Question

Why do the two-stage least squares estimators give biased but consistent estimators?

Answer 1

When the explanatory variables in the regression model are correlated with stochastic disturbance terms,the resulting estimates are biased as well as inconsistent.

In such a situation ,two stage least squares is an estimation procedure.

The basic idea behind 2SLS is to replace the (stochastic) endogenous explanatory variable by a
linear combination of the predetermined variables in the model and use this combination as the explanatory variable in lieu of the original endogenous variable.

In this technique the estimators are biased due to the violation of assumption of Linear model...but as the sample size tends to infinity estimators will be consistent.

The estimates obained are consistent, that is, as the sample size increases indefinitely, estimates converge to their true population values. The estimates may not

satisfy small-sample properties, such as unbiasedness and minimum variance. Therefore, the results obtained by applying these methods to small
samples and the inferences drawn from them should be interpreted with due caution.

The statistical justification of the 2SLS is of the large sample type. When there are no lagged endogenous variables, the 2SLS coefficient estimators are consis-
tent if the exogenous variables are constant in repeated samples and if the disturbance [appearing in the various behavioral or structural equations] . . . are independently and identically distributed with zero means and finite variances. . . .
If these two conditions are satisfied, the sampling distribution of 2SLS coefficient estimators becomes approximately normal for large samples. When the equation system contains lagged endogenous variables, the consistency and large-sample normality of the 2SLS coefficient estimators require additional condition, . that as the sample increases the mean square of the values taken by each lagged endogenous variable converges in probability to a positive limit. If the [disturbances appearing in the various structural equations are] not independently distributed, lagged endogenous variables are not independent of the current operation of the equation system , which means these variables are not really predetermined. If these variables are nevertheless treated as predetermined in the 2SLS procedure, the resulting estimators are not consistent.