# 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.

• I have only one instrument; the t-statistic of $\beta_z$ is $9.63$ in my first stage, so I guess I'm ok. I'll read the Stock & Yogo paper to gain more insight on the interpretation of this rule. – oDDsKooL Dec 3 '15 at 10:39