Re-draws the shocks by permutation (optionally within exchangeability `block`s) and reports where the observed statistic falls in the resulting placebo distribution, in the spirit of Adao-Kolesar-Morales (2019) and Borusyak & Hull.
Arguments
- design
An [ssb_design()] object.
- R
Number of permutation draws.
- block
Optional exchangeability blocks for shocks: a column name in the shocks table, or a vector of length equal to the number of shock-cells. Shocks are permuted only within blocks. In sector x period panels you almost always want blocks that separate periods, so shocks are not permuted across time.
- null
The null value \(\beta_0\) of the coefficient (default 0).
- seed
Optional RNG seed.
Value
A list (class `ssb_ri`) with the IV point estimate `beta`, the observed Anderson-Rubin `statistic`, `null`, `p_value`, `R`, and the vector `perm` of placebo statistics.
Details
The statistic is Anderson-Rubin-style: the reduced-form coefficient of \(y - \beta_0 x\) on the reconstructed instrument, with \(\beta_0 =\) `null`. Under the constant-effects null \(\beta = \beta_0\) (plus the exclusion restriction), \(y - \beta_0 x\) does not respond to how the shocks are assigned, so the permutation distribution of this statistic is *exact* given the exchangeability encoded in `block`. Permuting the IV ratio itself (holding the observed treatment fixed) would *not* be exact — the treatment also responds to the shocks through the first stage, and placebo draws with weak first stages give the ratio very heavy tails — so this function does not do that.
**The exchangeability assumption is stronger than shocks being as-good-as-random.** Permutation validity requires the shocks (within a `block`) to be *exchangeable* — in particular, identically distributed up to reordering. The Borusyak-Hull-Jaravel framework only assumes shocks are as-good-as-randomly assigned (mean-independent of the unobservables), which allows their variances and higher moments to differ across cells; under that weaker assumption randomization inference can over- or under-reject. Use `block` to group shocks that are plausibly comparable draws (and to keep permutations within periods in panels), and treat the RI p-value as a complement to — not a substitute for — the exposure-robust inference in [ssb_estimate()] / [ssb_shock_iv()].
Examples
sim <- ssb_simulate(n_loc = 80, n_sec = 10, seed = 1)
d <- ssb_design(sim$data, sim$shares, sim$shocks, exogenous = "share")
ssb_ri(d, R = 199, seed = 1)
#> <ssBartik randomization inference>
#> IV estimate : 1.4468 (H0: beta = 0.000)
#> AR statistic : 1.4664 (reduced form of y - beta0*x on the instrument)
#> RI p-value : 0.0300 (199 permutations)
#> note: valid if shocks are EXCHANGEABLE (within blocks) -- stronger than
#> the as-good-as-random assumption of Borusyak-Hull-Jaravel, which
#> allows shock variances to differ; see ?ssb_ri