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So in Mostly Harmless Econometrics, page 154, they analyse the bias of instrumental variables:

They consider the case of one endogenous variable $x$, multiple instruments $Z$, and $\eta$ is the structural error and $\xi$ is the error of the first-stage.

They arrive at this equation (equation 4.6.19 in the book):

$$ \begin{align} \beta_{2SLS}-\beta &= (x'P_{Z}x)^{-1}(\pi'Z'+\xi')P_{Z} \eta \\ &=((x'P_{Z}x)^{-1})\pi'Z' \eta + (x'P_{Z}x)^{-1}\xi'P_{Z}\eta. \end{align} $$

For a consistency argument, I guess, one needs to be able to argue that both terms on the right hand side are zero.

They then argue that bias can be studied by looking at

$$ \mathbb{E}[\hat{\beta}_{2SLS}-\beta]\approx(\mathbb{E}(x'P_{Z}x))^{-1}\mathbb{E}[\pi'Z' \eta] + (\mathbb{E}(x'P_{Z}x))^{-1} \mathbb{E}[\xi'P_{Z}\eta]. $$

They then say that $\mathbb{E}[\xi'P_{Z}\eta]$ is non-zero, but $\xi'P_{Z}\eta \rightarrow 0$. I have trouble seeing this.

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    $\begingroup$ A few questions regarding your (their?) notation: are $P_z$ and $P_Z$ different? Also, over which (error) distribution is $\mathbb{E}$ taken over, $\eta$ or $\xi$ (or both)? And finally: in your first equation, second line, why did the $P_z$ in the first expanded term vanish? $\endgroup$
    – Durden
    Jan 8 at 15:26
  • $\begingroup$ hey - so $P_{Z}=P_{z}$ - this is the projection matrix from the projection of $x$ onto $Z$. So I think the expectation is with respect to both - the random element here should be $[\eta, \xi \]$. Regarding your last question: I think $Z'P_{Z}=Z'$ from the properties of the projection matrix (projection onto itself). $\endgroup$
    – clog14
    Jan 8 at 20:04

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It will not be the first time that an expression that converges in probability to zero, has non-zero expected value in finite samples.

$$\xi'P_Z\eta = \xi'Z(Z'Z)^{-1}Z'\eta = \xi'Z\left(\frac 1 n Z'Z\right)^{-1}\left(\frac 1n Z'\eta\right). $$

Namely, we can decompose the initial vector into sub-components which, asymptotically, will converge unilaterally to constants. As $n\to \infty$, the $\eta$ vector is "stochastically separated" from the $\xi$ vector (with which it is correlated), and what remains is its relation with the instruments matrix $Z$, to which it is orthogonal, $$\left(\frac 1n Z'\eta\right) \to_p 0,$$

and we don't care that to the left we have $\xi$, a wall has been erected that does not allow the correlation between $\xi$ and $\eta$ to pass through and materialize. Hence, we get consistency.

But as regards finite-sample bias, this wall (the unstoppable effect of the sample size going to infinity) is not available: the related expected value cannot be decomposed, exactly because $\xi$ has statistical dependence with $\eta$,

$$\mathbb E \left[\xi'P_Z\eta\right] \neq \mathbb E \left[\xi'\right]\cdot \mathbb E \left[P_Z\eta\right] =0$$

And conditioning doesn't help either. For example, we could write

$$\mathbb E \left[\xi'P_Z\eta\right] = \mathbb E \left\{\xi' \mathbb E\left[P_Z\eta \mid \xi\right] \right\}$$

but we cannot argue that $\mathbb E\left[P_Z\eta \mid \xi\right] =0$.

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