Measurement Error

In parametrics, the assumption of fixed regressors is made mainly for mathematical convenience, if the regressors can be considered to be fixed in repeated samples, the desirable properties of the OLS estimator can be derived quite straightforwardly. The essence of this assumption is that, if the regressors are nonstochastic, they are distributed independently of the disturbances. If this assumption is weakened to allow the explanatory variables to be stochastic but to be distributed independently of error term, all the desirable properties of the OLS estimator are maintained; their algebraic derivation is more complicated, however, and their interpretation in some instances must be changed (for example, in this circumstance, βOLS is not, strictly speaking, a linear estimator).

If the regressors are only contemporaneously uncorrelated with with the disturbance vector, the OLS estimator is biased but retains its desirable asymptotic properties at the expense of the small-sample properties of βOLS. If the regressors are contemporaneously correlated with the error term, the OLS estimator is even asymptotically biased.

When there exists contemporaneous correlation between the disturbance and a regressor, alternative estimators with desirable small-sample properties can not in general be found; as a consequence, the search for alternative estimators is conducted on the basis of their asymptotic properties. The most common estimator used in this context is the instrumental variable (IV) estimator.