COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1486

"Empirical Similarity"

Itzhak Gilboa, Offer Lieberman, and David Schmeidler

October 2004

An agent is asked to assess a real-valued variable Y_p based on certain characteristics X_p = (X¹_p,...,X^m_p), and on a database consisting (X¹_i,...,X^m_i,Y_i) for i = 1,...,n. A possible approach to combine past observations of X and Y with the current values of X to generate an assessment of Y is similarity-weighted averaging. It suggests that the predicted value of Y, Y^s_p, be the weighted average of all previously observed values Y_i, where the weight of Y_i, for every i =1,...,n, is the similarity between the vector X¹_p,...,X^m_p, associated with Y_p, and the previously observed vector, X¹_i,...,X^m_i. We axiomatize this rule. We assume that, given every database, a predictor has a ranking over possible values, and we show that certain reasonable conditions on these rankings imply that they are determined by the proximity to a similarity-weighted average for a certain similarity function. The axiomatization does not suggest a particular similarity function, or even a particular functional form of this function. We therefore proceed to suggest that the similarity function be estimated from past observations. We develop tools of statistical inference for parametric estimation of the similarity function, for the case of a continuous as well as a discrete variable. Finally, we discuss the relationship of the proposed method to other methods of estimation and prediction.

Keywords: Similarity, Estimation

JEL Classification: C1, C8, D8