Efficient Estimation Of Missing Data Models Using Moment Conditions And Semiparametric Restrictions

This paper shows that the semiparametric efficiency bound for a parameter identified by an unconditional moment restriction with data missing at random (MAR) coincides with that of a particular augmented moment condition problem. The augmented system consists of the inverse probability weighted (IPW) original moment restriction and an additional conditional moment restriction which exhausts all other implications of the MAR assumption. The paper also investigates the value of additional semiparametric restrictions on the conditional expectation function (CEF) of the original moment function given always-observed covariates. In the missing outcome context, for example, such restrictions are implied by a semiparametric model for the outcome CEF given always-observed covariates. The efficiency bound associated with this model is shown to also coincide with that of a particular moment condition problem. Some implications of these results for estimation are briefly discussed.

This paper shows that the semiparametric efficiency bound for a parameter identified by an unconditional moment restriction with data missing at random (MAR) coincides with that of a particular augmented moment condition problem. The augmented system consists of the inverse probability weighted (IPW) original moment restriction and an additional conditional moment restriction which exhausts all other implications of the MAR assumption. The paper also investigates the value of additional semiparametric restrictions on the conditional expectation function (CEF) of the original moment function given always- observed covariates. In the program evaluation context, for example, such restrictions are implied by semiparametric models for the potential outcome CEFs given baseline covariates. The efficiency bound associated with this model is shown to also coincide with that of a particular moment condition problem. Some implications of these results for estimation are briefly discussed.

This paper proposes a simple and efficient estimation procedure for the model with non-ignorable missing data studied by Morikawa and Kim (2016). Their semiparametrically efficient estimator requires explicit nonparametric estimation and so suffers from the curse of dimensionality and requires a bandwidth selection. We propose an estimation method based on the Generalized Method of Moments (hereafter GMM). Our method is consistent and asymptotically normal regardless of the number of moments chosen. Furthermore, if the number of moments increases appropriately our estimator can achieve the semiparametric efficiency bound derived in Morikawa and Kim (2016), but under weaker regularity conditions. Moreover, our proposed estimator and its consistent covariance matrix are easily computed with the widely available GMM package. We propose two data-based methods for selection of the number of moments. A small scale simulation study reveals that the proposed estimation indeed out-performs the existing alternatives in finite samples.

This book evolved from lectures, courses and workshops on missing data and small-area estimation that I presented during my tenure as the ?rst C- pion Fellow (2000–2002). For the Fellowship I proposed these two topics as areas in which the academic statistics could contribute to the development of government statistics, in exchange for access to the operational details and background that would inform the direction and sharpen the focus of a- demic research. After a few years of involvement, I have come to realise that the separation of ‘academic’ and ‘industrial’ statistics is not well suited to either party, and their integration is the key to progress in both branches. Most of the work on this monograph was done while I was a visiting l- turer at Massey University, Palmerston North, New Zealand. The hospitality and stimulating academic environment of their Institute of Information S- ence and Technology is gratefully acknowledged. I could not name all those who commented on my lecture notes and on the presentations themselves; apart from them, I want to thank the organisers and silent attendees of all the events, and, with a modicum of reluctance, the ‘grey ?gures’ who kept inquiring whether I was any nearer the completion of whatever stage I had been foolish enough to attach a date.

This book summarizes current knowledge regarding the theory of estimation for semiparametric models with missing data, in an organized and comprehensive manner. It starts with the study of semiparametric methods when there are no missing data. The description of the theory of estimation for semiparametric models is both rigorous and intuitive, relying on geometric ideas to reinforce the intuition and understanding of the theory. These methods are then applied to problems with missing, censored, and coarsened data with the goal of deriving estimators that are as robust and efficient as possible.

Missing data is common in data sets in every field of science. In the past few decades, there has been interest in understanding the underlying pattern of missingness, formally known as the missingness mechanism. There are three types of missingness mechanisms: Missing Completely at Random (MCAR), Missing at Random (MAR) and Missing Not at Random (MNAR). These can also be classified into two main categories: Ignorable (MCAR and MAR) and Nonignorable (MNAR). Most likelihood or imputation-based methods developed assume the ignorable condition, which is the more well studied condition. We discuss the nonignorable condition which is less well studied and also the hardest to deal with. This dissertation consists of three chapters that address the issue of estimation under the nonignorable missing data setting. In the first chapter, we propose a robust estimator of a parameter or a summary quantity of the model parameters in the context where outcome is subject to nonignorable missingness. These estimators are robust to misspecification of the dependence on covariates. The robustness of the estimators are nonstandard and are established rigorously through theoretical derivations, and are supported by simulations and a data application. In the second chapter, we attempt the efficient estimation of a function of the response under nonignorable missingness. We briefly discuss efficiency and robustness of estimators under the ignorable missingness assumption which is well established. However, efficiency under the nonignorable setting requires more investigation. We derive the efficient score for a function of the response but it turns out to be very complex and infeasible. Therefore, we recommend trading efficiency in favor of feasibility and using an inefficient but consistent estimator. In the final chapter, we propose an efficient estimator for the parameter involved in the missingness propensity. We first estimate the dependence of the missingness on the covariates. We incorporate the above estimator to construct an efficient estimator for the parameter of interest. We study the theoretical properties of this estimator and also put forward an alternative estimator for the mean of the response.