In paired randomized experiments units are grouped in pairs, often based on covariate information, with random assignment within the pairs. Average treatment effects are then estimated by averaging the within-pair differences in outcomes. Typically the variance of the average treatment effect estimator is estimated using the sample variance of the within-pair differences. However, conditional on the covariates the variance of the average treatment effect estimator may be substantially smaller. Here we propose a simple way of estimating the conditional variance of the average treatment effect estimator by forming pairs-of-pairs with similar covariate values and estimating the variances within these pairs-of-pairs. Even though these within-pairs-of-pairs variance estimators are not consistent, their average is consistent for the conditional variance of the average treatment effect estimator and leads to asymptotically valid confidence intervals.
It is standard practice in empirical work to allow for clustering in the error covariance matrix if the explanatory variables of interest vary at a more aggregate level than the units of observation. Often, however, the structure of the error covariance matrix is more complex, with correlations varying in magnitude within clusters, and not vanishing between clusters. Here we explore the implications of such correlations for the actual and estimated precision of least squares estimators. We show that with equal sized clusters, if the covariate of interest is randomly assigned at the cluster level, only accounting for non-zero covariances at the cluster level, and ignoring correlations between clusters, leads to valid standard errors and confidence intervals. However, in many cases this may not suffice. For example, state policies exhibit substantial spatial correlations. As a result, ignoring spatial correlations in outcomes beyond that accounted for by the clustering at the state level, may well bias standard errors. We illustrate our findings using the 5% public use census data. Based on these results we recommend researchers assess the extent of spatial correlations in explanatory variables beyond state level clustering, and if such correlations are present, take into account spatial correlations beyond the clustering correlations typically accounted for.
We investigate the problem of optimal choice of the smoothing parameter (bandwidth) for the regression discontinuity estimator. We focus on estimation by local linear regression, which was shown to be rate optimal (Porter, 2003). Investigation of an expected-squared- error–loss criterion reveals the need for regularization. We propose an optimal, data dependent, bandwidth choice rule. We illustrate the proposed bandwidth choice using data previously analyzed by Lee (2008), as well as in a simulation study based on this data set. The simulations suggest that the proposed rule performs well.
We develop and analyze a tractable empirical model for strategic network formation that can be estimated with data from a single network at a single point in time. We model the network formation as a sequential process where in each period a single randomly selected pair of agents has the opportunity to form a link. Conditional on such an opportunity, a link will be formed if both agents view the link as beneficial to them. They base their decision on their own characteristics, the characteristics of the potential partner, and on features of the current state of the network, such as whether the the two potential partners already have friends in common. A key assumption is that agents do not take into account possible future changes to the network. This assumption avoids complications with the presence of multiple equilibria, and also greatly simplifies the computational burden of analyzing these models. We use Bayesian markov-chain-monte-carlo methods to obtain draws from the posterior distribution of interest. We apply our methods to a social network of 669 high school students, with, on average, 4.6 friends. We then use the model to evaluate the effect of an alternative assignment to classes on the topology of the network.
Two recent papers, Deaton (2009), and Heckman and Urzua (2009), argue against what they see as an excessive and inappropriate use of experimental and quasi-experimental methods in empirical work in economics in the last decade. They specifically question the increased use of instrumental variables and natural experiments in labor economics, and of randomized experiments in development economics. In these comments I will make the case that this move towards shoring up the internal validity of estimates, and towards clarifying the description of the population these estimates are relevant for, has been important and beneficial in increasing the credibility of empirical work in economics. I also address some other concerns raised by the Deaton and Heckman-Urzua papers.
This paper presents methods for evaluating the effects of reallocating an indivisible input across production units, taking into account resource constraints by keeping the marginal distribution of the input fixed. When the production technology is nonseparable, such reallocations, although leaving the marginal distribution of the reallocated input unchanged by construction, may nonetheless alter average output. Examples include reallocations of teachers across classrooms composed of students of varying mean ability. We focus on the effects of reallocating one input, while holding the assignment of another, potentially complementary, input fixed. We introduce a class of such reallocations – correlated matching rules – that includes the status quo allocation, a random allocation, and both the perfect positive and negative assortative matching allocations as special cases. We also characterize the effects of local (relative to the status quo) reallocations. For estimation we use a two-step approach. In the first step we nonparametrically estimate the production function. In the second step we average the estimated production function over the distribution of inputs induced by the new assignment rule. These methods build upon the partial mean literature, but require extensions involving boundary issues. We derive the large sample properties of our proposed estimators and assess their small sample properties via a limited set of Monte Carlo experiments.
We propose new semiparametric estimators for parameters that depend on the derivatives (up to any finite order) of unknown conditional expectations and densities. We consider two cases. In the first we average over all (conditioning) variables in the conditional expectation or density. In the second case we average over a strict subset of the conditioning variables. The unknown conditional expectations and densities are estimated by a first step kernel estimator. The kernel estimator has a boundary correction that makes it uniformly consistent if the distribution of the covariates has bounded support. The partial and full mean estimators therefore do not require trimming (asymptotic or fixed) as in the estimators developed by Newey (1994) and Powell, Stock, and Stoker (1989) and in many applications to specific settings. We provide a general formula for the influence function and the asymptotic variance for both full and partial averaging. We also specify a general set of regularity conditions that contains a new restriction on the kernel function to avoid bias in the case that the parameter depends on the derivatives of the conditional expectation or density.