Overidentification test across the share instruments (Sargan-Hansen)

Treats each sector's exposure share as a separate instrument — the exogenous-shares reading of a Bartik design (Goldsmith-Pinkham, Sorkin & Swift 2020) — estimates the corresponding overidentified IV, and reports the Sargan-Hansen J test of the overidentifying restrictions. Rejection points to a failure of shares exogeneity for some sectors **or** to treatment-effect heterogeneity across instruments. This is a share-route diagnostic; [ssb_pipeline()] runs it there.

Usage

ssb_overid(
  design,
  estimator = c("auto", "2sls", "gmm", "liml", "jive"),
  min_F = 0,
  level = 0.95
)

Arguments

design: An [ssb_design()] object.
estimator: `"auto"` (default), `"2sls"`, `"gmm"`, `"liml"`, or `"jive"`. See Details.
min_F: Drop instruments whose own first-stage F is below this before estimating and testing.
level: Confidence level for the reported interval.

Value

A list (class `ssb_overid`) with the chosen `estimator`, its `beta`, `se`, `conf.low`/`conf.high`, the Hansen `J` statistic, `df`, `p`, the number of instruments used `K` (plus `n_dropped`, `n_collinear`, `n_weak`), and the per-instrument table `instruments` of just-identified estimates for [ssb_plot_overid()].

Details

The J statistic comes from efficient two-step GMM with a heteroskedasticity-robust weight matrix (cluster-robust when the design has a `cluster` variable), so it accounts for the fact that the just-identified estimates are estimated on the *same* sample and are mutually correlated. Earlier versions reported a precision-weighted Cochran Q, which is only valid when those estimates are independent; it has been replaced.

Following the Borusyak-Hull-Jaravel JEP practical guide, several estimators of the common coefficient are available: `"2sls"` and efficient two-step `"gmm"` are natural when the number of instruments \(K\) is modest, while `"liml"` and `"jive"` (JIVE1) guard against many-instrument bias when \(K\) is large relative to the sample. `estimator = "auto"` (default) picks 2SLS when \(K \le \max(3, 0.05\,n)\) and LIML otherwise, with a message. When the shares sum to one (or the sum of shares is controlled) the residualised share instruments are exactly collinear; redundant columns are dropped automatically, so a complete-shares design with \(K\) sectors uses \(K-1\) instruments and the test has \(K-2\) degrees of freedom, matching the usual leave-one-share-out formulation.

Examples

sim <- ssb_simulate(n_loc = 80, n_sec = 10, seed = 1)
d <- ssb_design(sim$data, sim$shares, sim$shocks, exogenous = "share")
ssb_overid(d, estimator = "2sls")
#> <ssBartik overidentification test (Sargan-Hansen)>
#>   estimator : 2SLS over 9 share instruments (1 collinear dropped)
#>   beta = 1.5851   se = 0.1487   [1.294, 1.877]
#>   Hansen J = 5.77 on 8 df,  p = 0.6731
#>   instruments dropped: 0 (min_F / degenerate); weak (F<10): 9
#>   small p => reject joint validity of the share instruments
#>   (exclusion failure for some shares OR treatment-effect heterogeneity)