Semiparametric GEE analysis in partially linear single-index models for longitudinal data

In this article, we study a partially linear single-index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases. A semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations (GEE) is introduced to estimate the two parameter vectors as well as the unknown link function. Under some mild conditions, we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios, from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data. We also discuss the estimation of the covariance (or weight) matrices involved in the semiparametric GEE method. Furthermore, we provide some numerical studies including Monte Carlo simulation and an empirical application to illustrate our methodology and theory.Comment: Published at http://dx.doi.org/10.1214/15-AOS1320 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation of a rank-reduced functional-coefficient panel data model with serial correlation

We consider estimation of a functional-coefficient panel data model. This model is useful for modeling time varying and cross-sectionally heterogeneous relationships between economic variables. We allow for serial correlation and heteroscedasticity in the model. When the number of explanatory variables is large, we impose a rank-reduced structure on the model’s functional coefficients to reduce the number of functions to be estimated and thus improve estimation efficiency. To adjust for serial correlation and further improve estimation efficiency, we use a Cholesky decomposition on the serial covariance matrices to produce a transformation of the original panel data model. By applying the standard semiparametric profile least squares method to the transformed model, more efficient estimates of the coefficient functions can be obtained. Under some regularity conditions, we derive the asymptotic distribu- tion for the developed semiparametric estimators and show their efficiency improvement under correct specification of the serial covariance matrices. To attain this efficiency gain when the serial covariance structure is unknown, we propose approaches to consistently estimate the lower triangular matrix in the Cholesky decomposition for balanced panel data, and the serial covariance matrices for unbalanced panel data. Numerical studies, including Monte Carlo experiments and an empirical application to economic growth data, show that the developed semiparametric method works reasonably well in finite samples

Semiparametric Bayesian inference in smooth coefficient models

We describe procedures for Bayesian estimation and testing in cross-sectional, panel data and nonlinear smooth coefficient models. The smooth coefficient model is a generalization of the partially linear or additive model wherein coefficients on linear explanatory variables are treated as unknown functions of an observable covariate. In the approach we describe, points on the regression lines are regarded as unknown parameters and priors are placed on differences between adjacent points to introduce the potential for smoothing the curves. The algorithms we describe are quite simple to implement - for example, estimation, testing and smoothing parameter selection can be carried out analytically in the cross-sectional smooth coefficient model. We apply our methods using data from the National Longitudinal Survey of Youth (NLSY). Using the NLSY data we first explore the relationship between ability and log wages and flexibly model how returns to schooling vary with measured cognitive ability. We also examine a model of female labor supply and use this example to illustrate how the described techniques can been applied in nonlinear settings

Semiparametric Bayesian inference in multiple equation models

This paper outlines an approach to Bayesian semiparametric regression in multiple equation models which can be used to carry out inference in seemingly unrelated regressions or simultaneous equations models with nonparametric components. The approach treats the points on each nonparametric regression line as unknown parameters and uses a prior on the degree of smoothness of each line to ensure valid posterior inference despite the fact that the number of parameters is greater than the number of observations. We develop an empirical Bayesian approach that allows us to estimate the prior smoothing hyperparameters from the data. An advantage of our semiparametric model is that it is written as a seemingly unrelated regressions model with independent normal-Wishart prior. Since this model is a common one, textbook results for posterior inference, model comparison, prediction and posterior computation are immediately available. We use this model in an application involving a two-equation structural model drawn from the labour and returns to schooling literatures

Projected principal component analysis in factor models

This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employs principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accurately even when the sample size is finite. We propose a flexible semiparametric factor model, which decomposes the factor loading matrix into the component that can be explained by subject-specific covariates and the orthogonal residual component. The covariates' effects on the factor loadings are further modeled by the additive model via sieve approximations. By using the newly proposed Projected-PCA, the rates of convergence of the smooth factor loading matrices are obtained, which are much faster than those of the conventional factor analysis. The convergence is achieved even when the sample size is finite and is particularly appealing in the high-dimension-low-sample-size situation. This leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings. The proposed method is illustrated by both simulated data and the returns of the components of the S&P 500 index.Comment: Published at http://dx.doi.org/10.1214/15-AOS1364 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fused kernel-spline smoothing for repeatedly measured outcomes in a generalized partially linear model with functional single index

We propose a generalized partially linear functional single index risk score model for repeatedly measured outcomes where the index itself is a function of time. We fuse the nonparametric kernel method and regression spline method, and modify the generalized estimating equation to facilitate estimation and inference. We use local smoothing kernel to estimate the unspecified coefficient functions of time, and use B-splines to estimate the unspecified function of the single index component. The covariance structure is taken into account via a working model, which provides valid estimation and inference procedure whether or not it captures the true covariance. The estimation method is applicable to both continuous and discrete outcomes. We derive large sample properties of the estimation procedure and show a different convergence rate for each component of the model. The asymptotic properties when the kernel and regression spline methods are combined in a nested fashion has not been studied prior to this work, even in the independent data case.Comment: Published at http://dx.doi.org/10.1214/15-AOS1330 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Does aggregate relative risk aversion change countercyclically over time? evidence from the stock market

Using a semiparametric estimation technique, we show that the risk-return tradeoff and the Sharpe ratio of the stock market increases monotonically with the consumption wealth ratio (CAY) across time. While early studies have commonly interpreted such a finding as evidence of the countercyclical variation in aggregate relative risk aversion (RRA), we argue that it mainly reflects changes in investment opportunities for two reasons. First, we fail to reject the null hypothesis of constant RRA after controlling for CAY as a proxy for the hedge against changes in the investment opportunity set. Second, by contrast with habit formation models but consistent with ICAPM, we find that loadings on the conditional stock market variance scaled by CAY are negatively priced in the cross-sectional regressions. For illustration, we replicate the countercyclical stock market risk-return tradeoff using simulated data from Guo's (2004) limited stock market participation model, in which RRA is constant and CAY is a proxy for shareholders' liquidity conditions.Capital assets pricing model ; Stock market

Nonparametric estimation of varying coefficient dynamic panel models

This is the publisher's version, also available electronically from http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2059888&fileId=S0266466608080523.We suggest using a class of semiparametric dynamic panel data models to capture individual variations in panel data. The model assumes linearity in some continuous/discrete variables that can be exogenous/endogenous and allows for nonlinearity in other weakly exogenous variables. We propose a nonparametric generalized method of moments (NPGMM) procedure to estimate the functional coefficients, and we establish the consistency and asymptotic normality of the resulting estimators

Three Essays on Nonparametric and Semiparametric Methods and Their Applications

This dissertation contains three essays on nonparametric and semiparametric regression methods. In the first essay, we consider the problem of nonparametric regression with mixed discrete and continuous covariates using the k-nearest neighbor (k-nn) method. We derive the asymptotic normality of the proposed estimator and use Monte Carlo simulations to demonstrate its finite sample performance. We apply the method to estimate corn yields in Iowa as a function of agricultural district, temperature, and precipitation. In the second essay, we consider the problem of testing error serial correlation in fixed effects panel data models in a nonparametric framework. We show that our test statistic has a standard normal distribution under the null hypothesis of zero serial correlation. The test statistic diverges to infinity at the rate of √N under the alternative hypothesis that errors are serially correlated, where N is the cross-sectional sample size. We propose a bootstrap version of the test which we show to perform well in finite sample applications. In the third essay, we consider estimation of varying-coefficient single-index models with an endogenous regressor. We propose a multi-step instrumental variables procedure to estimate the coefficient function and the corresponding index parameters. We prove the consistency of the estimators, and we present Monte Carlo simulations demonstrating their finite sample performance. We then apply the proposed method to examine the determinants of aggregate illiquidity in the U.S. stock market

A selective overview of nonparametric methods in financial econometrics

This paper gives a brief overview on the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inferences of instantaneous returns and volatility functions of time-homogeneous and time-dependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.Comment: 32 pages include 7 figure