Simultaneous inference of the partially linear model with a multivariate unknown function

In this paper, we conduct simultaneous inference of the non-parametric part a partially linear model when component is multivariate unknown function. Based on semi-parametric estimates model, construct confidence region function for inference. The developed methodology applied to perform U.S. gasoline demand where income and price variables are contaminated by Berkson errors. empirical results strongly suggest that linearity rejected. also used propose an alternative form demand.