Splitting Instrumental Variable into Discrete Groups I have been reading Mostly Harmless Econometrics and have been won over by the virtues of splitting my instrument into groups. In particular, I have a continuous variable as an instrument representing percent and would like to split it into 3 groups (0-33%, 34-66%, 67-100%) and use dummies to indicate each group, i.e. if the continuous variable is between 0-33% $\text{dummy}_1=1$ and the others 0, if between 34-66% $\text{dummy}_2=1$ etc. If I run 2SLS would I need to drop the first group (0-33%) in the first stage? And if so, what would be the interpretation of the constant in the first stage and reduced form?
Also would I be able to over identification test? What would be the number of over identification restrictions?
Yes split into 3 groups and use a dummy for each one as a separate instrument. In particular a higher percentage represents a higher randomised treatment intensity, so this would be 3 randomised treatment intensities. I was hoping I could include all 3 in the first stage without facing linear dependence.
 A: Say you have your three group dummies $z_1$, $z_2$, and $z_3$. When you run your regressions, you can either keep all dummies and drop the constant or you exclude $z_1$ (or any of the other dummies; in the following suppose we drop $z_1$ if any). The first stages and reduced forms would be
$$
\begin{align}
D &= \beta_1 z_1 \, + \beta_2 z_2 + \beta_3 z_3 + e \\
D &= \beta_1 \quad + \pi_2 z_2 + \pi_3 z_3 + u \\ \newline
Y &= \phi_1 z_1 \, + \phi_2 z_2 + \phi_3 z_3 + \epsilon \\
Y &= \phi_1 \quad + \rho_2 z_2 + \rho_3 z_3 + \eta
\end{align}
$$
where I supress $i$ subscripts for simplicity. Note that $\beta_1$ is the same estimate in both first stages. If you omit $z_1$, this will be your intercept and $\pi_2 = \beta_2 - \beta_1$ will then be the deviation from the reference category (which is $z_1$). The same is true for the reduced form.
If you want a simple (but meaningless) example of this, here is some Stata code to try out:
webuse nlswork
gen d1 = (ttl_exp < 1)
gen d2 = (1 <= ttl_exp & ttl_exp < 5)
gen d3 = (5 <= ttl_exp)
bysort idcode: gen n = _n
keep if n==1
reg ln_wage d1 d2 d3, nocons
reg ln_wage d2 d3
reg collgrad d2 d3
reg collgrad d1 d2 d3, nocons
ivreg ln_wage (collgrad = d2 d3), first

When you discretize the instrument you gain efficiency but there is also a potential bias which gets introduced when some of your instruments end up being weak. This bias-efficiency trade-off should be kept in mind when splitting continuous instruments into group dummies.
Overidentification tests like the Sargan test in this context don't work because having discretized your instrument you already reject the idea of a linear constant effects model which is an underlying assumption of overidentification tests. Also overidentification tests are rarely useful in general due to their low power and the assumption that at least one instrument is valid. Given that your instruments all come from the same data generating process either all your instruments are valid or all of them are invalid but this is something the test can't tell you.
A side note: discretizing your instrument in this way kills the nice interpretation properties of the second stage. If you use the continuous instrument, the coefficient of the endogenous variable in the second stage can be interpreted as the causal effect of the endogenous variable due to a one percentage point increase in the instrument. This interpretation gets lost with multiple instruments because the coefficient of your endogenous variable becomes a weighted average of the IV estimates from each separate instrument.
