Influence in Testing Whether an Observation Follows a Given Regression

In econometric field, a question often arises whether the consumption pattern at a later time is the same as the one found before. This question is statistically answered by testing whether an additional observation belongs to a regression estimated by a group of observations obtained before. A diagnostic method of investigating the influence of observations in this testing problem is suggested. A perturbed test statistic for an appropriately defined perturbation scheme is computed, and then this and the perturbation vector form a surface in the Euclidean space as the perturbation vector varies over a certain region. The maximum direction vector associated with the largest curvature of the curve on this perturbation-formed surface at the null point provides information about influential observations for this testing problem. Some necessary computations are provided for finding curvatures and direction vectors. A numerical example is given for illustration.