Package {pathintdid}

pathintdid: Path-Integrated Difference-in-Differences

Description

Implements the identification, estimation, and inference framework of Salavi (2026), "Path-Integrated Difference-in-Differences (PI-DiD): Identification, Estimation, and Inference for Cumulative Treatment Effects." Instead of comparing treated and control outcomes at a single endline date, PI-DiD treats the treatment effect as a trajectory \tau(t) = c_1(t) - c_0(t) and integrates the baseline-differenced gap over the post-treatment window to obtain the cumulative causal effect \sigma and the path-integrated average treatment effect on the treated. This avoids the endpoint-subtraction bias: whenever a policy's effect is transitory and control/treated trajectories fully rejoin before the evaluation date, the conventional static difference-in-differences estimate can be exactly zero even though the cumulative benefit delivered to treated units was strictly positive.

Main functions

• [pathintdid()] – point estimates and inference for \sigma, the path-integrated ATT, and the conventional static DiD.

• [pathintdidplot()] – treated vs. counterfactual paths, and the running cumulative-effect plot.

• [pathintdidrobust()] – the section 5 robustness checks and specification tests: a dynamic placebo test for pre-treatment parallel trends, sensitivity to grid density and quadrature scheme, and an anticipation-robust bounding estimator.

Data

[trainingpanel] is a bundled 20-unit example panel (10 treated, 10 control; household consumption, periods 0-5) used throughout the documentation.

Author(s)

Maintainer: Comlan Apollinaire-Fabrice Salavi fsalavi@africanschoolofeconomics.com

References

Salavi, C. A.-F. (2026). "Path-Integrated Difference-in-Differences (PI-DiD): Identification, Estimation, and Inference for Cumulative Treatment Effects." Working paper, African School of Economics.

See Also

Useful links:

• https://github.com/FabriceSALAVI/PathIntDID

• Report bugs at https://github.com/FabriceSALAVI/PathIntDID/issues

Path-Integrated Difference-in-Differences (PI-DiD) estimator

Description

Implements the Path-Integrated Difference-in-Differences framework of Salavi (2026). Instead of comparing treated and control outcomes at a single endline date, the treatment effect is treated as a trajectory \tau(t) = c_1(t) - c_0(t), and the baseline-differenced gap \hat\tau(t) = \tau(t) - \tau(t_0) is integrated over the post-treatment window [t_0, t_1] (trapezoidal rule) to obtain the cumulative causal effect

\sigma = \int_{t_0}^{t_1} \hat\tau(t)\, dt,

and the path-integrated average treatment effect on the treated, \bar\tau = \sigma / (t_1 - t_0). For comparison, 'pathintdid' also reports the conventional static two-period DiD estimate, \hat\tau(t_1), which uses only the two boundary dates and can be zero (or misleadingly small) whenever a transitory intervention's effect has fully decayed by t_1, even though \sigma remains strictly positive (the "endpoint-subtraction bias").

Usage

pathintdid(
  data,
  yname,
  idname,
  tname,
  treatname,
  t0,
  t1 = NULL,
  level = 95,
  se = TRUE,
  subset = NULL
)

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

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

Arguments

Details

Unless 'se = FALSE', 'pathintdid' also computes a standard error, t-statistic, p-value, and confidence interval for \sigma, \bar\tau, and the static estimate, built from the individual-level, baseline-differenced trajectories \Delta Y_{i,k} = Y_{i,t_k} - Y_{i,t_0} of every unit observed at every date on the integration grid, using a group-specific (treated/control) sample covariance matrix combined as \hat\Omega = \hat\Sigma_1/\hat\pi + \hat\Sigma_0/(1 - \hat\pi) and the trapezoidal-rule weight vector w implied by the integration grid, giving \mathrm{Var}(\hat\sigma) = w'\hat\Omega w / N. This requires at least two treated and two control units with a complete record at every grid date within '[t0, t1]'; see Details.

'pathintdid' works with panel or repeated cross-section data for the point estimates (c_0(t), c_1(t), \sigma, \bar\tau are computed from group-time means, so 'id' does not need to be balanced across periods for those). Standard errors, however, require individual-level data and are computed only from units with a complete trajectory across the requested integration grid. If the panel is too unbalanced or too small to support inference, 'pathintdid' returns point estimates only, together with a message explaining why (rather than failing).

See Salavi (2026) for the full identification assumptions (SUTVA, random sampling, path-level parallel trends, no anticipation, path smoothness, moment conditions) under which this standard error is consistent. The companion function [pathintdidrobust()] implements the section 5 robustness checks and specification tests of the companion paper.

Value

An object of class '"pathintdid"', a list with components:

References

Salavi, C. A.-F. (2026). "Path-Integrated Difference-in-Differences (PI-DiD): Identification, Estimation, and Inference for Cumulative Treatment Effects." Working paper, African School of Economics.

See Also

[pathintdidplot()] to visualize the treated/counterfactual paths and the running cumulative effect; [pathintdidrobust()] for the companion specification tests.

Examples

data(trainingpanel)

fit <- pathintdid(trainingpanel, yname = "consumption", idname = "id",
                   tname = "time", treatname = "treat", t0 = 0, t1 = 5)
print(fit)

## point estimates only (faster; no inference)
pathintdid(trainingpanel, "consumption", "id", "time", "treat",
           t0 = 0, t1 = 5, se = FALSE)

Plot the causal impact of a Path-Integrated DiD analysis

Description

Companion plotting function to [pathintdid()]. Draws two panels: (1) the treated vs. counterfactual paths, with the region between them over '[t0, t1]' shaded to represent the cumulative effect \sigma; and (2) the running cumulative effect \sigma(t), which plateaus even after \tau(t) has decayed back to zero – the visual signature of the endpoint-subtraction bias documented in Salavi (2026) – with a shaded confidence band around the final plateau, pulled from [pathintdid()]'s standard error.

Usage

pathintdidplot(
  data,
  yname,
  idname,
  tname,
  treatname,
  t0,
  t1 = NULL,
  t2 = NULL,
  level = 95,
  se = TRUE,
  subset = NULL,
  xlab = "Time",
  main = "PI-DiD causal impact",
  col_control = "navy",
  col_treated = "firebrick",
  col_shade = grDevices::adjustcolor("grey70", alpha.f = 0.5)
)

Arguments

Value

Invisibly, a list with elements 'sigma_final', 'did_static', and (if standard errors were available) 'se_sigma', 'ci_lb', 'ci_ub'. Called for its side effect of drawing a two-panel base-graphics plot.

See Also

[pathintdid()]

Examples

data(trainingpanel)
pathintdidplot(trainingpanel, yname = "consumption", idname = "id",
               tname = "time", treatname = "treat",
               t0 = 0, t1 = 5, t2 = 5,
               main = "Training program: cumulative consumption impact")

Robustness checks and specification tests for PI-DiD

Description

Companion function to [pathintdid()] / [pathintdidplot()], implementing the three diagnostics of section 5 ("Robustness checks and specification tests") of Salavi (2026):

1. Dynamic placebo test for pre-treatment parallel trends – a joint Wald test on the vector of pre-treatment gaps, and a "pre-treatment cumulative envelope" (PCE) Z-test on the integrated pre-treatment gap. Requires at least one wave observed before 't0'; otherwise it is skipped with a note.

2. Sensitivity to grid density and quadrature scheme – compares the trapezoidal estimate of \sigma on the full post-treatment grid against a coarsened half-density grid, and against Simpson's 1/3 rule. Requires an even number of equally spaced post-treatment intervals between 't0' and 't1'.

3. Anticipation-robust bounding estimator – recomputes \sigma with the reference baseline shifted back by 0, 1, \dots, 'maxanticip' pre-treatment grid points, reporting a full sensitivity curve with standard errors.

All three diagnostics degrade gracefully – with an explicit message, never silently – when the data cannot support them.

Usage

pathintdidrobust(
  data,
  yname,
  idname,
  tname,
  treatname,
  t0,
  t1 = NULL,
  level = 95,
  maxanticip = 0,
  subset = NULL
)

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

Arguments

Value

An object of class '"pathintdidrobust"', a list with components 'pretest' (diagnostic A, or 'NULL'), 'grid' (diagnostic B, or 'NULL'), 'anticipation' (diagnostic C, or 'NULL'), plus bookkeeping ('t0', 't1', 'level', 'call').

References

Salavi, C. A.-F. (2026). "Path-Integrated Difference-in-Differences (PI-DiD): Identification, Estimation, and Inference for Cumulative Treatment Effects." Working paper, African School of Economics.

See Also

[pathintdid()], [pathintdidplot()]

Examples

data(trainingpanel)

## K = 4 (even) post-treatment intervals -> diagnostic (B) runs
rb <- pathintdidrobust(trainingpanel, yname = "consumption", idname = "id",
                        tname = "time", treatname = "treat",
                        t0 = 0, t1 = 4, maxanticip = 2)
print(rb)

Example training-program panel

Description

A simulated 20-unit individual-level panel (10 treated, 10 control) illustrating the Path-Integrated Difference-in-Differences framework of Salavi (2026): household consumption expenditure observed over 6 waves, with genuine unit-level sampling variation (i.e., not a pair of pre-aggregated cohort means), so that standard errors can be computed.

Usage

trainingpanel

Format

A data frame with 120 rows and 4 columns:

id

integer; household identifier, 1-20.

time

integer; period, 0 (baseline) through 5 (endline).

treat

integer; time-invariant treatment indicator (1 = treated, 0 = control).

consumption

numeric; daily per-capita household consumption expenditure (US dollars).

Source

Simulated data accompanying Salavi, C. A.-F. (2026), "Path-Integrated Difference-in-Differences (PI-DiD): Identification, Estimation, and Inference for Cumulative Treatment Effects," working paper, African School of Economics.

Examples

data(trainingpanel)
summary(trainingpanel)