# Using dependent variable's and exogenous variables' lags as instruments in 2SLS

Suppose we have the following structural equations: $$y=\beta_0+\beta_1x+\beta_2z_z+\beta_3z_3+\epsilon,$$ where $$x$$ is the endogenous variable and $$z_1,z_2$$ are exogenous variables. In the literature, as well as in software packages (ivreg in STATA), I find that it is quite common to add dependent variable's as well as lagged exogenous variables as intruments in the first stage of 2SLS (to capture exogenous variation of the endogenous variable). In our case, we can consider $$y_{t-1}, x_{t-1}, x_{t-k}$$ as instruments for $$x$$. Is this legit and what is the rational behind this? Also, please provide some resources to read more about this. Thanks!

Is this legit and what is the rational behind this?

Yes you will find this even as a recommendation in many textbooks (e.g. see Romer Advanced Macroeconomics pp 376) so it is legit although with a caveat.

A good instrument should be correlated with the endogenous variable and be able to through it exert an effect on dependent variable. Well lags are more often than not highly correlated with the contemporaneous observation of the same variable.

Next, instruments should not be correlated with residuals and for lags this will often hold (but not always) in economics. This is because in many cases when the residual reflects new information learned by people between $$t$$ and $$t-1$$, economic theory simply often tells us that any variable that is known as of time $$t − 1$$ is uncorrelated with the residual.

However, this being said lagged instruments can still turn out to be weak, or violate some of the other assumptions of IV (for overview of all assumptions required for IV see for example Verbeek, a Guide to Modern Econometrics or Angrist & Pischke Mostly Harmless Econometrics), and the above might not hold generally for any economic relationship. So these are no silver bullet, but they often do make sense in economics. Nonetheless, there is a criticism of their overuse in economics (see for example Wang & Bellemare 2019). However, this is to my best understanding not because they would not be legitimate instruments but rather many practitioners just apply them without even bothering to check if other conditions that good instrument should satisfy are actually really satisfied.

• Thank you very much. Just for clarification: if in structural equation the dependent variable/exogenous variable is in level, then in first-stage I use as an instrument lagged dependent/exogenous variable in level; if in structural equation dependent/exogenous variable is in first-difference, then I use it as in instrument in first-difference, right? I am asking this because somewhere I've noticed that it is used lagged dependent variable in level as an instrument while in structural equation it is in first-difference. Thanks!
– Duo
Nov 13 '20 at 15:18
• @Duo this is bit case dependent. In standard time series models if there is unit root in the variable then both endogenous and instrument have to be first differenced. However, there are some models that allow mixing of level and first differenced variables even in presence of unit root (e.g. cointegration models). Also many bigger structural models (e.g. something along the lines of DSGE) can have parts where everything is modeled on level and parts modeled on first differences - if you have specific example in mind perhaps consider asking it as a separate question
– 1muflon1
Nov 13 '20 at 15:24
• My question was different. As in my question $x_t$ is endogenous and it is in level. Should I in the first stage (of 2SLS) use $x_{t-1}$ or $\Delta x_{t-1}$? From another hand, if in structural model we have $\Delta x_t$, then as an instrument in the first stage should I use $\Delta x_{t-1}$ or $x_{t-1}$? I noticed in some paper that in structural equation endogenous variable is in first-stage (i.e. $\Delta x_t$), but in the first-stage as an instrument is considered lagged endogenous variable in level (i.e. $x$).
– Duo
Nov 13 '20 at 15:27
• @Duo in that case the first stage it would make more sense to use $x_{t-1}$ although $\Delta x_{t-1}$ should be still exogenous I would expect it would be less correlated. The same for the opposite case.
– 1muflon1
Nov 13 '20 at 15:31
• Thanks, got it!
– Duo
Nov 13 '20 at 15:32