## Sunday, March 23, 2014

### Instrumental Variables

 A graph of an instrumental variable (Z).
If there is one idea that econometricians are rightly proud, it is instrumental variables.  The econometrician, Guido Imbens, has a new NBER working paper, Instrumental Variables: An econometrican's perspective, in which he endeavors to explain instrumental variables to a broader statistical audience.  Note that the paper is technical and not aimed at the lay reader.

Consider the graph to the right.  We are interested in measuring the casual relationship between chemotherapy (X) and survival (Y) (in the graph it is the blue line from X to Y).  That causal relationship may be mediated by some unobserved characteristic of the patient (U) (the red line from U to Y).  It may be that for some patients the drug increases survival while for other patients the drug decreases survival.  The observed relationship between X and Y may also be confounded by U (the red line from U to X).  In observational data we may see some patients (or their doctors) choosing a particular chemotherapy treatment, while other patients who look similar choosing not to have that chemotherapy.  If patient/doctor choice is based on the unobserved characteristic (U) then the observed relationship between X and Y may be biased.

To solve the problem of confounding, economists have suggested using instrumental variables.  There may exist a variable (Z) that determines the treatment but is unrelated to the outcome and is also unrelated to the unobserved characteristics of the patient.  Random assignment in randomized control trials can be thought of as an instrument.  In fact, it may be such a powerful instrument that the red line from U to X actually disappears.  Under random assignment there is no confounding (unconfoundedness).  Imbens discusses various examples of instruments that have been used in economics from weather at sea, to distance from patient to hospital, to lottery number in the Vietnam War draft.  Weather at sea may affect the price of fish by affecting the supply of fish in the market but not the demand for fish.  Distance from patients to the hospital may affect the patients willingness to get a particular treatment but not the outcome of the treatment.  Lottery number in the draft may affect a potentially draftees willingness to go to college but not their wages post college (conditional on going to college).

Most instrumental variables do not completely remove the line from U to X.  They do not completely remove the confounding in the data.  Nevertheless, economists have shown that these variables can improve inference with confounded data.  Thus, instrumental variables may allow the researcher to make inference from observational data or RCTs affected by selection-into-study bias (participation bias) or attrition bias or non-compliance bias.