Bartik instruments

Causal inference, Instrumental variables estimation

• Bartik instruments: What, when, why, and how

Proposed in Bartik1991who and popularized in Blanchard1992regional.

A basic setting is: $$y_l = \rho + \beta_0 x_l + \epsilon_l,$$

where $y_l$ is wage growth in location $l$ between two time periods, $x_l$ is the employment growth rate, $\rho$ is a constant, and $\epsilon_l$ is a structural error term correlated with $x_l$. $\beta_0$ is what we try to estimate, which is the inverse elasticity of labor supply.

Then the Bartik instrument combines the following two identities. First, $$x_l = \sum_k z_{lk} g_{lk},$$ where $z_{lk}$ is the share of $l$‘s employment in industry $k$ and $g_{lk}$ is the growth rate of industry $k$ in $l$. The second is

$$g_{lk} = g_k + \tilde{g}_{lk}$$

where $g_k$ is the industry growth rate and $\tilde{g}_{lk}$ is an idiosyncratic industry-location growth rate. Now, the Bartik instrument is

$$B_l = \sum_k z_{lk}g_k$$

Examples

• Borusyak2024quasi-experimental