An application of directional quantiles to economic data with a multivariate response
Abstract
Quantile regression represents a popular and useful methodology for modeling quantiles of a response variable based on one or more independent variables. Directional quantiles represent an available extension to the linear regression model with a multivariate response. However, we are not aware of any application of directional quantiles to real data in the literature. An illustration of directional quantiles to an economic dataset is presented in this paper, particularly a modeling of a two-dimensional response in the classical Engel's dataset on household consumption from the 19th century. The results reveal the directional quantiles to yield meaningful results. They order individual observations according to their depth, i.e. from the most central to the most outlying. We compare their result with those of a (more standard) outlier detection. On the whole, we perceive directional quantiles as a potentially useful tool for the analysis of data, if accompanied by a thorough analysis by standard tools.
References
Azadi, S., & Karimi-Jashni, A. (2016). Verifying the performance of artificial neural network and multiple linear regression in predicting the mean seasonal municipal solid waste generation rate: A case study of Fars province, Iran. Waste Management 48, 14-23.
Boček, P., & Šiman, M. (2017a). Directional quantile regression in R. Kybernetika, 53 (3), 480-492.
Boček, P., & Šiman, M. (2017b). On weighted and locally polynomial directional quantile regression. Computational Statistics, 32 (3), 929-946.
Carlier, G., Chernozhukov, V., & Galichon, A. (2016). Vector quantile regression: An optimal transport approach. Annals of Statistics, 44 (3), 1165-1192.
De Andrés J., Landajo M., & Lorca P. (2018). Using Nonlinear Quantile Regression for the Estimation of Software Cost. In: de Cos Juez F. et al. (eds) Hybrid Artificial Intelligent Systems. HAIS 2018. Lecture Notes in Computer Science, 10870. Springer, Cham. 422-432.
Engel, E. (1857). The relations of production and consumption in the Kingdom of Saxony (in German). Zeitschrift des Statistischen Bureaus des Königlich Sächsischen Ministeriums des Inneren, 8, 1-54.
Filzmoser, P., & Gschwandtner, M. (2018). mvoutlier: Multivariate outlier detection based on robust methods. Retreived from https://CRAN.R-project.org/package=mvoutlier.
Galichon, A. (2017). Data file. Retrieved from http://alfredgalichon.com/wp-content/uploads/2017/03/.
Hallin, M., & Šiman, M. (2016). Elliptical multiple-output quantile regression and convex optimization. Statistics and Probability Letters, 109, 232-237.
Haughton, D., & Haughton, J. (2011). Living standards analytics. Development through the lens of household survey data. Springer. New York, NY, USA
Hlubinka, D., & Šiman, M. (2015). On generalized elliptical quantiles in the nonlinear quantile regression setup. Test, 24, 249-264.
Chernozhukov, V., Galichon, A., Hallin, M., & Henry, M. (2017). Monge-Kantorovich depth, quantiles, ranks and signs. Annals of Statistics, 45 (1), 223-256.
Jurečková, J., Picek, J., & Schindler, M. (2019). Robust statistical methods with R. 2nd edn. CRC Press. Boca Raton, FL, USA
Kalina, J. (2013). Highly robust methods in data mining. Serbian Journal of Management 8 (1), 9-24.
Kalina, J. (2014). On robust information extraction from high-dimensional data. Serbian Journal of Management, 9 (1), 131-144.
Kalina, J., & Schlenker, A. (2015). A robust supervised variable selection for noisy high-dimensional data. BioMed Research International, 2015, 320385.
Kalina, J., & Tichavský, J. (2020). On robust estimation of error variance in (highly) robust regression. Measurement Science Review, 20 (1), 6-14.
Koenker, R. (2005). Quantile regression. Cambridge, UK: Cambridge University Press.
Koenker, R., Chernozhukov, V., He, X., & Peng, L. (2017). Handbook of quantile regression. Chapman & Hall/CRC. Boca Raton, FL, USA.
Li, L., & Hwang, N.C.R. (2019). Do market participants value earnings management? An analysis using the quantile regression method. Managerial Finance, 45 (1), 103-123.
Liu, R., Parelius, J.M. & Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference. Annals of Statistics, 27 (3), 783-858.
Matallín-Sáez, J.C., Soler-Domínguez, A., & Tortosa-Ausina, E. (2019). Does active management add value? New evidence from a quantile regression approach. Journal of the Operational Research Society, 70 (10), 1734-1751.
Mumtaz, U., Ali, Y., & Petrillo, A. (2018). A linear regression approach to evaluate the green supply chain management impact on industrial organizational performance. Science of the Total Environment, 624, 162-169.
Paindaveine, D., & Šiman, M. (2011). On directional multiple-output quantile regression. Journal of Multivariate Analysis, 102 (2), 193-212.
Parente, P., & Silva, J.S. (2016). Quantile regression with clustered data. Journal of Econometric Methods, 5 (1), 1-15.
R Core Team (2019). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Version 3.5.1 Retrieved from https://www.R-project.org/.
Rahimi, R. (2017). Organizational culture and customer relationship management: A simple linear regression analysis. Journal of Hospitality Marketing & Management, 26 (4), 443-449.
Šiman, M., & Boček, P. (2016). modQR: Multiple-output directional quantile regression. R package version 0.1.1 [https://CRAN.R-project.org/package=modQR].
Thevaraja, M., & Rahman, A. (2020). Assessing robustness of regularized regression models with applications. In J. Xu, S.E. Ahmed, F.L. Cooke, G. Duca (Eds), Proceedings of the Thirteenth International Conference on Management Science and Engineering Management (ICMSEM 2019), Springer, Cham, 401-415.
Troster, V. (2018). Testing for Granger-causality in quantiles. Econometric Reviews, 37 (8), 850-866.
Víšek, J.Á. (2011). Consistency of the least weighted squares under heteroscedasticity. Kybernetika, 47 (2), 179-206.
Yuan, Y., Zhou, X., Man, J., Jiao, H., Jiang, Q., Xu, Q., Kong, S., & Gao, W. (2019). The safety evaluation of management in chemical enterprise with generalized regression neural network. IOP Conference Series Earth and Environmental Science, 295, 042010.
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