Checks for Generalized Linear Models

In this paper we propose and study non-parametric tests for the validity of (composite) Generalized Linear Models with a given parametric link structure, which are based on certain empirical processes marked by the residuals. When properly transformed to their innovation part the resulting test statistics are distribution-free. The method perfectly adapts to a situation, when also the input vector follows a dimension reducing model.     

Schwarz information criterion based tests for a change-point in regression models

Standard Schwarz information criterion for testing a change-point in regression models is considered and two new test procedures are evolved. The case of small sample size is investigated. Numerical approximations to the power against various alternatives are given and compared with powers of tests based on r-ahead recursive residuals and of the CUSUM of squares test. Application of these procedures to some real data is also provided.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x ²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach

We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end, we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov–Smirnov statistic based on the difference of the estimated innovation distributions built from the first ?ns?and the last n ? ?ns? residuals, respectively (0 ≤ s ≤ 1). Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change point. The result implies that the test is asymptotically distribution‐free. Consistency against fixed alternatives is shown. The small sample performance of the proposed test is investigated in a simulation study and the test is applied to a data example.     

On testing for causality in variance between two multivariate time series

Verifying the existence of a relationship between two multivariate time series represents an important consideration. In this article, the procedure developed by Cheung and Ng [A causality-in-variance test and its application to financial market prices, J. Econom. 72 (1996), pp. 33–48] designed to test causality in variance for univariate time series is generalized in several directions. A first approach proposes test statistics based on residual cross-covariance matrices of squared (standardized) residuals and cross products of (standardized) residuals. In a second approach, transformed residuals are defined for each residual vector time series, and test statistics are constructed based on the cross-correlations of these transformed residuals. Test statistics at individual lags and portmanteau-type test statistics are developed. Conditions are given under which the new test statistics converge in distribution towards chi-square distributions. The proposed methodology can be used to determine the directions of causality in variance, and appropriate test statistics are presented. Monte Carlo simulation results show that the new test statistics offer satisfactory empirical properties. An application with two bivariate financial time series illustrates the methods.     

Regression spline bivariate probit models: A practical approach to testing for exogeneity

Bivariate probit models can deal with a problem usually known as endogeneity. This issue is likely to arise in observational studies when confounders are unobserved. We are concerned with testing the hypothesis of exogeneity (or absence of endogeneity) when using regression spline recursive and sample selection bivariate probit models. Likelihood ratio and gradient tests are discussed in this context and their empirical properties investigated and compared with those of the Lagrange multiplier and Wald tests through a Monte Carlo study. The tests are illustrated using two datasets in which the hypothesis of exogeneity needs to be tested.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

On the Usefulness or Lack Thereof of Optimality Criteria for Structural Change Tests

Elliott and Müller (2006) considered the problem of testing for general types of parameter variations, including infrequent breaks. They developed a framework that yields optimal tests, in the sense that they nearly attain some local Gaussian power envelop. The main ingredient in their setup is that the variance of the process generating the changes in the parameters must go to zero at a fast rate. They recommended the so-called qL?L test, a partial sums type test based on the residuals obtained from the restricted model. We show that for breaks that are very small, its power is indeed higher than other tests, including the popular sup-Wald (SW) test. However, the differences are very minor. When the magnitude of change is moderate to large, the power of the test is very low in the context of a regression with lagged dependent variables or when a correction is applied to account for serial correlation in the errors. In many cases, the power goes to zero as the magnitude of change increases. The power of the SW test does not show this non-monotonicity and its power is far superior to the qL?L test when the break is not very small. We claim that the optimality of the qL?L test does not come from the properties of the test statistics but the criterion adopted, which is not useful to analyze structural change tests. Instead, we use fixed-break size asymptotic approximations to assess the relative efficiency or power of the two tests. When doing so, it is shown that the SW test indeed dominates the qL?L test and, in many cases, the latter has zero relative asymptotic efficiency.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.