Statistical decision rules map data into actions. Point estimators, inference procedures, and forecasting methods can be viewed as statistical decision rules. However, other types of rules are possible, such as rules for assigning individuals to treatments based on covariates, and methods for designing auctions. We discuss heuristics for constructing statistical decision rules, and survey results that characterize the properties of various classes of decision rules. Particular attention is paid to developing large-sample approximations to the distributions and associated risk properties of statistical decision rules.
This chapter surveys recent econometric methodologies for inference in large dimensional conditional factor models in finance. Changes in the business cycle and asset characteristics induce time variation in factor loadings and risk premia to be accounted for. The growing trend in the use of disaggregated data for individual securities motivates our focus on methodologies for a large number of assets. The chapter starts with an historical perspective on conditional factor models with a small number of assets for comparison purpose. Then, it outlines the concept of approximate factor structure in the presence of conditional information, and reviews an arbitrage pricing theory for large dimensional factor models in this framework. For inference, we distinguish between two different cases depending on whether factors are observable or not. We focus on diagnosing model specification, estimating conditional risk premia, and testing asset pricing restrictions under increasing cross-sectional and time series dimensions. At the end of the chapter, we provide new empirical findings based on a broad set of factor models and contrast analysis based on individual stocks and standard sets of portfolios. We also discuss the impact on computing time-varying cost of equity for a firm, and summarize differences between results for developed and emerging markets in an international setting.
This chapter sets out the extension of the scope of the classical IV model to cases in which unobserved variables are set-valued functions of observed variables. The resulting Generalized IV (GIV) models can be used when outcomes are discrete while unobserved variables are continuous, when there are rich specifications of heterogeneity as in random coefficient models, and when there are inequality restrictions constraining observed outcomes and unobserved variables. There are many other applications and classical IV models arise as a special case. The chapter provides characterizations of the identified sets delivered by GIV models. It gives details of the application of GIV analysis to models with an interval censored endogenous variable and to binary outcome models – for example probit models – with endogenous explanatory variables. It illustrates how the identified sets delivered by GIV models can be represented by moment inequality characterizations that have been the focus of recently developed methods for inference. An empirical application to a binary outcome model of female labor force participation is worked through in detail.
This chapter overviews the recent progress towards the identification and the estimation of models in which some of the variables are either imperfectly measured or even entirely unobserved, with a special focus on models with nonlinear, nonparametric, nonclassical or nonseparable features. Starting from the basic treatment of nonlinear models with measurement error assuming a known measurement error distribution or validation data availability, we then turn to the methods relying on more readily available auxiliary variables (such as repeated measurements, instrumental variables or general indicators). These models are then extended to fully nonlinear nonparametric and nonseparable factor models and to general latent (i.e. unobserved) variables models, in which the features of the latent variables are indirectly inferred from their effects on observables. We also identify important connections with related fields, such as nonlinear panel data, limited dependent variables, game theoretic models, dynamic models and set-identification.
This chapter reviews the microeconometrics literature on partial identification, focusing on the developments of the last thirty years. The topics presented illustrate that the available data combined with credible maintained assumptions may yield much information about a parameter of interest, even if they do not reveal it exactly. Special attention is devoted to discussing the challenges associated with, and some of the solutions put forward to, (1) obtain a tractable characterization of the values for the parameters of interest which are observationally equivalent, given the available data and maintained assumptions; (2) estimate this set of values; (3) conduct test of hypotheses and make confidence statements. The chapter reviews advances in partial identification analysis both as applied to learning (functionals of) probability distributions that are well-defined in the absence of models, as well as to learning parameters that are well-defined only in the context of particular models. A simple organizing principle is highlighted: the source of the identification problem can often be traced to a collection of random variables that are consistent with the available data and maintained assumptions. This collection may be part of the observed data or be a model implication. In either case, it can be formalized as a random set. Random set theory is then used as a mathematical framework to unify a number of special results and produce a general methodology to carry out partial identification analysis.
Many economic activities are embedded in networks: sets of agents and the (often) rivalrous relationships connecting them to one another. Input sourcing by firms, interbank lending, scientific research, and job search are four examples, among many, of networked economic activities. Motivated by the premise that networks' structures are consequential, this chapter describes econometric methods for analyzing them. I emphasize (i) dyadic regression analysis incorporating unobserved agent-specific heterogeneity and supporting causal inference, (ii) techniques for estimating, and conducting inference on, summary network parameters (e.g., the degree distribution or transitivity index); and (iii) empirical models of strategic network formation admitting interdependencies in preferences. Current research challenges and open questions are also discussed.
This chapter covers recent solutions to aggregation problems in three application areas: consumer demand analysis, consumption growth and wealth, and labor participation and wages. Each area involves treatment of heterogeneity and nonlinearity at the individual level. Three types of heterogeneity are highlighted: heterogeneity in individual tastes, heterogeneity in income and wealth risks, and heterogeneity in market participation. Work in each area is illustrated using results from empirical data. The overall aim is to present specific models that connect individual behavior with aggregate statistics, as well as to discuss the principles for constructing such models.
This chapter relates the literature on the econometric evaluation of social programs to the literature in statistics on "causal inference". In it, we develop a general evaluation framework that addresses well-posed economic questions and analyzes agent choice rules and subjective evaluations of outcomes as well as the standard objective evaluations of outcomes. The framework recognizes uncertainty faced by agents and ex ante and ex post evaluations of programs. It also considers distributions of treatment effects. These features are absent from the statistical literature on causal inference. A prototypical model of agent choice and outcomes is used to illustrate the main ideas.We formally develop models for counterfactuals and causality that build on Cowles Commission econometrics. These models anticipate and extend the literature on causal inference in statistics. The distinction between fixing and conditioning that has recently entered the statistical literature was first developed by Cowles economists. Models of simultaneous causality were also developed by the Cowles group, as were notions of invariance to policy interventions. These basic notions are updated to nonlinear and nonparametric frameworks for policy evaluation more general than anything in the current statistical literature on "causal inference". A formal discussion of identification is presented and applied to clearly formulated choice models used to evaluate social programs.
We survey recent microeconometric research on investment and employment that has used panel data on individual firms or plants. We focus on model specification and econometric estimation issues, but we also review some of the main empirical findings. We discuss advantages and limitations of microeconomic data in this context. We briefly review the neoclassical theory of the demand for capital and labour, on which most of the econometric models of investment and employment that we consider are based. We pay particular attention to dynamic factor demand models, based on the assumption that there are costs of adjustment, which have played a prominent role especially in the microeconometric literature on investment. With adjustment costs, current choices depend on expectations of future conditions. We discuss the challenges that this raises for econometric model specification, and some of the solutions that have been adopted. We also discuss estimation issues that arise for dynamic factor demand equations in the context of micro panel data for firms or plants. We then discuss a number of topics that have been the focus of recent microeconometric research on investment and employment. In particular, we review the literatures on investment and financing constraints, relative price effects on investment and employment, investment and uncertainty, investment in research and development (R&D), elasticities of substitution and complementarity between technology, capital and skilled and unskilled labour, and recent work on models with non-convex adjustment costs.
Often researchers find parametric models restrictive and sensitive to deviations from the parametric specifications; semi-nonparametric models are more flexible and robust, but lead to other complications such as introducing infinite-dimensional parameter spaces that may not be compact and the optimization problem may no longer be well-posed. The method of sieves provides one way to tackle such difficulties by optimizing an empirical criterion over a sequence of approximating parameter spaces (i.e., sieves); the sieves are less complex but are dense in the original space and the resulting optimization problem becomes well-posed. With different choices of criteria and sieves, the method of sieves is very flexible in estimating complicated semi-nonparametric models with (or without) endogeneity and latent heterogeneity. It can easily incorporate prior information and constraints, often derived from economic theory, such as monotonicity, convexity, additivity, multiplicity, exclusion and nonnegativity. It can simultaneously estimate the parametric and nonparametric parts in semi-nonparametric models, typically with optimal convergence rates for both parts. This chapter describes estimation of semi-nonparametric econometric models via the method of sieves. We present some general results on the large sample properties of the sieve estimates, including consistency of the sieve extremum estimates, convergence rates of the sieve M-estimates, pointwise normality of series estimates of regression functions, root-n asymptotic normality and efficiency of sieve estimates of smooth functionals of infinite-dimensional parameters. Examples are used to illustrate the general results.
This chapter covers the theory and methods for productivity measurement for nations. Labor, multifactor and total factor productivity measures are defined and are related to each other and to gross domestic product (GDP) per capita. Their growth over time and relative counterparts are defined as well.Different conceptual meanings have been proposed for a total factor productivity growth (TFPG) index. These are easiest to understand for the case in which the index number problem is absent: a production process that involves one input and one output (a 1-1 process). It is easily seen that four common concepts of TFPG all lead to the same result in the 1-1 case. Moving on to a general N input, M output production scenario, we demonstrate that a Paasche, Laspeyres or Fisher index number formula provides a measure for all four of the concepts of TFPG introduced for the 1-1 case. This is an advantage of the Paasche-Laspeyres-Fisher family of formulas.When multiple inputs or outputs are involved, there is the problem of choosing among alternative functional forms. The axiomatic and economic approaches to index formula choice are reviewed.In addition, we briefly cover the Divisia index number approach and growth accounting, including the KLEMS (capital, labor, energy, materials and services) approach. The gross output measures of the KLEMS approach are contrasted with value added output measures such as GDP. Also, an alternative family of revenue function based productivity growth indexes proposed by Diewert, Kohli and Morrison (DKM) is outlined. The DKM approach facilitates the decomposition of productivity growth into economically meaningful components. This approach is useful, for example, for examining the effects of changes in the terms of trade on productivity growth.
This chapter is concerned with the identification and estimation of models of labor supply. The focus is on the key issues that arise from unobserved heterogeneity, nonparticipation and dynamics. We examine the simple "static" labor supply model with proportional taxes and highlight the problems surrounding nonparticipation and missing wages. The difference-in-differences approach to estimation and identification is developed within the context of the labor supply model. We also consider the impact of incorporating nonlinear taxation and welfare program participation. Family labor supply is looked at from both the unitary and collective perspectives. Finally we consider intertemporal models focusing on the difficulties that arise with participation and heterogeneity.
We study structural models of stochastic discount factors and explore alternative methods of estimating such models using data on macroeconomic risk and asset returns. Particular attention is devoted to recursive utility models in which risk aversion can be modified without altering intertemporal substitution. We characterize the impact of changing the intertemporal substitution and risk aversion parameters on equilibrium short-run and long-run risk prices and on equilibrium wealth.
This chapter discusses structural econometric approaches to auctions. Remarkably, much of what can be learned from auction data can be learned without restrictions beyond those derived from the relevant economic model. This enables us to take a nonparametric perspective in discussing how the structure of auction models can be combined with observables to uncover (or test hypotheses about) primitives of interest in auction markets. We focus on first-price sealed-bid and ascending auctions, including extensions to Dutch auctions, Internet auctions, multi-unit auctions, and multi-object auctions. We consider a wide range of underlying structures of bidder demand and information, as well as a variety of types of data one may encounter in applications. We discuss identification and testable restrictions of these models and present a variety of estimation approaches.
We survey recent microeconometric research on investment and employment that has used panel data on individual firms or plants. We focus on model specification and econometric estimation issues, but we also review some of the main empirical findings. We discuss advantages and limitations of microeconomic data in this context.We briefly review the neoclassical theory of the demand for capital and labour, on which most of the econometric models of investment and employment that we consider are based. We pay particular attention to dynamic factor demand models, based on the assumption that there are costs of adjustment, which have played a prominent role especially in the microeconometric literature on investment. With adjustment costs, current choices depend on expectations of future conditions. We discuss the challenges that this raises for econometric model specification, and some of the solutions that have been adopted. We also discuss estimation issues that arise for dynamic factor demand equations in the context of micro panel data for firms or plants.We then discuss a number of topics that have been the focus of recent microeconometric research on investment and employment. In particular, we review the literatures on investment and financing constraints, relative price effects on investment and employment, investment and uncertainty, investment in research and development (R&D), elasticities of substitution and complementarity between technology, capital and skilled and unskilled labour, and recent work on models with non-convex adjustment costs.
The greatest problem for empirical analysis is how best to allow the context to affect the inferences. Econometric theory presupposes contextual “restrictions” that can be taken as given or assigned a probability distribution. These contextual inputs are rarely available. I illustrate this point with a review of the empirical work in international economics which has focused not on properties of estimators but instead how best to link the theory with the data. I argue that the two errors we should worry about are not rejecting a true null or accepting a false null but rather taking the theory too seriously and not taking the theory seriously enough.
Inverse problems can be described as functional equations where the value of the function is known or easily estimable but the argument is unknown. Many problems in econometrics can be stated in the form of inverse problems where the argument itself is a function. For example, consider a nonlinear regression where the functional form is the object of interest. One can readily estimate the conditional expectation of the dependent variable given a vector of instruments. From this estimate, one would like to recover the unknown functional form.This chapter provides an introduction to the estimation of the solution to inverse problems. It focuses mainly on integral equations of the first kind. Solving these equations is particularly challenging as the solution does not necessarily exist, may not be unique, and is not continuous. As a result, a regularized (or smoothed) solution needs to be implemented. We review different regularization methods and study the properties of the estimator. Integral equations of the first kind appear, for example, in the generalized method of moments when the number of moment conditions is infinite, and in the nonparametric estimation of instrumental variable regressions. In the last section of this chapter, we investigate integral equations of the second kind, whose solutions may not be unique but are continuous. Such equations arise when additive models and measurement error models are estimated nonparametrically.
This chapter discusses structural econometric approaches to auctions. Remarkably, much of what can be learned from auction data can be learned without restrictions beyond those derived from the relevant economic model. This enables us to take a nonparametric perspective in discussing how the structure of auction models can be combined with observables to uncover (or test hypotheses about) primitives of interest in auction markets. We focus on first-price sealed-bid and ascending auctions, including extensions to Dutch auctions, Internet auctions, multi-unit auctions, and multi-object auctions. We consider a wide range of underlying structures of bidder demand and information, as well as a variety of types of data one may encounter in applications. We discuss identification and testable restrictions of these models and present a variety of estimation approaches.
