A test for heteroscedasticity in functional linear models

Abstract

We propose a new test to validate the assumption of homoscedasticity in a functional linear model. We consider a minimum distance measure of heteroscedasticity in functional data, which is zero in the case where the variance is constant and positive otherwise. We derive an explicit form of the measure, propose an estimator for the quantity, and show that an appropriately standardized version of the estimator is asymptotically normally distributed under both the null (homoscedasticity) and alternative hypotheses. We extend this result for residuals from functional linear models and develop a bootstrap diagnostic test for the presence of heteroscedasticity under the postulated model. Moreover, our approach also allows testing for "relevant" deviations from the homoscedastic variance structure and constructing confidence intervals for the proposed measure. We investigate the performance of our method using extensive numerical simulations and a data example.

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“…Despite this issue, multicollinearity does not lead to bias in fit while determining the equation. Where R is high, above 0.80, the two variables are highly correlated and multicollinearity is likely to become an issue (Cameron & Bagchi, 2021). With the heteroscedasticity in the data noted, the Hausman test would be essential in checking the fixed and random effects and it would help decide on the best model to be used (Cameron & Bagchi, 2021).…”

“…Where R is high, above 0.80, the two variables are highly correlated and multicollinearity is likely to become an issue (Cameron & Bagchi, 2021). With the heteroscedasticity in the data noted, the Hausman test would be essential in checking the fixed and random effects and it would help decide on the best model to be used (Cameron & Bagchi, 2021). From the research by Cameron and Bagchi, 95% confidence level is recommended from the analysis of the previous studies in evaluating the significance of the coefficient.…”

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“…Test is used to know the classical assumption existence of heteroscedasticity. If the significant value is above alpha = 5%, then there is no heteroscedasticity problem (Cameron & Bagchi, 2021;Zhao et al, 2021;Sun & Wang, 2022). Tab.…”