Instrumental Variables in the Modern Age

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# Instrumental Variables in the Modern Age
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### Ian McCarthy | Emory University
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### Econ 771, Fall 2022
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# Common IV Designs

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# Judge Fixed Effects

- Many different possible decision makers
- Individuals randomly assigned to one decision maker
- Decision makers differ in leniency of assigning treatment
- Common in crime studies due to random assignment of judges to defendants

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# Judge Fixed Effects

Aizer and Doyle (2015), QJE, "Juvenile Incarceration, Human Capital, and Future Crime: Evidence from Randomly Assigned Judges"
- Proposed instrument: propensity to convict by the judge
- Idea: judge has some fixed leniency, and random assignment into judges introduces exogenous variation in probability of conviction
- In practice: judge assignment isn't truly random, but it **is** plausibly exogenous

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# Judge Fixed Effects

Constructing the instrument:
- Leave-one-out mean `$$z_{j} = \frac{1}{n_{j} - 1} \sum_{k \neq i}^{n_{j}-1} JI_{k}$$`
- This is the mean of incarceration rates for judge `$j$` when excluding the current defendant, `$i$`
- Could also residualize `$JI_{k}$` (remove effects due to day of week, month, etc.)

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# Judge Fixed Effects

- Common design: possible in settings where some influential decision-makers exercise discretion and where individuals can't control the match
- Practical issue: Use jackknife IV (JIVE) to "fix" small-sample bias
    - JIVE more general than judge fixed effects design
    - Idea: estimate first-stage without observation `$i$`, use coefficients for predicted endogeneous variable for observation `$i$`, repeat
    - May improve finite-sample bias but also loses efficiency
- Biggest threat: monotonicity...judges may be more/less lenient in different situations or for different defendants

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# Bartik (shift-share) Instruments

- Named after Timothy Bartik, traced back to Perlof (1957)
- Original idea: Estimate effect of employment growth rates on labor-market outcomes
    - Clear simultaneity problem
    - Seek IV to shift labor demand

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# Bartik Instruments

Decompose observed growth rate into:
1. "Share" (what extra growth would have occurred if each industry in an area grew at their industry national average)
2. "Shift" (extra growth due to differential growth locally versus nationally)

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# Bartik Instruments

- Want to estimate `$$y_{l} = \alpha + \delta I_{l} + \beta w_{l} + \varepsilon_{l}$$` for location `$l$` (possibly time `$t$` as well)
- `$I_{l}$` reflect immigration flows
- `$w_{l}$` captures other observables and region/time fixed effects

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# Bartik Instruments

Instrument:

`$$B_{l} = \sum_{k=1}^{K} z_{l,k} \Delta_{k},$$`

- `$l$` denotes market location (e.g., Atlanta), country, etc. (wherever flows are coming *into*)
- `$k$` reflects the source country (where flows are coming out)
- `$z_{lk}$` denotes the **share** of immigrants from source `$k$` in location `$l$` (in a base period)
- `$\Delta_{k}$` denotes the **shift** (i.e., change) from source country into the destination country as a whole (e.g., immigration into the U.S.)

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# Other Examples

`$$\begin{align}
B_{l} &= \sum_{k=1}^{K} z_{lk} \Delta_{k},\\
\Delta_{lk} &= \Delta_{k} + \tilde{\Delta_{lk}}
\end{align}$$`

China shock (Autor, Dorn and Hanson, 2013):
- `$z_{lk}$`: location, `$l$`, and industry, `$k$`, composition
- `$\Delta_{lk}$`: location, `$l$`, and industry, `$k$`, growth in imports from China
- `$\Delta_{k}$`: industry `$k$` growth of imports from China

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# Other Examples

`$$\begin{align}
B_{l} &= \sum_{k=1}^{K} z_{lk} \Delta_{k},\\
\Delta_{lk} &= \Delta_{k} + \tilde{\Delta_{lk}}
\end{align}$$`

Immigrant enclave (Altonji and Card, 1991):
- `$z_{lk}$`: share of people from foreign country `$k$` living in current location `$l$` (in base period)
- `$\Delta_{lk}$`: growth in the number of people from `$k$` to `$l$`
- `$\Delta_{k}$`: growth in people from `$k$` nationally

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# Other Examples

`$$\begin{align}
B_{l} &= \sum_{k=1}^{K} z_{lk} \Delta_{k},\\
\Delta_{lk} &= \Delta_{k} + \tilde{\Delta_{lk}}
\end{align}$$`

Market size and demography (Acemoglu and Linn, 2004):
- `$z_{lk}$`: spending share on drug `$l$` from age group `$k$`
- `$\Delta_{lk}$`: growth in spending of group `$k$` on drug `$l$`
- `$\Delta_{k}$`: national growth in spending of group `$k$` (e.g., due to population aging)

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# Bartik Instruments

- Goldsmith-Pinkham, Sorkin, and Swift (2020) show that using `$B_{l}$` as an instrument is equivalent to using local industry shares, `$z_{lk}$`, as IVs
- Can decompose Bartik-style IV estimates into weighted combination of estimates where each share is an instrument (Rotemberg weights)
- Borusyak, Hull, and Jaravel (2022), ReStud, instead focus on situation in which the shifts are exogenous and the shares are potentially endogenous
- Borusyak and Hull (2021), Econometrica (maybe), provide general approach when using exogenous shifts (recentering as in homework)

**key:** literature was vague as to the underlying source of variation in the instrument...recent papers help in understanding this (and thus defending your strategy)

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# Shift-Share (focusing on the shift)

- Did ACA medicaid expansion affect insurance status?
- Construct "simulated" IV (dummy for whether person `$i$` is newly eligible given state expansion)
- Instrument can be thought of as, `$z_{is} = f(x_{i}, e_{s})$`
- Want to separate variation due to `$e_{s}$` (the state policy changes) from variation in demographics, `$x_{i}$`
    - Identify `$p(x)$`, probability of eligibility on average across other states' laws
    - Recenter, `$\tilde{z}_{is} = z_{is} - p(x)$`. Can also just control for `$p(x)$` in regression.