Where Are We At?

Setting the Scene

Non-Experimental Data

IV provides a solution to OVB.

John Snow's (Non) Experiment: Cholera Hits the Town

Let's Go Watch a Movie

May the force be with you. (click here!)

Snow's Detective Work

The cholera package

cholera::snowMap()

cholera

Removal of the Broad Street Pump?

Mapping London's Water Supply

• Water supply came from the River Thames

• Different supply companies had different intake points

• Southwark and Vauxhall water companies took in water beneath a major sewage discharge.

• Lambeth water did not.

Snow's conclusion

• Snow collected the following data:

• And concluded

that if Southwark and Vauxhall water companies had moved their water intakes upstream to where Lambeth water was taking in their supply, roughly 1,000 lives could have been saved.

• For proponents of the miasma theory, this was still not evidence enough, because there were also many factors that led to poor air quality in those areas.

Snow's Model of Cholera Transmission

• Suppose that $c_i$ takes the value 1 if individual $i$ dies of cholera, 0 else.

• Let $w_i = 1$ mean that $i$'s water supply is impure and $w_i = 0$ vice versa. Water purity is assessed with a technology that cannot detect small microbes.

• Collect in $u_i$ all unobservable factors that impact $i$'s likelihood of dying from the disease: whether $i$ is poor, where exactly they reside, whether there is bad air quality in $i$'s surrounding, and other invidivual characteristics which impact the outcome (like genetic setup of $i$).

We can write:

$$c_i = \alpha + \delta w_i + u_i$$

Doing the Simple Thing is always right?

The Simple Thing

• It does not make sense to compare someone who drinks pure water with someone with impure water.

• because all else is not equal: pure water is correlated with being poor, living in bad area, bad air quality and so on - all factors that we encounter in $u_i$.

• This violates the crucial orthogonality assumption for valid OLS estimates, $E[u_i | w_i]=0$ in this context.

• Another way to say this, is that $Cov(w_i, u_i) \neq 0$, implying that $w_i$ is endogenous.

• There are factors in $u_i$ that affect both $w_i$ and $c_i$

Snow's Model and Some Algebra

Remember our simple model: $$c_i = \alpha + \delta w_i + u_i$$ Now let's condition on both values of $w$: $$\begin{align} E[c_i | w_i = 1] &= \alpha + \delta + E[u_i | w_i = 1] \\ E[c_i | w_i = 0] &= \alpha + \phantom{\delta} + E[u_i | w_i = 0] \end{align}$$

Now substract one line from the other:

$$\begin{equation} E[c_i | w_i = 1] - E[c_i | w_i = 0] = \delta + \left\{ E[u_i | w_i = 1] - E[u_i | w_i = 0]\right\} \end{equation}$$

• The last term $\left\{ E[u_i | w_i = 1] - E[u_i | w_i = 0]\right\}$ is not equal to zero (by what Deaton said!)

• A regression estimate for $\delta$ would be biased by that quantity.

John Snow Says

[...] the mixing of the supply is of the most intimate kind. The pipes of each Company go down all the streets, and into nearly all the courts and alleys. [...] The experiment, too, is on the grandest scale. No fewer than three hundred thousand people of both sexes, of every age and occupation, and of every rank and station, from gentlefolks down to the very poor, were divided into two groups without their choice, and in most cases, without their knowledge; one group supplied with water containing the sewage of London, and amongst it, whatever might have come from the cholera patients, the other group having water quite free from such impurity.

London Water Supply

Proposing an IV

• Snow is proposing an instrumental variable $z_i$, the identity of the water supplying company to household $i$:

More formally, let's define the instrument as follows:

$$\begin{align*} z_i &= \begin{cases} 1 & \text{if water supplied by Lambeth} \\ 0 & \text{if water supplied by Southwark or Vauxhall.} \\ \end{cases} \\ \end{align*}$$

• $z_i$ is highly correlated with the water purity $w_i$.

• However, it seems to be uncorrelated with all the other factors in $u_i$, which worried us before: Water supply was decided years before, and now houses on the same street have different suppliers!

Simple IV in a DAG

• $u$ affects both outcome and explanatory variable

Defining Snow's IV Formally

Here are the conditions for a valid instrument:

1. Relevance or First Stage: Water purity is indeed a function of supplier identity. We want that $$E[w_i | z_i = 1] \neq E[w_i | z_i = 0]$$ i.e. the average water purity differs across suppliers. We can verify this condition with observational data. We want this effect to be reliably causal.

2. Independence: Whether a household has $z_i = 1$ or $z_i = 0$ is unrelated to $u$, hence as good as random. Whether we condition $u$ on certain values of $z$ does not change the result - we want $$E[u_i | z_i = 1] = E[u_i | z_i = 0].$$

3. Excludability the instrument should affect the outcome $c$ only through the specified channel (i.e. via water purity $w$), and nothing else.

Defining the IV Estimator

We are now ready to define a simple IV estimator. Like before, let's condition on the values of $z$:

$$\begin{align} E[c_i | z_i = 1] &= \alpha + \delta E[w_i | z_i = 1] + E[u_i | z_i = 1] \\ E[c_i | z_i = 0] &= \alpha + \delta E[w_i | z_i = 0] + E[u_i | z_i = 0] \end{align}$$

which upon differencing both lines gives

$$\begin{align} E[c_i | z_i = 1] - E[c_i | z_i = 0] &= \delta \left\{ E[w_i | z_i = 1] - E[w_i | z_i = 0]\right\} \\ &+ \underbrace{\left\{ E[u_i | z_i = 1] - E[u_i | z_i = 0] \right\}}_{=0 \text{ by Exogeneity Assumption}} \end{align}$$

• Finally, if the IV is relevant, i.e. $E[w_i | z_i = 1] - E[w_i | z_i = 0] \neq 0$:

$$\begin{equation} \delta = \frac{E[c_i | z_i = 1] - E[c_i | z_i = 0]}{E[w_i | z_i = 1] - E[w_i | z_i = 0]} (\#eq:IV) \end{equation}$$

Special Case: Wald Estimator

Let's say that $x \mapsto y$ means that $x$ is an estimate for $y$:

1. $\overline{c}_1 \mapsto E[c_i | z_i = 1]$: the proportion of households supplied by Lambeth with cholera.
2. $\overline{w}_1 \mapsto E[w_i | z_i = 1]$: the proportion of households supplied by Lambeth with bad water.
3. $\overline{c}_0 \mapsto E[c_i | z_i = 0]$: the proportion of households not supplied by Lambeth with cholera.
4. $\overline{w}_0 \mapsto E[w_i | z_i = 0]$: the proportion of households not supplied by Lambeth with bad water.

The estimator would then be

$$\begin{equation} \hat{\delta} = \frac{\overline{c}_1 - \overline{c}_0}{\overline{w}_1 - \overline{w}_0} \end{equation}$$

In this special case where all involved variables $c,w,z$ are binary, the estimator is called the Wald estimator.

Summary: IVs are a powerful tool to establish causality in contexts with observational data only and where we are concerned that the conditional mean assumption $E[u_i | x_i]=0$ is violated, hence, we cannot say all else equal, as $x$ changes, $y$ changes like this and that. Then we say that $x$ is endogenous. The key features of IV $z$ are that

1. $z$ is relevant for $x$. For example, in a simple regression of $z$ on $x$, we want $z$ to have considerable predictive power. We can test this condition in data.
2. We need a theory according to which is reasonable to assume that $z$ is unrelated to other unobservable factors that might impact the outcome. Hence, $z$ is exogenous to $u$, or $E[u | z] = 0$. This is an assumption (i.e. we can not test this with data).