Before revision:  We derive the approximate mean squared error (MSE) that depends on the size of the complete subset and characterize the proposed estimator based on the approximate MSE. The size of the complete subset is chosen by minimizing the sample counterpart of the approximate MSE.
After revision:  We derive the approximate mean squared error (MSE) is derived as a function of the subset size k by the Nagar (1959) expansion. The CSA-2SLS estimator is obtained by choosing k minimizing the sample counterpart of the approximate MSE.