A Lagrange Multiplier Test for the Validity of Instruments in MNL Models: an Application to Residential Choice
C A Guevara, Massachusetts Institute of Technology, US and Universidad de Los Andes, CL; M Ben-Akiva, Massachusetts Institute of Technology, US
A statistical test for the validity of instruments in MNL models is derived and applied. Test is based in Sargan test of overidentifying restrictions. Test properties are studied using Monte Carlo experiments and real residential choice data
Endogeneity is a key issue in residential discrete choice modeling. It occurs because each dwelling unit is quasi-unique, making thus unavoidable the omission of attributes which would be correlated, particularly, with price. This problem causes an inconsistent and a severely upward biased estimation of the price coefficient in such type of models.
The effect of this model misspecification is indeed huge. Many empirical applications (Bhat and Guo, 2004; Sermonss and Koppelman, 2001; Levine, 1998; Waddell, 1992; Quigley, 1976) have reported statistically insignificant, small or even positive dwelling-unit price coefficients. Furthermore, even residential choice models in which estimated price coefficients are negative and significant, are not necessarily consistently estimated. The case is that, even if it would be possible to account for all relevant attributes which are correlated with price, the quasi-uniqueness of the alternatives in the choice set faced by each individual would make necessary to estimate each coefficient in the model with a single observation. This would lead again to inconsistent estimators in what is known as the incidental parameters problem (Wooldrige, 2002).
Guevara and Ben-Akiva (2006) stated the relevance of the endogeneity issue in residential choice modeling and argument that the control-function method is the more suitable to deal with that issue in this area. Complementing this line of research, present article describes the derivation and application of a novel Lagrange Multiplier test for the validity of instrumental variables required to build control-functions to correct for endogenity in MNL models.
The test proposed in this article is inspired by the Sargan (1958) test of overidentifying restrictions and is based in the asymptotics derived by McFadden (1987) and Engle (1984). It principally consists in the estimation of an auxiliary regression where the dependent variable is a weighed measure of the difference between fitted probabilities and actual choices, and where the independent variables correspond to weighed alternative variables and the instruments which exogeneity is to be tested. The intuition behind the test is that, if instruments are exogenous, the unadjusted multiple correlation coefficient of that auxiliary regression should be nearly zero. Concordantly, a Lagrange Multiplier statistical test, based on this unadjusted multiple correlation coefficient but corrected for the degrees of freedom, is constructed. Finally, using McFadden (1987) and Engle (1984) asymptotics, it can be claimed that the statistical is distributed chi-square with a number of degrees of freedom equal to number of independent variables.
Test properties are studied using a Monte Carlo simulated database and residential data from Santiago, Chile. The empirical results show that endogeneity is relevant in the studied sample and that the average price of dwelling units in adjacent zones is a valid instrument for this experiment. Power properties and suitability of the proposed statistical test are also discussed, and suggestions for further research are drawn.
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