Yanqin Fan

CONSISTENT MODEL SPECIFICATION TESTS: OMITTED VARIABLES AND SEMIPARAMETRIC FUNCTIONAL FORMS

In this paper, we develop several consistent tests in the context of a nonparametric regression model. These include tests for the significance of a subset of regressors and tests for the specification of the semiparametric functional form of the regression function, where the latter covers tests for a partially linear and a single index specification against a general nonparametric alternative. One common feature to the construction of all these tests is the use of the Central Limit Theorem for degenerate U-statistics of order higher than two. As a result, they share the same advantages over most of the corresponding existing tests in the literature: (a) They do not depend on any ad hoc modifications such as sample splitting, random weighting, etc. (b) Under the alternative hypotheses, the test statistics in this paper diverge to positive infinity at a faster rate than those based on ad hoc modifications.

Efficient Estimation of Semiparametric Multivariate Copula Models

We propose a sieve maximum likelihood estimation procedure for a broad class of semiparametric multivariate distributions. A joint distribution in this class is characterized by a parametric copula function evaluated at nonparametric marginal distributions. This class of distributions has gained popularity in diverse fields due to its flexibility in separately modeling the dependence structure and the marginal behaviors of a multivariate random variable, and its circumvention of the “curse of dimensionality” associated with purely nonparametric multivariate distributions. We show that the plug-in sieve maximum likelihood estimators (MLEs) of all smooth functionals, including the finite-dimensional copula parameters and the unknown marginal distributions, are semiparametrically efficient, and that their asymptotic variances can be estimated consistently. Moreover, prior restrictions on the marginal distributions can be easily incorporated into the sieve maximum likelihood estimation procedure to achieve further efficiency gains. Two such cases are studied: (a) the marginal distributions are equal but otherwise unspecified, and (b) some but not all marginal distributions are parametric. Monte Carlo studies indicate that the sieve MLEs perform well in finite samples, especially when prior information on the marginal distributions is incorporated.

Central limit theorem for degenerate U-Statistics of Absolutely Regular Processes with Applications to Model Specification Testing

Under quite general conditions we establish a central limit theorem for second order degenerate U-statistics of absolutely regular processes. The new central limit theorem is then used to establish the validity of an asymptotic test for the parametric functional form of a general regression model involving time series.

Testing the Goodness of Fit of a Parametric Density Function by Kernel Method

Let F denote a distribution function defined on the probability space (Ω,null, P ), which is absolutely continuous with respect to the Lebesgue measure in R with probability density function f . Let f 0 (·,β) be a parametric density function that depends on an unknown p × 1 vector β. In this paper, we consider tests of the goodness-of-fit of f 0 (·,β) for f (·) for some β based on (i) the integrated squared difference between a kernel estimate of f (·) and the quasimaximum likelihood estimate of f 0 (·,β) denoted by I and (ii) the integrated squared difference between a kernel estimate of f (·) and the corresponding kernel smoothed estimate of f 0 (·, β) denoted by J . It is shown in this paper that the amount of smoothing applied to the data in constructing the kernel estimate of f (·) determines the form of the test statistic based on I . For each test developed, we also examine its asymptotic properties including consistency and the local power property. In particular, we show that tests developed in this paper, except the first one, are more powerful than the Kolmogorov-Smirnov test under the sequence of local alternatives introduced in Rosenblatt [12], although they are less powerful than the Kolmogorov-Smirnov test under the sequence of Pitman alternatives. A small simulation study is carried out to examine the finite sample performance of one of these tests.

Semiparametric Estimation of Stochastic Production Frontier Models

This article extends the linear stochastic frontier model proposed by Aigner, Lovell, and Schmidt to a semiparametric frontier model in which the functional form of the production frontier is unspecified and the distributions of the composite error terms are of known form. Pseudolikelihood estimators of the parameters characterizing the two error terms of the model are constructed based on kernel estimation of the conditional mean function. The Monte Carlo results show that the proposed estimators perform well in finite samples. An empirical application is presented. Extensions to a partially linear frontier function and to more flexible one-sided error distributions than the half-normal are discussed

Simple Tests for Models of Dependence Between Multiple Financial Time Series, with Applications to U.S. Equity Returns and Exchange Rates

Evidence that asset returns are more highly correlated during volatile markets and during market downturns (see Longin and Solnik, 2001, and Ang and Chen, 2002) has lead some researchers to propose alternative models of dependence. In this paper we develop two simple goodness-of-fit tests for such models. We use these tests to determine whether the multivariate Normal or the Student’s t copula models are compatible with U.S. equity return and exchange rate data. Both tests are robust to specifications of marginal distributions, and are based on the multivariate probability integral transform and kernel density estimation. The first test is consistent but requires the estimation of a multivariate density function and is recommended for testing the dependence structure between a small number of assets. The second test may not be consistent against all alternatives but it requires kernel estimation of only a univariate density function, and hence is useful for testing the dependence structure between a large number of assets. We justify our tests for both observable multivariate strictly stationary time series and for standardized innovations of GARCH models. A simulation study demonstrates the efficacy of both tests. When applied to equity return data and exchange rate return data, we find strong evidence against the normal copula, but little evidence against the more flexible Student’s t copula.

Maximization by Parts in Likelihood Inference

This article presents and examines a new algorithm for solving a score equation for the maximum likelihood estimate in certain problems of practical interest. The method circumvents the need to compute second-order derivatives of the full likelihood function. It exploits the structure of certain models that yield a natural decomposition of a very complicated likelihood function. In this decomposition, the first part is a log-likelihood from a simply analyzed model, and the second part is used to update estimates from the first part. Convergence properties of this iterative (fixed-point) algorithm are examined, and asymptotics are derived for estimators obtained using only a finite number of iterations. Illustrative examples considered in the article include multivariate Gaussian copula models, nonnormal random-effects models, generalized linear mixed models, and state-space models. Properties of the algorithm and of estimators are evaluated in simulation studies on a bivariate copula model and a nonnormal linear random-effects model.

Unit Root Tests with Wavelets

This paper develops a wavelet (spectral) approach to test the presence of a unit root in a stochastic process. The wavelet approach is appealing, since it is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via the discrete wavelet transformation (DWT), we design unit root tests against near unit root alternatives. Since DWT is an energy preserving transformation and able to disbalance energy across high and low frequency components of a series, it is possible to isolate the most persistent component of a series in a small number of scaling coefficients. We demonstrate the size and power properties of our tests through Monte Carlo simulations.

Consistent hypothesis testing in semiparametric and nonparametric models for econometric time series

Abstract In this paper we modify the general hypothesis studied by Robinson (1989) for semi-/nonparametric time-series models, and present a consistent testing procedure for the modified hypothesis. As examples, we provide consistent tests for the portfolio conditional mean-variance efficiency hypothesis, for theomitted variables in a multivariate nonparametric time-series regression model, and for the two original examples in Robinson. The asymptotic distributions under the null and Pitman local alternatives are established by invoking central limit theorems for Hilbert-valued-dependent random arrays. To approximate the critical values of the general test, we modify the conditional Monte-Carlo approach of Hansen (1996) and the stationary bootstrap of Politis and Romano (1994a,b), and show that both work asymptotically.

On goodness-of-fit tests for weakly dependent processes using kernel method

In this paper, we extend some existing goodness-of-fit tests for independent observations using kernel method to tests for weakly dependent processes. The tests considered include: (i) a two sample goodness-of-fit test; (ii) a symmetry test; and (iii) a test for the goodness-of-fit of a parametric density function. We also develop a center-free test for the goodness-of-fit of a parametric density function. We establish the asymptotic normality of the tests under the corresponding null hypotheses and verify their consistency.