Constrained Regression for Interval-Valued Data

Abstract

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.

Set-valued and interval-valued stationary time series

“…Blanco-Fernandez et al [5] studied strong consistency and asymptotic distribution of the least square estimate. Gonzalez-Rivera and Lin [14] introduced a constrained condition for the regression models of upper and lower bounds of intervals, which guarantees the nature order of interval in the forecast problem. Beresteanu and Molinari [4] investigated the inference problem for partially observed models via an asymptotic approach; they assumed the observations to be uncertain and proposed an estimation method for the real-valued parameters.…”

Extreme returns and intensity of trading

“…In contrast to the modeling of a classical time series of returns, in which it is very difficult to find any time-dependence, the intervals formed by extreme returns have statistical properties that can be exploited. For instance, in González-Rivera and Lin (2013), the authors estimate a constrained bivariate linear system for the daily lowest/highest returns of the SP500 index and find that there is statistically significant dependence with adjusted R-squared (in-sample) of about 50%.…”

A spatial-type interval-valued median for random intervals

“…Data analysis involving random interval-valued data has already received some attention in the literature. Some of the problems that have been considered are regression analysis for interval-valued data [6][7][8][9][10][11][12][13][14], testing hypotheses with interval-valued data [15,16] (or [11] in a more general situation), clustering interval-valued data [17][18][19][20][21][22], principal component analysis with interval-valued data [5,23], modelling distributions for intervalvalued data [24,25], among others.…”