Residential segregation by race and income are enduring features of urban America. Understanding the effects of residential segregation on educational attainment, labor market outcomes, criminal activity, and other outcomes has been a leading project of the social sciences for over half a century. This paper describes techniques for measuring the effects of neighborhood of residence on long-run life outcomes. ( JEL C51, I24, J15, K42, R23)

I introduce a model of undirected dyadic link formation which allows for assortative matching on observed agent characteristics (homophily) as well as unrestricted agent-level heterogeneity in link surplus (degree heterogeneity). Like in fixed effects panel data analyses, the joint distribution of observed and unobserved agent-level characteristics is left unrestricted. Two estimators for the (common) homophily parameter, ?0, are developed and their properties studied under an asymptotic sequence involving a single network growing large. The first, tetrad logit (TL), estimator conditions on a sufficient statistic for the degree heterogeneity. The second, joint maximum likelihood (JML), estimator treats the degree heterogeneity \{Ai0\}i = 1N as additional (incidental) parameters to be estimated. The TL estimate is consistent under both sparse and dense graph sequences, whereas consistency of the JML estimate is shown only under dense graph sequences.

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.

The copyright to this Article is held by the Econometric Society. It may be downloaded, printed and reproduced only for educational or research purposes, including use in course packs. No downloading or copying may be done for any commercial purpose without the explicit permission of the Econometric Society. For such commercial purposes contact the Office of the Econometric Society (contact information may be found at the website

Consider a bipartite network where N consumers choose to buy or not to buy M different products. This paper considers the properties of the logit fit of the N × M array of “ i ‐buys‐ j ” purchase decisions, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="bold">Y</a:mi> <a:mo>=</a:mo> <a:msub> <a:mrow> <a:mo stretchy="false">[</a:mo> <a:msub> <a:mrow> <a:mi>Y</a:mi> </a:mrow> <a:mrow> <a:mi>i</a:mi> <a:mi>j</a:mi> </a:mrow> </a:msub> <a:mo stretchy="false">]</a:mo> </a:mrow> <a:mrow> <a:mn>1</a:mn> <a:mo>≤</a:mo> <a:mi>i</a:mi> <a:mo>≤</a:mo> <a:mi>N</a:mi> <a:mo>,</a:mo> <a:mn>1</a:mn> <a:mo>≤</a:mo> <a:mi>j</a:mi> <a:mo>≤</a:mo> <a:mi>M</a:mi> </a:mrow> </a:msub> </a:math>, onto a vector of known functions of consumer and product attributes under asymptotic sequences where (i) both N and M grow large, (ii) the average number of products purchased per consumer is finite in the limit, (iii) there exists dependence across elements in the same row or same column of Y (i.e., dyadic dependence), and (iv) the true conditional probability of making a purchase may, or may not, take the assumed logit form. Condition (ii) implies that the limiting network of purchases is sparse : only a vanishing fraction of all possible purchases are actually made. Under sparse network asymptotics, I show that the parameter indexing the logit approximation solves a particular Kullback–Leibler Information Criterion (KLIC) minimization problem (defined with respect to a certain Poisson population). This finding provides a simple characterization of the logit pseudo‐true parameter under general misspecification (analogous to a (mean squared error (MSE) minimizing) linear predictor approximation of a general conditional expectation function (CEF)). With respect to sampling theory, sparseness implies that the first and last terms in an extended Hoeffding‐type variance decomposition of the score of the logit pseudo composite log‐likelihood are of equal order. In contrast, under dense network asymptotics, the last term is asymptotically negligible. Asymptotic normality of the logistic regression coefficients is shown using a martingale central limit theorem (CLT) for triangular arrays. Unlike in the dense case, the normality result derived here also holds under degeneracy of the network graphon. Relatedly, when there “happens to be” no dyadic dependence in the data set in hand, it specializes to recently derived results on the behavior of logistic regression with rare events and i.i.d. data. Simulation results suggest that sparse network asymptotics better approximate the finite network distribution of the logit estimator. A short empirical illustration, and additional calibrated Monte Carlo experiments, further illustrate the main theoretical ideas.

In this paper we study identification and estimation of a correlated random coefficients (CRC) panel data model. The outcome of interest varies linearly with a vector of endogenous regressors. The coefficients on these regressors are heterogenous across units and may covary with them. We consider the average partial effect (APE) of a small change in the regressor vector on the outcome (cf. Chamberlain (1984), Wooldridge (2005a)). Chamberlain (1992) calculated the semiparametric efficiency bound for the APE in our model and proposed a √N-consistent estimator. Nonsingularity of the APE's information bound, and hence the appropriateness of Chamberlain's (1992) estimator, requires (i) the time dimension of the panel (T) to strictly exceed the number of random coefficients (p) and (ii) strong conditions on the time series properties of the regressor vector. We demonstrate irregular identification of the APE when T = p and for more persistent regressor processes. Our approach exploits the different identifying content of the subpopulations of stayers—or units whose regressor values change little across periods—and movers—or units whose regressor values change substantially across periods. We propose a feasible estimator based on our identification result and characterize its large sample properties. While irregularity precludes our estimator from attaining parametric rates of convergence, its limiting distribution is normal and inference is straightforward to conduct. Standard software may be used to compute point estimates and standard errors. We use our methods to estimate the average elasticity of calorie consumption with respect to total outlay for a sample of poor Nicaraguan households.

We consider estimation of population averages when data are missing at random. If some cells contain few observations, there can be substantial gains from imposing parametric restrictions on the cell means, but there is also a danger of misspecification. We develop a simple empirical Bayes estimator, which combines parametric and unadjusted estimates of cell means in a data-driven way. We also consider ways to use knowledge of the form of the propensity score to increase robustness. We develop an empirical Bayes extension of a double robust estimator. In a small simulation study, the empirical Bayes estimators perform well. They are similar to fully nonparametric methods and robust to misspecification when cells are moderate to large in size, and when cells are small they maintain the benefits of parametric methods and can have lower sampling variance.