Abstract
This chapter is focuses on how one might estimate a model in which the dependent variable is a point in the plane rather than a point on the real line. A situation that comes to mind is a market in which there are just two suppliers and the desire is to estimate the market shares of the two. An example would be determination of the respective shares of AT&T and MCI in the early days of competition in the long-distance telephone market.
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Notes
- 1.
See Nahin (1998, p. 67).
- 2.
See Taylor and Houthakker (2010), Chap. 5).
- 3.
Households with after-tax income less than $5,000 are excluded from the analysis.
- 4.
t-ratios are in parentheses. All calculations are done in SAS.
- 5.
Interestingly, a much improved fit is obtained in a model with total expenditure and the thirteen other principal components (which, by construction, are orthogonal to the principal component that is being explained) as predictors. The R 2 of this model is 0.46.
- 6.
- 7.
The elasticity for LEC minutes in the “cos θ” equation is calculated as \( \hat{c}\bar{h}\bar{z}/\bar{v} \), where h denotes the ratio of the LEC price to the OC price. The “aggregate” elasticities are calculated, not for the sum of LEC and OC minutes, but for the radius vector z (the positive square root of the sum of squares of LEC and OC minutes). The OC share elasticities are calculated from equations in which the dependent variable is sin θ.
References
Kridel DJ, Rappoport PN, Taylor LD (2002) IntraLATA long-distance demand: carrier choice, usage demand, and price elasticities. Int J Forecast 18(4):545–559
Nahin P (1998) An imaginary tale: the story of the square root of −1. Princeton University Press, New Jersey
Taylor LD, Houthakker HS (2010) Consumer demand in the United States: prices, income, and consumer behavior, 3rd edn. Springer, Berlin
Taylor LD, Rappoport PN (1997) Toll price elasticities from a sample of 6,500 residential telephone bills. Inf Econ Policy 9(1):51–70
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Taylor, L.D. (2014). Regression with a Two-Dimensional Dependent Variable. In: Alleman, J., Ní-Shúilleabháin, Á., Rappoport, P. (eds) Demand for Communications Services – Insights and Perspectives. The Economics of Information, Communication, and Entertainment. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-7993-2_1
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