In estimating parameters of the Cobb-Douglas production function, assuming competition and profit maximization, the estimator to be employed depends on the specification of the behavior of the disturbance term in the production function. If this disturbance term is not transmitted to inputs, that is, if inputs are independent of this disturbance, then the least squares estimator is consistent; if the disturbance is fully transmitted to inputs, then a consistent estimator is obtained if some restrictions are imposed on the second moments of the disturbances in the system. A more general case may be specified, however, encompassing the above specifications as subcases. In this general case, the disturbance term may be only partially transmitted. If this occurs, then neither of the estimators noted above are consistent. In fairly general situations, these estimators furnish upper and lower bounds for the production function elasticity (in a one-input case) or for the sum of the elasticities (in the Q input case; Q any number). The consequences of each of these specifications, in terms of probability limits, are examined in some detail. This is carried out, first, for the one input case, and then the Q input case is discussed.