Continuous Instrumental Variable?

Question

In the classical Instrumental Variable model:

$Y = \beta X + U \, $

an instrumental, binary variable ($Z$) is used to correct for confoundness between $Y$ and $U$ and must satisfy independence wrt the error term ($cov(Z,U$) = 0) while being correlated to the regressors ($r(X,Y) \neq 0$):

$(Z \perp\!\!\!\perp Y_x)\qquad (Z \not\!\!{\perp\!\!\!\perp} X)$

Now suppose we don't have access to such a binary variable but to a continuous one instead (that still satisfy independence to $Y$) and follows a known distribution.

In that case is there (and what would be) a method to estimate the difference in outcome conditioned on the variable of interest, similarly to the binary IV case: $E[Y|X=1] - E[Y|X=0]$

I guess it would be a kind of regression model with $Z$ explaining $\beta$.

My intuition is that it resembles the "what-if" studies one can do with Importance Sampling.

Answer 1

For the binary case (both treatment and instrument) estimating the local average treatment effect (LATE) is straightforward, and you can estimate it as $$E(Y_{i1} - Y_{i0}|D_{i0}=0, D_{i1}=1) = \frac{E[Y_i|Z_i=1] - E[Y_i|Z_i=0]}{P[D_i=1|Z_i = 1] - P[D_i=1|Z_i = 0]} $$

So how does this compare to the multivalued instrument case: first of all, the conditions for identification of a LATE are very similar to the binary case. One additional requirement is strict monotonicity. Suppose your $Z_i$ has a finite support and takes values from $0,...,J$, and you have a binary, endogenous treatment $D_i$, then the requirement on the first stage is $$P(D_i = 1|Z_i = j) > P(D_i = 1|Z_i = j-1)$$ so the higher your value of the instrument the higher is the probability that you get treated.

Also suppose that individuals with the lowest value of the instrument have $D_{i0}=0$ and conversely those with the highest value have $D_{iJ} = 1$. What your instrumental variables estimator will give you in this case is a weighted average of Wald ratios, $$E(Y_{i1} - Y_{i0}|D_{i0}=0,D_{iJ}=1) = \sum^J_{j=1}\mu_j \cdot \text{wald}_{j,j-1} $$ where $$\text{wald}_{j,j-1} = \frac{E[Y_i|Z_i = j] - E[Y_i|Z_i = j-1]}{P[D_i = 1|Z_i = j] - P[D_i = 1|Z_i = j-1]}$$ and $$\mu_j = \frac{P[D_i = 1|Z_i = j] - P[D_i = 1|Z_i = j-1]}{\sum^J_{j=1}P[D_i = 1|Z_i = j] - P[D_i = 1|Z_i = j-1]} $$ are the weights which sum to one.

So you do lots of pairwise comparisons between the $J$ subgroups of individuals where you always compare group $j$ with group $j-1$, which is why the above stated monotonicity condition is needed. The proof for all of this is rather lengthy and annoying so I would like to avoid it but from the statement you already see why multivalued instruments are not necessarily well liked because they are hard to interpret. That's because the average treatment effect you estimate here is the average of treatment effects in each of the $J$ subgroups of compliers.

A critical discussion of the LATE framework in general is given by Deaton (2009) and Heckman and Urzua (2009) with a response by Guido Imbens [link]. Another discussion regards whether discretizing even highly continuous instruments rather than estimating a weighted average of Wald ratios is better (in the sense of being less biased) but I haven't seen any paper which would settle this debate. Nonetheless I hope this helps to clear up for you what you are getting into when you use multivalued instruments in the LATE framework.

Answer 2

It is not true that instrumental variables are "usually binary".

The reason why many applications use binary variables is (1) that good instruments are usually scarce, and while a continuous would often be stronger, people have to go with what's available; (2) that binary randomized treatments in experimental designs naturally provide binary instruments.

There are however prominent examples for non-binary instruments

• Angrist & Krueger (1991) use the quarter of birth as an instrument (which, addmittedly is still only a collection of several binary varialbes, but any variable could be represented as such) for schooling.
• Acemoglu, Johnson & Robinson (2000) use settler mortality rates as instruments for different colonization policies in colonies.
• The Areano-Bond estimator uses lagged values of the dependent variable as instruments.
• Many other examples exists

To receive an answer to the remainder of your question I guess you need to clarify what you need, as the Wikipidia-article on this subject is rather detailed and not specific to the case of a binary instrument .

Answer 3

Below I try to shortly explain a simple version of linear IV with a continuous treatment. Details are in the references at the end of my answer.

Let $Y=g(X,U)$ (outcome) and $X=h(Z,V)$ (treatment) such that $Z$ (instrument) is independent of $(U,V)$ (statistical errors).

If we assume that $X$ is non-negative then $$Y=g(0, U)+\int_0^{\infty}g'_1(x,U)\textbf{1}(x\leq X)\,dx$$ so that $$\mathbb{C}(Z,Y)=\int_0^{\infty}\mathbb{E}(g'_1(x,U)\omega(x))\,dx$$ where $\omega(x):=\mathbb{E}(\textbf{1}(x\leq X)(Z-\mathbb{E}(Z))|V).$ Note that the weights $\omega(x)$ are positive if $X=h(Z,V)$ is increasing in $Z$ given $V$ (by, e.g., results for truncated distributions).

Now, similarly $$\mathbb{C}(Z,X)=\int_0^{\infty}\mathbb{E}(\omega(x))\,dx.$$ Hence, linear IV gives $$\frac{\mathbb{C}(Z,Y)}{\mathbb{C}(Z,X)}=\int_0^{\infty}\mathbb{E}(g'_1(x,U)\overline{\omega}(x))\,dx$$ if the denominator is non-zero, where $\overline{\omega}(x):=\frac{\omega(x)}{\int_0^{\infty}\mathbb{E}(\omega(x))\,dx}$.

Thus, linear IV gives a weighted average of causal marginal treatment effects $g'_1(x,U)$ such that most weight is given to observations where the CDF of the treatment is shifted most sharply by the instrument.

This result is very much similar to Yitzhaki's Theorem (see Heckman et al. 2006, 429-430).

REFERENCES

Angrist, J. D., Graddy, K., & Imbens, G. W. (2000). The interpretation of instrumental variables estimators in simultaneous equations models with an application to the demand for fish. The Review of Economic Studies, 67(3), 499-527.

Heckman, J. J., Urzua, S., & Vytlacil, E. (2006). Understanding instrumental variables in models with essential heterogeneity. The Review of Economics and Statistics, 88(3), 389-432.

Angrist, J. D. Jörn-Steffen Pischke. 2008. Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press.