Decomposes the shift-share 2SLS estimate into a weighted sum of the just-identified estimates that use each sector's share as a single instrument, following Goldsmith-Pinkham, Sorkin and Swift (2020): $$\hat\beta = \sum_n \hat\alpha_n \hat\beta_n,\qquad \hat\alpha_n = \frac{g_n\, \tilde s_n' \tilde x}{\sum_{n'} g_{n'}\, \tilde s_{n'}' \tilde x},$$ where tildes denote residualisation on the controls (and, in panels, sector-cells are sector \(\times\) period pairs). The weights \(\hat\alpha_n\) sum to one and measure the sensitivity of \(\hat\beta\) to misspecification of each sector's instrument; a small number of large weights is a warning sign. Unlike Goodman-Bacon weights, negative Rotemberg weights are not automatically problematic.
Value
A `data.frame` of class `ssb_rotemberg`, one row per sector-cell, with columns `sector`, `g` (shock, after any demeaning), `alpha` (Rotemberg weight), `beta` (just-identified estimate), `F` (first-stage F of that instrument), and `sign`. Carries the overall estimate `beta_hat` and the normalisation used (`demeaned`) as attributes. Pass it to [ssb_plot_rotemberg()] for the canonical figure.
Details
**Normalisation.** When the exposure shares sum to one (or the sum of shares is controlled, as it is automatically on the shift route), adding a constant to every shock leaves the instrument — and \(\hat\beta\) — unchanged but changes the individual weights: the decomposition is unique only up to a normalisation of the shocks. Following Goldsmith-Pinkham, Sorkin & Swift and Borusyak-Hull-Jaravel, `demean = TRUE` (default) resolves this by demeaning the shocks with exposure weights, **within periods in a panel** (when the corresponding per-period constant directions are absorbed by the controls). If the constant is *not* absorbed (incomplete shares without a sum-of-shares control) the decomposition is already pinned down by the raw shocks; demeaning would then change it, so the raw shocks are kept, with a message. The just-identified estimates \(\hat\beta_n\), the first-stage Fs and \(\hat\beta\) itself are unaffected by the normalisation — only the weights are.
Examples
sim <- ssb_simulate(n_loc = 80, n_sec = 10, seed = 1)
d <- ssb_design(sim$data, sim$shares, sim$shocks, exogenous = "share")
ssb_rotemberg(d)
#> <ssBartik Rotemberg weights>
#> overall beta_hat : 1.4468
#> sum positive alpha : 1.005 sum negative alpha : -0.005
#> largest weight : alpha = 0.585 (6)
#> ! one share instrument carries |alpha| = 0.58; check robustness via ssb_drop_top()
#> shocks demeaned (overall, exposure-weighted) before weighting -- GPSS/BHJ normalisation
#> top 6 sectors by |alpha|:
#> sector alpha beta F
#> 6 0.5850 1.486 12.24
#> 3 0.1736 1.472 5.53
#> 8 0.1109 0.631 1.54
#> 4 0.0875 1.956 2.83
#> 9 0.0165 2.041 1.22
#> 5 0.0130 1.651 1.94
#> (negative weights are not by themselves a red flag; see GPSS 2020)