Now we're going to talk about instrumental variables as another approach for helping to establish cause and effect as a source for a relationship between a X and a Y variable. For instrumental variables, we use these in a situation, again, where we think that there might be some omitted variable W. Which is influencing both our right hand side variable X, and our outcome variable Y. And the instrument, as it's referred to, is a variable that hopefully influences X, and has some effect on X without affecting Y necessarily. So in an instrumental variable analysis, what we seek to do is we essentially conduct an analysis where we look at the effect of the instrument on X. And then we look at the variation in X that is associated with the variation in the instrument. And then we see whether or not that variation in X that's actually attributable to variation in the instrument is systematically associated with changes in Y. So generally when we choose an instrument we try to find something that, for theoretical or empirical reasons, we are sure only affects X. But again, has no direct effect on Y. And then, by again looking at the changes in Y that seem to be associated with the changes in X that are produced by variations in the instrument. And then we're isolating out all of the other things that are going on with X that might be attributable to the influence of W. Then hopefully we get an estimate of the causal relationship between X and Y. That was a bit abstract, so let's think about a real, concrete example. The relationship between schooling and earnings. That's something that a lot of people are interested in, whether or not additional schooling leads to additional earnings. But we're worried that there are other factors, especially in an observational study, that might affect schooling as well as earnings. Perhaps parental characteristics, community characteristics, and so forth. However, there are classic studies that treat season of birth as an instrument that determines the amount of schooling, or influences the amount of schooling that people receive. And then by isolating out the amount of variation in schooling that people receive that is attributable to the season in which they were born. And then comparing people, essentially, according to their season of birth. We're using that as an instrument and then looking at the resulting variation in schooling and its association with earnings. Then they come up with an estimate of the causal influence of schooling on earnings. In a nut shell, we can do this because season of birth, at least in a lot of places, will affect the year in which people start school. Because there are many places where, whether or not you can start school in a particular year will depend on whether or not you've reached a particular age by some cutoff date. Perhaps December 31st or January 1st. And so, this leads to variation across the year, in terms of the calendar year in which people actually start school. However, school leaving dates, that is, the year at which people can quit school, tend to be tied specifically to age. So we get variation between the amount of schooling that people receive according to their season of birth. And then, using season of birth, we conduct an instrumental variable analysis to try to establish the causal relationship between schooling and earnings. There are lots of examples of applications of instrumental variable analysis to try to establish the causal relationship between variables. We just talked about the example of the use of season of birth as an instrument for the amount of schooling that people receive, in trying to measure the relationship between schooling and earnings. Again, very difficult normally, in an observational study. People have used distance to college as an instrument for college attendance to try to measure the relationship between college attendance and earning. The basic idea is that the person's distance to college may affect whether or not they actually go to college by essentially making it easier or more difficult to go to college without having a direct effect on their earnings. Other studies have looked at draft lottery number as a instrument for military service, to look at the relationship between military service and earnings. So obviously, people chose whether or not they want to go and serve in the military, at least in an all-volunteer army. But in earlier eras, whether or not people served in the military was affected by their draft lottery number. This was obviously random and it shouldn't have had any subsequent relationship with earnings. So if we think about the lottery number as a instrument for military service, we can try to estimate the causal relationship between military service and earnings. And then people wanting to understand the relationship between maternal smoking and birth weight have used state cigarette taxes as an instrument for maternal smoking. Because it turns out that people's smoking behavior is highly responsive to cigarette prices. Meanwhile cigarette taxes normally won't have a direct effect on birth weight. So people conduct instrumental variable analyses where again, the instrument for maternal smoking was state cigarette taxes. To try to assess the relationship between maternal smoking and birth weight. Overall, instrumental variable approaches have some strengths and some weaknesses. On the one hand, instrumental variables can be very useful in situations where our right-hand side variable X can't be manipulated. And where natural or quasi-experiments are not available. But there are some weaknesses, or at least challenges. One is that an instrument that we choose may only have a weak association with the X variable. So even though it is associated, the association is a weak one and that may make it harder to detect the effects that we're interested in. The instrument may be associated with the outcome Y in ways that are not anticipated. So whatever we've chosen as a instrument may actually have some association with Y that we didn't even think about. But might be pointed out to us when we present our research. Overall, choices of instruments are often controversial. So, published studies involving instrumental variable studies are often challenged or lead to discussion about whether or not the choice of instrument is an appropriate one. In the sense that really does affect only the X variable of interest without having any direct effects on Y.
