Car Ownership? Evidence from the Copenhagen Metropolitan Area

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area. Since it is difficult to choose for a particular nesting structure, the flexibility offered by the cross-nested logit model (see Vovsha, 1997; Bierlaire, 2006) is attractive. In the model the three types of correlation can be present simultaneously 23 and by estimating the model the data show the relative importance of each of them. However, estimation of (cross-)nested logit models is typically more involved than that of estimating a multinomial logit model. For instance, the likelihood function is no longer necessarily concave and the ‘logsum parameters’ may be difficult to estimate. A grid search over relevant values of these parameters may be a useful strategy, although probably time consuming. With respect to the instrument for the house price we noted some ambiguity with respect to the use of exogenous characteristics of areas in the vicinity. It is not always clear that such characteristics are excluded from the utility of a particular choice alternative. For instance, amenities in contiguous areas may be easy to visit as those of one’s own residential area and therefore contribute as much to the utility of living there. Bayer’s procedure for computing the instrument for the price uses exogenous characteristics of all alternatives and therefore does not rely solely on those of the neighbouring areas. But it does not exclude these characteristics from the computation of the instrument which raises questions about its appropriateness. It may therefore be useful to look for other ways to compute an instrument for the price. Reynaert and Verboven (2014) have recently emphasized the importance of using approximations to the optimal instruments. Their methodology appears to be flexible enough to avoid the use of characteristics of neighbouring areas in the construction of the instrument. 24 We are currently working on the incorporation of the nesting structures and the alternative procedures for finding an instrument for the house price in models like that of the present paper. 23 The cross-nested logit may be formulated as a weighted average of the various nested logit models where the weights are determined by some of the parameters to be estimated. 24 An alternative approach may be based on Belloni et al. (2012) who propose the use of Lasso methods. 36
References Anderson, M.L. 2014. Subways, strikes and slowdowns: The impact of public transit on traffic congestion. American Economic Review, 104, 2763-2796. Bayer, P., F. Ferreira and R. McMillan. 2007. A unified framework for measuring preferences for schools and neighborhoods. Journal of Political Economy, 115, 588-638. Bayer, P., R. McMillan and K. Rueben. 2004. An equilibrium sorting model of sorting in an urban housing market. NBERWorking Paper No. 10865. Bayer, P. and C. Timmins. 2007. Estimating equilibrium models of sorting across locations. Economic Journal, 117(518), 353-374. Belloni, A., D. Chen, V. Chernozhukov and C. Hansen (2012) Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain. Econometrica, 80, 2369-2429. Berry, S. 1994. Estimating discrete choice models of product differentiation. RAND Journal of Economics, 25, 242-262. Berry, S., J. Levinsohn and A. Pakes. 1995. Automobile prices in market equilibrium. Econometrica, 63, 841-890. Bierlaire, M. 2006 A Theoretical Analysis of the Cross-Nested Logit Model. Annals of Operations Research, 144, 287-300. Chamberlain, G. 1987 Asymptotic Efficiency in Estimation with Condtional Moment Restrictions. Journal of Econometrics, 34, 305-334. Dargay, J.M. 2002. Determinants of car ownership in rural and urban areas: a pseudo-panel analysis. Ttransportation Research E, 38, 351-366. de Jong, G.C. 1998. An indirect utility model of car ownership and private car use. European Economic Review, 34, 971-985. De Palma, A. and K. Kilani. 2003. (Un)conditional compensating variation in discrete choice models. Working paper, THEMA, University of Cergy-Pontoise. Dieleman, F.M., M. Dijst and G. Burghouwt. 2002. Urban form and travel behavior: micro level household attributes and residential context. Urban Studies, 39, 507-527. Epple, D. and G.J. Platt. 1998. Equilibrium and local redistribution in an urban economy when households differ in both preferences and incomes. Journal of Urban Economics, 43, 23-51. Fernandez, R. and R. Rogerson. 1996. Income distribution, communities, and the quality of public education. The Quarterly Journal of Economics, 111, 135–164. Goodwin, P.B. 1993. Car ownership and public transport use: revisiting the interaction. Transportation, 27, 21-33. 37
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Car Ownership? Evidence from the Copenhagen Metropolitan Area

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