How to construct an IV using time invariant census data interacted with year dummies? (Urgent)

[Alex Buchelli]

I am producing a thesis on the impact of immigration on wages. To address simultaneity I must use an instrumental variable approach. My intended IV is to use historical settlement patterns which is the regional share of immigrants in 1991 (with the rest of the data from 2005-2018). However, the time invariant nature of the IV is clearly problematic. I can think of two ways possibly to deal with this:

1. Take 1991 regional shares and scale for each year by multiplying this by national growth in immigration from 1991 to each year t. Then I will have a time varying IV. However, I do not know if this is valid? The idea is that immigrants are drawn to areas where immigrants have already settled. However, this historical settlement does not directly impact wages today. Further, regions with higher shares of immigrants will be appropriately more proportionally impacted by this scaling up to period t (i.e. each year 2005 to 2018).
2. The second approach has perplexed me. It comes from Dustmann et al. (2013), who states: "compute the 1991 ratio of immigrants to natives for each of these regions, from the Census of Population, interacted with year dummies". I do not understand the intuition of using year dummies. Would it still be time invariant as those interacted dummies would all be 1? Please could anyone clarify what I am missing?

Otherwise, I cannot come up with any other solutions currently. Due to data constraints I am unable to run just a simple lag of an immigrant-native ratio which would be an ideal IV for simplicity.

Thank you very much and I hope you may assist me as this is rather urgent.

[Jeff Wooldridge]

I assume your plan is to use the fixed effects two-stage least squares estimator, which is why the fact that the immigration share is constant poses a problem. I think interacting such variables with year dummies is reasonably common -- but certainly flawed. Your first proposal seems like it's the same as treating immigration as exogenous, but maybe I'm missing something.

I think this should do it.

xtreg immigration x1 ... xk i.year c.share#i.year, fe vce(cluster region)
xtivreg logwage x1 ... xk i.year (immigration = c.share91#i.year) fe vce(cluster region)

The first regression is the first stage; you should confirm that the IVs are sufficiently strong.

[Alex Buchelli]

Thank you for your fast response. I intend to use first differences in this case. However, I believe this is along the lines of what you suggest: 1. Use first stage regression to estimate the relationship between the IV (c#sharei.year) and the endogenous variable (immigration).

2. The second stage regression examines the impact of the predicted immigration share on wages, using only the variation in immigration caused by the instrument.

In terms of my first proposal, I am assuming 1991 regional immigrant shares are not directly related to wages across 2005 to 2018. However, they are related to immigrant self-selection decisions today. Which explains my motivation for proposal 1, however I suppose you think this is invalid?

Thank you for your assistance Jeff Wooldridge

[Jeff Wooldridge]

Edited after reading more carefully: You're using the national growth in immigration, not regional growth. So this has the flavor of a shift-share instrument. Since the national growth only varies by t, this IV is less general than interacting with year dummies. But should give similar answers if your reasoning is sound.

You can use the second strategy with differencing, although I don't know how Stata will handle factor notation with the fd option. You might have to create the IVs by hand.

[Alex Buchelli]

Indeed, it was originally intended to be a shift-share instrument; alas data limitations have prevented that from happening at this point in time. However, I believe this can still be employed with first differencing but I am not sure if this is what you are implying.

However, I am still confused by how interacting a time invariant figure with time dummies results in a useful IV. Clearly time dummies are either 0 or 1 and so I am unsure of the relative merit of this approach? I believe the most plausible IV is taking the 1991 census data and projecting forward to year t according to growth in national immigrant stocks. Therefore, this deals with the time invariant issue and delivers upon the required conditions for the IV. It may be further possible to interact this with year dummies; however, again, I am unsure of the relative merit of doing so.

Thank you again for your help Jeff Wooldridge