Testing for weak instrument: include intercept in regression of instrument? When you want to use the IV (instrumental variable) estimator, you typically first test if you have a strong instrument.
You do so by regressing the (endogenous) predictor against the instrument. With the regression coefficient, you can calculate the F statistic.
My question is: do you have to include an intercept in this regression model? In all the formulas I find, there is no intercept included.
But this would not be meaningful? Below is the regression of my endogenous variable on the instrument, with no intercept, which leads to a wrong conclusion about the regression coefficient?
Thanks in advance.

 A: It is rarely advisable to exclude the intercept unless you have very strong theoretical reasons for doing so. If you perform your first stage regression of the endogenous variable $y_{2}$ on the instrument vector $Z$,
$$y_{2} = \alpha + Z' \beta + \epsilon$$
and you omit the constant $\alpha$, your coefficient estimate for the instruments will be
$$
\begin{align}
E[\beta] &= E[(Z'Z)^{-1}Z'y_{2}] \newline
&=E[(Z'Z)^{-1}Z'(\alpha + \beta Z + \epsilon_i)] \newline
&= E[(Z'Z)^{-1}Z'\alpha + (Z'Z)^{-1}Z'Z\beta + (Z'Z)^{-1}Z'\epsilon] \newline
&= E[(Z'Z)^{-1}Z'\alpha] + \beta
\end{align}
$$
where the term $(Z'Z)^{-1}Z'\epsilon$ vanishes because $E[Z'\epsilon]=0$ by assumption. Hence your $\beta$ will be biased if $\alpha \neq 0$, which is even true if $\alpha$ is not significantly different from zero. In this case your F-test on the excluded instruments will equally be false.
What is mostly the case in books and articles is that if they specify a first stage like
$$y_2 = X'\beta + Z'\pi + \nu$$
it is implicitly assumed that the vector of covariates $X$ includes a constant, i.e. $X = (\alpha, x_1, x_2,..., x_k)$. So long story short: there should be an intercept in both the first and second stage (as well as the $X$ should be the same in both first and second stage, I just omitted them from the bias proof above for simplicity).
