Endogeneity in multinomial response models

The endogeneity issue in multinomial response model refers to the correlation between the explanatory variable and the unobservable variable.^[1] Take a choice-making model^[2] as an example for a multinomial response model with an unobservable variable:^[3]

$y_{i} =$ j if and only if $y_{j, i} = m a x {y_{1, i}, y_{2, i} . . . . . y_{j, i}};$

$y_{j, i} = x_{j, i} β + v_{i} + u_{j, i}$

In the choice-making context, i indexes the N individuals and j indexes the J choices, for instance, different brands of chocolates. $v_{i}$ represents an unobservable feature of individual i, for instance, the taste of individual i, and denotes the i.i.d response error, which might come from cognitive limitations, information analysis difficulties, and many other reasons. $y_{j, i}^{*}$ is a latent variable representing the utility of individual i when she chooses choice j. The response error $u_{j, i}$ is assumed to be i.i.d. with zero mean and has a density $f_{u} (\cdot)$ . When $f_{u} (\cdot)$ is normal, then the given model will be a multinomial probit model; when it is a Gumbel density, then the model will be a multinomial logit model. In usual cases, $v_{i}$ is assumed to be uncorrelated with $x_{j, i}$ , which usually is a group of variables describing the features of the choice, for instance, the weight of the chocolate; the features of the individual, for instance, the age of the individual; and the interaction between the choice and the individual, for instance, the quantity of chocolate consumed by the individual last year. Under this assumption, $v_{i} | x_{j, i} \sim f_{v} (\cdot)$ . Then the log-likelihood function of a typical multinomial response model with unobservable variable can be written as:

$\sum_{i = 1}^{N} \log {\int P [y_{i, j}^{*} > y_{i, k}^{*} \forall k \neq j, x_{1, i} . . . . x_{J, i}] f_{v} (v_{i}) d v_{i}}$

However, in many practical cases, the personal feature $v_{i}$ is correlated with $x_{j, i}$ . In this situation, the estimates from the model estimation without considering this correlation will be inconsistent. To fix this problem, the log-likelihood function should be revised as:

$\sum_{i = 1}^{N} \log {\int P [y_{i, j}^{*} > y_{i, k}^{*} \forall k \neq j, x_{1, i} . . . . x_{J, i}] f_{v} (v_{i}) d v_{i}}$

Then, the model can be estimated consistently by MLE.^[4] Because the construction of this correlation can be very non-standard, there is not a unified solution for this type of problem.^[5] One common practice is to impose some parametric assumption to model the distribution of the unobservable variable conditional on the observable explanatory variables and then implement MLE based on the new likelihood function.

References

↑ Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 652.
↑ For more details, refer to: J. Miguel Villas-Boas, Russell S. Winer, (1999) “Endogeneity in Brand Choice Models,” Management Science 45(10):1324-1338.
↑ For more examples, refer to: Ben-Akiva, M., Boccara, B. (1995). “Discrete choice models with latent choice sets,” International Journal of Research in Marketing, 12(1), pp9–24
↑ To deal with the integral in the loglikelihood function while computing the MLE, EM algorithm is usually needed. For more details of the algorithm, please refer to: Olivier Cappé, Eric Moulines, and Tobias Ryden. (2005): Inference in Hidden Markov Models. Springer-Verlag New York, Inc., Secaucus, NJ, USA.
↑ For a summary, refer to: Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 654.

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[1] Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 652.

[2] For more details, refer to: J. Miguel Villas-Boas, Russell S. Winer, (1999) “Endogeneity in Brand Choice Models,” Management Science 45(10):1324-1338.

[3] For more examples, refer to: Ben-Akiva, M., Boccara, B. (1995). “Discrete choice models with latent choice sets,” International Journal of Research in Marketing, 12(1), pp9–24

[4] To deal with the integral in the loglikelihood function while computing the MLE, EM algorithm is usually needed. For more details of the algorithm, please refer to: Olivier Cappé, Eric Moulines, and Tobias Ryden. (2005): Inference in Hidden Markov Models. Springer-Verlag New York, Inc., Secaucus, NJ, USA.

[5] For a summary, refer to: Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 654.

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