Maximum likelihood estimation of market share models with large numbers of shares

Maximum likelihood procedures for estimating market share models break down or produce very unstable estimates when the number of brands is large as compared with the number of observations. The reason behind this phenomenon is that the estimate of the contemporaneous covariance matrix of the error terms of the model becomes singular or almost singular. This problem may be resolved by imposing restrictions on the contemporaneous covariance matrix. The resulting estimation procedure suggests that the model may contain a large number of shares, while the variance of each share may be estimated freely.