A Quantitative Measure of Identifiability

The identifiability problem is studied through the establishment of upper and lower bounds for identification error for finite observation samples. Two cases are considered. In the first case, the parameter set is assumed to be finite and in the second case, the parameter set is assumed to be a metric space. An upper bound for the maximum likelihood estimation method and a lower bound for the optimum estimation method are established for each of the cases. It is shown that the behavior of the upper and lower bounds for both cases are described completely by a resolvability function which describes the degree of resolvability between different parameters in the parameter set. By investigating the asymptotic behavior of this function, one can deduce conditions for identifiability. Moreover, the resolvability function provides a quantitative measure of identifiability. An example on a consumption model is used to illustrate the applicability of the theory and point out the importance of identifiability question in analyzing new policy options.