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Durham University

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Publication details for Dr Anurag Narayan Banerjee

Banerjee, A.N. (2007). A method of estimating the average derivative. Journal of Econometrics 136(1): 65-88.

Author(s) from Durham


We derive a simple semi-parametric estimator of the “direct” Average Derivative, δ=E(D[m(x)]), where m(x) is the regression function and S, the support of the density of x is compact. We partition S into disjoint bins and the local slope D[m(x)] within these bins is estimated by using ordinary least squares. Our average derivative estimate , is then obtained by taking the weighted average of these least squares slopes. We show that this estimator is asymptotically normally distributed. We also propose a consistent estimator of the variance of . Using Monte-Carlo simulation experiments based on a censored regression model (with Tobit Model as a special case) we produce small sample results comparing our estimator with the Härdle–Stoker [1989. Investigating smooth multiple regression by the method of average derivatives. Journal of American Statistical Association 84, 408, 986–995] method. We conclude that performs better that the Härdle–Stoker estimator for bounded and discontinuous covariates.


Banerjee, A.N., 1994. A method of estimating the average derivative. CORE Discussion paper: 9403,
Universite Catholique de Louvain.
Fan, J., Gijbels, I., 1992. Variable bandwidth and local linear regression smoothers. The Annals of
Statistics 20, 2008–2036.
Greene, W., 1999. Marginal effects in the censored regression model. Economics Letters 64 (1), 43–49.
Ha¨ rdle, W., 1992. Applied of Non Parametric Regression, Econometric Society Monographs. Cambridge
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Ha¨ rdle, W., Stoker, T.M., 1989. Investigating smooth multiple regression by the method of average
derivatives. Journal of American Statistical Association 84 (408), 986–995.
Ha¨ rdle, W., Hilderbrand, W., Jerison, M., 1991. Empirical evidence for the law of demand. Econometrica
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Ha¨ rdle, W., Hart, J., Marron, J.S., Tsybakov, A.B., 1992. Bandwidth choice for average derivative
estimation. Journal of the American Statistical Association 87 (417), 218–226.
Horowitz, J.L., Ha¨ rdle, W., 1996. Direct semiparametric estimation of a single-index model with discrete
covariates. Journal of the American Statistical Association 91, 1632–1640.
Li, W., 1996. Asymptotic equivalence of estimators of average derivative. Economic Letters 52, 241–245.
Newey, W.K., Stoker, T.M., 1993. Efficiency of weighted average derivative estimators and index models.
Econometrica 61, 1199–1223.
Powell, J., Stock, J.H., Stoker, T.M., 1989. Semiparametric estimation of index coefficients. Econometrica
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Stoker, T.M., 1991a. Lectures on Semiparametric Econometrics, CORE Lecture Series. Core Foundation,
Louvain-la Neuve.
Stoker, T.M., 1991b. Equivalance of direct, indirect and slope estimators of average derivatives. In:
Barnett, W.A., Powell, J., Tauchen, G. (Eds.), Nonparametric and Semiparametric Methods in
Econometrics and Statistics. Cambridge University Press, Cambridge.
Ullah, A., Vinod, H., 1988. Flexible production function estimation by nonparametric kernel estimators.
Advances in Econometrics, JAI Press.
Zhang, S., Karuhamuni, R.J., 2000. On nonparametric density estimation at the boundary. Nonparametric
Statistics 12, 197–221.