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A PRIMER ON CAUSALITY Marc F. Bellemare∗

Introduction This is the second of two handouts written to help students understand quantitative methods in the social sciences. This handout is dedicated to discussing (some) of the ways in which one can identify causal relationships in the social sciences. In keeping with the notation introduced in the handout on linear regression, let 𝐷𝐷 be our variable of interest; 𝑦𝑦 be an outcome of interest; and the vector 𝑥𝑥 = (𝑥𝑥1, … , 𝑥𝑥𝐾𝐾) represent other factors – or control variables – for which we have data. For the purposes of this discussion, let 𝐷𝐷 measure a given policy, 𝑦𝑦 measure welfare, and the vector 𝑥𝑥 measure the various control variables the researcher has seen fit to include. See my “A Primer on Linear Regression” for a more basic handout.

Mechanics Recall that the regression of 𝑦𝑦 on (𝐷𝐷, 𝑥𝑥1, … , 𝑥𝑥𝐾𝐾) is written as

𝑦𝑦𝑖𝑖 = 𝛼𝛼 + 𝛽𝛽1𝑥𝑥1𝑖𝑖 + ⋯ + 𝛽𝛽𝐾𝐾𝑥𝑥𝐾𝐾𝑖𝑖 + 𝛾𝛾𝐷𝐷𝑖𝑖 + 𝜖𝜖𝑖𝑖, (1)

where i denotes a unit of observation. In the example of wages and education, the unit of observation would be an individual, but units of observations can be individuals, households, plots, firms, villages, communities, countries, etc. Just as the research question should drive the choice of what to measure for 𝑦𝑦, 𝐷𝐷, and 𝑥𝑥, the research question also drives the choice of the relevant unit of observation.

The problem is that unless the researcher runs an experiment in which she randomly assigns the level of 𝐷𝐷 to each unit of observation i, the relationship from 𝐷𝐷 to 𝑦𝑦 will not be causal. That is, 𝛾𝛾 will not truly capture the impact of 𝐷𝐷 on 𝑦𝑦, as it will be “contaminated” by the presence of unobservable factors. Some of those factors can be included in 𝑥𝑥 = (𝑥𝑥1, … , 𝑥𝑥𝐾𝐾), of course, but it is in general impossible to fully control for every relevant factor. This is especially true when unobservable or costly to observe factors (e.g., risk aversion, technical ability, soil quality, etc.) play an important role in determining 𝐷𝐷 and 𝑦𝑦. So even if we get an estimate of 𝛾𝛾 that is statistically significant, we cannot necessarily assume that the relationship between the variable of interest and the outcome variable is causal. In other words, correlation does not imply causation.

For example, suppose 𝐷𝐷 is an individual’s consumption of orange juice and 𝑦𝑦 is (some) indicator of health. We have often discussed in lecture how a simple regression of 𝑦𝑦 to 𝐷𝐷 would provide us with a biased estimate of 𝛾𝛾 because orange juice consumption is nonrandom and not exogenous to health. That is, there are factors other than orange juice consumption which determine health. Some are observable (e.g., how much someone exercises; whether they smoke; their diet; etc.), but several are unobservable (e.g., their willingness to pay for orange juice; their subjective valuation of health; their level of risk aversion; their genes; etc.) Thus, it really isn’t sufficient to run a kitchen-sink regression (i.e., a regression in which everything observable is thrown in as a control) to properly identify the causal impact of 𝐷𝐷 on 𝑦𝑦.

∗ Associate Professor, Department of Applied Economics, and Director, Center for International Food and Agricultural Policy, University of Minnesota, 1994 Buford Ave, Saint Paul, MN 55113, [email protected]. This is the August 2017 version of this handout.

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Identification So how do we identify causality? The best way to do so is to run a randomized controlled trial (RCT), which we have discussed in lecture. In this case, the idea would be to get a random sample of individuals of size 𝑁𝑁 and to assign half of the sample (i.e., 𝑁𝑁 2⁄ ) to a control group and half to a treatment group. The latter group would be told to consume, say, one glass of orange juice every morning, and the other half would be told not to do so. Then, after a suitable period of time, we would compare the mean of 𝑦𝑦 between groups. The null hypothesis would of course be that the mean health of the treatment group is equal to the mean health of the control group. A rejection of the null in favor of finding that the mean health of the treatment group is higher than the mean health of the control group would then be evidence in favor of the hypothesis that orange juice is good for one’s health. More than that – it would be evidence in favor that orange juice consumption causes good health.

The problem is that it is not always possible to run an RCT, and even the simple example described above would be subject to important problems. For example, the individuals in the treatment group may not comply with the experimenters instructions, especially if they don’t like orange juice. More generally, they may simply forget to consume orange juice every morning. Likewise, the individuals in the control group may end up inadvertently consuming orange juice when they are not supposed to. These reasons – and others – would contaminate one’s estimate of 𝛾𝛾 in equation 1 and would invalidate the test of equality of means described above. So what is one to do?

Instrumental Variables Estimation When one only has observational (i.e., nonexperimental) data at one’s disposal, the best way to identify causality is to find an instrumental variable (IV) for the endogenous variable. In the example above, the endogenous variable is 𝐷𝐷, which is said to be endogenous to 𝑦𝑦.

What is an IV? It is a variable 𝑧𝑧 that is (i) correlated with 𝐷𝐷; but (ii) uncorrelated with 𝜖𝜖 and which is used to make 𝐷𝐷 exogenous to 𝑦𝑦. How does an IV exogenize an endogenous variable? By virtue of being correlated with the endogenous variable, yet uncorrelated with the error term, which is the definition of an instrument.

I realize that this sounds tautological, so for example, Angrist (1990) studies the impact of education (𝐷𝐷) on wages (𝑦𝑦). The problem is that education is endogenous to wage, if anything because people acquire education in expectation of the wage they think this will get them. In other words, even if we find a positive coefficient for education in a regression of wage on explanatory variables, this is merely a correlation, and it does not necessarily indicate that education causally affects wages.

To instrument for this, Angrist had to find a variable that would be correlated with how much education someone would get, but uncorrelated with anything unobserved and would affect wage only through how much education they acquire. The instrument he settled upon was an individual’s Vietnam draft lottery number, since this correlates with whether one goes to war and is then subject to the GI Bill, but since those numbers are randomly generated, they are uncorrelated with unobservables.

How does IV estimation work, mechanically speaking? Recall that our equation of interest is

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𝑦𝑦𝑖𝑖 = 𝛼𝛼 + 𝛽𝛽1𝑥𝑥1𝑖𝑖 + ⋯ + 𝛽𝛽𝐾𝐾𝑥𝑥𝐾𝐾𝑖𝑖 + 𝛾𝛾𝐷𝐷𝑖𝑖 + 𝜖𝜖𝑖𝑖. (1)

The way IV estimation proceeds is to first regress the endogenous variable 𝐷𝐷 on the instrument 𝑧𝑧 as well as on the control variables in 𝑥𝑥 = (𝑥𝑥1, … , 𝑥𝑥𝐾𝐾), such that

𝐷𝐷𝑖𝑖 = 𝛿𝛿 + 𝜋𝜋1𝑥𝑥1𝑖𝑖 + ⋯ + 𝜋𝜋𝐾𝐾𝑥𝑥𝐾𝐾𝑖𝑖 + 𝜃𝜃𝑧𝑧𝑖𝑖 + 𝜖𝜖𝑖𝑖. (2)

Once equation 2 is estimated, it is possible to predict the variable 𝐷𝐷, whose prediction we label 𝐷𝐷� (the circumflex accent – or “hat” – denotes a predicted variable in econometrics) and to then estimate equation 1 as follows

𝑦𝑦𝑖𝑖 = 𝛼𝛼 + 𝛽𝛽1𝑥𝑥1𝑖𝑖 + ⋯ + 𝛽𝛽𝐾𝐾𝑥𝑥𝐾𝐾𝑖𝑖 + 𝛾𝛾𝐷𝐷�𝑖𝑖 + 𝜖𝜖𝑖𝑖. (1’)

Note what has been done here: we have replaced the endogenous variable with an exogenized version of the same variable. The way it has been exogenized has been by regressing it on the IV, which is exogenous to the outcome of interest, and to obtain its predicted value, which we then use in lieu of the original endogenous variable.

The first requirement of an instrument – i.e., that it be correlated with 𝐷𝐷 – is easily testable: we only need to check that the coefficient 𝜃𝜃 in equation 2 is significantly different enough from zero. The second requirement of an instrument – i.e., that it only affect the outcome of interest 𝑦𝑦 through the treatment variable – cannot be tested for. Rather, one must make the case that it is truly exogenous to the outcome of interest. This is easier said than done in most cases, as some people have devoted entire careers to finding good IVs.

References Angrist, Joshua D. (1990), “Lifetime Earnings and the Vietnam Era Draft Lottery: Evidence from the Social Security Administrative Records,” American Economic Review 80(3): 313-336.