Asymptotic Bias Research Articles

Augmented inverse probability weighted estimation for conditional treatment effect

The Augmented Inverse Probability Weighted (AIPW) estimation has been extensively studied in various scenarios across diverse research areas. This investigation explores the asymptotic properties of the AIPW estimator for the conditional average treatment effect. Under various combinations of the parametric, semiparametric, and nonparametric structure in the nuisance propensity score and outcome regression models, we discuss the asymptotic bias and compare the asymptotic variances among the corresponding estimators. The study covers the asymptotic properties with no model misspecified; with either propensity score or outcome regressions locally/globally misspecified; and with all models locally/globally misspecified. To provide a deeper insight into the nature of these estimators and out of curiosity, we reveal the phenomenon that the asymptotic variances, with model-misspecification, could sometimes be even smaller than those with all models correctly specified. We also conduct a numerical study to validate the theoretical results.

A Subsampling Strategy for AIC-based Model Averaging with Generalized Linear Models

Subsampling is an effective approach to address computational challenges associated with massive datasets. However, existing subsampling methods do not consider model uncertainty. In this paper, we investigate the subsampling technique for the Akaike information criterion (AIC) and extend the subsampling method to the smoothed AIC model-averaging framework in the context of generalized linear models. By correcting the asymptotic bias of the maximized subsample objective function used to approximate the Kullback–Leibler divergence, we derive the form of the AIC based on the subsample. We then provide a subsampling strategy for the smoothed AIC model-averaging estimator and study the corresponding asymptotic properties of the loss and the resulting estimator. A practically implementable algorithm is developed, and its performance is evaluated through numerical experiments on both real and simulated datasets.

Asymptotic bias of inexact Markov chain Monte Carlo methods in high dimension

Asymptotic bias of inexact Markov chain Monte Carlo methods in high dimension

Exogeneity tests and weak identification in IV regressions: Asymptotic theory and point estimation

This paper provides new insights on exogeneity tests in linear IV models and their use for estimation, when identification fails or may not be strong. We make two main contributions. First, we show that Durbin-Wu-Hausman (DWH) and Revankar-Hartley (RH) exogeneity tests have correct level asymptotically, even when the first-stage coefficient matrix (which controls identification) is rank-deficient. We provide necessary and sufficient conditions under which these tests are consistent. In particular, we show that test consistency can hold even when identification fails, provided at least one component of the structural parameter vector is identifiable. Second, we study point estimation after estimator (or model) selection, when the outcome of a DWH/RH test determines whether OLS or an IV method is employed in the second-stage. For this purpose, we use (non-local) concepts of asymptotic bias, asymptotic mean squared error (AMSE), and asymptotic relative efficiency (ARE), which remain applicable even when the estimators considered do not have moments (as can happen for 2SLS) or may be inconsistent. We study the asymptotic properties of OLS, 2SLS, and pretest estimators which select OLS or 2SLS based on the outcome of a DWH/RH test. We show that: (i) OLS typically dominates 2SLS estimator asymptotically for MSE across a broad spectrum of cases, including weak identification and moderate endogeneity; (ii) exogeneity-pretest estimators exhibit consistently good performance and asymptotically dominate both OLS and 2SLS. The proposed theoretical findings are documented by Monte Carlo simulations.

Non-classical measurement error in instrumental variables estimation: An application to the medical care costs of obesity.

Estimates of the impact of body mass index and obesity on health and labor market outcomes often use instrumental variables estimation (IV) to mitigate bias due to endogeneity. When these studies rely on survey data that include self- or proxy-reported height and weight, there is non-classical measurement error due to the tendency of individuals to under-report their own weight. Mean reverting errors in weight do not cause IV to be asymptotically biased per se, but may result in bias if instruments are correlated with additive error in weight. We demonstrate the conditions under which IV is biased when there is non-classical measurement error and derive bounds for this bias conditional on instrument strength and the severity of mean-reverting error. We show that improvements in instrument relevance alone cannot eliminate IV bias, but reducing the correlation between weight and reporting error mitigates the bias. A solution we consider is regression calibration (RC) of endogenous variables with external validation data. In simulations, we find IV estimation paired with RC can produce consistent estimates when correctly specified. Even when RC fails to match the covariance structure of reporting error, there is still a reduction in asymptotic bias.

Iterated Data Sharpening

Data sharpening in kernel regression has been shown to be an effective method of reducing bias while having minimal effects on variance. Earlier efforts to iterate the data sharpening procedure have been less effective, due to the employment of an inappropriate sharpening transformation. In this article, an iterated data sharpening algorithm is proposed which reduces the asymptotic bias at each iteration, while having modest effects on the variance. The efficacy of the iterative approach is demonstrated theoretically and via a simulation study. Boundary effects persist and the affected region successively grows when the iteration is applied to local constant regression. By contrast, boundary bias successively decreases for each iteration step when applied to local linear regression. This study also shows that after iteration, the resulting estimates are less sensitive to bandwidth choice, and a further simulation study demonstrates that iterated data sharpening with data-driven bandwidth selection via cross-validation can lead to more accurate regression function estimation. Examples with real data are used to illustrate the scope of change made possible by using iterated data sharpening and to also identify its limitations. Supplementary materials for this article are available online.

The nonparametric GARCH model estimation using intraday high-frequency data

Most of nonparametric GARCH models typically employ daily frequency data to forecast the returns, correlations, and risk indicators of financial assets, without incorporating alternative frequency data. As a result, valuable financial market information may remain underutilized during the estimation process. To partially mitigate this issue, we introduce the intraday high-frequency data to enhance the estimation of the volatility function in a nonparametric GARCH model. To achieve this objective, we introduce a nonparametric proxy model for volatility. Under mild assumptions, we derive the asymptotic bias and variance of the estimator and further investigate the impact of various volatility proxies on estimation accuracy. Our findings from both simulations and empirical analysis indicate a considerable improvement in the estimation of the volatility function through the introduction of high-frequency data.

Prelimit Coupling and Steady-State Convergence of Constant-stepsize Nonsmooth Contractive SA

Motivated by Q-learning, we study nonsmooth contractive stochastic approximation (SA) with constant stepsize. We focus on two important classes of dynamics: 1) nonsmooth contractive SA with additive noise, and 2) synchronous and asynchronous Q-learning, which features both additive and multiplicative noise. For both dynamics, we establish weak convergence of the iterates to a stationary limit distribution in Wasserstein distance. Furthermore, we propose a prelimit coupling technique for establishing steady-state convergence and characterize the limit of the stationary distribution as the stepsize goes to zero. Using this result, we derive that the asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize, which stands in sharp contrast to smooth SA. This bias characterization allows for the use of Richardson-Romberg extrapolation for bias reduction in nonsmooth SA.

Extreme Value Statistics in Semi-Supervised Models

We consider extreme value analysis in a semi-supervised setting, where we observe, next to the n data on the target variable, n + m data on one or more covariates. This is called the semi-supervised model with n labeled and m unlabeled data. By exploiting the tail dependence between the target variable and the covariates, we derive estimators for the extreme value index and extreme quantiles of the target variable in this setting and establish their asymptotic behavior. Our estimators substantially improve the univariate estimators, based on only the n target variable data, in terms of asymptotic variances whereas the asymptotic biases remain unchanged. A simulation study confirms the substantially improved behavior of both estimators. Finally the estimation method is applied to rainfall data in France. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Improved estimators in bell regression model with application

In this paper, we propose the application of shrinkage strategies to estimate coefficients in the Bell regression models when prior information about the coefficients is available. The Bell regression models are well-suited for modelling count data with multiple covariates. Furthermore, we provide a detailed explanation of the asymptotic properties of the proposed estimators, including asymptotic biases and mean squared errors. To assess the performance of the estimators, we conduct numerical studies using Monte Carlo simulations and evaluate their simulated relative efficiency. The results demonstrate that the suggested estimators outperform the unrestricted estimator when prior information is taken into account. Additionally, we present an empirical application to demonstrate the practical utility of the suggested estimators.

Recursive generalized gamma kernel density estimation for nonnegative dependent data

In this article, we consider the recursive generalized gamma (GG) kernel density estimator (KDE) for nonnegative dependent data from stationary α-mixing process. Asymptotic bias, variance and mean integrated squared error are provided. Simulation experiments from an autoregressive conditional duration model and a stochastic volatility model are conducted to compare the recursive GG KDE with non-recursive GG KDE. Two applications are provided for nonnegative time series data.

No star is good news: A unified look at rerandomization based on [formula omitted]-values from covariate balance tests

Randomized experiments balance all covariates on average and are considered the gold standard for estimating treatment effects. Chance imbalances are nonetheless common in realized treatment allocations. To inform readers of the comparability of treatment groups at baseline, contemporary scientific publications often report covariate balance tables with not only covariate means by treatment group but also the associated p-values from significance tests of their differences. The practical need to avoid small p-values as indicators of poor balance motivates balance check and rerandomization based on these p-values from covariate balance tests (ReP) as an attractive tool for improving covariate balance in designing randomized experiments. Despite the intuitiveness of such strategy and its possibly already widespread use in practice, the literature lacks results about its implications on subsequent inference, subjecting many effectively rerandomized experiments to possibly inefficient analyses. To fill this gap, we examine a variety of potentially useful schemes for ReP and quantify their impact on subsequent inference. Specifically, we focus on three estimators of the average treatment effect from the unadjusted, additive, and interacted linear regressions of the outcome on treatment, respectively, and derive their asymptotic sampling properties under ReP. The main findings are threefold. First, the estimator from the interacted regression is asymptotically the most efficient under all ReP schemes examined, and permits convenient regression-assisted inference identical to that under complete randomization. Second, ReP, in contrast to complete randomization, improves the asymptotic efficiency of the estimators from the unadjusted and additive regressions. Standard regression analyses are accordingly still valid but in general overconservative. Third, ReP reduces the asymptotic conditional biases of the three estimators and improves their coherence in terms of mean squared difference. These results establish ReP as a convenient tool for improving covariate balance in designing randomized experiments, and we recommend using the interacted regression for analyzing data from ReP designs.

Modified score functions for von Mises regressions

We derive and evaluate a novel estimation approach for the von Mises regression model based on a modified score function whose solution ensures an estimator with a smaller asymptotic bias than the original maximum likelihood estimator. We consider Monte Carlo simulation experiments to show that the new estimation approach yields nearly unbiased estimates. An application to real data is also considered for illustrative purposes.

Bias assessment and correction for Levin’s population attributable fraction in the presence of confounding

In 1953, Morton Levin introduced a simple approach to estimating population attributable fractions (PAF) depending only on risk factor prevalence and relative risk. This formula and its extensions are still in widespread use today, particularly to estimate PAF in populations where individual data is unavailable. Unfortunately, Levin’s approach is known to be asymptotically biased for the PAF when the risk factor-disease relationship is confounded even if relative risks that are correctly adjusted for confounding are used in the estimator. Here we describe a simple re-expression of Miettinen’s estimand that depends on the causal relative risk, the unadjusted relative risk and the population risk factor prevalence. While this re-expression is not new, it has been underappreciated in the literature, and the associated estimator may be useful in estimating PAF in populations when individual data is unavailable provided estimated adjusted and unadjusted relative risks can be transported to the population of interest. Using the re-expressed estimand, we develop novel analytic formulae for the relative and absolute asymptotic bias in Levin’s formula, solidifying earlier work by Darrow and Steenland that used simulations to investigate this bias. We extend all results to settings with non-binary valued risk factors and continuous exposures and discuss the utility of these results in estimating PAF in practice.

High-dimensional IV cointegration estimation and inference

A semiparametric triangular systems approach shows how multicointegrating linkages occur naturally in an I(1) cointegrated regression model when the long run error variance matrix in the system is singular. Under such singularity, cointegrated I(1) systems embody a multicointegrated structure that makes them useful in many empirical settings. Earlier work shows that such systems may be analyzed and estimated without appealing to the associated I(2) system but with suboptimal convergence rates and potential asymptotic bias. The present paper develops a robust approach to estimation and inference of such systems using high dimensional IV methods that have appealing asymptotic properties like those known to apply in the optimal estimation of cointegrated systems (Phillips, 1991). The approach uses an extended version of high-dimensional trend IV (Phillips, 2006, 2014) estimation with deterministic orthonormal instruments. The methods and derivations involve new results on high-dimensional IV techniques and matrix normalization in the limit theory that are of independent interest. Wald tests of general linear restrictions are constructed using a fixed-b long run variance estimator that leads to robust pivotal HAR inference in both cointegrated and multicointegrated cases. Simulations show good properties of the estimation and inferential procedures in finite samples. An empirical illustration to housing stocks, starts and completions is provided.

A correction to Begg’s test for publication bias

Begg and Mazumdar proposed using a rank correlation test to test for publication bias when carrying out meta-analyses. The asymptotic variance of the rank correlation test statistic was derived under assumptions unmet by this application, often resulting in a loss of power. Low power when Begg’s test is used to screen for publication bias may lead to false positives in a subsequent meta-analysis. We obtain the asymptotic bias under the common conditionally normal model as a function of the distribution of primary study variances. In simulations, we consider the performance of Begg’s test using an approximation to the correct asymptotic variance. We consider this performance under the common fixed effects and random effects frameworks. We then examine several meta-analyses drawn from the literature where the standard and bias-corrected versions of Begg’s test lead to different conclusions.

Non parametric multivariate distribution estimation under right censoring

A non parametric estimator of the joint distribution function of a positive bivariate random vector is introduced. The case where one of the two variables is subject to right censoring is considered. To construct the proposed estimator, Poisson distributions are used for smoothing the empirical estimator of Stute (1993). The strong uniform convergence is established. Also, by stating the asymptotic i.i.d. representation, the asymptotic bias, variance, and normality are deduced. The smooth estimator is applied for analyzing survival data from patients with advanced lung cancer.

INTERACTIVE EFFECTS PANEL DATA MODELS WITH GENERAL FACTORS AND REGRESSORS

This paper considers a model with general regressors and unobservable common factors. An estimator based on iterated principal component analysis is proposed, which is shown to be not only asymptotically normal, but under certain conditions also free of the otherwise so common asymptotic incidental parameters bias. Interestingly, the conditions required to achieve unbiasedness become weaker the stronger the trends in the factors, and if the trending is strong enough, unbiasedness comes at no cost at all. The approach does not require any knowledge of how many factors there are, or whether they are deterministic or stochastic. The order of integration of the factors is also treated as unknown, as is the order of integration of the regressors, which means that there is no need to pre-test for unit roots, or to decide on which deterministic terms to include in the model.

James-Stein type estimators in beta regression model: simulation and application

Recently, the beta regression model has been used in several fields of science to model data in the form of rate or proportion. In this paper, some novel and improved methods to estimate parameters in the beta regression model are proposed. We consider a sub-space on the regression coefficients of the beta regression model and combine the unrestricted and restricted estimators then we present Stein-type and preliminary estimators. We develop the expressions for the proposed estimators' asymptotic biases and their quadratic risks. Numerical studies through Monte Carlo simulations are used to evaluate the performance of the proposed estimators in terms of their simulated relative efficiency. The results show that the proposed estimators outperform the unrestricted estimator when the restrictions hold. Finally, an empirical application is given to show how useful the proposed estimators are in the practical area.

Fitting and Testing Log-Linear Subpopulation Models with Known Support.

In this paper, the support of the joint probability distribution of categorical variables in the total population is treated as unknown. From a general total population model with unknown support, a general subpopulation model with its support equal to the set of all observed score patterns is derived. In maximum likelihood estimation of the parameters of any such subpopulation model, the evaluation of the log-likelihood function only requires the summation over a number of terms equal to at most the sample size. It is made clear that the parameters of a hypothesized total population model are consistently and asymptotically efficiently estimated by the values that maximize the log-likelihood function of the corresponding subpopulation model. Next, new likelihood ratio goodness-of-fit tests are proposed as alternatives to the Pearson chi-square goodness-of-fit test and the likelihood ratio test against the saturated model. In a simulation study, the asymptotic bias and efficiency of maximum likelihood estimators and the asymptotic performance of the goodness-of-fit tests are investigated.