A review of the development and application of recursive residuals in linear models

Recursive residuals have been shown to be useful in a variety of applications in linear models. Unlike the more familiar ordinary least squares residuals or studentized residuals, recursive residuals are independent as well as homoscedastic under the model. Their independence is particularly appealing for use in developing test statistics. They are not uniquely defined; their values depend on the order in which they are calculated, although their properties do not. In some applications one can exploit this order dependence, coupled with the fact that they are in clear one-to-one correspondence with the observations for which they are calculated. Uses for recursive residuals have been suggested in almost all areas of regression model validation. Regression diagnostics have been constructed from recursive residuals for detecting serial correlation, heteroscedasticity, functional misspecification, and structural change. Other statistics based on recursive residuals have focused on detection of outliers or observations that are influential or have high leverage. Recent work has explored properties and possible uses of recursive residuals in models with a general covariance matrix, multivariate linear models, and nonlinear models. Computing routines are available for obtaining recursive residuals accurately and efficiently.