On the use of higher order bias approximations for 2SLS and k-class estimators with non-normal disturbances and many instruments

The first and second moment approximations for the k-class of estimators were originally obtained in a general static simultaneous equation model under the assumption that the structural disturbances were i.i.d. and normally distributed. Later, higher-order bias approximations were obtained and were shown to be important especially in highly over identified cases. It is shown that the higher order bias approximation continues to be valid under symmetric, but not necessarily normal, disturbances with an arbitrary degree of kurtosis, but not when the disturbances are asymmetric. A modified higher-order approximation for the bias is then obtained which includes the case of asymmetric disturbances. The effect of asymmetry in the disturbances is explored in the context of a two equation model where it is shown that the bias of 2SLS may be substantially changed when the skewness factor increases. The use of the bias approximation is illustrated using empirical applications relating to the return to schooling, where a model with many instruments is employed, and to higher education wage premia.