# 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.

• Could you make your title more explicit and descriptive? It seems your question pertains to trichotomizing an item/score (splitting into 3 groups). – Patrick Coulombe Jul 11 '14 at 1:41
• Sorry, is this any clearer? 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. – Adam Smith Jul 11 '14 at 1:56

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