Estimation of Equilibrium Models Using a Likelihood ... - CiteSeerX

Estimation of Equilibrium Models Using a Likelihood Based Approach Maria Ana Vitorino∗ Graduate School of Business University of Chicago January 20, 2004

Abstract Random coefficient models are increasingly being used to estimate market demand and joint demand-supply equations in situations with differentiated goods where individual level data do not exist. The parameters of these models are usually estimated by instrumental variables methods to handle the price endogeneity problem. However, the lack of appropriate instruments and also the fact that instrumental variable methods are not invariant to the choice of the instruments increases the interest in using more efficient estimation methods. In this paper, we derive a likelihood approach for estimating the market equilibrium in the case of single-product firms for both the simple logit model and the random coefficients logit model. We apply this estimation procedure to price and sales from the industry for Personal Digital Assistants (PDAs) and compare the results of the full information maximum likelihood estimation procedure with the results of the generalized method of moments estimation procedure. The preliminary results analyzed in this paper seem to suggest that there are gains in efficiency when the model parameters are estimated using a likelihood based approach.

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The author would like to thank J.P. Dubé and Pradeep Chintagunta for helpful discussions and suggestions. Of course, responsability for the errors remains with the author.

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Introduction

Random coefficient models are increasingly being used to estimate market demand and joint demand-supply equations in situations with differentiated goods where individual level data do not exist. The parameters of these models are usually estimated by instrumental variables methods to handle the price endogeneity problem. However, the lack of appropriate instruments and also the fact that instrumental variable methods are not invariant to the choice of the instruments increases the interest in using more efficient estimation methods. In this paper, we derive a likelihood approach for estimating the market equilibrium in the case of single-product firms for both the simple logit model and the random coefficients logit model. We apply this estimation procedure to price and sales from the industry for Personal Digital Assistants (PDAs) and compare the results of the full information maximum likelihood estimation procedure with the results of the generalized method of moments estimation procedure. We find that the Jacobian term in the likelihood function for the simple logit case does not depend on the model parameters and therefore can be omitted in this case. However, the same result does not hold in the random coefficients model. We also find that, although the calculation of the Jacobian term makes the full-information maximum likelihood more complicated than the instrumental variables procedures, our empirical results indicate that there are significant gains in efficiency when a likelihood based approach is used. The remainder of the paper is organized as follows. In the second section we describe the data used to estimate demand and supply using different estimation procedures. In section three we discuss the specification of the demand and supply sides. In the fourth section we discuss how the demand and supply sides can be estimated using generalized method of moments estimation and non-linear full information maximum likelihood estimation and in the sixth section we discuss our empirical results. We conclude in section six.

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Description of the data

The aggregate data available consists of SKU-level monthly sales and prices of all PDA models sold through the retail channel in the US from January 1999 to July 2002. These data are collected by NPD Techworld using pointof-sale scanners linked to over 80% of the consumer-electronics retail ACV in the US. The life-cycle of each PDA model tends to be characterized, similarly to the life cycle of other high tech products, by a rapid increase in sales in the first months after the introduction, a quick drop in sales after the product reaches its maturity and by a very slow decay during the remaining of its decline phase (see Figure 1). Since, in the decline phase, there are several months in which, for some models, the number of PDAs sold is negligible we decided to exclude these observations from the data since they caused unrealistic variations in the average prices.

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Figure 1: Sales evolution for one PDA model

Therefore, after removing models corresponding to brands with insignificant overall market shares (