Robust penalized quantile regression estimation for panel data

This paper investigates a class of penalized quantile regression estimators for panel data. The penalty serves to shrink a vector of individual specific effects toward a common value. The degree of this shrinkage is controlled by a tuning parameter λ$\lambda$. It is shown that the class of estimators is asymptotically unbiased and Gaussian, when the individual effects are drawn from a class of zero-median distribution functions. The tuning parameter, λ$\lambda$, can thus be selected to minimize estimated asymptotic variance. Monte Carlo evidence reveals that the estimator can significantly reduce the variability of the fixed-effect version of the estimator without introducing bias.     

Nonparametric estimation and inference on conditional quantile processes

This paper presents estimation methods and asymptotic theory for the analysis of a nonparametrically specified conditional quantile process. Two estimators based on local linear regressions are proposed. The first estimator applies simple inequality constraints while the second uses rearrangement to maintain quantile monotonicity. The bandwidth parameter is allowed to vary across quantiles to adapt to data sparsity. For inference, the paper first establishes a uniform Bahadur representation and then shows that the two estimators converge weakly to the same limiting Gaussian process. As an empirical illustration, the paper considers a dataset from Project STAR and delivers two new findings.     

Efficient estimation in dynamic conditional quantile models

In this paper we consider the problem of semiparametric efficient estimation in conditional quantile models with time series data. We construct an M-estimator which achieves the semiparametric efficiency bound recently derived by Komunjer and Vuong (forthcoming). Our efficient M-estimator is obtained by minimizing an objective function which depends on a nonparametric estimator of the conditional distribution of the variable of interest rather than its density. The estimator is new and not yet seen in the literature. We illustrate its performance through a Monte Carlo experiment.     

Two-stage rank estimation of quantile index models

This paper estimates a class of models which satisfy a monotonicity condition on the conditional quantile function of the response variable. This class includes as a special case the monotonic transformation model with the error term satisfying a conditional quantile restriction, thus allowing for very general forms of conditional heteroscedasticity. A two-stage approach is adopted to estimate the relevant parameters. In the first stage the conditional quantile function is estimated nonparametrically by the local polynomial estimator discussed in Chaudhuri (Journal of Multivariate Analysis 39 (1991a) 246–269; Annals of Statistics 19 (1991b) 760–777) and Cavanagh (1996, Preprint). In the second stage, the monotonicity of the quantile function is exploited to estimate the parameters of interest by maximizing a rank-based objective function. The proposed estimator is shown to have desirable asymptotic properties and can then also be used for dimensionality reduction or to estimate the unknown structural function in the context of a transformation model.     

Semiparametric quantile regression estimation in dynamic models with partially varying coefficients

We study quantile regression estimation for dynamic models with partially varying coefficients so that the values of some coefficients may be functions of informative covariates. Estimation of both parametric and nonparametric functional coefficients are proposed. In particular, we propose a three stage semiparametric procedure. Both consistency and asymptotic normality of the proposed estimators are derived. We demonstrate that the parametric estimators are root-n$n$ consistent and the estimation of the functional coefficients is oracle. In addition, efficiency of parameter estimation is discussed and a simple efficient estimator is proposed. A simple and easily implemented test for the hypothesis of a varying-coefficient is proposed. A Monte Carlo experiment is conducted to evaluate the performance of the proposed estimators.     

Small sample considerations in estimation from panel data

Recent interest in statistical inference for panel data has focused on the problem of unobservable, individual-specific, random effects and the inconsistencies they introduce in estimation when they are correlated with other exogenous variables. Analysis of this problem has always assumed the variance components to be known. In this paper, we re-examine some of these questions in finite samples when the variance components must be estimated. In particular, when the effects are uncorrelated with other explanatory variables, we show that (i) the feasible Gauss-Markov estimator is more efficient than the within groups estimator for all but the fewest degrees of freedom and its variance is never more than 17% above the Cramer-Rao bound, (ii) the asymptotic approximation to the variance of the feasible Gauss-Markov estimator is similarly within 17% of the true variance but remains significantly smaller for moderately large samples sizes, and (iii) more efficient estimators for the variance components do not necessarily yield more efficient feasible Gauss-Markov estimators.     

Health and political regimes: Evidence from quantile regression

This study investigates how political regimes affect health conditions such as infant and child mortality rates and life expectancy using data from 180 countries observed between 1960 and 2013. Panel quantile regression is used to examine the effects at different intervals throughout the distribution of health outcomes. The estimation results indicate that democracy has significant positive effects on health outcomes and that its impacts are greater when health outcomes are worse. These results are robust to different democracy and health indices. The effects of different types of democracies and dictatorships are also considered, that is parliamentary, mixed (semi-presidential) and presidential democracies, and civilian, military and royal dictatorships. The parliamentary form of democracy has the largest positive impact on health outcomes at the worst quantile of health outcomes, although the difference in the impacts of the three types of democracies is not necessarily large. Furthermore, all types of dictatorships have a negative impact on health outcomes, with military dictatorship having the worst outcome when health outcomes are worse. Finally, the effects of democratization on health outcomes are significantly positive when the health outcomes are worse.     

Design and estimation problems when estimating a regression coefficient from survey data

Summary The values of a variablex are assumed known for all elements in a finite population. Between this variable and another variableY, whose values are registered in a sample survey, there is the usual linear regression relationship. This paper considers problems of design and of estimation of the regression coefficienta and the interceptb. The followingGodambe type theorem is proved: There exists no minimum variance unbiased linear estimator ofa andb. We also derive that the usual estimators ofa andb have minimum variance if attention is restricted to the class of linear estimators unbiased in any given sample.     

A minimum chi-squared method for indirect parameter estimation from Poisson data

In this paper, we introduce a new algorithm for estimating non-negative parameters from Poisson observations of a linear transformation of the parameters. The proposed objective function fits both a weighted least squares (WLS) and a minimum χ² estimation framework, and results in a convex optimization problem. Unlike conventional WLS methods, the weights do not need to be estimated from the datas, but are incorporated in the objective function. The iterative algorithm is derived from an alternating projection procedure in which "distance" is determined by the chi-squared test statistic, which is interpreted as a measure of the discrepancy between two distributions. This may be viewed as an alternative to the Kullback-Leibler divergence which corresponds to the maximum likelihood (ML) estimation. The algorithm is similar in form to, and shares many properties with, the expectation maximization algorithm for ML estimation. In particular, we show that every limit point of the algorithm is an estimator, and the sequence of projected (by the linear transformation into the data space) means converge. Despite the similarities, we show that the new estimators are quite distinct from ML estimators, and obtain conditions under which they are identical.     

On the estimation of multinomial logit models from relative frequency data

Standard estimators for the binomial logit model and for the multinomial logit model allow for an error arising from the use of relative frequencies instead of the true probabilities as the dependent variable. Recently Amemiya and Nold (1975) have considered the effect of the presence of an additional specification error in the binomial logit model and have proposed a modified logit estimation scheme to take the additional error variance into account. This paper extends their idea to the multinomial logit model and proposes an estimator that is consistent and asymptotically more efficient than the standard multinomial logit estimator. The paper presents a comparison of the results of applying the new estimator and existing estimators to a logit model for the choice of automobile ownership in the United States.     

Non-parametric, unconditional quantile estimation for efficiency analysis with an application to Federal Reserve check processing operations

This paper examines the technical efficiency of US Federal Reserve check processing offices over 1980–2003. We extend results from Park et al. [Park, B., Simar, L., Weiner, C., 2000. FDH efficiency scores from a stochastic point of view. Econometric Theory 16, 855–877] and Daouia and Simar [Daouia, A., Simar, L., 2007. Nonparametric efficiency analysis: a multivariate conditional quantile approach. Journal of Econometrics 140, 375–400] to develop an unconditional, hyperbolic, α-quantile estimator of efficiency. Our new estimator is fully non-parametric and robust with respect to outliers; when used to estimate distance to quantiles lying close to the full frontier, it is strongly consistent and converges at rate root-n, thus avoiding the curse of dimensionality that plagues data envelopment analysis (DEA) estimators. Our methods could be used by policymakers to compare inefficiency levels across offices or by managers of individual offices to identify peer offices.     

Identification and estimation of statistical functionals using incomplete data

Incomplete data, due to missing observations or interval measurement of variables, usually cause parameters of interest in applications to be unidentified except under untestable and often controversial assumptions. However, it is often possible to identify sharp bounds on parameters without making untestable assumptions about the process through which data become incomplete. The bounds contain all logically possible values of the parameters and can be estimated consistently by replacing the population distribution of the data with the empirical distribution. This is straightforward in some circumstances but computationally burdensome in others. This paper describes the general problem and presents an empirical illustration.     

Efficient estimation and inference in linear pseudo-panel data models

We consider pseudo-panel data models constructed from repeated cross sections in which the number of individuals per group is large relative to the number of groups and time periods. First, we show that, when time-invariant group fixed effects are neglected, the OLS estimator does not converge in probability to a constant but rather to a random variable. Second, we show that, while the fixed-effects (FE) estimator is consistent, the usual t statistic is not asymptotically normally distributed, and we propose a new robust t statistic whose asymptotic distribution is standard normal. Third, we propose efficient GMM estimators using the orthogonality conditions implied by grouping and we provide t tests that are valid even in the presence of time-invariant group effects. Our Monte Carlo results show that the proposed GMM estimator is more precise than the FE estimator and that our new t test has good size and is powerful.     

Formulation and estimation of dynamic models using panel data

This paper presents a statistical analysis of time series regression models for longitudinal data with and without lagged dependent variables under a variety of assumptions about the initial conditions of the processes being analyzed. The analysis demonstrates how the asymptotic properties of estimators of longitudinal models are critically dependent on the manner in which samples become large: by expanding the number of observations per person, holding the number of people fixed, or by expanding the number of persons, holding the number of observations per person fixed. The paper demonstrates which parameters can and cannot be identified from data produced by different sampling plans.     

Kernel density estimation for time series data

A time-varying probability density function, or the corresponding cumulative distribution function, may be estimated nonparametrically by using a kernel and weighting the observations using schemes derived from time series modelling. The parameters, including the bandwidth, may be estimated by maximum likelihood or cross-validation. Diagnostic checks may be carried out directly on residuals given by the predictive cumulative distribution function. Since tracking the distribution is only viable if it changes relatively slowly, the technique may need to be combined with a filter for scale and/or location. The methods are applied to data on the NASDAQ index and the Hong Kong and Korean stock market indices.     

Longitudinal LISREL model estimation from incomplete panel data using the EM algorithm and the Kalman smoother

Longitudinal data sets with the structure T (time points) × N (subjects) are often incomplete because of data missing for certain subjects at certain time points. The EM algorithm is applied in conjunction with the Kalman smoother for computing maximum likelihood estimates of longitudinal LISREL models from varying missing data patterns. The iterative procedure uses the LISREL program in the M-step and the Kalman smoother in the E-step. The application of the method is illustrated by simulating missing data on a data set from educational research.     

Identification and estimation with contaminated data: When do covariate data sharpen inference?

Contaminated or corrupted data typically require strong assumptions to identify parameters of interest. However, weaker assumptions often identify bounds on these parameters. This paper addresses when covariate data—variables in addition to the one of interest—tighten these bounds. First, we construct the identification region for the distribution of the variable of interest. This region demonstrates that covariate data are useless without knowledge about the distribution of erroneous data conditional on the covariates. Then, we develop bounds both on probabilities and on parameters of this distribution that respect stochastic dominance.     

Hedonic house prices and spatial quantile regression

Despite its long history, hedonic pricing for housing valuation remains an active research area, and applications of new estimation methods continually push research frontiers. However, housing studies regarding Chinese cities are limited because of the short history of China’s free housing market. Such studies may, nonetheless, provide new insights given the nation’s current transitional stage of economic development. Therefore, this research makes use of publicly accessible sources to construct a new micro-dataset for an emerging Chinese city, Changsha, and it incorporates quantile regression with spatial econometric modeling to examine how implicit prices of housing characteristics may vary across the conditional distribution of house prices. Substantial variations are found, and the intuitions and implications are discussed. Additionally, the spatial dependence exhibits a U-shape pattern. The dependence is strong in the upper and lower parts of the response distribution, but it is little in the medium range.     

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.