Nonparametric estimation and testing of fixed effects panel data models

In this paper we consider the problem of estimating nonparametric panel data models with fixed effects. We introduce an iterative nonparametric kernel estimator. We also extend the estimation method to the case of a semiparametric partially linear fixed effects model. To determine whether a parametric, semiparametric or nonparametric model is appropriate, we propose test statistics to test between the three alternatives in practice. We further propose a test statistic for testing the null hypothesis of random effects against fixed effects in a nonparametric panel data regression model. Simulations are used to examine the finite sample performance of the proposed estimators and the test statistics.     

Local asymptotic normality for randomly censored models with applications to rank tests

For randomly right censored models we study the asymptotic behaviour of linear (rank) statistics under local alternatives. The results can be used to evaluate the asymptotic power of the corresponding tests. For instance we treat the question how to choose the best scores in order to derive asymptotically optimal (rank) tests under certain alternatives.     

Semiparametric Estimation of a Duration Model

Within the framework of the proportional hazard model proposed in Cox (1972), Han and Hausman (1990) consider the logarithm of the integrated baseline hazard function as constant in each time period. We, however, proposed an alternative semiparametric estimator of the parameters of the covariate part. The estimator is considered as semiparametric since no prespecified functional form for the error terms (or certain convolution) is needed. This estimator, proposed in Lewbel (2000) in another context, shows at least four advantages. The distribution of the latent variable error is unknown and may be related to the regressors. It takes into account censored observations, it allows for heterogeneity of unknown form and it is quite easy to implement since the estimator does not require numerical searches. Using the Spanish Labour Force Survey, we compare empirically the results of estimating several alternative models, basically on the estimator proposed in Han and Hausman (1990) and our semiparametric estimator.     

Nonparametric inference based on conditional moment inequalities

This paper develops methods of inference for nonparametric and semiparametric parameters defined by conditional moment inequalities and/or equalities. The parameters need not be identified. Confidence sets and tests are introduced. The correct uniform asymptotic size of these procedures is established. The false coverage probabilities and power of the CS’s and tests are established for fixed alternatives and some local alternatives. Finite-sample simulation results are given for a nonparametric conditional quantile model with censoring and a nonparametric conditional treatment effect model. The recommended CS/test uses a Cramér–von-Mises-type test statistic and employs a generalized moment selection critical value.     

A score test under logistic regression models based on case–control data

We propose a score statistic to test the vector of odds ratio parameters under the logistic regression model based on case–control data. The proposed score test is based on the semiparametric profile loglikelihood function under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to the Wald test under the logistic regression model based on case–control data. In addition, we demonstrate that the proposed score statistic and its asymptotic distribution may be obtained by fitting the prospective logistic regression model to case–control data. We present some results on simulation and on the analysis of two real datasets.     

Precedence-type tests based on record values

Precedence-type tests based on order statistics are simple and efficient nonparametric tests that are very useful in the context of life-testing, and they have been studied quite extensively in the literature; see Balakrishnan and Ng (Precedence-type tests and applications. Wiley, Hoboken, 2006). In this paper, we consider precedence-type tests based on record values and develop specifically record precedence test, record maximal precedence test and record-rank-sum test. We derive their exact null distributions and tabulate some critical values. Then, under the general Lehmann alternative, we derive the exact power functions of these tests and discuss their power under the location-shift alternative. We also establish that the record precedence test is the uniformly most powerful test for testing against the one-parameter family of Lehmann alternatives. Finally, we discuss the situation when we have insufficient number of records to apply the record precedence test and then make some concluding remarks.     

Semi-nonparametric estimation and misspecification testing of diffusion models

Novel transition-based misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in [Kristensen, D., 2010. Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models. Journal of Econometrics 156, 239-259] are proposed. It is demonstrated that transition-based tests in general lack power in detecting certain departures from the null since they integrate out local features of the drift and volatility. As a solution to this, tests that directly compare drift and volatility estimators under the relevant null and alternative are also developed which exhibit better power against local alternatives.     

Sieve estimation of panel data models with cross section dependence

In this paper we consider the problem of estimating semiparametric panel data models with cross section dependence, where the individual-specific regressors enter the model nonparametrically whereas the common factors enter the model linearly. We consider both heterogeneous and homogeneous regression relationships when both the time and cross-section dimensions are large. We propose sieve estimators for the nonparametric regression functions by extending Pesaran’s (2006) common correlated effect (CCE) estimator to our semiparametric framework. Asymptotic normal distributions for the proposed estimators are derived and asymptotic variance estimators are provided. Monte Carlo simulations indicate that our estimators perform well in finite samples.     

A semiparametric cointegrating regression: Investigating the effects of age distributions on consumption and saving

We consider a semiparametric cointegrating regression model, for which the disequilibrium error is further explained nonparametrically by a functional of distributions changing over time. The paper develops the statistical theories of the model. We propose an efficient econometric estimator and obtain its asymptotic distribution. A specification test for the model is also investigated. The model and methodology are applied to analyze how an aging population in the US influences the consumption level and the savings rate. We find that the impact of age distribution on the consumption level and the savings rate is consistent with the life-cycle hypothesis.     

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.     

Efficient semiparametric estimation for endogenously stratified regression via smoothed likelihood

This paper presents efficient semiparametric estimators for endogenously stratified regression with two strata, in the case where the error distribution is unknown and the regressors are independent of the error term. The method is based on the use of a kernel-smoothed likelihood function which provides an explicit solution for the maximization problem for the unknown density function without losing information in the asymptotic limit. We consider both standard stratified sampling and variable probability sampling, and allow for the population shares of the strata to be either unknown or known a priori.     

Semiparametric estimation of logistic regression model with missing covariates and outcome

We consider a semiparametric method to estimate logistic regression models with missing both covariates and an outcome variable, and propose two new estimators. The first, which is based solely on the validation set, is an extension of the validation likelihood estimator of Breslow and Cain (Biometrika 75:11–20, 1988). The second is a joint conditional likelihood estimator based on the validation and non-validation data sets. Both estimators are semiparametric as they do not require any model assumptions regarding the missing data mechanism nor the specification of the conditional distribution of the missing covariates given the observed covariates. The asymptotic distribution theory is developed under the assumption that all covariate variables are categorical. The finite-sample properties of the proposed estimators are investigated through simulation studies showing that the joint conditional likelihood estimator is the most efficient. A cable TV survey data set from Taiwan is used to illustrate the practical use of the proposed methodology.     

Nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters

We consider nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters. Most of the existing works on asymptotic distributions of a nonparametric/semiparametric estimator or a test statistic are based on some deterministic smoothing parameters, while in practice it is important to use data-driven methods to select the smoothing parameters. In this paper we give a simple sufficient condition that can be used to establish the first order asymptotic equivalence of a nonparametric estimator or a test statistic with stochastic smoothing parameters to those using deterministic smoothing parameters. We also allow for general weakly dependent data.     

Monitoring structural change in dynamic econometric models

The classical approach to testing for structural change employs retrospective tests using a historical data set of a given length. Here we consider a wide array of fluctuation‐type tests in a monitoring situation—given a history period for which a regression relationship is known to be stable, we test whether incoming data are consistent with the previously established relationship. Procedures based on estimates of the regression coefficients are extended in three directions: we introduce (a) procedures based on OLS residuals, (b) rescaled statistics and (c) alternative asymptotic boundaries. Compared to the existing tests our extensions offer ease of computation, improved size in finite samples for dynamic models and better power against certain alternatives, respectively. We apply our methods to three data sets, German M1 money demand, US labour productivity and S&P 500 stock returns. Copyright © 2005 John Wiley & Sons, Ltd.     

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 changing mean with monotonic power

Several widely used tests for a changing mean exhibit nonmonotonic power in finite samples, due to “incorrect” estimation of nuisance parameters under the alternative. In this paper, we study the issue of nonmonotonic power in testing for changing mean. We investigate the asymptotic power properties of the tests, using a new framework where alternatives are characterized as having “large” changes. The asymptotic analysis provides a theoretical explanation to the power problem. Modified tests that have monotonic power against a wide range of alternatives of structural change are proposed. Instead of estimating the nuisance parameters based on ordinary least squares residuals, the proposed tests use modified estimators, based on nonparametric regression residuals. It is shown that tests based on the modified long-run variance estimator provide an improved rate of divergence of the tests under the alternative of a change in mean. Tests for structural breaks based on such an estimator are able to remain consistent, while still retaining the same asymptotic distribution under the null hypothesis of constant mean.     

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.     

An improved bootstrap test of stochastic dominance

We propose a new method of testing stochastic dominance that improves on existing tests based on the standard bootstrap or subsampling. The method admits prospects involving infinite as well as finite dimensional unknown parameters, so that the variables are allowed to be residuals from nonparametric and semiparametric models. The proposed bootstrap tests have asymptotic sizes that are less than or equal to the nominal level uniformly over probabilities in the null hypothesis under regularity conditions. This paper also characterizes the set of probabilities so that the asymptotic size is exactly equal to the nominal level uniformly. As our simulation results show, these characteristics of our tests lead to an improved power property in general. The improvement stems from the design of the bootstrap test whose limiting behavior mimics the discontinuity of the original test’s limiting distribution.     

Specification testing in discretized diffusion models: Theory and practice

We propose two new tests for the specification of both the drift and the diffusion functions in a discretized version of a semiparametric continuous-time financial econometric model. Theoretically, we establish some asymptotic consistency results for the proposed tests. Practically, a simple selection procedure for the bandwidth parameter involved in each of the proposed tests is established based on the assessment of the power function of the test under study. To the best of our knowledge, this is the first approach of this kind in specification of continuous-time financial econometrics. The proposed theory is supported by good small and medium-sample studies.     

Conducting inference in semiparametric duration models under inequality restrictions on the shape of the hazard implied by job search theory

Using a four-month panel of revised Current Population Survey data from September–December 1993, we extend the class of semiparametric hazard models of the type first studied by Prentice and Gloeckler ( 1978 ), and brought to the attention of economists by Meyer ( 1988 , 1990 ), to incorporate inequality restrictions on the shape of the hazard. This extension enables us to test hypotheses regarding the shape of the hazard implied by search theory using duration data alone. These tests provide another link between the empirical and theoretical literatures on unemployment duration and job search. The GHK probability simulator makes it straightforward to generate approximate hypothesis test results, as simulation estimates of the probability under the null hypothesis are generated using the asymptotic normal approximation to the distribution of the hazard parameters obtained from maximum likelihood estimation. Importance sampling is used to conduct inference under the null and obtain exact finite sample estimates of the probability the null is satisfied. A new algorithm for maintaining stability of the importance weights is also developed. Copyright © 1999 John Wiley & Sons, Ltd.