We are interested in estimating the average e#ect of a binary treatment on a scalar outcome. If assignment to the treatment is independent of the potential outcomes given pretreatment variables, biases associated with simple treatment-control average comparisons can be removed by adjusting for di#erences in the pre-treatmentvariables. Rosenbaum and Rubin #1983, 1984# show that adjusting solely for di#erences between treated and control units in a scalar function of the pre-treatment variables, the propensity score, also removes the entire bias associated with di#erences in pre-treatment variables. Thus it is possible to obtain unbiased estimates of the treatment e#ect without conditioning on a possibly highdimensional vector of pre-treatment variables. Although adjusting for the propensity score removes all the bias, this can come at the expense of e#ciency. We show that weighting with the inverse of a nonparametric estimate of the propensity score, rather than the true propensity scor...

. A class of weighted-bootstrap techniques, called biasedbootstrap methods, is proposed. It is motivated by the need to adjust more conventional, uniform-bootstrap methods in a surgical way, so as to alter some of their features while leaving others unchanged. Depending on the nature of the adjustment, the biased bootstrap can be used to reduce bias, or reduce variance, or render some characteristic equal to a predetermined quantity. More specifically, applications of bootstrap methods include hypothesis testing, variance stabilisation, both density estimation and nonparametric regression under constraints, `robustification ' of general statistical procedures, sensitivity analysis, generalised method of moments, shrinkage, and many more. 1991 Mathematics Subject Classification: Primary 62G09, Secondary 62G05 Keywords and Phrases: Bias reduction, empirical likelihood, hypothesis testing, local-linear smoothing, nonparametric curve estimation, variance stabilisation, weighted bootstrap 1...

Recent developments in empirical likelihood (EL) methods are reviewed. First, to put the method in perspective, two interpretations of empirical likelihood are presented, one as a nonparametric maximum likelihood estimation method (NPMLE) and the other as a generalized minimum contrast estimator (GMC). The latter interpretation provides a clear connection between EL, GMM, GEL and other related estimators. Second, EL is shown to have various advantages over other methods. The theory of large deviations demonstrates that EL emerges naturally in achieving asymptotic optimality both for estimation and testing. Interestingly, higher order asymptotic analysis also suggests that EL is generally a preferred method. Third, extensions of EL are discussed in various settings, including estimation of conditional moment restriction models, nonparametric specification testing and time series models. Finally, practical issues in applying EL to real data, such as computational algorithms for EL, are discussed. Numerical examples to illustrate the efficacy of the method are presented.

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We introduce a general testing procedure in models with possible identification failure that has exact asymptotic rejection probability under the null hypothesis. The procedure is widely applicable and in this paper we apply it to tests of arbitrary linear parameter hypotheses and to tests of overidentifying restrictions in time series models given by unconditional moment conditions. The main idea is to subsample classical tests, like for example the Wald or the J test. More precisely, instead of using these tests with critical values based on asymptotic theory, we compute data—dependent critical values based on the subsampling technique. We show that the resulting tests have exact asymptotic rejection probabilities under the null hypothesis independent of identification failure. Furthermore, the subsampling tests of parameter hypotheses are shown to be consistent against fixed alternatives and to have the same local power as the original tests under full identification. The subsampling test of overidentifying restrictions is shown

This paper highlights a problem in using the first-differenced GMM panel data estimator to estimate cross-country growth regressions. When the time series are persistent, the first-differenced GMM estimator can be poorly behaved, since lagged levels of the series provide only weak instruments for subsequent first- differences. Revisiting the work of Caselli, Esquivel and Lefort (1996), we show that this problem may be serious in practice. We suggest using a more efficient GMM estimator that exploits stationarity restrictions, and this approach is shown to give more reasonable results than first-differenced GMM in our estimation of an empirical growth model.

Newey and Smith (2001) have recently shown that Empirical Likelihood (EL) exhibits desirable higher-order asymptotic properties, namely, that its O ¡ n −1 ¢ bias is particularly small and that biascorrected EL is higher-order efficient. Although EL possesses these properties when the model is correctly specified, this paper shows that the asymptotic variance of EL in the presence of model misspecification may become undefined when the functions defining the moment conditions are unbounded. In contrast, the Exponential Tilting (ET) estimator avoids this problem under mild regularity conditions. Since ET does not share the higher-order asymptotic properties of EL, there is a need for an estimator that combines the qualities of both estimators. This paper introduces a new estimator called Exponentially Tilted Empirical Likelihood (ETEL) that is shown to have the same O ¡ n −1 ¢ bias and the same O ¡ n −2¢ variance as EL, while maintaining a well-defined asymptotic variance under model misspecification.

How large would consumer search costs have to be in order for them to rationalize online price dispersion? We evaluate the ability of equilibrium search models to explain observed patterns of price dispersion in online markets by developing a methodology for estimating equilibrium search-based price dispersion models. We consider models of both sequential and nonsequential search strategies, and exploit equilibrium restrictions to recover estimates of search cost heterogeneity which are theoretically consistent with the search models. For a number of online electronics and new book markets, our estimates show that the nonsequential search model uniformly yields more realistic results for the search costs. However, the large magnitudes for the search costs suggests that the search models may be incomplete as a descriptor of consumer behavior and firm pricing in online markets.

Economic models typically involve a set of moment conditions. The most widely used tests for moment conditions are the J-test associated with the generalized method of moments (GMM) and discrepancy tests associated with the family of generalized empirical likelihood (GEL). It is known that all of these tests have the same asymptotic properties under the null and local alternatives. This paper studies the Hodges and Lehmann (1956) optimality of these tests: evaluate these tests in terms of the exponential rate of growth to one of the power functions evaluated at a fixed alternative while keeping the asymptotic sizes of the tests bounded by some constant. We derive an optimal rate for this exponential rate of convergence and provide general conditions for a test to achieve this optimal rate. The results are applied to show that the GMM and GEL tests are Hodges-Lehmann optimal under mild conditions.

This paper studies the Bahadur efficiency of empirical likelihood for testing moment condition models. It is shown that under mild regularity conditions, the empirical likelihood overidentifying restriction test is Bahadur efficient, i.e., its p-value attains the fastest convergence rate under each fixed alternative hypothesis. Analogous results are derived for parameter hypothesis testing and set inference problems. 1