Abstract
Nonparametric Kernel methods have been discussed by Vinod and Ullah (1988), who develop an estimator of the partial derivatives of conditional expectations. These have been applied in Econometrics for estimation of elasticities and other constructs related to partial derivatives. Applications of those ideas need not be restricted to the first pariials. This paper is concerned with estimation of second partials with similar kernel methods. We attempt to empirically study Marshall’s ‘law’ of the elasticity of demand which claims that elasticity increases with price.
John McMillan and Aman Ullah thank the National Science Foundation and SSHRCC, respectively, for research support. This paper is an extension of a paper presented at the Third Canadian Econometric Study Group Conference in Montreal, Canada, September 1986. The authors thank Robin Carter and Preston McAfee for comments.
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Notes
Marshall (1920), p. 87. An account of Marshall’s invention of the concept of elasticity is given by Keynes (1963), who described it as one of Marshall’s most important contributions to economics.
In an earlier paper Gasser and Müller (1984) considered the second derivative in the case of regression with fixed design variables. In the related work Rilstone (1987) and Rilstone and Ullah (1987) have considered the derivative estimation by using the finite difference method.
We ignore here possible simultaneity problems arising from the interaction of supply and demand. This can be given both an economic and a statistical justification. First, the existence of a production lag may mean that supply does not depend on current price; instead, it depends on a prediction about price made some time in the past. Second, Kramer (1984) has shown that there is no asymptotic simultaneous equations bias when the regressors are trended.
We note that the estimates of second order partial may be biased due to the normal kernel considered in this paper. This bias could however be eliminated by using kernels which are not necessarily positive; see Bartlett (1963). One such kernel is g 3(w) = 2-1 K(w) (3-w 2), another kernel among many in Vinod (1987) is g 7(w) = (1 /48)(105-105w 2+21w 4-w 6)K(w), where K(w) is the same as in (5). It can be shown that g 3 satisfies μ′1 = μk = 0 for its first three moments; and similarly g 7 satisfies μ′1 = μk = 0 for k = 2,3,4,5,6,7. These will be the subject of future study.
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© 1989 Springer Science+Business Media Dordrecht
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McMillan, J., Ullah, A., Vinod, H.D. (1989). Estimation of the Shape of the Demand Curve by Nonparametric Kernel Methods. In: Raj, B. (eds) Advances in Econometrics and Modelling. Advanced Studies in Theoretical and Applied Econometrics, vol 15. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7819-6_6
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