Conditional versus unconditional analysis in some regression models

This paper discusses extensions of the variability of the parameters (or functions of parameters) in a recursive system of regression models, and shows that conditioning on the carriers may lead to drastically different conclusions than when the carriers are viewed as stochastic. The relationships among the variables in these models are derived by a sequence of regressions, in which the dependent variable of one equation may reappear as a carrier in a later equation. The model to be fitted need not be identical with the generating equations. In these recursive systems of equations, when the models are miss-specified, or when functions of parameters from different equations are to be estimated, the variability of the estimators is shown to depend critically on the level of conditioning assumed. Various jackknife and bootstrap methods of estimating the variability of the estimators are suggested. In particular the bootstrap estimators of variability can be adopted to captured the correct level of conditioning, by mimicking the conditioning in their design. Two problems in which the level of conditioning matters are described and analysed under the general chained regression models. A real data problem. Omission of variables is sometimes advocated for reducting the variance of the remaining estimators. In both cases the effectiveness of the nonparametric variance estimators is demonstrated using simulation studies.