# Is exogeneity guaranteed for a lottery-generated instrumental variable?

If using $$z$$ as an instrument for $$x$$, to study the effect $$x$$ has on $$y$$ and given that $$z$$ was indeed generated through a lottery, is it definite that exogeneity of $$z$$ will hold?

I have looked at this as two-stage OLS and I believe that the fact $$z$$ is lottery-generated can satisfy the exogeneity condition, i.e. $$\text{corr}(z_i, \epsilon_i) = 0$$. Is that true?

• do you mean by lottery simple random sampling or some real life program that used some sort of lottery? If some real life program then you should link to it/provide more details on how people were selected/drawn
– 1muflon1
May 10, 2021 at 19:06
• The instrument should only affect y through x. Saying that z is generated by lottery does not in itself seem to garanty that the outcome of the lottery - that is z - do not somehow affect y (even if not through x). May 10, 2021 at 19:12
• @1muflon1 Yes, it was meant to be simple random sampling. May 10, 2021 at 23:21
• @JesperHybel I can see the reasoning, but is it possible to formulate that more rigorously? May 10, 2021 at 23:21

Randomness does not imply exogeneity.

In my opinion, the following example from Deaton: "Randomization in the tropics, and the search for the elusive keys to economic development" is quite illuminating.

Assume that you want to estimate the effect of having a railway-station on poverty. We have data on the poverty rate $$P_c$$ and the presence of a station $$R_c \in \{0,1\}$$ for a large number of cities $$c$$. Assume that the true data generating process is given by: $$P_c = \alpha + \beta_c R_c + \varepsilon_c,$$ Notice here that the slope $$\beta_c$$ is city specific, so the effect of a railroad is heterogeneous across cities.

The coefficient $$\beta_c$$ cannot be estimated, but you might wonder whether it is possible to estimate the mean of $$\beta_c$$ over all cities that decide to build a railroad. $$\overline{\beta} = \mathbb{E}(\beta_c|R_c = 1)$$. Then: $$P_c = \alpha + \overline{\beta} R_c + (\varepsilon_c + (\beta_c - \overline{\beta})R_c) = \alpha + \overline{\beta} R_c + \upsilon_c,$$ where $$\upsilon_c = \varepsilon_c + (\beta_c - \overline{\beta})R_c$$ is the new error. Although $$\mathbb{E}(\upsilon_c) = 0$$, you might be worried that: $$\mathbb{E}(R_c \upsilon_c) \ne 0.$$ Indeed, we have: $$\mathbb{E}(R_c \upsilon_c) = \mathbb{E}(\varepsilon_c R_c = 1) + \mathbb{E}(\beta_c - \overline{\beta}|R_c = 1) \Pr(R_c = 1) = \mathbb{E}(\varepsilon_c R_c).$$ This will only be zero if the decision to build a station is uncorrelated to anything that might otherwise influence the poverty rate in the city. As this is probably not satisfied, we need an instrument.

### A random instrument

Assume that the government randomly allocates cities to development areas, captured by the instrument $$Z_c$$ (equal to 1 if city $$c$$ is a development area and zero otherwise).

If designations to development areas motivate the building of railways, we have that there is positive correlation between $$Z_c$$ and $$R_c$$. Also as $$Z_c$$ is random, we can assume that $$\mathbb{E}(Z_c \varepsilon_c) = 0$$. However: $$\mathbb{E}(Z_c \upsilon_c) = \mathbb{E}(\varepsilon_c Z_c) + \mathbb{E}(\beta_c - \overline{\beta}|Z_c = 1, R_c = 1)\Pr(R_c = 1, Z_c = 1).$$ The first term on the right hand side is zero by the randomness of $$Z_c$$. As such, $$Z_c$$ will be uncorrelated to $$\upsilon_c$$ iff: $$\mathbb{E}(\beta_c - \overline{\beta}|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 1, R_c = 1) - \mathbb{E}(\beta_c|R_c = 1) = 0.$$ This will be the case if the mean effect of building a station over all cities that are a development area equals the mean effect over all cities that build the railroad (whether they are a development area or not).

The last condition can be rewritten as: \begin{align*} &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|R_c = 1),\\ \to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_C = 1, R_c = 1) \Pr(Z_c = 1) + \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0),\\ \to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0),\\ \to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0|R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1)\Pr(Z_c = 0|R_c = 1),\\ \to &\mathbb{E}(\beta_c|Z_c = 1, R_c = 1)\Pr(Z_c = 0, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1) \Pr(Z_c = 0, R_c = 1). \end{align*} This will be the case if either:

1. $$\mathbb{E}(\beta_c|Z_c = 1, R_c = 1) = \mathbb{E}(\beta_c|Z_c = 0, R_c = 1)$$ which means that the mean effect among all cities who are a development area and decided to build a railroad should be equal to the mean effect among those who build a railroad and are not a development area. So if the behaviour (i.e. the decision to build a station) changes due to being designated as a development area, this condition will not be satisfied.
2. Or $$Pr(Z_c = 0, R_c = 1) = 0$$ which means that no city that is not a development area builds a station.

In short: randomness of the instrument makes sure that $$\mathbb{E}(\varepsilon_c Z_c) = 0$$ but does not guarantee that $$\mathbb{E}(\upsilon_c Z_c) = 0$$.