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I see in a lot of instrumental variables papers that authors will often discuss first stage $R^2$ values or $F$ statistics to assuage concerns that they are working with a weak instrument. This seems to me to be potentially misleading as the high $R^2$ or $F$ statistics could just be an indicator of the predictive power of the control variables. Is this concern valid? If not, why not? If so, what is a more principled way to demonstrate that weak instruments is not a problem?

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The concern is still valid, since the $F$ that you refer to is more than just "an indicator of the predictive power of the control variables."

In fact, consider the standard model with many instruments $\mathbf{Z}$ and homoskedastic $\nu$. Then the first stage is

$$ \begin{align} \mathbf{x}&=\mathbf{Z}\Pi+\nu \end{align} $$ and you can show that the bias of the 2SLS estimator is $$ \begin{align} \mathbb{E}\left(\hat{\beta}_{2sls}-\beta\right)&\approx\frac{\sigma^{2}_{\nu,\varepsilon}}{\sigma^2_{\nu}}\left(\frac{1}{F+1}\right) \end{align} $$ where $F$ is the population F-statistic for the joint significance of all regressors in the first stage regression, i.e. $$ F=\frac{\mathbb{E}\left(\Pi'\mathbf{Z}'\mathbf{Z}\Pi\right)/q}{\sigma^2_{\nu}} $$ The $2SLS$ estimator is biased in the presence of weak instruments in small samples, but this problem is magnified as you add more weak instruments. Consider the case when all these new instruments are irrelevant, then their coefficients in $\Pi$ will be zero, leaving $\mathbf{Z}\Pi$ unchanged. Since the first stage will be the same as before, $\sigma^2_{\nu}$ will be the same as well. However, now $q$ is larger, which implies that the $F$ statistic is smaller, and the bias will become worse.

So, to answer your questions, I would say the following:

  1. The importance of $F$ relies on its close connection to the bias of the $2SLS$. It is important because it helps us to detect weak instruments.

  2. The rule of thumb is: $F<10$ suggests the presence of weak instruments.

  3. However, this rule and this example is for the case of 1 endogenous variable.

  4. Quite fundamentally, and something you can use when arguing about its relevance, this test for weak IV only works when $\nu$ is assumed to be homoskedastic, which is a strong assumption.

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