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does anybody know how I can show that this estimator is typically biased (and consistent)

PS: It is just the standard OLS-estimator. However a constant Matrix $A$ has been added within $X'X$. And sorry for the $^{-1}$ - I did not figure out yet how to do that properly ;)

$ = (X'X + A)^{-1}X'y$

Best, Daniel

nonuser
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daniel von ahlen
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1 Answers1

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I assume we have the usual linear regression model $y= Xb+u$. Then

$$\hat b = (X'X + A)^{-1}X'Xb + (X'X + A)^{-1}X'u$$

Assuming strict exogeneity of regressors (i.e. mean-independence, $E(u\mid X) =0$), we have

$$E(\hat b \mid X) = (X'X + A)^{-1}X'Xb + (X'X + A)^{-1}X'E(u\mid X) = (X'X + A)^{-1}X'Xb$$

$$\implies E(\hat b) = E[E(\hat b \mid X)] = E[(X'X + A)^{-1}X'X]b$$

$$E[(X'X + A)^{-1}X'X] \neq I$$

So it is biased.

For consistency, we examine

$$\text{plim} \hat b = \text{plim} (n^{-1}X'X + n^{-1}A)^{-1}\cdot \text{plim} (n^{-1}X'X)b + \text{plim} (n^{-1}X'X + n^{-1}A)^{-1}\cdot \text{plim}(n^{-1}X'u)$$

Since $A$ is a constant matrix we have that $n^{-1}A \to 0$, and the rest goes as usual.

Alecos Papadopoulos
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