Invariant estimators for market share systems and their finite sample behavior

The sales-marketing mix relationships for brands of a product class are often modelled as a multiple equation system. Whenever sales are expressed as shares, such systems are sum-constrained and therefore singular. Singular systems can be calibrated by deleting one equation from the model and estimating the remaining equations using the seemingly unrelated regressions (SUR) technique. A two-step procedure is typically adopted whenever the disturbance covariance structure is unknown. This study highlights an issue often overlooked by marketing researchers. Traditional two-step estimators that delete one equation at the initial step usually lack invariance to the equation deleted. Invariant estimates are available using maximum likelihood estimation (MLE) or iterated least squares. However, relative to simple, two-step procedures, these approaches require more computation and are often less accurate in small samples. The authors describe several ways to obtain invariant, two-step estimators and conduct a sampling experiment to compare the finite sample behavior of these and other estimators to an iterated solution. The present results show that a ‘balanced’ two-step estimator which deletes no equations initially outperforms the iterated solution over a wide range of conditions. The results also show that a constrained, single-step estimator which deletes no equations from the system frequently outperforms all two-step methods examined. Further, the results of the present study also apply to nonlinear specifications of market share models since the invariance issue arises in this context as well.