# Solved – Is E(u|x)=0 is a required condition for estimator consistency

For OLS parameter estimates to be consistent it must be the case that
E(u|x)=0. Is it true?
E(u|x)=0 is a required condition for unbiasedness. But as far as I understand, unbiasedness does not necessarily mean consistency. Therefore I am really confused.

Contents

Ok. The model is, in matrix notation and conformable dimensions \$\$mathbf y = mathbf Xbeta + mathbf u \$\$

The \$OLS\$ estimator is

\$\$hat beta = (mathbf X'mathbf X)^{-1}mathbf X' mathbf y = (mathbf X'mathbf X)^{-1}mathbf X' (mathbf Xbeta + mathbf u) \$\$

\$\$= (mathbf X'mathbf X)^{-1}mathbf X' mathbf Xbeta + (mathbf X'mathbf X)^{-1}mathbf X'mathbf u = beta + (mathbf X'mathbf X)^{-1}mathbf X'mathbf u\$\$

For consistency we examine

\$\$operatorname{plim}hat beta = operatorname{plim}beta + operatorname{plim}left[(mathbf X'mathbf X)^{-1}mathbf X'mathbf uright] = beta + operatorname{plim}left[left(frac 1nmathbf X'mathbf Xright)^{-1}left(frac 1nmathbf X'mathbf uright)right] \$\$

And here is the crucial point that makes us need a weaker assumption for consistency compared to unbiasedness: for unbiasedness we would face \$Eleft[(mathbf X'mathbf X)^{-1}mathbf X'mathbf uright]\$, and in order to "insert" the expected value into the expression we have to condition on \$mathbf X\$, which leads us to the expression \$E(mathbf umid mathbf X)\$ and the need to assume it as being equal to zero, i.e. assume "mean-independence" between the error term and the regressors.

But \$operatorname{plim}\$ is a more "flexible" operator than \$E\$: under \$operatorname{plim}\$ expressions and products can be decomposed (something that under the expected value requires independence), and also \$operatorname{plim}\$ can "go inside the expression" (while \$E\$ cannot except if it is an affine function), as long as the function is a continuous transformation (and it very rarely isn't) – so

\$\$operatorname{plim}left[left(frac 1nmathbf X'mathbf Xright)^{-1}left(frac 1nmathbf X'mathbf uright)right] = operatorname{plim}left(frac 1nmathbf X'mathbf Xright)^{-1}operatorname{plim}left(frac 1nmathbf X'mathbf uright)\$\$

For consistency we need to assume that the first \$operatorname{plim}\$ is finite -but this is an assumption on the properties of the regressor matrix, unrelated to the error term. So we are left with the second \$operatorname{plim}\$ which, written for clarity using sums it is \$\$operatorname{plim}left(frac 1nmathbf X'mathbf uright) = left[begin{matrix} operatorname{plim}frac 1nsum_{i=1}^nx_{1i}u_i \ .\ .\ operatorname{plim}frac 1nsum_{i=1}^nx_{ki}u_i \ end{matrix}right] rightarrowleft[begin{matrix} frac 1nsum_{i=1}^nE(x_{1i}u_i) \ .\ .\ frac 1nsum_{i=1}^nE(x_{ki}u_i) \ end{matrix}right] \$\$ …the last transformation due to the usual assumptions that permit the application of the law of large numbers.

Exactly because we have been able to "separate" \$(mathbf X'mathbf X)^{-1}\$ from \$mathbf X'mathbf u\$ (due to the fact that we are examining the \$operatorname{plim}\$ and not \$E\$) we ended up looking only at the contemporaneous relation between each regressor and the error term. And so what we need to assume for consistency of the \$OLS\$ estimator is only that \$E(x_{1i}u_i) =0 ; forall k, ; forall i\$, (contemporaneous uncorrelatedness) which is much weaker than \$E(mathbf umid mathbf X)\$, the latter requiring mean-independence, and moreover, not only contemporaneous independence, but across time too (since we condition the whole error vector on the whole regressor matrix).

Rate this post