Testing linear independence in linear models with interval-valued data

Fuzzy sets are often used to handle the imprecision/vagueness that affects some characteristics in environmental sciences. A determination coefficient is introduced in order to quantify the degree of relationship between an imprecise response variable and a scalar explanatory predictor in a linear regression problem. An estimator of such coefficient useful to measure the goodness of fit of the model is proposed and its strong consistency is proved. Moreover, a specific linear independence testing procedure is established and both the asymptotic significance level and the power under local alternatives are established. Since the asymptotic results require large samples, a consistent bootstrap approach is developed. The empirical behavior of the suggested methods is illustrated by means of some simulations and real-life examples. Copyright (C) 2010 John Wiley & Sons, Ltd

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“…Thus, an asymptotic approach and a bootstrap one are employed (see, for instance, Gil et al, 2006Gil et al, , 2007.…”

Section: Gh-linear Independence Testmentioning

confidence: 99%

A determination coefficient for a linear regression model with imprecise response

et al. 2010

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“…In this paper, we formulate linear regression models for convex compact random sets in R p by following a set arithmetic approach (other approaches are discussed in Gil et al 2006). In fact, we present here a generalization of the models considered in Gil et al (2001Gil et al ( , 2002Gil et al ( , 2006 and González-Rodríguez et al (2006) for the particular case of interval data in R to the more general case of convex compact sets in R p .…”

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 98%

“…This scenario has been generalized, from different points of view, to the case where the variables take intervals as their values (see, e.g., Diamond 1990;Gil et al 2001Gil et al , 2002Gil et al , 2006Billard and Diday 2003;Lima Neto et al 2004;De Carvalho et al 2004;González-Rodríguez et al 2006). In this paper, we formulate linear regression models for convex compact random sets in R p by following a set arithmetic approach (other approaches are discussed in Gil et al 2006). In fact, we present here a generalization of the models considered in Gil et al (2001Gil et al ( , 2002Gil et al ( , 2006 and González-Rodríguez et al (2006) for the particular case of interval data in R to the more general case of convex compact sets in R p .…”

Section: Introductionmentioning

confidence: 98%

“…On the other hand, Gil et al (2006) has considered the simple linear regression model (that will be analyzed in this paper) for the particular case of interval data, and they have shown that testing for linear independence in the underlying linear model reduces to testing for the covariances of certain real-valued variables which characterize the interval-valued random elements (concretely, the mid and the spread values).…”

Section: Introductionmentioning

confidence: 99%

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Least squares estimation of linear regression models for convex compact random sets

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ADAC

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Simple and multiple linear regression models are considered between variables whose "values" are convex compact random sets in R p , (that is, hypercubes, spheres, and so on). We analyze such models within a set-arithmetic approach. Contrary to what happens for random variables, the least squares optimal solutions for the basic affine transformation model do not produce suitable estimates for the linear regression model. First, we derive least squares estimators for the simple linear regression model and examine them from a theoretical perspective. Moreover, the multiple linear regression model is dealt with and a stepwise algorithm is developed in order to find the estimates in this case. The particular problem of the linear regression with interval-valued data is also considered and illustrated by means of a real-life example.

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Predicting Stock Return Volatility: Can We Benefit from Regression Models for Return Intervals?

Fischer

Blanco-Fernández

Winker

2015

Journal of Forecasting

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We study the performance of recently developed linear regression models for interval data when it comes to forecasting the uncertainty surrounding future stock returns. These interval data models use easy-to-compute daily return intervals during the modeling, estimation and forecasting stage. They have to stand up to comparable point-data models of the well-known capital asset pricing model type-which employ single daily returns based on successive closing prices and might allow for GARCH effects-in a comprehensive out-of-sample forecasting competition. The latter comprises roughly 1000 daily observations on all 30 stocks that constitute the DAX, Germany's main stock index, for a period covering both the calm market phase before and the more turbulent times during the recent financial crisis. The interval data models clearly outperform simple random walk benchmarks as well as the point-data competitors in the great majority of cases. This result does not only hold when one-day-ahead forecasts of the conditional variance are considered, but is even more evident when the focus is on forecasting the width or the exact location of the next day's return interval. Regression models based on interval arithmetic thus prove to be a promising alternative to established point-data volatility forecasting tools.

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Testing linear independence in linear models with interval-valued data

Cited by 58 publications

References 14 publications

A determination coefficient for a linear regression model with imprecise response

A determination coefficient for a linear regression model with imprecise response

Least squares estimation of linear regression models for convex compact random sets

Predicting Stock Return Volatility: Can We Benefit from Regression Models for Return Intervals?

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