This proposal considers estimation and inference methods for models that exhibit problems with identification. Such models are common in many areas of economics and other social sciences, as well as biological sciences. When a model is well identified, standard methods available in the literature can be used to carry out inference. However, when identification is weak or when identification is only partial, such methods are not reliable. Typically, they lead to invalid and potentially misleading inference. We propose to investigate the property of standard methods when identification is only weak. We will consider a general class of estimation and inference methods and determine when the methods are reliable and when they are not. The results will apply to maximum likelihood, least squares, and generalized method of moments estimators and tests. Next, we will develop methods that are robust to the existence of weak identification in parts of the parameter space. These results will apply to a broad class of cross-section and time series models used in economics. For example, they will apply to the work-horse autoregressive-moving average (1, 1) time series model. The results also will apply to nonlinear regression models, smooth transition autoregressive models, binary choice models, and instrumental variables models. These models are employed routinely in macroeconomics times series applications and labor, public finance, and development applications. Considerable time and effort will be spent in applying the general results to specific models. In this proposal, we will also develop new methods for carrying out inference when there is a complete breakdown of identification in a model. In this case, it is not possible to consistently estimate the unknown parameters in the model. However, it still is possible to construct valid tests and confidence intervals. We will do this when the models under consideration specify a number of conditional moment inequalities and/or equalities, as is common in many incomplete economic models. For example, game theory models with multiple equilibria, which are used in industrial organization, often exhibit this feature. We will consider the case where there are a large, possibly infinite, number of moment conditions, as well as the case where the unknown quantity of interest is a nonparametric quantity. In all cases, we will establish the uniform large sample validity of the proposed methods. We will also address long-standing issues of testing subsequent to model selection. This is a common scenario in empirical applications in economics. It is well-known that methods that ignore model selection do not exhibit correct size. In previous research, we have developed some methods that circumvent this problem. Here we aim to go a step further and determine methods that have correct size and are optimal in a specific sense. Recently in the literature, there have been a number of new methods introduced that improve estimators in problems where there are a large number of parameters, such as a large number of regression parameters, with only a small number of non-zero parameters. Model scenarios with these properties are called sparse. These estimation results are quite useful and are being employed increasingly in economics. However, few methods are available for carrying out tests and constructing confidence intervals in sparse models. This proposal will investigate optimal tests in sparse models. Such results will yield either useful new methods or impossibility results showing that existing methods cannot be improved upon significantly. The proposed research will benefit society through improved empirical methods that lead to more accurate empirical research and, consequently, better informed policy analysis. The research will promote teaching and training through the use of graduate students as research assistants and collaborative researchers. The research will enhance infrastructure by making new computer software available for use by the profession. The results of the research will be disseminated broadly via presentation at international conferences.
