Formula For Asymptotic Variance Research Articles

Improving instrumental variable estimators with poststratification

Abstract Experiments studying get-out-the-vote (GOTV) efforts estimate causal effects of various mobilization efforts on voter turnout. However, there is often substantial noncompliance in these studies. A usual approach is to use an instrumental variable (IV) analysis to estimate impacts for compliers, here being those actually contacted by the investigators. Unfortunately, popular IV estimators can be unstable in studies with a small fraction of compliers. We explore poststratifying the data (e.g. taking a weighted average of IV estimates within each stratum) using variables that predict complier status (and, potentially, the outcome) to mitigate this. We present the benefits of poststratification in terms of bias, variance, and improved standard error estimates, and provide a finite-sample asymptotic variance formula. We also compare the performance of different IV approaches and discuss the advantages of our design-based poststratification approach over incorporating compliance-predictive covariates into the two-stage least squares (2SLS) estimator. In the end, we show that covariates predictive of compliance can increase precision, but only if one is willing to make a bias-variance trade-off by down-weighting or dropping strata with few compliers. By contrast, standard approaches such as 2SLS fail to use such information. We finally examine the benefits of our approach in two GOTV applications.

Graphical tools for selecting conditional instrumental sets

Summary We consider the efficient estimation of total causal effects in the presence of unmeasured confounding using conditional instrumental sets. Specifically, we consider the two-stage least-squares estimator in the setting of a linear structural equation model with correlated errors that is compatible with a known acyclic directed mixed graph. To set the stage for our results, we characterize the class of linearly valid conditional instrumental sets that yield consistent two-stage least-squares estimators for the target total effect and derive a new asymptotic variance formula for these estimators. Equipped with these results, we provide three graphical tools for selecting more efficient linearly valid conditional instrumental sets: first, a graphical criterion that, for certain pairs of linearly valid conditional instrumental sets, identifies which of the two corresponding estimators has the smaller asymptotic variance second, an algorithm that greedily adds covariates that reduce the asymptotic variance to a given linearly valid conditional instrumental set and, third, a linearly valid conditional instrumental set for which the corresponding estimator has the smallest asymptotic variance that can be ensured with a graphical criterion.

Causal mediation analysis: From simple to more robust strategies for estimation of marginal natural (in)direct effects.

This paper aims to provide practitioners of causal mediation analysis with a better understanding of estimation options. We take as inputs two familiar strategies (weighting and model-based prediction) and a simple way of combining them (weighted models), and show how a range of estimators can be generated, with different modeling requirements and robustness properties. The primary goal is to help build intuitive appreciation for robust estimation that is conducive to sound practice. We do this by visualizing the target estimand and the estimation strategies. A second goal is to provide a "menu" of estimators that practitioners can choose from for the estimation of marginal natural (in)direct effects. The estimators generated from this exercise include some that coincide or are similar to existing estimators and others that have not previously appeared in the literature. We note several different ways to estimate the weights for cross-world weighting based on three expressions of the weighting function, including one that is novel; and show how to check the resulting covariate and mediator balance. We use a random continuous weights bootstrap to obtain confidence intervals, and also derive general asymptotic variance formulas for the estimators. The estimators are illustrated using data from an adolescent alcohol use prevention study. R-code is provided.

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.

Assess predictive values of a binary diagnostic test under a nested case–control design

ABSTRACT Predictive values of a binary diagnostic test are often evaluated under a random sample design. When the disease is rare, however, such a design might not be as efficient as a nested case–control design where the cases are oversampled from a large existing cohort. Under a nested case–control design, direct proportion estimators of predictive values are biased because cases are oversampled. Consistent estimates of predictive values can be easily obtained by inverse probability weighting (IPW) method. The only difficulty with these IPW estimators has been the absence of expressions for their variances. To fill this gap, in the current paper, we obtain the asymptotic variance formulas for the IPW estimators of predictive values. Unlike their counterparts from weighted logistic regression, our variance formulas take into account the variance of the estimated weights in the IPW estimators of predictive values. We further use the proposed variance formulas to examine the gain in efficiency under a nested case–control design compared with a simple random sampling design. Our results clearly show that when the disease is rare, a nested case–control design can achieve a substantial amount of variance reduction by oversampling cases, compared with a random sample design. Finally, we compare via simulation the accuracy of the proposed variance formulas with the existing methods and illustrate the proposed method by a real data example evaluating the accuracy of D-dimer test.

Combining Non-Probability and Probability Survey Samples Through Mass Imputation

Abstract Analysis of non-probability survey samples requires auxiliary information at the population level. Such information may also be obtained from an existing probability survey sample from the same finite population. Mass imputation has been used in practice for combining non-probability and probability survey samples and making inferences on the parameters of interest using the information collected only in the non-probability sample for the study variables. Under the assumption that the conditional mean function from the non-probability sample can be transported to the probability sample, we establish the consistency of the mass imputation estimator and derive its asymptotic variance formula. Variance estimators are developed using either linearization or bootstrap. Finite sample performances of the mass imputation estimator are investigated through simulation studies. We also address important practical issues of the method through the analysis of a real-world non-probability survey sample collected by the Pew Research Centre.

Better nonparametric confidence intervals via robust bias correction for quantile regression

In this article, we revisit the problem of how to construct better nonparametric confidence intervals for the conditional quantile function from an optimization perspective. We apply the fully data‐driven bias correction procedure based on local polynomial smoothing to estimate the conditional quantile. To account for the effect of the estimated bias, we apply an asymptotic framework that the ratio of the bandwidth to the pilot bandwidth tends to some positive constant rather than zero as the sample size grows. We derive an alternative asymptotic normality of the proposed bias‐corrected quantile estimator as well as a new asymptotic variance formula. Based on theoretical results, two new pointwise confidence intervals are constructed through resampling strategies. Extensive simulation studies show that our proposed confidence intervals enjoy better performance than other competitors in terms of coverage probabilities and interval lengths and are not sensitive to the choice of bandwidth. Finally, our proposed procedure is further illustrated through United States' natality birth data in 2017.

Influence Functions for Risk and Performance Estimators

A new general method for computing standard errors of risk and performance estimators is developed. The method relies on the fact that the influence function of an estimator, the Gateaux derivative of the estimator functional in the direction of point mass distributions, may be used to represent the asymptotic variance of the estimator as the expected value of the squared influence function. The law of large numbers shows that the asymptotic variance of an estimator can be estimated as the time series average of the squared influence function, thereby yielding a very simple estimator standard error calculation that does not require knowledge of the asymptotic variance formula. We derive formulas for the influence functions of six risk estimators and seven performance estimators, thereby providing a convenient portfolio performance and risk management tool to easily compute standard errors for most risk and performance estimators of interest or practical importance. We conduct a simulation study to evaluate the quality of the standard errors and confidence interval error rates for the Sharpe ratio and downside Sharpe ratio estimators. Software implementations of our proposed method in the R packages RPEIF and RPESE are publicly available on CRAN.

Asymptotic Variance of Test Statistics in the ML and QML Frameworks

In this study, we consider the test statistics that can be written as the sample average of data and derive their limiting distributions under the maximum likelihood (ML) and the quasi-maximum likelihood (QML) frameworks. We first generalize the asymptotic variance formula suggested in Pierce (Ann Stat 10(2):475–478, 1982) in the ML framework and illustrate its applications through some well-known test statistics: (1) the skewness statistic, (2) the kurtosis statistic, (3) the Cox statistic, (4) the information matrix test statistic, and (5) the Durbin’s h-statistic. We next provide a similar result in the QML setting and illustrate its applications by providing two examples. Illustrations show the simplicity and the effectiveness of our results for the asymptotic variance of test statistics, and therefore, they are recommended for practical applications.

Finite sample variance estimation for optimal dynamic treatment regimes of survival outcomes.

Deriving valid confidence intervals for complex estimators is a challenging task in practice. Estimators of dynamic weighted survival modeling (DWSurv), a method to estimate an optimal dynamic treatment regime of censored outcomes, are asymptotically normal and consistent for their target parameters when at least a subset of the nuisance models is correctly specified. However, their behavior in finite samples and the impact of model misspecification on inferences remain unclear. In addition, the estimators' nonregularity may negatively affect the inferences under some specific data generating mechanisms. Our objective was to compare five methods, two asymptotic variance formulas (adjusting or not for the estimation of nuisance parameters) to three bootstrap approaches, to construct confidence intervals for the DWSurv parameters in finite samples. Via simulations, we considered practical scenarios, for example, when some nuisance models are misspecified or when nonregularity is problematic. We also compared the five methods in an application about the treatment of rheumatoid arthritis. We found that the bootstrap approaches performed consistently well at the cost of longer computational times. The asymptotic variance with adjustments generally yielded conservative confidence intervals. The asymptotic variance without adjustments yielded nominal coverages for large sample sizes. We recommend using the asymptotic variance with adjustments in small samples and the bootstrap if computationally feasible. Caution should be taken when nonregularity may be an issue.

Leaves on the line and in the plane

The dead leaves model (DLM) provides a random tessellation of $d$-space, representing the visible portions of fallen leaves on the ground when $d=2$. For $d=1$, we establish formulae for the intensity, two-point correlations, and asymptotic covariances for the point process of cell boundaries, along with a functional CLT. For $d=2$ we establish analogous results for the random surface measure of cell boundaries, and also determine the intensity of cells in a more general setting than in earlier work of Cowan and Tsang. We introduce a general notion of dead leaves random measures and give formulae for means, asymptotic variances and functional CLTs for these measures; this has applications to various other quantities associated with the DLM.

Estimation of dynamic panel threshold model using Stata

In this article, we develop a command, xthenreg, that implements the first-differenced generalized method of moments estimation of the dynamic panel threshold model that Seo and Shin (2016, Journal of Econometrics 195: 169–186) proposed. Furthermore, we derive the asymptotic variance formula for a kink-constrained generalized method of moments estimator of the dynamic threshold model and provide an estimation algorithm. We also propose a fast bootstrap algorithm to implement the bootstrap for the linearity test. We illustrate the use of xthenreg through a Monte Carlo simulation and an economic application.

Asymptotic Variance of Test Statistics in ML and QML Frameworks

In this study, we consider the test statistics that can be written as the sample average of data and derive their limiting distribution under the maximum likelihood (ML) and the quasi-maximum likelihood (QML) frameworks. We first generalize the asymptotic variance formula suggested in Pierce (1982) in the ML framework and illustrate its applications through some well-known test statistics: (i) the skewness statistic, (ii) the kurtosis statistic, (iii) the Cox statistic, (iv) the information matrix test statistic, and (v) the Durbin's h-statistic. We next provide a similar result in the QML setting and illustrate its applications by providing two examples. Illustrations show the simplicity and the effectiveness of our results for the asymptotic variance of test statistics, and therefore, they are recommended for practical applications.

Model robust inference with two-stage maximum likelihood estimation for copulas

This article is concerned with inference in parametric copula setups, where both the marginals and the copula have parametric forms. For such models, two-stage maximum likelihood estimation, often referred to as inference function for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Previous studies of the two-stage maximum likelihood estimator have largely been based on the assumption that the chosen parametric model captures the true model that generated data. We study the impact of dropping this true model assumption, both theoretically and numerically. We first show that the two-stage maximum likelihood estimator is consistent for a well-defined least false parameter value, different from the analogous least false parameter associated with the full maximum likelihood procedure. Then we demonstrate limiting normality of the full vector of estimators, with concise matrix notation for the variance matrices involved. Along with consistent estimators for these, we have built a model-robust machinery for inference in parametric copula models. The special case where the parametric model is assumed to hold corresponds to situations studied earlier in the literature, with simpler formulas for variance matrices. As a numerical illustration, we perform a set of simulations. We also analyze five-dimensional Norwegian precipitation data. We find that the variance of the copula parameter estimate can both increase and decrease, by dropping the true model assumption. In addition, we observe that the two-stage maximum likelihood estimator is still highly efficient when the true model assumption is dropped and thus the model robust asymptotic variance formulas are used. Additionally, we discover that using highly misspecified models can lead to situations where the asymptotic variance of the two-stage maximum likelihood estimator is lower than that of full maximum likelihood estimator. Our results are also used to analyze the mean squared error properties for both the full and the two-stage maximum likelihood estimators of any focus parameter.

Inefficiency and bias of modified value-at-risk and expected shortfall

Modified value-at-risk (mVaR) and modified expected shortfall (mES) are risk estimators that can be calculated without modeling the distribution of asset returns. These modified estimators use skewness and kurtosis corrections to normal distribution parametric VaR and ES formulas to obtain more accurate risk measurement for non-normal return distributions. Use of skewness and kurtosis corrections can result in reduced bias, but these also lead to inflated mVaR and mES estimator standard errors. We compare modified estimators with their respective parametric counterparts in three ways. First, we assess the magnitude of standard error inflation by deriving formulas for the large-sample standard errors of mVaR and mES using the multivariate delta method. Monte Carlo simulation is then used to determine sample sizes and tail probabilities for which our asymptotic variance formula can be reliably used to compute finite-sample standard errors. Second, to evaluate the large-sample bias, we derive formulas for the asymptotic bias of modified estimators for t-distributions. Third, we analyze the finite-sample performance of the modified estimators for normal and t -distributions using their root-mean-squared-error efficiency relative to the parametric VaR and ES maximum likelihood estimators using Monte Carlo simulation. Our results show that the modified estimators are inefficient for both normal and t-distributions: the more so for t-distributions.

Evaluating principal surrogate markers in vaccine trials in the presence of multiphase sampling.

This article focuses on the evaluation of vaccine-induced immune responses as principal surrogate markers for predicting a given vaccine's effect on the clinical endpoint of interest. To address the problem of missing potential outcomes under the principal surrogate framework, we can utilize baseline predictors of the immune biomarker(s) or vaccinate uninfected placebo recipients at the end of the trial and measure their immune biomarkers. Examples of good baseline predictors are baseline immune responses when subjects enrolled in the trial have been previously exposed to the same antigen, as in our motivating application of the Zostavax Efficacy and Safety Trial (ZEST). However, laboratory assays of these baseline predictors are expensive and therefore their subsampling among participants is commonly performed. In this article, we develop a methodology for estimating principal surrogate values in the presence of baseline predictor subsampling. Under a multiphase sampling framework, we propose a semiparametric pseudo-score estimator based on conditional likelihood and also develop several alternative semiparametric pseudo-score or estimated likelihood estimators. We derive corresponding asymptotic theories and analytic variance formulas for these estimators. Through extensive numeric studies, we demonstrate good finite sample performance of these estimators and the efficiency advantage of the proposed pseudo-score estimator in various sampling schemes. We illustrate the application of our proposed estimators using data from an immune biomarker study nested within the ZEST trial.

Cross-sectional design with a short-term follow-up for prognostic imaging biomarkers

Medical imaging techniques are being rapidly developed and used for diagnosis and prognostic predictions. To validate a prognostic predictive utility of a new imaging marker, a temporal association needs to be established to show an association between its baseline value with a subsequent chance of having the relevant clinical outcome. Validation of such techniques has several difficulties. First, different from techniques based on blood or tissue specimen, imaging techniques often have no historical samples to study and require new studies to collect data. For rare events, it can be costly. Second, the rapid technology evolution requires such validation studies to be short in order to keep the evaluation relevant. A new statistical design is proposed that extends traditional prospective cohort study by adding cases with known time of events and including a short-term follow-up to estimate the prospective odds ratio for the clinical endpoint of interest within a reasonably short duration of time. Under a Markov model, this new design can deliver a consistent estimate of the odds ratio and a formula for asymptotic variance. Numerical studies suggest that the new design induces a smaller variance than the corresponding prospective cohort study within three follow-ups. An application to Alzheimer’s disease data demonstrates that the proposed design has a potential to be useful to rapidly establish a prognostic validity of a new imaging marker within a reasonable time, with a small sample size.

Fluctuations, stability and instability of a distributed particle filter with local exchange

We study a distributed particle filter proposed by Bolić et al. (2005). This algorithm involves m groups of M particles, with interaction between groups occurring through a “local exchange” mechanism. We establish a central limit theorem in the regime where M is fixed and m→∞. A formula we obtain for the asymptotic variance can be interpreted in terms of colliding Markov chains, enabling analytic and numerical evaluations of how the asymptotic variance behaves over time, with comparison to a benchmark algorithm consisting of m independent particle filters. We prove that subject to regularity conditions, when m is fixed both algorithms converge time-uniformly at rate M−1/2. Through use of our asymptotic variance formula we give counter-examples satisfying the same regularity conditions to show that when M is fixed neither algorithm, in general, converges time-uniformly at rate m−1/2.

Robust inference for the Two-Sample 2SLS estimator

The Two-Sample Two-Stage Least Squares (TS2SLS) data combination estimator is a popular estimator for the parameters in linear models when not all variables are observed jointly in one single data set. Although the limiting normal distribution has been established, the asymptotic variance formula has only been stated explicitly in the literature for the case of conditional homoskedasticity. By using the fact that the TS2SLS estimator is a function of reduced form and first-stage OLS estimators, we derive the variance of the limiting normal distribution under conditional heteroskedasticity. A robust variance estimator is obtained, which generalises to cases with more general patterns of variable (non-)availability. Stata code and some Monte Carlo results are provided in an Appendix. Stata code for a nonlinear GMM estimator that is identical to the TS2SLS estimator in just identified models and asymptotically equivalent to the TS2SLS estimator in overidentified models is also provided there.

Nonparametric versus Parametric Expected Shortfall

The Basel III recommendation to replace value-at-risk with expected shortfall (ES) for capital adequacy risk management purposes is a welcome event that recognizes the intuitive and theoretical advantages of ES, and is likely to increase use of ES in portfolio construction and risk management generally. Correspondingly it seems timely to obtain a deeper understanding of the statistical properties ES estimators than currently exists, and in particular the comparative statistical properties of the simple non-parametric estimators versus the more complex parametric ES maximum-likelihood estimators. We analyze the statistical differences between these two basic types ES estimators in three ways for both normal and t-distributed returns. First we derive formulas for statistical influence functions for the two types of ES estimators for both normal and t-distributions and provide and use influence function displays to clearly reveal how asset returns impact the risk estimates. The influence function analysis immediately reveals the undesirable feature of the parametric ES estimators that positive returns contribute very substantially to risk, a behavior not present in the non-parametric ES estimator. We show how this undesirable feature of the parametric ES estimator can be substantially ameliorated through use of a semi-standard deviation (SSD) estimator as a scale estimator. Then we use the influence functions to derive asymptotic variance formulas of the ES estimators and compare their large sample variabilities and statistical efficiencies. The asymptotic variance formulas yield finite-sample standard error (S.E.) approximations, and our third analysis uses Monte Carlo to assess the accuracy of finite sample size standard error approximations, which turn out to be rather good for example using the Basel III recommended 2.5% tail probability and sample sizes of interest in risk management applications.