Interval Least Squares - A Diagnostic Tool.

Linear least-squares models are often used to describe how an endogenous variable depends on some exogenous variables and to make forecasts based on that description. In using such models, one should consider various sources of uncertainty in the parameter estimates and forecasts. Often there are uncertainties in the exogenous variables. When these uncertainties are confined to small intervals and symmetrically distributed, interval least-squares estimates of the model's parameters can furnish either bounds on the component of parameter estimation or forecast error contributed by errors in the exogenous variables, along with an assurance that there is little bias, or else a warning that linear least-squares estimates and forecasts may be subject to significant bias. Indeed, one byproduct is an index of nonlinearity that can warn of possible bias; a similar measure is available from singular value analysis. On problems where bias is insignificant, interval least-squares solutions provide a more detailed collinearity diagnostic than the scaled condition number espoused in the book of Belsley, Kuh, and Welsch.