# Continuous Instrumental Variable?

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.

• I don't understand what you mean by "natural extension." What is that intended to be?
– whuber
Aug 21, 2015 at 15:39
• I mean that usually instrumental variables are binary; is it possible to use a similar method if we have a continuous variable Z instead ? Aug 23, 2015 at 10:16
• I am puzzled because the definition of instrumental variable has nothing to do with whether it is binary or not.
– whuber
Aug 23, 2015 at 22:14
• yes, you're right as sheB answer shows - it's just that I've only exposed to the binary variant. Aug 25, 2015 at 8:49

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.

• The equations you provide in the multivalued case are a good step; I guess I can bucketize a continuous variable to fall back to this case; I guess I could also replace the sums by continuous integrals to handle continuous instruments, keeping the monotonicity constraint ? Aug 25, 2015 at 8:53
• Also, something that is not clear to me is the relationship between these equations and the two step least squares (2SLS) method - which seems intuitive but disconnected from those Aug 25, 2015 at 8:56
• Well, the question mentioned nothing about 2SLS. With this information before I would have written the answer differently :-) the answer is essentially the same just that the weights are different. Also 2SLS captures a weighted average of Wald ratios. However, this average is not equal to the average effect of the treatment among the whole population of compliers because it counts some groups more often than others. The 2SLS LATE is then only the same as the IV LATE if the treatment effect is the same among all subgroups of compliers.
– Andy
Aug 25, 2015 at 9:42
• Your idea regarding the use of integrals is technically correct but I don't see its use in practice. No variable is ever really continuous.
– Andy
Aug 25, 2015 at 9:44
• If you prefer a reference you can find all these concepts in Angrist and Pischke (2009) "Mostly Harmless Econometrics".
– Andy
Aug 25, 2015 at 10:29

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 .

• Thanks for mentioning these works; Andy's answer made it quite clear how to extend the method to multivalued cases. Aug 25, 2015 at 8:57

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.