Simple estimators for monotone index models


ABSTRACT In this paper, estimation of the coe¢ cients in a "single-index" regression model is considered under the assumption that the regression function is a smooth and strictly monotonic function of the index. The estimation method follows a "two-step" approach, where the …rst step uses a nonparametric regression estimator for the dependent variable, and the second step estimates the unknown index coe¢ cients (up to scale) by an eigenvector of a matrix de…ned in terms of this …rst-step estimator. The paper gives conditions under which the proposed estimator is root-n-consistent and asymptotically normal.

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    ABSTRACT: The analysis of data with endogenous regressors - that is, observable explana- tory variables that are correlated with unobservable error terms - is arguably the main contribution of econometrics to statistical science. Although "endo- geneity" can arise from a number of different sources, including mismeasured regressors, sample selection, heterogeneous treatment effects, and correlated random effects in panel data, the term originally arose in the context of "simul- taneity, "i nw hich the explanatory variables were, with the dependent variable, determined through a system of equations, so that their correlation with error terms arose from feedback from the dependent to the explanatory variables. Analysi so f linear supply-and-demand systems (with normal errors) yielded the familiar rank and order conditions for identification, two- and three-stage esti- mation methods, and analysis of structural interventions. Although these multi- step estimation procedures have been extended to nonlinear parametric models with additive nonnormal errors (e.g., Amemiya, 1974 and Hansen 1982), ex- tensions to nonparametric and semiparametric models have only recently been considered. The ai mo f th is chapter is to examine the existing literature on estimation of some "nonparametric" models with endogenous explanatory variables, and to compare the different identifying assumptions and estimation approaches for particular models and determine their applicability to others. To maintain a manageable scope for the chapter, we restrict our attention to nonparamet- ric and semiparametric extensions of the usual simultaneous equations models (with endogenous regressors that are continuously distributed). We consider the identification and estimation of the "average structural function" and argue that this parameter is one parameter of central interest in the analysi so f semi- parametric and nonparametric models with endogenous regressors. The two leading cases we consider are additive nonparametric specifications i nw h ich the regression function is unknown, and nonadditive models i nw hich there is some known transformation function that is monotone but not invertible. An im- portant example of the latter, and one that we use as an empirical illustration, is
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    ABSTRACT: This paper considers estimation of the coe¢ cients in a semiparametric multinomial choice model with linear indirect utility functions (with common coe¢ cients but diering regressors) and errors that are assumed to be independent of the regressors. This implies that the conditional mean of the vector of dependent indicator variables is a smooth and invertible function of a corresponding vector of linear indices. The estimation method is an extension of an approach proposed by Ahn, Ichimura, and Powell (2004) for monotone single-index regression models to a multi-index setting, estimating the unknown index coe¢ cients (up to scale) by an eigenvector of a matrix de…ned in terms of a …rst- step nonparametric estimator of the conditional choice probabilities. Under suitable conditions, the proposed estimator is root-n-consistent and asymptotically normal.
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    ABSTRACT: This paper contributes to the literature on econometric estimation of incomplete information games with Nash equilibrium behavior by introducing a two-step estimation procedure that makes no parametric assumptions about the distribution of unobservable payoffs shocks. Instead, its asymptotic properties rely on assuming only that these distributions satisfy an invertibility condition, and that the underlying equilibrium selection mechanism is degenerate. Our methodology relies on a pairwise-differencing procedure which, unlike Aradillas-Lopez (2008), does not require computing the equilibria of the game. Furthermore, if normal-form payoffs are linear in the parameters of interest, our procedure results in an estimator with a closed-form expression. We contribute to the pairwise-differencing econometric literature by introducing the first model where both the control variables being matched and the regressors in the index function parameterized by contain nonparametric functions. In particular, the asymptotic theory developed in Aradillas-Lopez, Honoré, and Powell (2007) does not cover this setting. We describe conditions under which nonparametrically estimated plug-ins yield a √ −consistent and asymptotically normal estimator for the parameter of interest. A consistent specification test based on semiparametric residuals is also developed. It appears to be the first test of this type for a model involving nonparametric or "generated" regressors. Several extensions of our method are also discussed. A series of Monte carlo experiments are used to investigate the properties of our estimator and our specification test.
    Journal of Econometrics 09/2009; · 1.53 Impact Factor


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