Package 'pathintdid'

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

data	a data frame (or object coercible to one) containing the panel.
yname	character scalar; name of the numeric outcome column.
idname	character scalar; name of the unit/panel-id column.
tname	character scalar; name of the calendar-time/period column.
treatname	character scalar; name of the time-invariant 0/1 treatment-group indicator column (1 = treated, 0 = control).
t0	numeric scalar; baseline period marking the start of the integration window. Under parallel pre-trends, $\hat\tau(t_0) \approx 0$; pathintdid reports it as a diagnostic and, regardless of its value, nets it out of $\sigma$, $\bar\tau$, and the static estimate by construction.
t1	numeric scalar; terminal evaluation date (endline). Defaults to the maximum observed value of tname.
level	confidence level, in percent, used for the reported confidence intervals and the critical value in the t-test. Default 95.
se	logical; if FALSE, skip the standard-error/t-statistic/confidence-interval calculation and report point estimates only (faster; also useful if the panel is too small or too unbalanced for inference). Default TRUE.
subset	optional logical vector (of length nrow(data)), or an expression evaluated in data, restricting the estimation sample – analogous to Stata's if/in qualifiers.
x	an object of class "pathintdid".
...	further arguments (currently unused).
object	an object of class "pathintdid".

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]⁠.

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:

sigma	cumulative causal effect.
att_path	path-integrated ATT, sigma / (t1 - t0).
did_static	conventional static DiD, tauhat(t1).
tau_t0	pre-treatment gap at t0 (parallel-trends diagnostic).
t0, t1, level	the window and confidence level used.
path	a data frame with columns time, c0, c1, taud, giving the group-time paths on the estimation window.
have_se	logical; were standard errors computed.
se_sigma, t_sigma, p_sigma, ci_sigma	standard error, t-stat, p-value, and confidence interval for sigma (when have_se).
se_att, t_att, p_att, ci_att	ditto for att_path.
se_did, t_did, p_did, ci_did	ditto for did_static.
N1, N0, df	number of treated/control units used for inference, and residual degrees of freedom (when have_se).
yname, idname, tname, treatname, call	bookkeeping.

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

data	a data frame (or object coercible to one) containing the panel.
yname	character scalar; name of the numeric outcome column.
idname	character scalar; name of the unit/panel-id column.
tname	character scalar; name of the calendar-time/period column.
treatname	character scalar; name of the time-invariant 0/1 treatment-group indicator column (1 = treated, 0 = control).
t0	numeric scalar; baseline period.
t1	numeric scalar; terminal evaluation date. Defaults to the maximum observed value of tname.
t2	optional numeric scalar; an intermediate "rejoining" date, marked with a vertical reference line alongside t0 and t1. Must satisfy t0 < t2 <= t1.
level	confidence level, in percent. Default 95.
se	logical; compute and shade the confidence band for the plateau of $\sigma(t)$? Default TRUE.
subset	optional logical vector, or an expression evaluated in data, restricting the estimation sample.
xlab	x-axis label. Default "Time".
main	overall plot title.
col_control, col_treated, col_shade	colors for the counterfactual path, the treated path, and the shaded cumulative-effect region.

Value

Invisibly, a list with elements sigma_final, did_static, and (if standard errors were available) se_sigma, ci_lb, and 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

data	a data frame (or object coercible to one) containing the panel.
yname	character scalar; name of the numeric outcome column.
idname	character scalar; name of the unit/panel-id column.
tname	character scalar; name of the calendar-time/period column.
treatname	character scalar; name of the time-invariant 0/1 treatment-group indicator column (1 = treated, 0 = control).
t0	numeric scalar; baseline period.
t1	numeric scalar; terminal evaluation date. Defaults to the maximum observed value of tname.
level	confidence level, in percent. Default 95.
maxanticip	non-negative integer; maximum number of pre-treatment grid points by which to shift the reference baseline back, for diagnostic (C). If 0 (the default), diagnostic (C) is skipped.
subset	optional logical vector, or an expression evaluated in data, restricting the estimation sample.
x	an object of class "pathintdidrobust".
...	further arguments (currently unused).

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)

---