Summary

The package ShiftShareSE implements confidence intervals proposed by Adão, Kolesár, and Morales (2019) for inference in shift-share least squares and instrumental variables regressions, in which the regressor of interest (or the instrument) has a shift-share structure, as in Bartik (1991). A shift-share variable has the structure $X_{i}=\sum_{s=1}^{S}w_{is}\Xs_{s}$, where i indexes regions, s indexes sectors, $\Xs_{s}$ are sectoral shifters (or shocks), and w_is are shares, such as initial share of region i’s employment in sector s.

This vignette illustrates the use of the package using a dataset from Autor, Dorn, and Hanson (2013) (ADH hereafter). The dataset is included in the package as the list ADH. The first element of the list, ADH$reg is a data-frame with regional variables, the second element, ADH$sic is a vector of SIC codes for the sectors, and ADH$W is a matrix of shares. See ?ADH for a description of the dataset.

Examples

We now replicate column (1) of Table V in Adão, Kolesár, and Morales (2019). First we load the package, define the vector of controls, and define a vector of 3-digit SIC codes:

library("ShiftShareSE")
ctrls <- paste("t2 + l_shind_manuf_cbp + l_sh_popedu_c +",
    "l_sh_popfborn + l_sh_empl_f + l_sh_routine33", " + l_task_outsource + division")
sic <- floor(ADH$sic/10)

We cluster the standard errors at the 3-digit SIC code (using the option sector_cvar), and, following ADH, weight the data using the weights ADH$reg$weights. See ?reg_ss and ?ivreg_ss for full description of the options.

The first-stage regression:

reg_ss(as.formula(paste("shock ~ ", ctrls)), W = ADH$W,
    X = IV, data = ADH$reg, weights = weights, region_cvar = statefip,
    sector_cvar = sic, method = "all")
#> Estimate: 0.6310409
#> 
#> Inference:
#>               Std. Error      p-value  Lower CI  Upper CI
#> Homoscedastic 0.02732516 0.000000e+00 0.5774846 0.6845973
#> EHW           0.08700719 4.083400e-13 0.4605100 0.8015719
#> Reg. cluster  0.09142372 5.113909e-12 0.4518537 0.8102281
#> AKM           0.05296055 0.000000e+00 0.5272402 0.7348417
#> AKM0          0.07671358 1.282891e-03 0.5375710 0.8382827

Note that for "AKM0", "Std. Error" corresponds to the normalized standard error, i.e. the length of the confidence interval divided by 2z_{1 − α/2}.

The reduced-form and IV regressions:

reg_ss(as.formula(paste("d_sh_empl ~", ctrls)), W = ADH$W,
    X = IV, data = ADH$reg, region_cvar = statefip, weights = weights,
    sector_cvar = sic, method = "all")
#> Estimate: -0.4885687
#> 
#> Inference:
#>               Std. Error      p-value   Lower CI   Upper CI
#> Homoscedastic 0.06332778 1.221245e-14 -0.6126889 -0.3644485
#> EHW           0.11244360 1.392685e-05 -0.7089541 -0.2681833
#> Reg. cluster  0.07578147 1.140306e-10 -0.6370977 -0.3400398
#> AKM           0.16419445 2.924641e-03 -0.8103839 -0.1667535
#> AKM0          0.25437489 4.218033e-04 -1.2368853 -0.2397541
ivreg_ss(as.formula(paste("d_sh_empl ~", ctrls, "| shock")),
    W = ADH$W, X = IV, data = ADH$reg, region_cvar = statefip,
    weights = weights, sector_cvar = sic, method = "all")
#> Estimate: -0.7742267
#> 
#> Inference:
#>               Std. Error      p-value   Lower CI   Upper CI
#> Homoscedastic  0.1069532 4.523049e-13 -0.9838511 -0.5646022
#> EHW            0.1647892 2.623532e-06 -1.0972075 -0.4512459
#> Reg. cluster   0.1758096 1.063809e-05 -1.1188071 -0.4296462
#> AKM            0.2403730 1.277718e-03 -1.2453492 -0.3031041
#> AKM0           0.3318966 4.218033e-04 -1.6903240 -0.3893132

Collinear share matrix

Let W denote the share matrix with the (i, s) element given by w_is and sth column w_s. Suppose that columns of W are collinear, so that it has rank S₀ < S. Without loss of generality, suppose that the first S₀ columns of the matrix are full rank, so that the collinearity is caused by the last S − S₀ sectors. In this case, it is not possible to recover, $\tilde{\Xs}_s$, the sectoral shifters with the controls partialled without further assumptions. The reg_ss and ivreg_ss functions will return a warning message "Share matrix is collinear". To compute the standard errors, the commands implement a default solution to this issue based on aggregating the shocks to the collinear sectors, which we describe in Section below. However, there are other ways of dealing with collinearity in the share matrix, as we describe in below. Depending on the the setting, researchers may wish to instead use one of these alternatives.

Default way of dealing with collinear sectors

We use a QR factorization of W with column pivoting (see Chapter 5.4.2 in Golub and Van Loan (2013)) to drop the collinear columns in W. That is, we decompose W = QRP′, where Q is an N × S orthogonal matrix, the matrix R takes the form $R=\bigl(\begin{smallmatrix}R_1 &R_2\\0&0\end{smallmatrix}\bigr)$, where R₁ is an S₀ × S₀ upper triangular matrix, R₂ has dimensions S₀ × (S − S₀), and P is a permutation matrix such that the diagonal elements of R are decreasing. We then drop S₀ − S columns of W that correspond to the last S − S₀ columns of QR, as indicated by the permutation matrix, obtaining a new share matrix W_new. Most software implementations of ordinary least squares, including LAPACK used by R, use this algorithm to drop collinear columns of the regressor matrix.

This solution keeps the regional shocks X_i the same, so that the point estimates do not change, while implicitly redefining the sectoral shocks $\Xs_s$. In particular, by definition of collinearity, each column w_s of W that we drop can be written as a linear combination of the new share matrix W_new. We can determine the coefficients γ_s in this linear combination by regressing w_s onto W_new. Observe that since dropping the collinear columns of W doesn’t change the regional shocks X_i if we implicitly define a new sectoral shock vector $\Xs_{new}$ as Here $\Xs_0$ corresponds to the first S₀ entries of the S-vector of shocks $\Xs$.

Note that re-ordering the columns of W will generally result in different columns being dropped, so that the standard errors will generally depend on the order of the sectors.

Extensions to multiple shifters and multiple endogenous variables

We now discuss how the methods in Adão, Kolesár, and Morales (2019) extend to the case where there are multiple shifters, or, in the case of an IV regression, multiple endogenous variables. Currently, these extensions are not implemented in the package.

OLS

Suppose that we’re interested in the effect of a k-vector of shift-share regressors, $X_i=\sum_{s}w_{is}\Xs_{s}$, where $\Xs_{s}$ is a vector of length k. For inference on the coefficient on the jth element of X_i, we proceed as if this was the only shift-share regressor, treating the remaining shifters as part of the controls.

IV with a single endogenous regressor and multiple shift-share instruments

Now suppose that the k-vector X_i defined in section is a k-vector of instruments. Let X denote the $\N\times k$ matrix with rows given by X_i′. Consider the setup in Section IV.C of Adão, Kolesár, and Morales (2019), with the first-stage coefficients β_is in eq. (31) now a k-vector, and α being the scalar treatment effect of Y₂ on Y₁ as in eq. (30). Letting $\ddot{X}=X-Z(Z' Z)^{-1}Z' X$ denote the $\N\times k$ matrix of instruments with the covariates partialled out, the two-stage least squares estimator is given by where $\hat{\beta}=(\ddot{X}'\ddot{X})^{-1}\ddot{X}'Y_{2}$ is a k-vector of first-stage coefficients.

Thus,

Now, letting Y₁(0) = Z′δ + ϵ, we have, as in the proof of Proposition 4 in the paper, Thus, using arguments in Proposition 4 in the paper, we obtain the infeasible standard error formula where $\tilde{\Xs}_{s}$ is a (vector) residual from the population regression of the vector $\Xs_{s}$ onto the controls.

This suggests the feasible standard error formula where $\widehat{\Xs}=(W' W)^{-1}W'\ddot{X}$ are the regression coefficients from the regression of $\ddot{X}$ onto W (as in Remark 6, except now a $\widehat{\Xs}$ is an S × k matrix), and ϵ̂_i are estimates of structural residual. For AKM, ϵ̂ = Y₁ − Y₂α̂ − Z(Z′Z)⁻¹Z′(Y₁ − Y₂α̂).

For AKM0, the construction is more complicated. Let $\hat{\gamma}=(\ddot{X}'\ddot{X})^{-1}\ddot{X}'Y_{1}$ denote the reduced-form coefficient. Let R̂_s, α₀ = ∑_iw_isϵ̂_α₀, where ϵ̂_α₀ = (I − Z(Z′Z)⁻¹Z′)(Y₁ − Y₂α₀). Then will be distributed χ_k² in large samples, because $(\ddot{X}'\ddot{X})^{-1}\sum_{s}\widehat{\Xs}_{s} \widehat{\Xs}_{s}'\hat{R}_{s,\alpha_{0}}^{2} (\ddot{X}'\ddot{X})^{-1}$ consistently estimates the asymptotic variance of γ̂ − β̂α₀ under the null. Therefore, we reject the null H₀: α = α₀ if Q(α₀) > χ_{k, 1 − α}², where χ_{k, 1 − α}² is the 1 − α quantile of a χ_k². A confidence set is collected by all nulls that are not rejected, Note that (i) unlike the case with a single instrument (Remark 6, step (iv)), there is not a closed form solution to the confidence set anymore: one needs to do a grid search over the real line, collecting all values of α for which the test doesn’t reject, and (ii) the confidence set will be valid even if the instruments are weak; however, if the instruments are strong, the AKM0 test is less powerful than the AKM test, and consequently the AKM0 confidence set will tend to be bigger than the AKM confidence interval.

Not that properties (i) and (ii) are inherited from the properties of the heteroskedasticity-robust version of the Anderson-Rubin test when there is more than one instrument (see, for example, Section 5.1 in Andrews, Stock, and Sun (2019) for a discussion). The AKM0 method adapts this test to the current setting with shift-share instruments, inheriting these properties.

If we do not require validity under weak instruments, we can also use a different version of AKM0, namely computing the confidence set as This form of the confidence can be thought of as the analog to the Lagrange multiplier confidence set in likelihood models, rather than the analog of the Anderson-Rubin test. In the case with a single instrument, these concepts coincide, but they are different in general. In this case, the inequality defining the set is just a quadratic inequality in α, and we can solve it explicitly as in Remark 6 in the paper to obtain a closed-form solution. If the instruments are strong, it will take the form of an interval.

IV with multiple endogenous variables

Consider a general setup with eqs. (30) and (31) in the paper replaced by with $\Xs$ and Y₂ now both vectors, and B_is has dimensions $\dim(\Xs)\times \dim(Y_{2})$. If $\Xs=Y_{2}$, the setup reduces to that in section . If Y₂ is scalar, the setup reduces to that in section . The two-stage least squares estimator of α is given by With scalar X_i and Y_2i, this expression reduces to eq. (33) in the paper. Now, Suppose that where $\mathcal{F}_{0}=(Y_{1}(0),Y_{2}(0),W,\Zs,U,B)$. Let δ be the coefficient on Z in the regression of Y_1i − Y_2i′α onto Z_i, and let ϵ_i = Y_1i − Y_2i′α − Z_i′δ = Y_1i(0) − Z_i′δ. Then, as in proof of Proposition 4 in the paper, where the second line follows by arguments in that proof. Now, since $\Xs_{s}$ is independent across s conditional on ℱ₀, it follows that conditional on ℱ₀, where R_s = ∑_iw_isϵ_i. This leads to variance formula where R̂_s = ∑_iw_isϵ̂_i, $\widehat{\Xs}=(W' W)^{-1}W'\ddot{X}$ as in eq. (36) in the paper, with rows $\Xs_{s}'$, and $\hat{B}=(\ddot{X}'\ddot{X})^{-1}\ddot{X}'Y_{2}$ is a matrix of the first-stage coefficients. Here ϵ̂_i is an estimate of the structural residual, such as For standard errors, take square root of the appropriate diagonal element.

The AKM0 version is a little tricky here if dim (α) > 1 and we’re only interested in inference on one element of α, say the first: this is analogous to issues with using the Anderson-Rubin test in a setting with multiple endogenous variables.

If we do not require validity under weak instruments, then the analog of the `alternative AKM0’ procedure from the preceding subsection uses the estimate (α₁₀, α̂₋₁(α₁₀)) in place of α̂ in (), where α₁₀ is the null hypothesized value, and is the estimate of the remaining elements of α with the null H₀: α₁ = α₁₀ imposed.

References

Adão, Rodrigo, Michal Kolesár, and Eduardo Morales. 2019. “Inference in Shift-Share Designs: Theory and Inference.” Quarterly Journal of Economics 134 (4): 1949–2010. https://doi.org/10.1093/qje/qjz025.

Andrews, Isaiah, James H. Stock, and Liyang Sun. 2019. “Weak Instruments in Instrumental Variables Regression: Theory and Practice.” Annual Review of Economics 11 (1): 727–53. https://doi.org/10.1146/annurev-economics-080218-025643.

Autor, David H., David Dorn, and Gordon H. Hanson. 2013. “The China Syndrome: Local Labor Market Effects of Import Competition in the United States.” American Economic Review 103 (6): 2121–68. https://doi.org/10.1257/aer.103.6.2121.

Bartik, Timothy J. 1991. Who Benefits from State and Local Economic Development Policies? Kalamazoo, MI: W.E. Upjohn Institute for Employment Research.

Golub, Gene H., and Charles F. Van Loan. 2013. Matrix Computations. Fourth. Johns Hopkins Studies in the Mathematical Sciences. Baltimore, MD: The Johns Hopkins University Press.

Standard Errors in Shift-Share Regressions