Finite-dimensional Parameter Research Articles

Two-step series estimation and specification testing of (partially) linear models with generated regressors

This paper studies three semiparametric models that are useful and frequently encountered in applied econometric work—a linear and two partially linear specifications with generated regressors, i.e., the regressors that are unobserved, but can be nonparametrically estimated from the data. Our framework allows for generated regressors to appear in linear or nonlinear components of partially linear models. We propose two-step series estimators for the finite-dimensional parameters, establish their -consistency (with sample size n) and asymptotic normality, and provide the asymptotic variance formulae that take into account the estimation error of generated regressors. Moreover, we develop a nonparametric specification test for the models considered. Numerical performances of the proposed estimators and test via simulation experiments and an empirical application illustrate the utility of our approach.

Identification and Estimation of Multinomial Choice Models with Latent Special Covariates

Identification of multinomial choice models is often established by using special covariates that have full support. This article shows how these identification results can be extended to a large class of multinomial choice models when all covariates are bounded. I also provide a new -consistent asymptotically normal estimator of the finite-dimensional parameters of the model.

Semiparametric Bayesian inference for optimal dynamic treatment regimes via dynamic marginal structural models.

Considerable statistical work done on dynamic treatment regimes (DTRs) is in the frequentist paradigm, but Bayesian methods may have much to offer in this setting as they allow for the appropriate representation and propagation of uncertainty, including at the individual level. In this work, we extend the use of recently developed Bayesian methods for Marginal Structural Models to arrive at inference of DTRs. We do this (i) by linking the observational world with a world in which all patients are randomized to a DTR, thereby allowing for causal inference and then (ii) by maximizing a posterior predictive utility, where the posterior distribution has been obtained from nonparametric prior assumptions on the observational world data-generating process. Our approach relies on Bayesian semiparametric inference, where inference about a finite-dimensional parameter is made all while working within an infinite-dimensional space of distributions. We further study Bayesian inference of DTRs in the double robust setting by using posterior predictive inference and the nonparametric Bayesian bootstrap. The proposed methods allow for uncertainty quantification at the individual level, thereby enabling personalized decision-making. We examine the performance of these methods via simulation and demonstrate their utility by exploring whether to adapt HIV therapy to a measure of patients' liver health, in order to minimize liver scarring.

LOCAL ASYMPTOTIC NORMALITY OF GENERAL CONDITIONALLY HETEROSKEDASTIC AND SCORE-DRIVEN TIME-SERIES MODELS

The paper establishes the local asymptotic normality property for general conditionally heteroskedastic time series models of multiplicative form, $\epsilon _t=\sigma _t(\boldsymbol {\theta }_0)\eta _t$ , where the volatility $\sigma _t(\boldsymbol {\theta }_0)$ is a parametric function of $\{\epsilon _{s}, s< t\}$ , and $(\eta _t)$ is a sequence of i.i.d. random variables with common density $f_{\boldsymbol {\theta }_0}$ . In contrast with earlier results, the finite dimensional parameter $\boldsymbol {\theta }_0$ enters in both the volatility and the density specifications. To deal with nondifferentiable functions, we introduce a conditional notion of the familiar quadratic mean differentiability condition which takes into account parameter variation in both the volatility and the errors density. Our results are illustrated on two particular models: the APARCH with asymmetric Student-t distribution, and the Beta-t-GARCH model, and are extended to handle a conditional mean.

Trading information, price discreteness, and volatility estimation

This paper investigates the challenging problem of volatility estimation using high-frequency financial data when both price discreteness and structural random market microstructure noise, i.e., a known function of trading information and a finite-dimensional parameter vector, are present. Under this setup, we propose a novel approach that features both nonlinear least squares and particle filters for the estimation of integrated volatility. Theoretical underpinnings are provided justifying the effectiveness of our method. Both simulation and empirical results show the clear advantage of our approach over a number of existing methods.

Parameter space study of optimal scale-dependent weights in TV image denoising

We investigate the choice of finite-dimensional parameter spaces within a bilevel optimization framework for selecting scale-dependent weights in total variation image denoising with non-uniform noise. Due to the pointwise box constraints on the parameter function, we prove existence of Lagrange multipliers in low regularity spaces, and derive a first-order optimality system. To cope with the difficulties related to the lack of regularity, a Moreau-Yosida regularization is introduced and convergence of the regularized optimal parameters towards the optimal weight for the original problem is verified. For each regularized bilevel problem a second-order quasi-Newton algorithm is proposed, together with a semismooth Newton scheme for solving the lower-level problem. Finally, several numerical tests are carried out to compare the different parameter space choices and draw some conclusions.

Feedback in panel data models

Much of the analysis of panel data has been based on an assumption of strict exogeneity. Distributions are specified for outcome variables conditional on a latent individual effect and conditional on observed predictor variables at all dates, with the future values of the predictor variables assumed to have no effect on the conditional distribution. The paper relaxes this assumption in order to allow for lagged dependent variables and, more generally, for feedback from lagged dependent variables to current values of the predictor variables. Such feedback would arise in an evaluation study if the treatment variable is randomly assigned only conditional on the individual effect and on previous outcomes.An information bound is derived for a semiparametric regression model with sequential moment restrictions, with the information set increasing over time. The bound is then applied to a model with a (scalar) multiplicative random effect. The mean of the random effect conditional on the predictor variables is not restricted, so that the random effect can control for various omitted variables. This conditional mean is the nonparametric component of the semiparametric regression model. There is a transformation that eliminates the random effect and leads to a set of sequential moment restrictions in which the moment function depends on only a finite-dimensional parameter. The information bound for this simpler problem coincides with that of the original problem. The form of the optimal instrumental variables is derived.The paper also considers the identification problems that arise when the random effect is a vector with two or more components.

Methods Based on Semiparametric Theory for Analysis in the Presence of Missing Data

A statistical model is a class of probability distributions assumed to contain the true distribution generating the data. In parametric models, the distributions are indexed by a finite-dimensional parameter characterizing the scientific question of interest. Semiparametric models describe the distributions in terms of a finite-dimensional parameter and an infinite-dimensional component, offering more flexibility. Ordinarily, the statistical model represents distributions for the full data intended to be collected. When elements of these full data are missing, the goal is to make valid inference on the full-data-model parameter using the observed data. In a series of fundamental works, Robins, Rotnitzky, and colleagues derived the class of observed-data estimators under a semiparametric model assuming that the missingness mechanism is at random, which leads to practical, robust methodology for many familiar data-analytic challenges. This article reviews semiparametric theory and the key steps in this derivation.

An IV estimator for a functional coefficient model with endogenous discrete treatments

We propose instrumental variables (IV) estimators for averaged conditional treatment effects and the parameters upon which they depend in the context of a semiparametric outcome model with endogenous discrete treatment variables. For this model, the treatment impacts are unknown functions of a vector of indices that depend on a finite dimensional parameter vector. We develop the theory for an estimator of these impacts when they are averaged over regions of interest. We prove identification, consistency and -asymptotic normality of the estimators. We also show that they are efficient under correct model specification. Further, we show that they are robust to misspecification of the propensity score model. In the Monte Carlo study, the estimators perform well over a wide variety of designs covering both correct and incorrect propensity score model specification.

Parametric regression analysis of bivariate the proportional hazards model with current status data

In this paper, we show the maximum sieved probability of each of the finite Dimensional parameters in a marginal Proportional Hazards risk model with bivariate current position data. We used the copula model to model the combined distribution of bivariate survival times. Simulation studies reveal that the proposed estimations for it have good finite sample properties.

Sample selection models with monotone control functions

The celebrated Heckman selection model yields a selection correction function (control function) proportional to the inverse Mills ratio, which is monotone. This paper studies a sample selection model that does not impose parametric distributional assumptions on the latent error terms, while maintaining the monotonicity of the control function. We show that a positive (negative) dependence condition on the latent error terms is sufficient for the monotonicity of the control function. The condition is equivalent to a restriction on the copula function of latent error terms. Using the monotonicity, we propose a tuning-parameter-free semiparametric estimation method and establish root n-consistency and asymptotic normality for the estimates of finite-dimensional parameters. A new test for selectivity is also developed in the presence of the shape restriction. Simulations and an empirical application are conducted to illustrate the usefulness of the proposed methods.

Inference for partially linear additive higher-order spatial autoregressive model with spatial autoregressive error and unknown heteroskedasticity

This article extends spatial autoregressive model with spatial autoregressive disturbances (SARAR(1,1)) which is the most popular spatial econometric model to the case of an arbitrary finite number of nonparametric additive terms and spatial autoregressive models with spatial autoregressive disturbances of arbitrary finite order (SARAR(R,S)). We propose a sieve two-stage least squares (S2SLS) regression and generalized method of moments (GMM) procedure of the high-order spatial autoregressive parameters of the disturbance process. Under some sufficient conditions, we show that the proposed estimator for the finite dimensional parameter is consistent and asymptotically normally distributed. We show that each proposed estimator for the additive terms is consistent and also asymptotically normally distributed at a rate slower than Consistent estimators for the asymptotic variances of the proposed estimators are provided. In addition, using asymptotic properties to make statistical inference for the parametric and additive is also considered. Monte Carlo evidence suggests that the estimation procedure performs reasonably well in small samples and the proposed approach has some practical value. The proposed method is applied to analyzing factors which affect haze pollution.

Predicting the popularity of tweets using internal and external knowledge: an empirical Bayes type approach

The problem of tweet popularity prediction, or forecasting the total number of retweets stemming from an ancestral tweet, has attracted considerable interest recently. The prediction can be accomplished by fitting a point process model to the sequence of retweet times up to a certain censoring time and project the fitted model to a future time point. However, models employing such approach tend to have inferior prediction accuracy when the censoring time is too short before sufficient information can accumulate. To overcome this, we propose an empirical Bayes type approach of parameter estimation to combine internal knowledge on the times of historical retweets up to the censoring time and external knowledge on complete retweet sequences in the training data. We demonstrate the approach using several point process models with finite-dimensional parameters, where the prior distribution for the parameter of each model is constructed based on the external knowledge, and the likelihood is calculated based on the internal knowledge. The mode of the posterior distribution is used as the estimator of the finite-dimensional parameter, and the mean of the predictive distribution for the number of retweets implied by each of the estimated models is used to predict the tweet popularity. Using a large Twitter data set, we reveal that the proposed methodology not only enables prediction at time zero before the arrival of any retweet event, but also substantially improves the prediction performances of existing models, especially at earlier censoring times.

Nonignorable Missing Data, Single Index Propensity Score and Profile Synthetic Distribution Function

In missing data problems, missing not at random is difficult to handle since the response probability or propensity score is confounded with the outcome data model in the likelihood. Existing works often assume the propensity score is known up to a finite dimensional parameter. We relax this assumption and consider an unspecified single index model for the propensity score. A pseudo-likelihood based on the complete data is constructed by profiling out a synthetic distribution function that involves the unknown propensity score. The pseudo-likelihood gives asymptotically normal estimates. Simulations show the method compares favorably with existing methods.

Nonparametric Estimation and Testing for Positive Quadrant Dependent Bivariate Copula

In many practical scenarios (e.g., finance, system reliability, etc.), it is often of interest to estimate a bivariate distribution and test for some desired association properties like positive quadrant dependent (PQD) or negative quadrant dependent (NQD). Often estimation and testing for PQD/NQD property are performed using copula models as it then eliminates the need for estimating marginal distributions. Many parametric copula families have been used that allow for controlling the PQD/NQD property by a finite dimensional parameter (often just real-valued) and the problem reduces to the straightforward estimation and testing for fixed dimensional parameter using standard statistical methodologies (e.g., maximum likelihood). This article extends such a line of work by dropping any parametric assumptions and provides a fully data-dependent automated approach to estimate a copula and test for PQD property. The estimator is shown to be large-sample consistent under a set of mild regularity conditions. Numerical illustrations based on simulated data are also provided to compare the performance of the proposed testing procedure with some available methods and applications to real case studies are also provided.

Joint analysis of multivariate interval-censored survival data and a time-dependent covariate.

We develop a joint modeling method for multivariate interval-censored survival data and a time-dependent covariate that is intermittently measured with error. The joint model is estimated using nonparametric maximum likelihood estimation, which is carried out via an expectation-maximization algorithm, and the inference for finite-dimensional parameters is performed using bootstrap. We also develop a similar joint modeling method for univariate interval-censored survival data and a time-dependent covariate, which excels the existing methods in terms of model flexibility and interpretation. Simulation studies show that the model fitting and inference approaches perform very well under realistic sample sizes. We apply the method to a longitudinal study of dental caries in African-American children from low-income families in the city of Detroit, Michigan.

Large Sample Justifications for the Bayesian Empirical Likelihood

This study investigates the asymptotic properties of the Bayesian empirical likelihood (BEL), which uses the empirical likelihood as an alternative to a parametric likelihood for Bayesian inference. We establish two asymptotic equivalence results based on the Bernstein-von Mises (BvM) theorem by introducing a new formulation of the moment restriction model. First, the limiting posterior distribution of the BEL is the same as that of a parametric Bayesian method that uses the likelihood of a least favorable model of the moment restriction model. Second, the limiting posterior distribution is also the same as that of a semi-parametric Bayesian method that places priors on both a finite-dimensional parameter of interest and an infinite-dimensional nuisance parameter. Because parametric and semi-parametric Bayesian methods are legitimate Bayesian procedures, the equivalence results provide a large sample justification for the BEL as a Bayesian inference method. Moreover, the BvM theorem guarantees the validity of the BEL as a frequentist estimator.

Construction of leading economic index for recession prediction using vine copulas

Abstract This paper constructs a composite leading index for business cycle prediction based on vine copulas that capture the complex pattern of dependence among individual predictors. This approach is optimal in the sense that the resulting index possesses the highest discriminatory power as measured by the receiver operating characteristic (ROC) curve. The model specification is semi-parametric in nature, suggesting a two-step estimation procedure, with the second-step finite dimensional parameter being estimated by QMLE given the first-step non-parametric estimate. To illustrate its usefulness, we apply this methodology to optimally aggregate the 10 leading indicators selected by The Conference Board (TCB) to predict economic recessions in the United States. In terms of the discriminatory power, our method is significantly better than the Index used by TCB.

Sensitivity Analysis in Semiparametric Likelihood Models

We provide methods for inference on a finite dimensional parameter of interest, � 2 < d�, in a semiparametric probability model when an infinite dimensional nuisance parameter, g, is present. We depart from the semiparametric literature in that we do not require that the pair (�,g) is point identified and so we construct confidence regions forthat are robust to non-point identification. This allows practitioners to examine the sensitivity of their estimates ofto specification of g in a likelihood setup. To construct these confidence regions for �, we invert a profiled sieve likelihood ratio (LR) statistic. We derive the asymptotic null distribution of this profiled sieve LR, which is nonstandard whenis not point identified (but is � 2 distributed under point identifica- tion). We show that a simple weighted bootstrap procedure consistently estimates this complicated distribution's quantiles. Monte Carlo studies of a semiparametric dynamic binary response panel data model indicate that our weighted bootstrap procedures per- forms adequately in finite samples. We provide three empirical illustrations where we compare our results to the ones obtained using standard (less robust) methods.

Two-step combined nonparametric likelihood estimation of misspecified semiparametric models

This paper proposes to estimate possibly misspecified semiparametric estimating equations models using a two-step combined nonparametric likelihood method. The method uses in the first step the plug in principle and replaces the infinite dimensional parameter with a consistent estimator. In the second step an estimator for the finite dimensional parameter is obtained by combining exponential tilting with a another member of the empirical Cressie-Read discrepancy. The resulting class of semiparametric estimators are robust to misspecification and have the same asymptotic variance as that of the efficient semiparametric generalised method of moment estimator under correct specification. It is also shown that the asymptotic distributions of the proposed estimators can be consistently estimated by a multiplier bootstrap procedure. The results of the paper are illustrated with a quadratic inference function model and an instrumental variable partially linear additive model. Monte Carlo evidence suggests that the proposed estimators have competitive finite sample properties.