Review Articles

Verifiable identification condition for nonignorable nonresponse data with categorical instrumental variables

Kenji Beppu ,

Graduate School of Engineering Science, Osaka University, Osaka, Japan

Kosuke Morikawa

Graduate School of Engineering Science, Osaka University, Osaka, Japan

Pages | Received 01 Apr. 2023, Accepted 25 Dec. 2023, Published online: 04 Jan. 2024,
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We consider a model identification problem in which an outcome variable contains nonignorable missing values. Statistical inference requires a guarantee of the model identifiability to obtain estimators enjoying theoretically reasonable properties such as consistency and asymptotic normality. Recently, instrumental or shadow variables, combined with the completeness condition in the outcome model, have been highlighted to make a model identifiable. In this paper, we elucidate the relationship between the completeness condition and model identifiability when the instrumental variable is categorical. We first show that when both the outcome and instrumental variables are categorical, the two conditions are equivalent. However, when one of the outcome and instrumental variables is continuous, the completeness condition may not necessarily hold, even for simple models. Consequently, we provide a sufficient condition that guarantees the identifiability of models exhibiting a monotone-likelihood property, a condition particularly useful in instances where establishing the completeness condition poses significant challenges. Using observed data, we demonstrate that the proposed conditions are easy to check for many practical models and outline their usefulness in numerical experiments and real data analysis.


  • Amini-Seresht, E., Zhang, Y., & Li, X. (2019). On asset allocation for a threshold model with dependent returns. European Actuarial Journal9(2), 559–574.
  • Ariyafar, S., Tata, M., Rezapour, M., & Madadi, M. (2020). Comparison of aggregation, minimum and maximum of two risky portfolios with dependent claims. Journal of Multivariate Analysis178, 104620.
  • Balakrishnan, N., & Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: A review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences27(4), 403–443.
  • Barmalzan, G., Najafabadi, A. T. P., & Balakrishnan, N. (2015). Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications. Insurance: Mathematics and Economics61, 235–241.
  • Boonen, T. J., Cheung, K. C., & Zhang, Y. (2021). Bowley reinsurance with asymmetric information on the insurer's risk preferences. Scandinavian Actuarial Journal2021(7), 623–644.
  • Cai, J., & Wei, W. (2014). Some new notions of dependence with applications in optimal allocation problems. Insurance: Mathematics and Economics55, 200–209.
  • Cai, J., & Wei, W. (2015). Notions of multivariate dependence and their applications in optimal portfolio selections with dependent risks. Journal of Multivariate Analysis138, 156–169.
  • Chen, R. (2003). Information economics. Nankai University Press.
  • Chen, Z., & Hu, T. (2008). Asset proportions in optimal portfolios with dependent default risks. Insurance: Mathematics and Economics43(2), 223–226.
  • Cheung, K. C., & Yang, H. (2004). Ordering optimal proportions in the asset allocation problem with dependent default risks. Insurance: Mathematics and Economics35(3), 595–609.
  • Denuit, M., Dhaene, J., Goovaerts, M., & Kaas, R. (2006). Actuarial theory for dependent risks: Measures, orders and models. John Wiley & Sons.
  • Ding, W., Wang, C., & Zhang, Y. (2021). Ordering properties of generalized aggregation with applications. Applied Stochastic Models in Business and Industry37(2), 282–302.
  • Frazzini, A., & Pedersen, L. H. (2014). Betting against beta. Journal of Financial Economics111(1), 1–25.
  • Giovagnoli, A., & Wynn, H. P. (2011). (u, v)-ordering and a duality theorem for risk aversion and Lorenz-type orderings. Preprint, arXiv:1108.1019.
  • Hadar, J., & Seo, T. K. (1988). Asset proportions in optimal portfolios. The Review of Economic Studies55(3), 459–468.
  • Hagen, O. (1979). Towards a positive theory of preferences under risk. Springer.
  • Hennessy, D. A., & Lapan, H. E. (2002). The use of Archimedean copulas to model portfolio allocations. Mathematical Finance12(2), 143–154.
  • Kijima, M., & Ohnishi, M. (1996). Portfolio selection problems via the bivariate characterization of stochastic dominance relations. Mathematical Finance6(3), 237–277.
  • Landsberger, M., & Meilijson, I. (1990). Demand for risky financial assets: A portfolio analysis. Journal of Economic Theory50(1), 204–213.
  • Li, H., & Li, X. (2013). Stochastic orders in reliability and risk. Springer.
  • Li, X., & Li, C. (2016). On allocations to portfolios of assets with statistically dependent potential risk returns. Insurance: Mathematics and Economics68, 178–186.
  • Li, X., & You, Y. (2014). A note on allocation of portfolio shares of random assets with Archimedean copula. Annals of Operations Research212(1), 155–167.
  • Li, X., & You, Y. (2015). Permutation monotone functions of random vectors with applications in financial and actuarial risk management. Advances in Applied Probability47(1), 270–291.
  • Ma, C. (2000). Convex orders for linear combinations of random variables. Journal of Statistical Planning and Inference84(1-2), 11–25.
  • Marshall, A. W., Olkin, I., & Arnold, B. C. (1979). Inequalities: Theory of majorization and its applications (Vol. 143). Springer.
  • Rinott, Y., Scarsini, M., & Yu, Y. (2012). A colonel blotto gladiator game. Mathematics of Operations Research37(4), 574–590.
  • Scholes, M. S. (2000). Crisis and risk management. American Economic Review90(2), 17–21.
  • Shaked, M., & Shanthikumar, G. (2007). Stochastic orders. Springer Science Business Media.
  • Shane, S., & Venkataraman, S. (2000). The promise of entrepreneurship as a field of research. Academy of Management Review25(1), 217–226.
  • Xu, M., & Hu, T. (2012). Stochastic comparisons of capital allocations with applications. Insurance: Mathematics and Economics50(3), 293–298.
  • Yan, R., Zhang, J., & Zhang, Y. (2021). Optimal allocation of relevations in coherent systems. Journal of Applied Probability58(4), 1152–1169.
  • You, Y., & Li, X. (2015). Functional characterizations of bivariate weak SAI with an application. Insurance: Mathematics and Economics64, 225–231.
  • Zhang, J., Yan, R., & Wang, J. (2022). Reliability optimization of parallel-series and series-parallel systems with statistically dependent components. Applied Mathematical Modelling102, 618–639.
  • Zhang, J., Yan, R., & Zhang, Y. (2023a). Reliability analysis of fail-safe systems with heterogeneous and dependent components subject to random shocks. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability237(6), 1073–1087.
  • Zhang, J., Yan, R., & Zhang, Y. (2023b). Stochastic comparisons of largest claim amount from heterogeneous and dependent insurance portfolios. Journal of Computational and Applied Mathematics431, 115265.
  • Zhang, J., & Zhang, Y. (2022). A copula-based approach on optimal allocation of hot standbys in series systems. Naval Research Logistics (NRL)69(6), 902–913.
  • Zhang, J., & Zhang, Y. (2023). Stochastic comparisons of relevation allocation policies in coherent systems. TEST32, 865–907.
  • Zhang, Y. (2022). Stochastic comparisons on total capacity of weighted k-out-of-n systems with heterogeneous components. Statistical Theory and Related Fields6(1), 72–80.
  • Zhang, Y., & Cheung, K. C. (2020). On the increasing convex order of generalized aggregation of dependent random variables. Insurance: Mathematics and Economics92, 61–69.
  • Zhang, Y., Ding, W., & Zhao, P. (2018). On total capacity of k-out-of-n systems with random weights. Naval Research Logistics65(4), 347–359.
  • Zhang, Y., & Zhao, P. (2015). Comparisons on aggregate risks from two sets of heterogeneous portfolios. Insurance: Mathematics and Economics65, 124–135.

To cite this article: Kenji Beppu & Kosuke Morikawa (2024) Verifiable identification condition for nonignorable nonresponse data with categorical instrumental variables, Statistical Theory and Related Fields, 8:1, 40-50, DOI: 10.1080/24754269.2023.2300407

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