Testing cointegration relationship in a semiparametric varying coefficient model

In this paper, we develop two cointegration tests for two varying coefficient cointegration regression models, respectively. Our test statistics are residual based. We derive the asymptotic distributions of test statistics under the null hypothesis of cointegration and show that they are consistent against the alternative hypotheses. We also propose a wild bootstrap procedure companioned with the continuous moving block bootstrap method proposed in  Paparoditis and Politis (2001) and  Phillips (2010) to rectify severe distortions found in simulations when the sample size is small. We apply the proposed test statistic to examine the purchasing power parity (PPP) hypothesis between the US and Canada. In contrast to the existing results from linear cointegration tests, our varying coefficient cointegration test does not reject that PPP holds between the US and Canada.     

Model specification test with correlated but not cointegrated variables

Many macroeconomic and financial variables show highly persistent and correlated patterns but are not necessarily cointegrated. Recently,  Sun et al. (2011) propose using a semiparametric varying coefficient approach to capture correlations between integrated but non cointegrated variables. Due to the complication arising from the integrated disturbance term and the semiparametric functional form, consistent estimation of such a semiparametric model requires stronger conditions than usually needed for consistent estimation for a linear (spurious) regression model, or a semiparametric varying coefficient model with a stationary disturbance. Therefore, it is important to develop a testing procedure to examine for a given data set, whether linear relationship holds or not, while allowing for the disturbance being an integrated process. In this paper we propose two test statistics for detecting linearity against semiparametric varying coefficient alternative specification. Monte Carlo simulations are used to examine the finite sample performances of the proposed tests.     

The power of four tests of autocorrelation in the linear regression model

The power of each of four tests of first-order autocorrelation in the linear regression model is determined for a simple and multiple regression model whose parameters are presumed to be known. The tests are: Durbin-Watson bounds test, a test based on Theil's best linear unbiased scalar estimator, a test devised by Abrahamse, Koerts and Louter, and an exact test devised by Durbin.For positive values of the coefficient of autocorrelation the Durbin-Watson bounds test is generally better than the tests based on the estimator proposed by Abrahamse, Koerts and Louter, the best linear unbiased scalar estimator, and the Durbin exact test. For negative values of the coefficient of autocorrelation, the pattern of results is mixed for all four test procedures. A byproduct of these experiments is the demonstrated feasibility of enumerating the distribution of the Durbin-Watson test statistic for any regression matrix and thus eliminating the region of indeterminacy from the Durbin-Watson test procedure.     

Testing the adequacy of varying coefficient models with missing responses at random

In this paper, we investigate checking the adequacy of varying coefficient models with response missing at random. In doing so, we first construct two completed data sets based on imputation and marginal inverse probability weighted methods, respectively. The empirical process-based tests by using these two completed data sets are suggested and the asymptotic properties of the test statistics under the null and local alternative hypotheses are studied. Because the limiting null distribution is intractable, a Monte Carlo approach is applied to approximate the distribution to determine critical values. Simulation studies are carried out to examine the performance of our method, and a real data set from an environmental study is analyzed for illustration.     

Testing for a unit root in a random coefficient panel data model

This paper proposes new unit root tests in the context of a random autoregressive coefficient panel data model, in which the null of a unit root corresponds to the joint restriction that the autoregressive coefficient has unit mean and zero variance. The asymptotic distributions of the test statistics are derived and simulation results are provided to suggest that they perform very well in small samples.     

On testing the correlation coefficient of a bivariate normal distribution

Summary Here we propose two tests for testing in the bivariate normal population assuming that the ratio of the variances in it is known. The first test (U.M.P.U.) is derived by using the Neyman-Pearson lemma, whereas the second test is obtained through testing the scale parameter of the Cauchy distribution. The powers of the first and second tests are compared with a well-known test, based on the sample correlation coefficient for small and large samples respectively.     

Bootstrap specification tests for diffusion processes

This paper discusses specification tests for diffusion processes. In the one-dimensional case, our proposed test is closest to the nonparametric test of Aı̈t-Sahalia (Rev. Financ. Stud. 9 (1996) 385). However, we compare CDFs instead of densities. In the multidimensional and/or multifactor case, our proposed test is based on comparison of the empirical CDF of actual data and the empirical CDF of simulated data. Asymptotically valid critical values are obtained using an empirical process version of the block bootstrap which accounts for parameter estimation error. An example based on a simple version of the Cox et al. (Econometrica 53 (1985) 385) model is outlined and related Monte Carlo experiments are carried out.     

Evaluating the accuracy of value-at-risk forecasts: New multilevel tests

We propose independence and conditional coverage tests which are aimed at evaluating the accuracy of Value-at-Risk (VaR) forecasts from the same model at different confidence levels. The proposed procedures are multilevel tests, i.e., joint tests of several quantiles corresponding to different confidence levels. In a comprehensive Monte Carlo exercise, we document the superiority of the proposed tests with respect to existing multilevel tests. In an empirical application, we illustrate the implementation of the tests using several VaR models and daily data for 15 MSCI world indices.     

Certificate-of-need regulation and the diffusion of innovations: A random coefficient model

In this paper, we examine empirically the effect that certificate-of-need regulation by state health planning organizations has had on the speed of diffusion of a relatively new medical technology—haemodialysis. Specifically, we test the hypothesis that a requirement that investments be subject to certificate-of-need review has significantly slowed the rate of adoption of this particular treatment modality. In subjecting this hypothesis to empirical verification, we estimate a random coefficient model. This approach allows us to make more efficient use of the available data than the traditional two-stage approach to modelling diffusion processes wherein separate logistic functions are first estimated over the time series observations followed by hypothesis tests conducted over the cross-sectional observations. We find evidence that certificate-of-need regulation slows the spread of haemodialysis technology.     

Diagnostic Tests for the Necessity of Weight in Regression With Survey Data

To weight or not to weight in regression analyses with survey data has been debated in the literature. The problem is essentially a tradeoff between the bias and the variance of the regression coefficient estimator. An array of diagnostic tests for informative weights have been developed. Nonetheless, studies comparing the performance of the tests, especially for finite samples, are scarce, and the theoretical equivalence of some tests has not been investigated. Focusing on the linear regression setting, we review a collection of such tests and propose enhanced versions of some of them that require an auxiliary regression model for the weight. Further, the equivalence of two popular tests is established which has not been reported before. In contrast to existing reviews with no empirical comparison, we compare the sizes and powers of the tests in simulation studies. The reviewed tests are applied to a regression analysis of the family expenditure using the data from the China Family Panel Study.     

Exact tests for contemporaneous correlation of disturbances in seemingly unrelated regressions

This paper proposes finite-sample procedures for testing the SURE specification in multi-equation regression models, i.e. whether the disturbances in different equations are contemporaneously uncorrelated or not. We apply the technique of Monte Carlo (MC) tests, see Dwass and Barnard, respectively (Ann. Math. Statist. 28 (1957) 181; J.R. Statist. Soc. Ser. B 25 (1963) 294) to obtain exact tests based on standard LR and LM zero correlation tests. We also suggest a MC quasi-LR (QLR) test based on feasible generalized least squares (FGLS). We show that the latter statistics are pivotal under the null, which provides the justification for applying MC tests. Furthermore, we extend the exact independence test proposed by Harvey and Phillips (Bull. Econ. Res. 34 (2) (1982) 79) to the multi-equation framework. Specifically, we introduce several induced tests based on a set of simultaneous Harvey/Phillips-type tests and suggest a simulation-based solution to the associated combination problem. The properties of the proposed tests are studied in a Monte Carlo experiment which shows that standard asymptotic tests exhibit important size distortions, while MC tests achieve complete size control and display good power. Moreover, MC-QLR tests performed best in terms of power, a result of interest from the point of view of simulation-based tests. The power of the MC induced tests improves appreciably in comparison to standard Bonferroni tests and in certain cases outperform the likelihood-based MC tests. The tests are applied to data used by Fischer (J. Monetary Econ. 32, 485) (1993) to analyze the macroeconomic determinants of growth.     

Functional coefficient regression models with time trend

We consider the problem of estimating a varying coefficient regression model when regressors include a time trend. We show that the commonly used local constant kernel estimation method leads to an inconsistent estimation result, while a local polynomial estimator yields a consistent estimation result. We establish the asymptotic normality result for the proposed estimator. We also provide asymptotic analysis of the data-driven (least squares cross validation) method of selecting the smoothing parameters. In addition, we consider a partially linear time trend model and establish the asymptotic distribution of our proposed estimator. Two test statistics are proposed to test the null hypotheses of a linear and of a partially linear time trend models. Simulations are reported to examine the finite sample performances of the proposed estimators and the test statistics.     

A predictability test for a small number of nested models

We introduce quasi-likelihood ratio tests for one sided multivariate hypotheses to evaluate the null that a parsimonious model performs equally well as a small number of models which nest the benchmark. The limiting distributions of the test statistics are non-standard. For critical values we consider: (i) bootstrapping and (ii) simulations assuming normality of the mean square prediction error difference. The proposed tests have good size and power properties compared with existing equal and superior predictive ability tests for multiple model comparison. We apply our tests to study the predictive ability of a Phillips curve type for the US core inflation.     

A nonparametric test for equality of distributions with mixed categorical and continuous data

In this paper we consider the problem of testing for equality of two density or two conditional density functions defined over mixed discrete and continuous variables. We smooth both the discrete and continuous variables, with the smoothing parameters chosen via least-squares cross-validation. The test statistics are shown to have (asymptotic) normal null distributions. However, we advocate the use of bootstrap methods in order to better approximate their null distribution in finite-sample settings and we provide asymptotic validity of the proposed bootstrap method. Simulations show that the proposed tests have better power than both conventional frequency-based tests and smoothing tests based on ad hoc smoothing parameter selection, while a demonstrative empirical application to the joint distribution of earnings and educational attainment underscores the utility of the proposed approach in mixed data settings.     

Extended Neyman smooth goodness-of-fit tests,applied to competing heavy-tailed distributions

A simplified version of the Neyman (1937) “Smooth” goodness-of-fit test is extended to account for the presence of estimated model parameters, thereby removing overfitting bias. Using a Lagrange Multiplier approach rather than the Likelihood Ratio statistic proposed by Neyman greatly simplifies the calculations. Polynomials, splines, and the step function of Pearson’s test are compared as alternative perturbations to the theoretical uniform distribution. The extended tests have negligible size distortion and more power than standard tests. The tests are applied to competing symmetric leptokurtic distributions with US stock return data. These are generally rejected, primarily because of the presence of skewness.     

Relative Efficiency of the Fuzzy p-Value Approach to Hypothesis Testing

In missing data problems, it is often the case that there is a natural test statistic for testing a statistical hypothesis had all the data been observed. A fuzzy  p -value approach to hypothesis testing has recently been proposed which is implemented by imputing the missing values in the "complete data" test statistic by values simulated from the conditional null distribution given the observed data. We argue that imputing data in this way will inevitably lead to loss in power. For the case of scalar parameter, we show that the asymptotic efficiency of the score test based on the imputed "complete data" relative to the score test based on the observed data is given by the ratio of the observed data information to the complete data information. Three examples involving probit regression, normal random effects model, and unidentified paired data are used for illustration. For testing linkage disequilibrium based on pooled genotype data, simulation results show that the imputed Neyman Pearson and Fisher exact tests are less powerful than a Wald-type test based on the observed data maximum likelihood estimator. In conclusion, we caution against the routine use of the fuzzy  p -value approach in latent variable or missing data problems and suggest some viable alternatives.     

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.     

Testing equality of shape parameters in several inverse Gaussian populations

Due to the strikingly resemblance to the normal theory and inference methods, the inverse Gaussian (IG) distribution is commonly applied to model positive and right-skewed data. As the shape parameter in the IG distribution is greatly related to other important quantities such as the mean, skewness, kurtosis and the coefficient of variation, it plays an important role in distribution theory. This paper focuses on testing the equality of shape parameters in several inverse Gaussian distributions. Three tests are suggested: the exact generalized inference-based test, the asymptotic test and a test that is based on parametric bootstrap approximation. Simulation studies are undertaken to examine the performances of the these methods, and three real data examples are analyzed for illustration.     

Restricted forecasting with VAR models: An analysis of a test for joint compatibility between restrictions and forecasts

A restricted forecasting compatibility test for Vector Autoregressive Error Correction models is analyzed in this work. It is shown that a variance–covariance matrix associated with the restrictions can be used to cancel out model dynamics and interactions between restrictions. This allows us to interpret the joint compatibility test as a composition of the corresponding single restriction compatibility tests. These tests are useful for appreciating the contribution of each and every restriction to the joint compatibility between the whole set of restrictions and the unrestricted forecasts. An estimated process adjustment for the test is derived and the resulting feasible joint compatibility test turns out to have better performance than the original one. An empirical illustration of the usefulness of the proposed test makes use of Mexican macroeconomic data and the targets proposed by the Mexican Government for the year 2003.     

Linear rank tests for comparing treatments with a control when data are subject to unequal patterns of censorship

A class of asymptotically distribution-free tests is considered for comparing several treatments with a control when the, data are subject to unequal right-censorship. A particular member of this class is proposed for use in practice and an illustrative numerical example is given. A general result for the Pitman efficacy of a test based on an asymptotically normal test statistic is proved for the multiparameter case and using this result the efficacy of the proposed class of tests is obtained under sequences of translation and proportional hazards alternatives. Simulation studies are conducted to compare the performance of the proposed test, in terms of power, with some other tests.