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Constrained Regression for Interval-Valued Data


  • Gloria González-Rivera
  • Wei Lin


Current regression models for interval-valued data do not guarantee that the predicted lower bound of the interval is always smaller than its upper bound. We propose a constrained regression model that preserves the natural order of the interval in all instances, either for in-sample fitted intervals or for interval forecasts. Within the framework of interval time series, we specify a general dynamic bivariate system for the upper and lower bounds of the intervals. By imposing the order of the interval bounds into the model, the bivariate probability density function of the errors becomes conditionally truncated. In this context, the ordinary least squares (OLS) estimators of the parameters of the system are inconsistent. Estimation by maximum likelihood is possible but it is computationally burdensome due to the nonlinearity of the estimator when there is truncation. We propose a two-step procedure that combines maximum likelihood and least squares estimation and a modified two-step procedure that combines maximum likelihood and minimum-distance estimation. In both instances, the estimators are consistent. However, when multicollinearity arises in the second step of the estimation, the modified two-step procedure is superior at identifying the model regardless of the severity of the truncation. Monte Carlo simulations show good finite sample properties of the proposed estimators. A comparison with the current methods in the literature shows that our proposed methods are superior by delivering smaller losses and better estimators (no bias and low mean squared errors) than those from competing approaches. We illustrate our approach with the daily interval of low/high SP500 returns and find that truncation is very severe during and after the financial crisis of 2008, so OLS estimates should not be trusted and a modified two-step procedure should be implemented. Supplementary materials for this article are available online.

Suggested Citation

  • Gloria González-Rivera & Wei Lin, 2013. "Constrained Regression for Interval-Valued Data," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(4), pages 473-490, October.
  • Handle: RePEc:taf:jnlbes:v:31:y:2013:i:4:p:473-490
    DOI: 10.1080/07350015.2013.818004

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    References listed on IDEAS

    1. White,Halbert, 1996. "Estimation, Inference and Specification Analysis," Cambridge Books, Cambridge University Press, number 9780521574464, October.
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    Cited by:

    1. Lin, Wei & González-Rivera, Gloria, 2016. "Interval-valued time series models: Estimation based on order statistics exploring the Agriculture Marketing Service data," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 694-711.
    2. Wang, Xun & Zhang, Zhongzhan & Li, Shoumei, 2016. "Set-valued and interval-valued stationary time series," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 208-223.
    3. Wilson Ye Chen & Gareth W. Peters & Richard H. Gerlach & Scott A. Sisson, 2017. "Dynamic Quantile Function Models," Papers 1707.02587,, revised Sep 2017.
    4. Gloria Gonzalez-Rivera & Yun Luo & Esther Ruiz, 2018. "Prediction Regions for Interval-valued Time Series," Working Papers 201817, University of California at Riverside, Department of Economics.
    5. Gloria Gonzalez-Rivera & Wei Lin, 2016. "Extreme Returns and Intensity of Trading," Working Papers 201607, University of California at Riverside, Department of Economics.

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