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In the Present Book Chapter-I is an introductory one. Chapter - II deals the various types of residuals discussed in the literature for the linear regression analysis. Different types of residuals such as OLS, BLUS, Recursive, Internally and Externally studentized, predicted, and weighted residuals have been explained with their properties. Chapter - III presents some new applications of residuals in linear regression models under the problem of heteroscedasticity.Chapter - IV proposes some criteria for testing the equality between sets of regression coefficients in two linear models under two different specifications of error variance using studentized residuals. Chapter - V depicts the conclusion of the present research study. Various selected references regarding present study have been documented under a separate title "BIBLIOGRAPHY".