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.     

Semiparametric estimation of a bivariate Tobit model

The existing semiparametric estimation literature has mainly focused on univariate Tobit models and no semiparametric estimation has been considered for bivariate Tobit models. In this paper, we consider semiparametric estimation of the bivariate Tobit model proposed by Amemiya (1974), under the independence condition without imposing any parametric restriction on the error distribution. Our estimator is shown to be consistent and asymptotically normal, and simulation results show that our estimator performs well in finite samples. It is also worth noting that while Amemiya’s (1974) instrumental variables estimator (IV) requires the normality assumption, our semiparametric estimator actually outperforms his IV estimator even when normality holds. Our approach can be extended to higher dimensional multivariate Tobit models.     

Characterization of the asymptotic distribution of semiparametric M-estimators

This paper develops a concrete formula for the asymptotic distribution of two-step, possibly non-smooth semiparametric M-estimators under general misspecification. Our regularity conditions are relatively straightforward to verify and also weaker than those available in the literature. The first-stage nonparametric estimation may depend on finite dimensional parameters. We characterize: (1) conditions under which the first-stage estimation of nonparametric components do not affect the asymptotic distribution, (2) conditions under which the asymptotic distribution is affected by the derivatives of the first-stage nonparametric estimator with respect to the finite-dimensional parameters, and (3) conditions under which one can allow non-smooth objective functions. Our framework is illustrated by applying it to three examples: (1) profiled estimation of a single index quantile regression model, (2) semiparametric least squares estimation under model misspecification, and (3) a smoothed matching estimator.     

Estimation and model selection of semiparametric multivariate survival functions under general censorship

We study estimation and model selection of semiparametric models of multivariate survival functions for censored data, which are characterized by possibly misspecified parametric copulas and nonparametric marginal survivals. We obtain the consistency and root-n$n$ asymptotic normality of a two-step copula estimator to the pseudo-true copula parameter value according to KLIC, and provide a simple consistent estimator of its asymptotic variance, allowing for a first-step nonparametric estimation of the marginal survivals. We establish the asymptotic distribution of the penalized pseudo-likelihood ratio statistic for comparing multiple semiparametric multivariate survival functions subject to copula misspecification and general censorship. An empirical application is provided.     

Estimating panel data models in the presence of endogeneity and selection

We consider estimation of panel data models with sample selection when the equation of interest contains endogenous explanatory variables as well as unobserved heterogeneity. Assuming that appropriate instruments are available, we propose several tests for selection bias and two estimation procedures that correct for selection in the presence of endogenous regressors. The tests are based on the fixed effects two-stage least squares estimator, thereby permitting arbitrary correlation between unobserved heterogeneity and explanatory variables. The first correction procedure is parametric and is valid under the assumption that the errors in the selection equation are normally distributed. The second procedure estimates the model parameters semiparametrically using series estimators. In the proposed testing and correction procedures, the error terms may be heterogeneously distributed and serially dependent in both selection and primary equations. Because these methods allow for a rather flexible structure of the error variance and do not impose any nonstandard assumptions on the conditional distributions of explanatory variables, they provide a useful alternative to the existing approaches presented in the literature.     

Semiparametric efficient estimation of dynamic panel data models

This paper extends the semiparametric efficient treatment of panel data models pursued by Park and Simar [Park, B.U., Simar, L., 1994. Efficient semiparametric estimation in stochastic frontier models. Journal of the American Statistical Association 89, 929–936] and Park et al. [Park, B.U., Sickles, R.C., Simar, L., 1998. Stochastic frontiers: a semiparametric approach. Journal of Econometrics 84, 273–301; Park, B.U., Sickles, R.C., Simar, L., 2003. Semiparametric efficient estimation of AR(1) panel data models. Journal of Econometrics 117, 279–309] to a dynamic panel setting. We develop a semiparametric efficient estimator under minimal assumptions when the panel model contains a lagged dependent variable. We apply this new estimator to analyze the structure of demand between city pairs for selected U.S. airlines during the period 1979 I–1992 IV.     

Identification and information in monotone binary models

This paper considers binary response models where errors are uncorrelated with a set of instrumental variables and are independent of a continuous regressor v$v$, conditional on all other variables. It is shown that these exclusion restrictions are not sufficient for identification and that additional identifying assumptions are needed. Such an assumption, introduced by Lewbel [Semiparametric qualitative response model estimation with unknown heteroskedasticity or instrumental variables. Journal of Econometrics 97, 145–177], is that the support of the continuous regressor is large, but we show that it significantly restricts the class of binary phenomena which can be analysed. We propose an alternative additional assumption under which β$\beta$ remains just identified and the estimation unchanged. This alternative assumption does not impose specific restrictions on the data, which broadens the scope of the estimation method in empirical work. The semiparametric efficiency bound of the model is also established and an existing estimator is shown to achieve that bound. The efficient estimator uses a plug-in density estimate. It is shown that plugging in the true density rather than an estimate is inefficient. Extensions to ordered choice models are provided.     

Model‐Assisted Regression Estimators for Longitudinal Data with Nonignorable Dropout

Estimation with longitudinal Y having nonignorable dropout is considered when the joint distribution of Y and covariate X is nonparametric and the dropout propensity conditional on (Y,X) is parametric. We apply the generalised method of moments to estimate the parameters in the nonignorable dropout propensity based on estimating equations constructed using an instrument Z, which is part of X related to Y but unrelated to the dropout propensity conditioned on Y and other covariates. Population means and other parameters in the nonparametric distribution of Y can be estimated based on inverse propensity weighting with estimated propensity. To improve efficiency, we derive a model‐assisted regression estimator making use of information provided by the covariates and previously observed Y‐values in the longitudinal setting. The model‐assisted regression estimator is protected from model misspecification and is asymptotically normal and more efficient when the working models are correct and some other conditions are satisfied. The finite‐sample performance of the estimators is studied through simulation, and an application to the HIV‐CD4 data set is also presented as illustration.     

Semiparametric bounds on treatment effects

We present a variety of semiparametric models that produce bounds on the average causal effect of a binary treatment on a binary outcome. The semiparametric assumptions exploit variation in observable covariates to narrow the bounds. In our main model, the outcome is determined by a generalized linear model, but the treatment may be arbitrarily endogenous. Our bounding strategy does not require the existence of an instrument, but incorporating an instrument narrows the bounds. The bounds are further improved by combining the semiparametric model with the joint threshold-crossing assumption of Shaikh and Vytlacil (2005).     

Robustness issues in structural equation modeling: a review of recent developments

In structural equation modeling the statistician needs assumptions inorder (1) to guarantee that the estimates are consistent for the parameters of interest, and (2) to evaluate precision of the estimates and significance level of test statistics. With respect to purpose (1), the typical type of analyses (ML and WLS) are robust against violation of distributional assumptions; i.e., estimates remain consistent or any type of WLS analysis and distribution of z. (It should be noted, however, that (1) is sensitive to structural misspecification.) A typical assumption used for purpose (2), is the assumption that the vector z of observable follows a multivariate normal distribution.In relation to purpose (2), distributional misspecification may have consequences for efficiency, as well as power of test statistics (see Satorra, 1989a); that is, some estimation methods may bemore precise than others for a given specific distribution of z. For instance, ADF-WLS is asymptotically optimal under a variety of distributions of z, while the asymptotic optimality of NT-WLS may be lost when the data is non-normal     

Nonparametric IV estimation of local average treatment effects with covariates

In this paper nonparametric instrumental variable estimation of local average treatment effects (LATE) is extended to incorporate covariates. Estimation of LATE is appealing since identification relies on much weaker assumptions than the identification of average treatment effects in other nonparametric instrumental variable models. Including covariates in the estimation of LATE is necessary when the instrumental variable itself is confounded, such that the IV assumptions are valid only conditional on covariates. Previous approaches to handle covariates in the estimation of LATE relied on parametric or semiparametric methods. In this paper, a nonparametric estimator for the estimation of LATE with covariates is suggested that is root-n asymptotically normal and efficient.     

Pricing used books on Amazon.com: a spatial approach to price dispersion

This paper estimates a spatial autoregressive (SAR) model of price dispersion using publicly available internet bookselling data. It uses a semiparametric adaptive estimator that does not require the usual Gaussian assumption of maximum likelihood (ML) estimators. The results suggest that both price competition and seller heterogeneity are key drivers of the observed price dispersion. The paper finds that sellers with large sales volume, newly established sellers and US mainland states-based sellers tend to price lower. The identified significant spatial interaction is evidence of spatial price competition. Controlling for everything else, a seller asks a lower price when large sellers charge relatively high prices, which is also evidence of price-based selling and undercutting.     

Testing the capital asset pricing model efficiently under elliptical symmetry: a semiparametric approach

We develop new tests of the capital asset pricing model that take account of and are valid under the assumption that the distribution generating returns is elliptically symmetric; this assumption is necessary and sufficient for the validity of the CAPM. Our test is based on semiparametric efficient estimation procedures for a seemingly unrelated regression model where the multivariate error density is elliptically symmetric, but otherwise unrestricted. The elliptical symmetry assumption allows us to avoid the curse of dimensionality problem that typically arises in multivariate semiparametric estimation procedures, because the multivariate elliptically symmetric density function can be written as a function of a scalar transformation of the observed multivariate data. The elliptically symmetric family includes a number of thick‐tailed distributions and so is potentially relevant in financial applications. Our estimated betas are lower than the OLS estimates, and our parameter estimates are much less consistent with the CAPM restrictions than the corresponding OLS estimates. Copyright © 2002 John Wiley & Sons, Ltd.     

Model misspecification effects for biased samples

The model misspecification effects on the maximum likelihood estimator are studied when a biased sample is treated as a random one as well as when a random sample is treated as a biased one. The relation between the existence of a consistent estimator under model misspecification and the completeness of the distribution is also considered. The cases of the weight invariant distribution and the scale parameter distribution are examined and finally an example is presented to illustrate the results.     

Gaussian semiparametric estimation of multivariate fractionally integrated processes

This paper analyzes the semiparametric estimation of multivariate long-range dependent processes. The class of spectral densities considered is motivated by and includes those of multivariate fractionally integrated processes. The paper establishes the consistency of the multivariate Gaussian semiparametric estimator (GSE), which has not been shown in other work, and the asymptotic normality of the GSE estimator. The proposed GSE estimator is shown to have a smaller limiting variance than the two-step GSE estimator studied by Lobato [1999. A semiparametric two-step estimator in a multivariate long memory model. Journal of Econometrics 90, 129–153]. Gaussianity is not assumed in the asymptotic theory. Some simulations confirm the relevance of the asymptotic results in samples of the size used in practical work.     

Distribution free estimation of heteroskedastic binary response models using Probit/Logit criterion functions

In this paper estimators for distribution free heteroskedastic binary response models are proposed. The estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models (such as Probit/Logit) exhibiting (multiplicative) heteroskedasticity. The first proposed estimator is based on the observational equivalence between the two models, and is a semiparametric sieve estimator (see, e.g. Gallant and Nychka (1987), Ai and Chen (2003) and Chen et al. (2005)) for the regression coefficients, based on maximizing standard Logit/Probit criterion functions, such as NLLS and MLE. This procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously. The second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model (,  and ) and the standard parametric (Probit/Logit) NLLS estimator. This estimator has the advantage of being implementable with standard software packages such as Stata. Distribution theory is developed for both estimators and a Monte Carlo study indicates they both perform well in finite samples.     

A Predictive Estimator of the Mean with Missing Data

One of the most difficult problems confronting investigators who analyze data from surveys is how treat missing data. Many statistical procedures can not be used immediately if any values are missing. This paper considers the problem of estimating the population mean using auxiliary information when some observations on the sample are missing and the population mean of the auxiliary variable is not available. We use tools of classical statistical estimation theory to find a suitable estimator. We study the model and design properties of the proposed estimator. We also report the results of a broad-based simulation study of the efficiency of the estimator, which reveals very promising results.     

Covariate Measurement Error in Quadratic Regression

We consider quadratic regression models where the explanatory variable is measured with error. The effect of classical measurement error is to flatten the curvature of the estimated function. The effect on the observed turning point depends on the location of the true turning point relative to the population mean of the true predictor. Two methods for adjusting parameter estimates for the measurement error are compared. First, two versions of regression calibration estimation are considered. This approximates the model between the observed variables using the moments of the true explanatory variable given its surrogate measurement. For certain models an expanded regression calibration approximation is exact. The second approach uses moment-based methods which require no assumptions about the distribution of the covariates measured with error. The estimates are compared in a simulation study, and used to examine the sensitivity to measurement error in models relating income inequality to the level of economic development. The simulations indicate that the expanded regression calibration estimator dominates the other estimators when its distributional assumptions are satisfied. When they fail, a small-sample modification of the method-of-moments estimator performs best. Both estimators are sensitive to misspecification of the measurement error model.     

Nonparametric estimation for self-selected interval data collected through a two-stage approach

Self-selected interval data arise in questionnaire surveys when respondents are free to answer with any interval without having pre-specified ranges. This type of data is a special case of interval-censored data in which the assumption of noninformative censoring is violated, and thus the standard methods for interval-censored data (e.g. Turnbull’s estimator) are not appropriate because they can produce biased results. Based on a certain sampling scheme, this paper suggests a nonparametric maximum likelihood estimator of the underlying distribution function. The consistency of the estimator is proven under general assumptions, and an iterative procedure for finding the estimate is proposed. The performance of the method is investigated in a simulation study.     

Optimal convergence rates,Bahadur representation,and asymptotic normality of partitioning estimators

This paper studies the asymptotic properties of partitioning estimators of the conditional expectation function and its derivatives. Mean-square and uniform convergence rates are established and shown to be optimal under simple and intuitive conditions. The uniform rate explicitly accounts for the effect of moment assumptions, which is useful in semiparametric inference. A general asymptotic integrated mean-square error approximation is obtained and used to derive an optimal plug-in tuning parameter selector. A uniform Bahadur representation is developed for linear functionals of the estimator. Using this representation, asymptotic normality is established, along with consistency of a standard-error estimator. The finite-sample performance of the partitioning estimator is examined and compared to other nonparametric techniques in an extensive simulation study.