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

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

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.)

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

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

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)

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

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.)

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

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

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)

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)

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.