Weak first stage in 2SLS I have a simple IV model with 1D variables:


*

*$N = \alpha_z + \beta_z Z + \epsilon_z$

*$S = \alpha_s + \beta_s N + \epsilon_s$


$N$ is an integer, while $S$ is dummy. 
$Z$ is by construction independent of $N$ and $S$, it is a random, continuous factor that is instrumented in the system. I verified $Z$ is correlated to $N$ : $\rho(Z,N) = .01$ (and its only impact on the system is through $N$). The variance of $Z$ was set so as to meet practical considerations.
I first regressed $N$ using $Z$ and found $\hat{\beta_z} = .28$ Then using $\hat{N}$ I regressed $S$ and found $\hat{\beta_s} = .0045$ (compared to the OLS biased estimate of $.0006$). Impact of a few tenths of percentage points are relevant in my application.
However, I'm concerned that my first stage may be too weak due to the limited impact of $Z$. The $R^2$ of the first stage is $.00036$ which I suspect is too small.
So are there any checks that I could use to qualify the weakness of my first stage ? I read mentions of the $F$ statistic but I don't know what to do in practice.
 A: To test for weak instruments you would test the joint significance of your instruments' coefficients via an F-test. The typical rule of thumb is that an F-statistic of more than 10 is fine (see Stock and Yogo, 2002), however, this is not a theorem and people may still give you a hard time if your test statistic is close to 10.
In case you only have one instrument, this F-statistic is equivalent to the square of the t-statistic of your instrument's coefficient in the first stage.
Should you find that your instrument is weak, i.e. if you get an F-statistic of less than 10 (or close to it), then you can use alternative estimators which are somewhat more robust to this weak instruments problem. One of such estimators is limited information maximum likelihood (LIML), which is programmed in pre-canned packages in most of the available statistics softwares.
CITATION
Stock, James H., Jonathan H. Wright, and Motohiro Yogo. "A survey of weak instruments and weak identification in generalized method of moments." Journal of Business & Economic Statistics 20.4 (2002): 518-529.
A: Since you have only one instrument and one treatment, you can use the Anderson-Rubin (AR) confidence interval by inverting the AR test. 
The AR test is uniformly most powerful unbiased in this setting, and it has correct coverage rates regardless of the strength of the instrument. This review might be useful.
This and other "weak-instrument robust" methods are implemented int he ivmodel R package.
