2SLS: Is reverse causality in first stage a problem?

[Michael Schuster]

Dear all,

I may have found a good instrument (male circumcision) for my reverse causality problem in my research project (effect of HIV on economic growth in panel data set from 2000 to 2017). However, since I am doing the first stage right now (HIV on male circumcision), I wonder if it is also a problem if there is reverse causality already in the first stage? The requirement for a valid instrument just says that Z must be correlated with X.

Thanks in advance!

[John Riveros]

Usually, by the mere fact when you assume that your instrument is fully exogenous, it implies that it is not changing when the endogenous regressor does change. Therefore if your model is something like:

x_it = c*z_it + e_it (1st stage)
y_it = B*x_it + u_it (2nd stage)

Where x is the endogenous regressor, and z is your instrument. From this, if you can come up with a good argument of why z is exogenous (that is not changing when x changes). It would be a good instrument. Beware that we use the expression that Cov(z,x) is different from 0 because we want that the actual instrument has some kind of explanatory power towards the endogenous regressor (the other way around would imply that the instrument is not good).

In general, the assumptions we do about the instrument is that 1) Cov(z,e)=0 (that is instrument is fully exogenous to the unobserved, often quite violated in applied research and hard to argue against but people and reviewers ignore it). 2) Cov(z,x) must be different from 0 (implying that instrument has some correlation with the endogenous regressor), and 3) z must impact y only through x.

[Michael Schuster]

Thanks for your reply John!

2) Cov(z,x) must be different from 0 (implying that instrument has some correlation with the endogenous regressor), and 3) z must impact y only through x.

This means that if Z-->X-->Y holds and assuming that Cov(z,e)=0 is true, we have a valid instrument.

But what if Z<-->X-->Y and Cov(z,e)=0 is true? Is this still a valid instrument?

[Fei Wang]

But what if Z<-->X-->Y and Cov(z,e)=0 is true? Is this still a valid instrument?

Z<--X essentially contradicts Cov(z,u)=0 (the exogeneity condition should be that z is uncorrelated with the second-stage error u.). As X is endogenous, X is correlated with u. If Z is a function of X (Z<--X), then Z is likely to be correlated with u via X.