• Use a different cross-sectional exposure to the aggregate shock.
• Card (2009).
• Use 1980, 1990, and 2000 US Census data.
• \(y_{lj}\): the residual log wage gap between immigrant and native men in skill group \(j\) in city \(l\).
• \(x_{lj}\): the ratio of immigrant to native hours in skill group \(j\) in city \(l\).
• \(X_l\): a vector of city-level controls.
• Estimate \(\beta\) in: \[ y_{lj} = \beta_0 + \beta \ln x_{lj} + \beta_2 X_l + \epsilon_{lj}. \]
• However, \(x_{lj}\) and \(\epsilon_{lj}\) can be correlated due to a labor demand shock specitic to the skill group and the city.

• Consider aggregate shocks, \(g_{kj}\), the number of people arriving in the United States from 1990 to 2000 from country group \(k\) and skill group \(j\).
• Consider immigrant shares: \[ z_{lk, 1980} = \frac{N_{lk, 1980}}{N_{k, 1980}} \cdot \frac{1}{P_{l, 2000}}, \] where:
• \(N_{lk, 1980}\): the number of immigrants from country \(k\) in the United States in 1980 in city \(l\).
• \(N_{k, 1980}\): the number of immigrants from country \(k\) in the United States in 1980.
• \(P_{l, 2000}\): population of city \(l\) in 2000.

• Construct a Bartik instrument (Bartik 1991): \[ B_{lj} = \sum_{k = 1}^K z_{lk, 1980} \cdot g_{kj}. \]
• Consider a first stage of: \[ \ln x_{lj} = \gamma_0 + \gamma_1 B_{lj} + \gamma_2 X_l + \eta_{lj}. \]
• Then, consider a two-stage least-squares (TSLS) estimator of \(\beta\).

• What are identifying assumptions forf this estimator?
• Share-vew: assume the shares \(z_{kl}\) are exogenous (Goldsmith-Pinkham et al. 2020).
• Shock-view: assume the shocks \(g_{kj}\) are exogenous (Borusyak et al 2021).
• They differ in the assumptions and valid inference.

• There are \(K\) industries and \(T\) times.
• Consider a structural model: \[ y_{lt} = x_{lt}\beta_0 + W_{lt}\rho + \epsilon_{lt}, \] where:
• \(y_{lt}\): the outcome in location \(l\) in time \(t\).
• \(x_{lt}\): the treatment in location \(l\) in time \(t\).
• \(W_{lt}\): a vector of observed controls in location \(l\) in tine \(t\).
• \(\epsilon_{lt}\): an error potentially correlated wit \(x_{lt}\).
• \(\beta_0\): parameter of interest.

• Suppose that \(x_{lt}\) is decomposed as: \[ x_{lt} = Z_{lt} \cdot G_{lt} = \sum_{k = 1}^K z_{lkt} g_{lkt}. \]
• For example, \(x_{lt}\) is the employment growth in location \(l\) in time \(t\), \(z_{lkt}\) is the employment share of industry \(k\) in location \(k\) at the beginning of time \(t\), and \(g_{lkt}\) is the employment growth of industry \(k\) in location \(l\) in time \(t\).

• The industry-location-time-specific shock \(g_{lkt}\) is further decomposed into: \[ g_{lkt} = g_{kt} + \tilde{g}_{lkt}, \] where \(g_{kt}\) is the average shock of industry \(k\) in time \(t\) and \(\tilde{g}_{lkt}\) is an idiosyncratic shock.
• We want to address the correlation between \(\epsilon_{lt}\) and \(\tilde{g}_{lkt}\).

• Define Bartik instrument as: \[ B_{lt} = Z_{l0} \cdot G_t = \sum_{k = 1}^K z_{lk0} g_{kt}, \] where \(G_t = (G_{1t}, \cdots, G_{Kt})\).
• Consider a linear regression of \(x_{lt}\) on \(B_{lt}\) and \(W_{lt}\) as: \[ x_{lt} \equiv B_{lt} \gamma + W_{lt} \tau + \eta_{lt}. \]

• We have data: \[ \{y_l, W_l, \widetilde{G}_l, Z_l, Z_{l0}\}_{l = 1}^L, \] that are i.i.d. across \(l\) where:
• \(y_l = (y_{l1}, \cdots, y_{lt})\);
• \(W_l = (W_{l1}, \cdots, W_{lT})\);
• \(\widetilde{G}_l = (\widetilde{G}_{l1}, \cdots, \widetilde{G}_{lT})\);
• \(Z_l = (Z_{l1}, \cdots, Z_{lT})\).
• We view \(G = (G_{1}, \cdots, G_T)\) as fixed.
• We arrow for non-i.i.d. data within \(l\).

• The Bartik estimator of \(\beta\): \[ \hat{\beta}_{bartik} = (B'M_WX)^{-1} B'M_WY \]
• If there is no control: \[ \begin{split} \hat{\beta}_{bartik} &= (B'X)^{-1} B'Y\\ &= \frac{\sum_{l = 1}^L \sum_{t = 1}^T \sum_{k = 1}^K z_{lk0} g_{kt} y_{lt}}{\sum_{l = 1}^L \sum_{t = 1}^T \sum_{k = 1}^K z_{lk0} g_{kt} x_{lt}}. \end{split} \]

• Bartik instrument: \[ B_l = z_{l1} g_1 + (1 - z_{l1}) g_2 = g_2 + (g_1 - g_2) z_{l1}. \]
• The first-stage regression: \[ \begin{split} x_l & = \gamma_0 + \gamma_1 B_t + \eta_{lt}\\ &= \gamma_0 + \gamma_1 g_2 + \gamma (g_1 - g_2) z_{l1} + \eta_l\\ &= \tilde{\gamma}_0 + \tilde{\gamma}_1 z_{l1} + \eta_l \end{split} \]
• Thus, this is a TSLS where \(z_{l1}\) is used as an instrument for \(x_l\).
• \(g_1 - g_2\) is the policy size.
• \(z_{l1}\) measures the exposure to the policy that affect industry 1.

• Bartik instrument: \[ B_{lt} = z_{l10} g_{1t} + z_{l20} g_{2t} = g_{2t} + (g_{1t} - g_{2t}) z_{l10}. \]
• The first-stage regression: \[ \begin{split} x_{lt} &= \tau_t + B_{lt} \gamma + \eta_{lt}\\ &= (\tau_t + g_{2t} \gamma) + z_{l10} (g_{1t} - g_{2t})\gamma + \eta_{lt}\\ &=(\tau_t + g_{2t} \gamma) + z_{l10}1\{t = 1\}(g_{11} - g_{21}) \gamma\\ &+ z_{l10} 1\{t = 2\}(g_{12} - g_{22}) \gamma + \eta_{lt}\\ &=\tilde{\tau}_t + z_{l10} 1\{t = 1\} \tilde{\gamma}_1 + z_{l10} 1\{t = 2\} \tilde{\gamma}_2 + \eta_{lt}. \end{split} \]
• \(z_{l10}\) measures the exposure to the policy that affect industry 1.
• There are two policy size \(g_{1t} - g_{2t}\) for \(t = 1, 2\).

• Policy size: \(g_{1t} - g_{2t}\) for \(t = 1, 2\).
• If we know that the policy is introduced at the end of period 1, we expect that the pre-trend is common: \(g_{11} - g_{11} = 0\).
• Then, we should test \(\tilde{\gamma}_1 = 0\).

• The Bartik estimator of \(\beta\): \[ \hat{\beta}_{bartik} = (B'M_WX)^{-1} B'M_WY \] is equivalent to a GMM estimator of \(\beta\): \[ \hat{\beta}_{GMM} = (X'M_W \tilde{Z} \Omega \tilde{Z}' M_W X)^{-1} X'M_W \tilde{Z} \Omega \tilde{Z}' M_W Y, \] where the instrument is: \[ \tilde{Z} = \text{diag}(Z, \cdots, Z) \] and the weight is: \[ \Omega = G'G \] where \(M_W = I - W(W'W)^{-1}W'\).

• Assumption: Relevance
  • For all \(k = 1, \cdots, K\) and \(s = 1, \cdots, T\): \[ x_{lt} = W_{lt} \tau + z_{lk0} 1\{t = s\} C_{k, s} + \eta_{lt}, \] where \(\mathbb{E}[\eta_{lt}|z_{lk0}, W_{lt}] = 0\), \(C_{ks}\) is finite for all \(k\) and \(s\), and \(\sum_s \sum_k g_{ks} C_{ks} \neq 0\).
• There must be an industry and time period when the indsustry share has predictive power for \(x_{lt}\) conditional on the controls.
• The growth rate \(g_{kt}\) cannot weight the covariances in such a way that they exactly cancel out.

• Assumption: Strict exogeneity
  • For all \(k\) where \(g_k \neq 0\): \[ \mathbb{E}[\epsilon_{lt} z_{lk0}| W_{lt}] = 0. \]
• Then, the Bartik estimator is consistent with \(L \to \infty\).
• Usually, we take changes in \(y_{lt}\) and \(x_{lt}\).
• Then, the exogeneity means that the initial industry share is exogenous to the changes in the error term conditional on controls.

• Check the correlation between \(Z_l\) and the other initial characteristics \(W_{l0}\).
  • If correlated, consider whether it poses any doubt on the exogeneity assumption.
• Check the correlation between \(Z_l\) and placebo outcomes, that is likely to be correlated with the error in the main outcome, but should be unrelated to \(Z_l\).
• If there is a pre-treatment period, test the pre-trend.
• Check the robustness by using each \(z_{k0}\) as \(K\) instruments, rather than summarizing to \(B_{lt}\).

• In the share-view, the exogeneity condition was for all \(k\) where \(g_k \neq 0\): \[ \mathbb{E}[\epsilon_{lt} z_{lk0}| W_{lt}] = 0. \]
• This is sufficient but not necessary.
• We just need for all \(t\): \[ \mathbb{E}[\sum_{l = 1}^L \epsilon_{lt} B_{lt}| W_{lt}] = 0. \]

• This can be written as: \[ \begin{split} 0 &= \mathbb{E}[\sum_{l = 1}^L \epsilon_{lt} \sum_{k = 1}^K z_{kl0} g_{kt}]\\ &= \mathbb{E}[\sum_{k = 1}^K g_{kt} \sum_{l = 1}^L z_{kl0} \epsilon_{lt}]\\ &= \mathbb{E}[\sum_{k = 1}^K g_{kt} \overline{\epsilon}_{kt}]. \end{split} \]
• It is satisfied as long as \(g_{kt}\) is uncorrelated with \(\overline{\epsilon}_{kt}\) even if \(z_{kl0}\) is correlated with \(\epsilon_{lt}\).

• Assumption: Quasi-random shock assignment
  • For all \(k\) and \(t\): \[ \mathbb{E}[g_{kt}| \overline{\epsilon}, z] = \mu. \]
• If this is the case: \[ \begin{split} \mathbb{E}[\sum_{k = 1}^K g_{kt} \overline{\epsilon}_{kt}] &= \mu \mathbb{E}[\sum_{k = 1}^K \overline{\epsilon}_{kt}] = \mu \mathbb{E}[\sum_{l = 1}^L \epsilon_{lt} \sum_{k = 1}^K z_{kl0}]\\ &= \mu \mathbb{E}[\sum_{l = 1}^L \epsilon_{lt}] = 0. \end{split} \]

• In the shock-view, we use asymptotics of \(K \to \infty\).
• Assumption: Many uncorrelated shocks
  • For all \(k \neq k'\) and \(t\): \[ \mathbb{E}[\sum_{k = 1}^K z_{kl0}^2] \to \infty. \] \[ Cov[g_{kt}, g_{k't}| \overline{\epsilon}, z] = 0. \]
• The expected shock exposure is asymptotically completely disperesed.
• The shocks are mutually uncorrelated.

• The estimator is the same with the share-view of shift-share design.
• However, the inference is involved, because:
  • \(g_{kt}\) is regarded as random variable.
  • This creates correlation across locations.
• Adao et al. (2019) propose a exposure-based clustering.
• Simpler way is to construct a shock-level main equation: \[ \overline{y}_{kt} = \overline{x}_{kt}\beta_0 + \overline{W}_{kt}\rho + \overline{\epsilon}_{kt}, \] where \(\overline{y}_{kt} = \sum_{l = 1}^L z_{kl0} y_{lt}\) and others are the same.
• Then, its heteroskedasticity-robust standard error yields asymptotically valid confidence interval.

• Hummels et al. (2014):
  • Estimate the wage effects of offshoring across Danish importin firm \(l\).
  • The shock index \(n\) refers to a type of intermediate inputs and an origin country; titanium hinges from Japan.
• Peri et al. (2016):
  • Estimate the effecet of immigrant STEM workers on the labour market outcomes of natives across U.S. cities \(l\).
  • The shock index \(n\) refers to a migration origin country.

set.seed(2)
L <- 100
T <- 2
K <- 2
beta <- 1
sigma <-c(0.1, 0.1, 1)

z_lk0 <-
  tidyr::expand_grid(
    l = 1:L,
    k = 1:K
  ) %>%
  dplyr::mutate(
    z_lk0 = runif(length(l))
  ) 
g_kt <-
  tidyr::expand_grid(
    t = 1:T,
    k = 1:K
  ) %>%
  dplyr::mutate(g_kt = rnorm(t))

df <- 
  tidyr::expand_grid(
    l = 1:L,
    t = 1:T,
    k = 1:K
  ) %>%
  dplyr::left_join(
    z_lk0,
    by = c("l", "k")
  ) %>%
  dplyr::left_join(
    g_kt,
    by = c("k", "t")
  ) %>%
  dplyr::mutate(
    z_lkt = z_lk0 + sigma[1] * runif(length(l)),
    g_lkt = g_kt + sigma[2] * rnorm(length(l))
  ) 

df <-
  df %>%
  dplyr::group_by(t, l) %>%
  dplyr::mutate(
    z_lk0 = z_lk0 / sum(z_lk0),
    z_lkt = z_lkt / sum(z_lkt),
    x_lt = sum(z_lkt * g_lkt)
  ) 

df_lt <-
  df %>%
  dplyr::mutate(g_tilde_lkt = g_lkt - g_kt) %>%
  dplyr::group_by(l, t) %>%
  dplyr::summarise(
    dplyr::across(
      c(x_lt, g_tilde_lkt),
      mean
    ),
    .groups = "drop"
  ) %>%
  dplyr::ungroup() %>%
  dplyr::mutate(
    y_lt = beta * x_lt + sigma[3] * g_tilde_lkt
  )

result_ols <-
  df_lt %>%
  lm(
    formula = y_lt ~ x_lt
  )
modelsummary::modelsummary(result_ols)

b_lt <-
  df %>%
  dplyr::group_by(l, t) %>%
  dplyr::summarise(b_lt = sum(z_lk0 * g_kt)) %>%
  dplyr::ungroup()

## `summarise()` has grouped output by 'l'. You can override using the `.groups` argument.

result_first_stage <-
  df_lt %>%
  dplyr::left_join(b_lt, by = c("l", "t")) %>%
  lm(
    formula = x_lt ~ b_lt  
  )
modelsummary::modelsummary(result_first_stage)

result_tsls <-
  df_lt %>%
  dplyr::left_join(b_lt, by = c("l", "t")) %>%
  estimatr::iv_robust(
    formula = y_lt ~ x_lt | b_lt
  )
modelsummary::modelsummary(result_tsls)

• Adao, Rodrigo, Michal Kolesár, and Eduardo Morales. 2019. “Shift-Share Designs: Theory and Inference.” The Quarterly Journal of Economics 134 (4): 1949–2010.
• Borusyak, Kirill, Peter Hull, and Xavier Jaravel. 2021. “Quasi-Experimental Shift-Share Research Designs.” The Review of Economic Studies, June. https://doi.org/10.1093/restud/rdab030.
• Goldsmith-Pinkham, Paul, Isaac Sorkin, and Henry Swift. 2020. “Bartik Instruments: What, When, Why, and How.” The American Economic Review 110 (8): 2586–2624.
• Hummels, David, Rasmus Jørgensen, Jakob Munch, and Chong Xiang. 2014. “The Wage Effects of Offshoring: Evidence from Danish Matched Worker-Firm Data.” The American Economic Review 104 (6): 1597–1629.
• Peri, Giovanni, Kevin Shih, and Chad Sparber. 2015. “STEM Workers, H-1B Visas, and Productivity in US Cities.” Journal of Labor Economics 33 (S1): S225–55.