This video is on weak instruments. By the end of the video you should be able to understand how to measure the strength of an instrumental variable and also you should be a lot understand why weak instruments cause problems. So the strength of an instrumental variable is basically how well it predicts treatment received. So remember, the instrumental variable we can think of as encouragement, some encouragement to receive treatment. And so if it's a strong instrument, then this encouragement works well and most people who are encouraged will actually take the treatment. So a strong instrument is highly predictive of treatment which means encouragement greatly increases the probability of treatment. A weak instrument is, on the other hand, is weakly predictive of treatment. So an encouragement might increase the probability of treatment a little bit but not very much. So we can actually quantify this so we can measure how strong the instrumental variable is and a way to do that is essentially to estimate the proportion of compliers. So this is something that we can identify from observed data, so we can simply remember that the proportion of compliers is just simply the probability of receiving treatment given that you were encouraged, so that's this part here. Expect the value of A|Z=1 so that is a probability that A=1|D=1 which is the probability of receiving treatment given that you were encouraged minus the probability of receiving treatment given that you weren't encouraged so that difference is the proportion of compliers. So these are just, you can estimate those from the sample means in your data. So the proportion of compliers is estimated as the observed proportion of treated subjects for Z=1 minus the observed proportions of treated subjects for Z=0. So it's a contrast of two proportions, and then we'll get an actual number, so we'll be able to estimate what proportion of this population have compliers and that will tell us then how strong our instrument is. So if you, if this is, if this number is close to one it's a strong instrument and if it's close to zero it's a weak instrument. So if you have a weak instrument this can cause problems. So as an example, I'm imagining that only one percent of the population are compliers. So this, getting encouraged only bumps up your probability at least of actually receiving treatment by just a very small percentage. So now remember that in instrumental variable analysis we're interested in estimating local treatment effects, so local causal effects or complier average causal effects. So we're sort of focused on the subset of compliers but if the proportion of compliers and the population is very small, then essentially our effective sample size is also very small. So if our original sample size, the sample size of our dataset is n, then one percent of n would be how many people really are contributing information to the causal effect estimate. So you could imagine that if you essentially have a small sample size, a small number of compliers, then we'll end up with noisy estimates of causal effects. So the causal effect estimates will have a very large variance associated with it. So we have these unstable estimates of causal facts and also weak instruments can also lead to bias. But, I think probably the larger issue is that these estimates will be very unstable, so if you, your confidence interval would be very wide if you have a weak instrument. And, you know, so then, if you did have a weak instrument then an IV analysis might not be the best option. So a lot of times the first thing that people do when they have a proposed instrument is to see is it really predicting treatment. So that's, a lot of times, a first step you might take is you have an idea and I think this variable might be a good instrument and then you want to check is it really predictive of treatment. And if it's not, you might want to, you know, consider trying to find a different instrument or doing some other kind of analysis. There's also an area of active research is that, are methods to strengthen instrumental variables, so you might have a instrumental variable that's weak overall but there might be ways to strengthen it. So one example is this methodology called near/far matching and the idea is that you're matching on covariates, so those kinds of things that you think of as confounder. So you're making, by matching you're making them people as similar as can be on these on these covariates but you're also matching in such a way that the instruments as different as possible. So a matched pair would include people who, all their covariates are similar except they differ by a lot on this instrument. And so that's actually similar to what you would see in randomized trials where you should, you know, people should, sort of, have similar covariate distributions but their level of encouragement would differ greatly. So there's this near/far matching approach and there's also other newer statistical methods that are focused on ways to strengthen the instrument. So those are a couple of options. There's one, if you have a weak instrument, you might want to just consider a different kind of analysis. Alternatively you might think about ways to strengthen the instrument.
