Package 'staggeredGMM'

State-level panel: bank branch deregulation and income inequality

Description

A panel of 49 US states over 1976–2006, used to study intrastate bank branching deregulation and the distribution of income.

Usage

beck_banks

Format

A data frame with 1519 rows and 8 variables:

state

Two-letter state abbreviation.

state_name

State name.

statefip

State FIPS code.

wrkyr

Calendar year, 1976–2006.

gini

Income Gini coefficient.

branch_reform

Year the state deregulated intrastate bank branching. 1960 marks states already deregulated before the sample begins.

ln_gini

Natural log of gini.

D_branch

Treatment indicator.

Always-treated states

This dataset is a worked example of a coding pitfall as much as of a design. Thirteen of the 49 states deregulated at or before 1976, the first observed year: AK, AZ, CA, DC, ID, MD, ME, NC, NV, NY, RI, SC and VT, ten of them carrying branch_reform == 1960. These units have no pre-treatment period, so none of their effects is identified. No state is coded 0, so the dataset also contains no never-treated group.

Passed to gmm_staggered() as it stands, the always-treated states raise an error naming them. That is deliberate: silently including them would return a plausible-looking aggregate contaminated by cells that are not estimated at all. Restrict to states adopting inside the window before estimating:

  dat <- beck_banks[beck_banks$branch_reform > 1976, ]
  fit <- gmm_staggered(dat, yname = "ln_gini", tname = "wrkyr",
                       idname = "state", gname = "branch_reform")
  

This leaves 36 states across cohorts 1977–1999, identified from not-yet-treated controls alone.

Source

Beck, T., Levine, R. and Levkov, A. (2010). "Big Bad Banks? The Winners and Losers from Bank Deregulation in the United States." The Journal of Finance 65(5), 1637–1667. doi:10.1111/j.1540-6261.2010.01589.x

Replication data deposited by the authors at DataverseNL, doi:10.34894/B1K9OU, and made available under the Creative Commons Attribution 4.0 International licence, https://creativecommons.org/licenses/by/4.0/.

Modified from the deposited data: a subset of columns was selected and the remainder discarded. No values were altered, recoded or derived. The full licence note is installed with the package; see system.file("LICENSE.note", package = "staggeredGMM").

---

Extract cohort-by-time effect estimates

Description

Extract cohort-by-time effect estimates

Usage

## S3 method for class 'staggered_gmm'
coef(object, ...)

Arguments

object	A staggered_gmm object, as returned by gmm_staggered().
...	Ignored.

Value

A named numeric vector of cohort-by-time effects, one per cell, named "g<cohort>:t<period>". Cells with no clean comparison are NA.

Examples

fit <- gmm_staggered(sim_panel, yname = "y", tname = "year",
                     idname = "unit_id", gname = "cohort")
head(coef(fit))

---

Confidence intervals for cohort-by-time effects

Description

Pointwise normal-approximation intervals. They are not adjusted for multiplicity, so reading a whole event-study path off them overstates confidence.

Usage

## S3 method for class 'staggered_gmm'
confint(object, parm, level = 0.95, ...)

Arguments

object	A staggered_gmm object, as returned by gmm_staggered().
parm	Optional character vector of coefficient names. Defaults to all of them.
level	Confidence level, strictly between 0 and 1.
...	Ignored.

Value

A two-column matrix of lower and upper limits, with one row per requested coefficient. Unidentified cells are NA.

Examples

fit <- gmm_staggered(sim_panel, yname = "y", tname = "year",
                     idname = "unit_id", gname = "cohort")
head(confint(fit))

---

Specification test for parallel trends and no anticipation

Description

A serial-correlation robust joint test of the identifying assumptions, following Section 4.4 of Arora and Bijani (2026).

Usage

gmm_j_test(object, type = c("local", "full"), window = 3L)

Arguments

object	A fitted gmm_staggered() object.
type	Character. "local" (default) restricts the test to placebo moments within window periods before each cohort's adoption; "full" uses all of them.
window	Integer. Width of the local pre-window, in periods. Ignored when type = "full".

Details

The test is built on the never-treated event-study basis of Eq. (27) of the paper. Fixing a base pre-period $t_0$, each treated cohort $g$ and period $t \ne t_0$ defines

$\Delta_{g,t} = (\bar{Y}_{g,t} - \bar{Y}_{g,t_0}) - (\bar{Y}_{\infty,t} - \bar{Y}_{\infty,t_0}).$

Those with $t \ge g$ identify the cohort-by-time effects. Those with $t < g$ are placebo restrictions whose expectation is zero only under parallel trends and no anticipation. Collecting a set of $J$ placebo moments into $\hat{p}$ with covariance $\hat\Omega_p$ implied by the fitted serial-correlation model,

$J = \hat{p}' \hat\Omega_p^{-1} \hat{p} \;\to\; \chi^2_J$

under the null. Because the placebo moments do not involve the treatment effects, no degrees of freedom are subtracted for estimated parameters.

The base period is $t_0 = g_{\min} - 1$, the last period before any cohort adopts, so that every moment has an untreated reference.

A large number of restrictions degrades the chi-squared approximation in small panels. type = "local" therefore uses only the placebo moments within window periods of each cohort's adoption, which holds its nominal size across sample sizes at essentially no cost in power, since a differential trend or an anticipation effect is already visible just before treatment. type = "full" uses every placebo moment and is well-sized only when the number of units is comfortably large.

Value

An object of class staggered_gmm_jtest, a list with components statistic, df, p_value, type, window, base_period, n_moments and moments (the placebo moment table).

References

Arora, P. and Bijani, R. (2026). "Estimating Treatment Effects under Staggered Timing and Non-Spherical Errors." doi:10.2139/ssrn.6558759

Examples

fit <- gmm_staggered(sim_panel, yname = "y", tname = "year",
                     idname = "unit_id", gname = "cohort")
gmm_j_test(fit)

---

GMM estimation of treatment effects under staggered adoption

Description

Estimates cohort-by-time average treatment effects (CATTs) under staggered treatment adoption by the generalized method of moments, following Arora and Bijani (2026).

Usage

gmm_staggered(
  data,
  yname,
  tname,
  idname,
  gname,
  weighting = c("pooled_toeplitz", "cohort_toeplitz", "unrestricted"),
  never_treated = NULL,
  covar = NULL,
  max_iter = 100L,
  tol = 1e-06
)

Arguments

data	A data frame with one row per unit-period.
yname	Character. Name of the outcome column. Must be numeric; factor, character and logical outcomes are rejected rather than coerced.
tname	Character. Name of the calendar-time column. Must be integer-valued, and the set of periods present in data (pooled across units) must be consecutive.
idname	Character. Name of the unit identifier column.
gname	Character. Name of the cohort column, giving each unit's first treated period, with 0 for units that are never treated. Must be constant within unit.
weighting	Character. The covariance model used to form the optimal GMM weight. One of "pooled_toeplitz" (the default: a single stationary autocovariance sequence shared by all groups), "cohort_toeplitz" (a separate stationary sequence per cohort), or "unrestricted" (a full, unrestricted within-cohort covariance). These correspond to GMM-T, GMM-HT and GMM-U in the paper.
never_treated	Logical or NULL. Whether to use never-treated units as controls in addition to not-yet-treated cohorts. NULL (the default) detects them automatically. TRUE requires that at least one exists and errors otherwise. FALSE excludes them, and warns if any are present.
covar	Optional character vector of baseline (pre-treatment, time-invariant) covariate columns. When supplied, the outcome-regression adjustment of Section 4.5 of the paper is applied, so that parallel trends need hold only conditional on covar.
max_iter	Integer. Maximum number of iterated-GMM reweighting steps.
tol	Numeric. Convergence tolerance on the largest absolute change in any CATT estimate between successive iterations.

Value

An object of class staggered_gmm: a list with components call, weighting, coefficients, vcov, catt, aggregate, weights, convergence, design, controls and validation. See summary.staggered_gmm() and vignette("staggeredGMM").

Data requirements

The following conditions are checked before any estimation, and each raises an error naming the offending rows, units, periods or cohorts:

• data is a data frame and every named column exists.

• The outcome is numeric; time and cohort are integer-valued.

• No missing unit id, period or cohort. Missing outcomes are permitted.

• Each unit-period combination appears exactly once.

• The cohort is constant within unit.

• No period is absent from the whole panel. Individual units may be missing individual periods.

• No unit is treated at or before the first observed period. Such units have no pre-treatment period, so none of their effects is identified, and silently including them contaminates the aggregate.

• At least one cohort adopts strictly inside the observed window.

Units whose cohort falls after the last observed period are never treated within the window and are therefore pooled with the never-treated group as clean controls, rather than discarded.

Unbalanced panels

Missing outcomes are supported. Group-period means are taken over the units observed at that period; autocovariances are divided by the number of residual pairs actually observed at each lag; and the variance of a group-period mean accounts for the overlap between the unit sets contributing at each pair of periods. When the panel is balanced all three reduce exactly to the complete-data formulas. Validity requires that missingness be independent of the outcome given group and period; selection on outcomes is not addressed by any weighting scheme.

Partial identification

A cohort-by-time effect with no clean comparison is not estimable. Such cells are reported with estimate = NA and identified = FALSE. Where an unidentified cell carries positive aggregation weight the corresponding aggregate is NA, and an identified-subset aggregate renormalised over the estimable cells is reported alongside it, with the share of weight it covers.

Convergence

Three states are reported separately. converged is TRUE only when the largest change in any CATT fell below tol. solve_ok records the weaker fact that at least one weighted solve succeeded. termination gives the reason the iteration stopped.

When solve_ok is FALSE, no weighted step was ever completed and the returned estimates are the identity-weighted seed rather than a GMM estimate under the requested weighting. That always raises a warning.

When solve_ok is TRUE but converged is FALSE, the iteration ran and did not settle; the estimates are the last valid iterate. That warns for the two Toeplitz weightings but not for "unrestricted", where slow convergence is routine.

References

Arora, P. and Bijani, R. (2026). "Estimating Treatment Effects under Staggered Timing and Non-Spherical Errors." doi:10.2139/ssrn.6558759

See Also

gmm_j_test() for the over-identification test of parallel trends and no anticipation.

Examples

fit <- gmm_staggered(sim_panel, yname = "y", tname = "year",
                     idname = "unit_id", gname = "cohort")
fit

---

Print a fitted staggered GMM model

Description

Print a fitted staggered GMM model

Usage

## S3 method for class 'staggered_gmm'
print(x, ...)

Arguments

x	A staggered_gmm object, as returned by gmm_staggered().
...	Ignored.

Value

x, invisibly. Called for the side effect of printing.

Examples

fit <- gmm_staggered(sim_panel, yname = "y", tname = "year",
                     idname = "unit_id", gname = "cohort")
print(fit)

---

Print a specification test result

Description

Print a specification test result

Usage

## S3 method for class 'staggered_gmm_jtest'
print(x, ...)

Arguments

x	A staggered_gmm_jtest object, as returned by gmm_j_test().
...	Ignored.

Value

x, invisibly. Called for the side effect of printing.

Examples

fit <- gmm_staggered(sim_panel, yname = "y", tname = "year",
                     idname = "unit_id", gname = "cohort")
print(gmm_j_test(fit))

---

Simulated staggered adoption panel

Description

A balanced synthetic panel used in the examples, the vignette and the tests: 60 units observed over 33 periods, with five treatment cohorts (first treated in periods 10, 13, 16, 19 or 22; 10 units each) and a never-treated group of 10 units. True effects are deterministic given (cohort, period), so both aggregate targets are known exactly: ATT_CW = -16.79375 under treated-observation weighting and ATT_EW = -15.82020 under cohort-equal weighting. The two differ because earlier cohorts are observed for more post-treatment periods; see gmm_staggered() for the definitions.

Usage

sim_panel

Format

A data frame with 1980 rows and 6 variables:

unit_id

Unit identifier, integer 1–60.

year

Period, integer 1–33.

cohort

0 = never treated; 10, 13, 16, 19 or 22 = first treated period.

y

Observed outcome.

x1

Continuous baseline covariate.

x2

Binary baseline covariate.

Details

The data-generating process, for unit $i$ in cohort $g_i$ at period $t$:

$y_{it} = \alpha_i + \lambda_t + \tau_{it} + x_{1i}\theta_{1t} + x_{2i}\theta_{2t} + \varepsilon_{it}$

• $\alpha_i \sim N(0, 1)$: unit fixed effect.

• $\lambda_t \sim N(0, 1)$: period fixed effect.

• $\tau_{it} = \beta_{g_i}(1 + r_{g_i})^{t - g_i}$ for $t \ge g_i$ and 0 otherwise, with $\beta = (-16, -12, -10, -9, -2)$ and $r = (0.01, 0.04, 0.08, 0.10, 0.07)$ for cohorts $(10, 13, 16, 19, 22)$.

• $\varepsilon_{it}$: AR(1) within unit, $\rho = 0.5$.

• $x_{1i}$, $x_{2i}$: baseline covariates with cohort-correlated means and linear-in-time loadings ($\theta_{1t}$, $\theta_{2t}$), so an unconditional difference-in-differences is genuinely confounded by them.

set.seed(312844) throughout. See data-raw/sim_panel.R for the exact implementation.

This is the design of Appendix E of the paper, so the specification test has 70 full-set and 14 local-window restrictions on it.

Source

Simulated for Arora, P. and Bijani, R. (2026). "Estimating Treatment Effects under Staggered Timing and Non-Spherical Errors." doi:10.2139/ssrn.6558759

---

Summarise a fitted staggered GMM model

Description

Prints everything print.staggered_gmm() shows, followed by the full table of cohort-by-time effects and the covariate-adjustment status.

Usage

## S3 method for class 'staggered_gmm'
summary(object, ...)

Arguments

object	A staggered_gmm object, as returned by gmm_staggered().
...	Ignored.

Value

object, invisibly. Called for the side effect of printing.

Examples

fit <- gmm_staggered(sim_panel, yname = "y", tname = "year",
                     idname = "unit_id", gname = "cohort")
summary(fit)

---

Covariance matrix of the estimated cohort-by-time effects

Description

Covariance matrix of the estimated cohort-by-time effects

Usage

## S3 method for class 'staggered_gmm'
vcov(object, ...)

Arguments

object	A staggered_gmm object, as returned by gmm_staggered().
...	Ignored.

Value

A square numeric matrix covering the identified cells only, with row and column names matching the corresponding entries of coef().

Examples

fit <- gmm_staggered(sim_panel, yname = "y", tname = "year",
                     idname = "unit_id", gname = "cohort")
dim(vcov(fit))

---