Okay, so now we're going to talk about instrumental variables when the treatment variable is continuous, which is a set up you often find in economics. A special case of mediation occurs when the treatment assignment variable Z affects the outcome only through the mediator M, i.e Z does not directly affect the outcome Y. And we formalized that notion before, remember? Now, in a randomized double blinded encouragement study, such an assumption can be reasonable. In randomized studies where subjects know which treatment group they belong to, such an assumption is less likely to hold. For example, if Z denotes assignment to a job training program or a control group and M denotes whether or not the treatment is taken. A subject assigned a treatment who does not take up treatment may feel demoralized. Because he or she didn't take up the treatment and therefore feel more depressed than he or she would feel had she been assigned to the control group, which would get you a direct effect. Now, natural experiments are observational studies where an event that is essentially randomized is used to assign units to a treatment or control group. And the assignment is presumed to affect an intermediate outcome that in turn affects the outcome Y. So in analyses of data from natural experiments, it's often assumed that assignment does not directly affect the outcome. So, as an example, Angrist, Imbens and Rubin used the draft lottery as an instrument to study the effect of service on access civilian mortality. Now under the assumption which is sometimes called the exclusion restriction, we called it no direct effect. Exclusion restriction below that Z affects the response only through the intermediate outcome M. The controlled direct effects and the natural direct effects are 0. I mean, that's fairly straightforward. And it's easy to see that the average indirect effects do not depend upon the decomposition that you use. Remember, in general, we had two different decompositions which led to two different indirect effects. And the average indirect effect is now equal to the total effect, of course, because there is no direct effect. So, total was equal to direct post indirect when there's no direct, total is equal to indirect. Now, if we make the additional assumption, okay, and this is a kind of thing we've been assuming anyways, when we did direct and indirect effects. And under which the total effect was identified, remember. The average indirect effect is clearly identified because it's the total. So, importantly, if the exclusion restriction holds, that is the direct effect is 0. It's not necessary as it was before to make additional assumptions about unconfoundedness of mediator outcome relationships that are often unreasonable to identify this effect. So at least we can get our hands on the indirect effect without making a whole lot of assumptions other than the one above. Okay, and the one above in a randomized experiment presuming that we think that it's irreasonable to define these kinds of potential outcomes, then that's pretty reasonable. However, it's often the case that such indirect effects are not of primary interest. In the examples above, scientific interest focuses on the controlled direct effects, where you're comparing different levels of M and of given Z. So what do those guys do? They capture the direct effect of the mediator on the outcome. But the exclusion restriction, and the condition that there is a non-zero effect on the mediator, don't suffice to identify these effects. So, now we're going to look at some additional conditions under which these effects. Okay, remember, we're interested in the effect of the mediator on the outcome. So, we're going to look at some conditions under which such effects and related effects are identified. Okay, so let's start to formalize this. We'll say that Z is an instrument with respect to the outcome Y if Z has a non-zero effect on the mediator, given X. And here's our randomization assumption. And now we've made this exclusion restriction assumption. So we've got these three assumptions at this point. I want to say that there's a lot of literature and economic sense statistics on instrumental variables, which are often called IVs, and this is pretty active area. Now Angrist, Imbens, and Rubin, in this article I've already talked about, and it's a great article, require also that an instrument satisfies a so-called monotonicity condition, which we'll get to. So what it says is that the value of the mediator outcome under treatment is greater than or equal to the value of the outcome under control. And I also want to say that in econometrics literature, an instrument is often defined using notation for observed outcomes only. And instruments are used in other sort of non-causal contexts as well. So if you're interested in all this stuff, there's a nice textbook by Greene on econometrics that gives a nice introduction. And there is a review by Imbens that covers work at the interface of this subject with causal inference. Often times when Z is the following, since there is no direct effect sometimes they just get rid of the Z and write the potential outcomes as Yim. because it depends on whatever you've set M to, you can set M to or thought about. Okay, so now we're going to consider three prominent cases. Z is binary, like treatment or not, with M and Y continuous. Now both Z and M are binary. And 3, Z is continuous and M is binary. And these are the three cases we're going to really consider. Take some time to do so but we'll take them all up. So I'm going to begin with the first case. And I'm going yo use a linear model for the potential outcomes to illustrate, which that ought to be a pretty familiar model. We just essentially looked at it but of course now we're going to have what? No direct effect of Z, okay, you can see that in the second equation. And we've rewritten Yi only with M, no with Z and M. Otherwise M can take out the interaction and otherwise it's identical to what we saw a couple of lessons ago when we talked about estimating mediated effects. Okay, so the model, as I said, is just a special case in which the exclusion restriction is assumed of the linear model studied in lesson one of the module. And the C superscript is causal model and I'll remove the C when we talk about models for the observed data, okay. All, so here is our model, okay. Now, we've assumed, remember we decided that an instrument ought to have, there should be some effect of treatment on the mediator. So, under our assumptions, we have this beta M, is the effect of the treatment on the mediator, the average effect. And we have this tall C, which is going to be the indirect effect, and it's equal to this product or these coefficient plus this error term. So we'd like, think about structural equations, we'd like it to be just the beta Y times the beta M. In order for that to be the case, we need this expectation of the errors to vanish, okay? All right, now how would we get that to vanish? If we made this very strong constant effect assumption that the errors are all equal for all M. Then the ratio of the total over beta Y, which is a so-called IV estimand would be the average controlled direct effect of M on Y for all X. Now this constant effect assumption is really strong because it basically says that the error doesn't depend upon M at all. Sometimes this is referred to as the assumption of no treatment heterogeneity in economics. Now you can also of course get this result when treatment effect heterogeneity is present just by assuming that the expectation, that annoying expectation, is 0. Now in a paper I wrote many years ago I gave some conditions for this to hold, and there's other conditions in the literature like this. And the conditions are more general. You can also do this in the case where you do have direct effects, okay? But the conditions are weaker than the condition previously studied in the mediation literature that the mediator outcome relationship is unconfounded given the covariates and treatment assignment. That's what we did assume. That's an assumption that we've made for identification mediator effects and I criticize that. But the assumption of constant effect, I showed that in that article, is neither stronger nor weaker than the unconfounded in this assumption above. In related conditions, you can also find in the econometrics literature, there's a nice text by Woodbridge. And economists have also studied IVs in the context of nonlinear semi-parametric and non-parametric models. Much more fancy that what we've done, but I just want to get the basic ideas across. And if you're interested in these sort of things, these sort of nonlinear, semi-parametric non-parametric, you can look at the paper by Imbens that I talked about before. I guess the one thing that I do want to say is that we can weaken those assumptions, but that still I consider the assumption about the errors above. It's about the difference in errors as opposed to the absolute errors, because the Yi, Z, M above can be written, I could write it in terms of errors epsilon iZM. I still consider this to be pretty strong on the conditional and the difference of the errors. It's weaker, well, you can show that it's more or less weaker. But it's still pretty strong. Okay, in the next lesson, I'm going to begin with case two, where the instrument Z and the treatment M are both binary. And as this is the case where Z denotes assignment to a treatment group or a control group. And M often indexes whether or not the assigned treatment was taken or not. Okay, and this is going into be, this is basically if you've heard of the subject of compliance, this is compliance. And it's going to lead us more generally to the subject of principle stratification, okay.