Identification and Estimation of Multinomial Choice Models with Latent Special Covariates

Abstract

Identification of multinomial choice models is often established by using special covariates that have full support. This article shows how these identification results can be extended to a large class of multinomial choice models when all covariates are bounded. I also provide a new $\sqrt{n}$-consistent asymptotically normal estimator of the finite-dimensional parameters of the model.

KEYWORDS:

• Bundles
• Identification at infinity
• Multinomial choice
• Random coefficients
• Special covariate

Supplementary Materials

The supplementary materials contain the replication package for all simulation and estimation results.

Acknowledgments

I thank the editor and two anonymous referees for comments and suggestions that have greatly improved the article. I also thank Roy Allen, Victor H. Aguiar, Tim Conley, Nirav Mehta, Salvador Navarro, Joris Pinkse, David Rivers, and Bruno Salcedo for their useful comments and discussions.

Notes

1 Completeness of a family of distributions is a well-known concept in the Statistics and Econometrics literature. See, for example, Mattner et al. (1993), Newey and Powell (2003), Chernozhukov and Hansen (2005), Blundell, Chen, and Kristensen (2007), Chernozhukov, Imbens, and Newey (2007), Hu and Schennach (2008), Andrews (2011), Darolles, Fan, Florens, and Renault (2011), and d’Haultfoeuille (2011).

2 See, for example, Manski (1985), Manski (1988), Heckman (1990), Matzkin (1992), Ichimura and Thompson (1998), Lewbel (1998), Lewbel (2000), Tamer (2003), Matzkin (2007), Berry and Haile (2009), Bajari, Hong, and Ryan (2010), Gautier and Kitamura (2013), Gautier and Hoderlein (2015), Fox and Gandhi (2016), Dunker, Hoderlein, and Kaido (2017), Fox and Lazzati (2017), Fox, Yang, and Hsu (2018), Fox (2020), and Kashaev and Salcedo (2021).

3 See, for instance, Magnac and Maurin (2007), Chen, Khan, and Tang (2016), Kline (2016), and Lewbel, Yan, and Zhou (2021).

4 Deterministic vectors are denoted by lower-case regular font Latin letters (e.g., x) and random objects by bold letters (e.g., x). Capital letters are usually used to denote supports of random variables (e.g., $\mathit{x}\in X$ ). I denote the support of a conditional distribution of x conditional on $\mathit{z}=z$ by X_z. The cumulative distribution function (cdf) and the probability density function (pdf) of x are denoted by $F_{\mathit{x}}$ and $f_{\mathit{x}}$. $F_{\mathit{x}|\mathit{z}}$ ( $f_{\mathit{x}|\mathit{z}}$ ) denotes the cdf (pdf) of x conditional on $\mathit{z}=z$.

5 Since $\mathrm{Pr}({\beta }_0(\mathit{w})+{\beta }_1(\mathit{w})\mathit{d}+\mathit{e}>0|\mathit{x}=x)=1-F_{\mathit{e}|\mathit{x}}(-{\beta }_0(w)-{\beta }_1(w)d|x)$ and there are no restrictions on ${\beta }_0(·)$ , the random coefficient ${\beta }_0(\mathit{w})+{\beta }_1(\mathit{w})\mathit{d}+\mathit{e}$ can be positive (negative) with probability that is arbitrarily close to 1 if the support of e conditional on $\mathit{x}=x$ is unbounded.

6 The outcome y = 0 can be replaced by any outcome. In this case, one will just need to renormalize the utility from that outcome to zero.

7 Since, for identification and estimation, I require the average structural function p₀, some forms of endogeneity (i.e., correlation between x and $\mathit{\epsilon }$ ) can be addressed using suitable instruments and control function residuals as in Blundell and Powell (2004) (see also Berry 1994; Berry, Levinsohn, and Pakes 1995; Berry and Haile 2014 for identification of structural demand function using aggregate data and instruments). I leave the detailed analysis of this case for future research.

8 For more detailed discussion of the problem of identification of the distribution from its moments see, for instance, Kleiber and Stoyanov (2013) and references therein.

9 For testability of the completeness assumptions see Canay, Santos, and Shaikh (2013).

10 Proposition 3.1 also provides a constructive identification for β₀ and β₁. However, Assumption 3(iii) fails to hold in my illustrative application presented in Section 6. Additionally, Proposition 3.1 uses limits of derivatives of identifiable functions at a single point, thus, most likely, leading to a consistent estimator with nonparametric rate of convergence.

11 The normality of e implies that p₀ has continuous derivatives of any order. See Appendix A.3 for details.

12 Both β and $F_{\mathit{\epsilon }}$ can be estimated in one step by the sieve maximum-likelihood estimator. In this case, however, the estimator of β may not be $\sqrt{n}$ consistent.

13 The results are qualitatively the same for higher order polynomials.

14 The mean absolute deviation of the estimator also decreases with the sample size. See, Appendix B for further details.

15 For comparison of my estimator with two alternative potentially misspecified parametric estimators, see Appendix B.

16 See Allenby and Rossi (1991) for specific details of the dataset construction.

17 Income and prices are measured in thousands of U.S. dollars and U.S. dollars, respectively.

18 Estimation using $\text{log}\hspace{0.17em}({\mathit{p}}_y)$ instead of ${\mathit{p}}_y$ gives qualitatively similar results.

19 For example, in U.K. households spend about 1% of their grocery expenditures on margarine and butter (Griffith, Nesheim, and O’Connell 2018).

20 I use the tensor product of the fourth degree Chebyshev polynomials for d and the first degree Chebyshev polynomials for every z_y.

21 This empirical finding is in line with the simulation results, presented in Appendix B.

22 I can differentiate under the integral sign since (i) h₀ being bounded implies that F₀ is bounded, (ii) all derivatives of the standard normal pdf are bounded.