Introduction to plausibounds: Computing Plausible Bounds for dynamic effect plots

Dynamic Effect Plots

To understand the dynamic effects of a policy, it is common to summarize estimated effects over time in a dynamic effect (or event-study) plot. For example, a macroeconomist may estimate the response of the Consumer Price Index to a monetary shock via local projections, or a microeconomist may estimate the response of labor market outcomes to a change in welfare policies via distributed lag models.

Overview

The plausibounds package provides simple additions to standard visualizations of dynamic treatment effects aimed at enhancing their informativeness following Freyaldenhoven and Hansen (2026). In particular, it provides functionality to:

• include joint hypothesis tests and summary measures that are often of direct interest alongside the standard event-study plot.

• compute a smooth approximation to the treatment-effect path from a prespecified, economically motivated class and produces corresponding confidence regions. The resulting intervals can be substantially tighter than conventional uniform bands while maintaining formal coverage guarantees through post-selection inference (Berk et al. [2013]).

Background

We write \(\theta=(\beta^{pre'},\beta')'\) for the \(p \times 1\) vector that collects the elements of the dynamic treatment effect path depicted on the dynamic effect plot. Here, \(\beta_{pre} = \{\beta^{pre}_h\}_{h={-L}}^{-1}\) denotes the displayed pre-event effects, \(\beta=\{\beta_h\}_{h={1}}^{H}\) collects the dynamic treatment effect path after the event up to the fixed maximum horizon of interest \(H\), and thus \(p = H+L\). We assume access to point estimates of the parameters \(\theta_h\), denoted by \(\hat{\theta}_h\), that correspond to the cumulative effect of the policy at horizon \(h\) relative to period 0. Throughout, we assume the vector that collects the estimated dynamic treatment effect path, \(\hat\theta\), satisfies \(\hat\theta \sim N(\theta,V_{\theta})\) and that we have access to the covariance matrix \(V_{\theta}\). Leading examples to obtain such estimates include distributed lag models, local projections, and event studies.

The package takes \(\hat{\theta}\) and \(\hat{V}_{\theta}\) as input.

Choosing the surrogate model

The main contribution is to include new visual elements meant to depict approximations to the underlying effect path constrained to exhibit economically and statistically plausible shapes. Intuitively, these elements formalize “visual data snooping”: Our proposal approximates looking at a dynamic effect plot and then choosing a specification that “captures the main dynamics.” This specification should a) provide a notion of valid inference that explicitly accounts for the data-dependent choice of specification and b) use the full covariance structure of the estimates when choosing the specification.

Specifically, we construct restricted estimates and plausible bounds by using data-driven regularization to select an approximation to the treatment path from a pre-specified class of models. We consider a default universe of approximations motivated by a preference for (1) smooth dynamics that (2) eventually die out. The restricted estimates are the resulting regularized point estimates of the treatment path. Our plausible bounds are sup-t bands under the selected approximation, adjusted to account for data-dependent selection using Berk et al.’s [2013] post-selection inference (PoSI).

Intuitively, the plausible bounds represent a compromise between inference under no restrictions — which may yield very wide confidence intervals — and inference based on a single, prespecified model that does not adapt to the data and may therefore be overly restrictive.

For the technical details, see Section 3 of Freyaldenhoven and Hansen (2026).

Basic Usage

Let’s start by examining the structure of the example datasets included in the package:

# Load example datasets
data(estimates_smooth)
data(var_smooth)

# Examine the data structures
print(estimates_smooth)
#>  [1] -0.17482676 -0.02004318 -0.10031844  0.07229334 -0.31769306  0.01454694  0.06322594
#>  [8] -0.04186633  0.12340034 -0.04534908 -0.16169097 -0.20259667 -0.03298893 -0.02878808
#> [15] -0.11865547 -0.33781817 -0.04988252 -0.23286451 -0.34721788 -0.31989801 -0.37015145
#> [22] -0.18439333 -0.08283743 -0.06628430  0.07793498 -0.16041782  0.08020676 -0.26365956
#> [29] -0.09156697 -0.08646877  0.17552747 -0.26331948  0.26817909  0.21390666  0.02886865
#> [36] -0.03854033 -0.17947818 -0.04629705 -0.26782818  0.03586062  0.17248994 -0.01214121
#> [43] -0.08781408  0.12013731
print(var_smooth[1:4, 1:4])
#>            [,1]       [,2]       [,3]       [,4]
#> [1,] 0.01981023 0.00000000 0.00000000 0.00000000
#> [2,] 0.00000000 0.01981023 0.00000000 0.00000000
#> [3,] 0.00000000 0.00000000 0.01981023 0.00000000
#> [4,] 0.00000000 0.00000000 0.00000000 0.01981023

The estimates_smooth vector contains 36 treatment effect estimates at different horizons and 8 preperiods. The var_smooth matrix is a diagonal variance matrix with no correlation between periods.

# Remove pretrends from the example dataset
estimates_smooth_nopretrends <- estimates_smooth[9:44]
var_smooth_nopretrends <- var_smooth[9:44, 9:44]

set.seed(916)

pb <- plausible_bounds(
  estimates = estimates_smooth_nopretrends,
  var = var_smooth_nopretrends
)

summary(pb)
#> Summary of Plausible Bounds Results
#> -----------------------------------
#> 
#>  horizon unrestr_est   restr_est restr_lower   restr_upper
#>        1  0.12340034  0.05029379  -0.2819062  0.3824937460
#>        2 -0.04534908 -0.01242527  -0.2383862  0.2135356269
#>        3 -0.16169097 -0.06083759  -0.2497435  0.1280683432
#>        4 -0.20259667 -0.09671456  -0.2846543  0.0912251951
#>        5 -0.03298893 -0.12450103  -0.3145739  0.0655718329
#>        6 -0.02878808 -0.14990419  -0.3364671  0.0366587226
#>        7 -0.11865547 -0.17627685  -0.3569812  0.0044275218
#>        8 -0.33781817 -0.20310332  -0.3800159 -0.0261907316
#>        9 -0.04988252 -0.22699284  -0.4032329 -0.0507527284
#>       10 -0.23286451 -0.24373054  -0.4200712 -0.0673898602
#>       11 -0.34721788 -0.24771659  -0.4212750 -0.0741582231
#>       12 -0.31989801 -0.23460638  -0.3994920 -0.0697207151
#>       13 -0.37015145 -0.20399763  -0.3533966 -0.0545986686
#>       14 -0.18439333 -0.16031714  -0.2899628 -0.0306714682
#>       15 -0.08283743 -0.11228840  -0.2247302  0.0001534093
#>       16 -0.06628430 -0.06984725  -0.1753613  0.0356668159
#>       17  0.07793498 -0.04085456  -0.1491331  0.0674239406
#>       18 -0.16041782 -0.02840299  -0.1404036  0.0835976484
#>       19  0.08020676 -0.02831138  -0.1403457  0.0837229598
#>       20 -0.26365956 -0.02835239  -0.1403724  0.0836676028
#>       21 -0.09156697 -0.02833664  -0.1403591  0.0836857875
#>       22 -0.08646877 -0.02831674  -0.1403418  0.0837083322
#>       23  0.17552747 -0.02829484  -0.1403223  0.0837325884
#>       24 -0.26331948 -0.02828204  -0.1403116  0.0837475459
#>       25  0.26817909 -0.02825973  -0.1402913  0.0837717986
#>       26  0.21390666 -0.02824996  -0.1402832  0.0837833347
#>       27  0.02886865 -0.02825026  -0.1402851  0.0837846091
#>       28 -0.03854033 -0.02825289  -0.1402892  0.0837833833
#>       29 -0.17947818 -0.02825500  -0.1402925  0.0837824990
#>       30 -0.04629705 -0.02825160  -0.1402902  0.0837869721
#>       31 -0.26782818 -0.02824739  -0.1402869  0.0837920882
#>       32  0.03586062 -0.02823508  -0.1402753  0.0838051515
#>       33  0.17248994 -0.02822505  -0.1402659  0.0838157745
#>       34 -0.01214121 -0.02822171  -0.1402630  0.0838195628
#>       35 -0.08781408 -0.02821898  -0.1402605  0.0838225786
#>       36  0.12013731 -0.02821446  -0.1402562  0.0838272497

str(pb)
#> List of 8
#>  $ alpha                     : num 0.05
#>  $ preperiods                : num 0
#>  $ wald_test                 :List of 1
#>   ..$ post:List of 2
#>   .. ..$ statistic: num 62.9
#>   .. ..$ p_value  : num 0.00368
#>  $ restricted_bounds         :'data.frame':  36 obs. of  5 variables:
#>   ..$ horizon    : int [1:36] 1 2 3 4 5 6 7 8 9 10 ...
#>   ..$ unrestr_est: num [1:36] 0.1234 -0.0453 -0.1617 -0.2026 -0.033 ...
#>   ..$ restr_est  : num [1:36] 0.0503 -0.0124 -0.0608 -0.0967 -0.1245 ...
#>   ..$ lower      : num [1:36] -0.282 -0.238 -0.25 -0.285 -0.315 ...
#>   ..$ upper      : num [1:36] 0.3825 0.2135 0.1281 0.0912 0.0656 ...
#>  $ restricted_bounds_metadata:List of 8
#>   ..$ supt_critval      : num 3.17
#>   ..$ supt_b            : num 3.59
#>   ..$ degrees_of_freedom: num 4.14
#>   ..$ K                 : int 18
#>   ..$ lambda1           : num 22026
#>   ..$ lambda2           : num 77
#>   ..$ restr_class       : chr "M"
#>   ..$ best_fit_model    :List of 5
#>   .. ..$ estimates_proj: num [1:36] 0.0503 -0.0124 -0.0608 -0.0967 -0.1245 ...
#>   .. ..$ var_proj      : num [1:36, 1:36] 0.008577 0.005264 0.002701 0.000868 -0.000297 ...
#>   .. ..$ bic           : num 32
#>   .. ..$ model_fit_pval: num 0.459
#>   .. ..$ df            : num 4.14
#>  $ avg_treatment_effect      :List of 4
#>   ..$ estimate: num -0.0773
#>   ..$ se      : num 0.0235
#>   ..$ lower   : num -0.123
#>   ..$ upper   : num -0.0313
#>  $ pointwise_bounds          :List of 3
#>   ..$ lower  : num [1:36] -0.111 -0.282 -0.401 -0.444 -0.276 ...
#>   ..$ upper  : num [1:36] 0.3576 0.1912 0.0772 0.0386 0.2105 ...
#>   ..$ critval: num 1.96
#>  $ supt_bounds               :List of 3
#>   ..$ lower  : num [1:36] -0.259 -0.431 -0.552 -0.596 -0.43 ...
#>   ..$ upper  : num [1:36] 0.506 0.341 0.228 0.191 0.364 ...
#>   ..$ critval: num 3.2
#>  - attr(*, "class")= chr "plausible_bounds"

Visualization with create_plot()

Next let’s plot the results:

create_plot(pb)

Plausible bounds for treatment path estimates with no correlation

The plot displays several types of bounds and statistics, each serving a different inferential purpose:

Unrestricted estimates (black dots): The original treatment effect estimates \(\hat{\beta}_h\) at each horizon.

Plausible bounds (blue dashed lines): These bounds provide uniform \((1-\alpha)\) coverage for the selected restricted estimates, not the true treatment path itself. The restricted path is chosen via BIC from the pre-specified model universe and represents a data-dependent smoothed approximation to the true treatment path.

Restricted estimates (blue line): The restricted estimates are the smoothed point estimates under the selected regularization.

Pointwise bounds (gray horizontal bars): Standard confidence intervals for each horizon separately, constructed as \(\hat{\beta}_h \pm z_{1-\alpha/2} \sqrt{V_{\beta,hh}}\).

Sup-t bounds (gray vertical lines): Uniform confidence bands that provide \((1-\alpha)\) joint coverage of the entire treatment path. These are constructed using the supremum of \(t\)-statistics from individual estimates. Guarantee full-path coverage but can become quite wide.

Annotations: The bottom of the plot shows several summary statistics. The “No effect p-value” comes from a Wald test of the null hypothesis that all treatment effects are zero. The “Average effect [CI]” provides the average effect \(\bar{\beta} = \frac{1}{H}\sum_h \beta_h\) along with its confidence interval.

Example 2: Correlated Estimates

When estimates are correlated across horizons (common in dynamic settings), the plausible bounds adapt to the correlation structure and will tend to become wider:

pb_corr <- plausible_bounds(
  estimates = estimates_bighump[1:12],
  var = var_bighump[1:12, 1:12],
  alpha = 0.05
)

create_plot(pb_corr)

Plausible bounds for treatment path estimates with correlated estimates

Pre-Treatment Periods

The package supports designs where some periods occur before treatment. This allows testing for pre-trends and distinguishing them from treatment effects. When specifying the preperiods argument, the function automatically computes two Wald tests: one testing \(H_0\): no pre-trends in the pre-treatment periods, and another testing \(H_0\): no treatment effect in the post-treatment periods. Both p-values appear in the plot annotations.

Specifying Pre-Treatment Periods

Use the preperiods argument to indicate how many of the initial estimates correspond to pre-treatment periods. Note that period 0 (the period immediately before treatment) is assumed to be normalized to zero and should not be included in the estimates vector:

pb_pre <- plausible_bounds(
  estimates = estimates_smooth,
  var = var_smooth,
  alpha = 0.05,
  preperiods = 8  # First 8 elements are pre-treatment periods
)

create_plot(pb_pre)

Event study with pre-treatment periods showing pre-trends

The vertical dashed line at event time 0 separates pre- and post-treatment periods. Pre-treatment estimates appear at negative horizons, revealing any pre-existing trends. The plausible bounds and restricted estimates are only computed for the post-treatment periods, since the goal is inference on the treatment effect path.

The annotations now include both Wald test p-values. The “Pretrends p-value” tests whether the pre-treatment coefficients are jointly zero—a rejection indicates violations of parallel trends. The “No effect p-value” tests whether the post-treatment coefficients are jointly zero.

Customizing plots

The create_plot() function provides a few arguments to customize which elements appear in the plot. The show_annotations argument controls whether the Wald test results and average treatment effect are displayed at the bottom. The show_supt and show_pointwise arguments control whether sup-t and pointwise bounds are included. For example, displaying only the plausible and pointwise bounds:

create_plot(pb, show_annotations = FALSE, show_pointwise = FALSE, show_supt = FALSE)

Customized plot showing only point estimates and restricted bounds

Parallel Processing

For datasets with many horizons, the restricted bounds calculation can be time-consuming due to the model selection procedure. The parallel argument enables parallel computation across the model universe. Use n_cores to specify the number of cores (the default uses all available cores minus one):

pb_parallel <- plausible_bounds(
  estimates = estimates_smooth,
  var = var_smooth,
  alpha = 0.05,
  parallel = TRUE,
  n_cores = 4
)

References

• Freyaldenhoven, S. and Hansen, C. (2026). “(Visualizing) Plausible Treatment Effect Paths.” Federal Reserve Bank of Philadelphia and University of Chicago.
• Berk, R., Brown, L., Buja, A., Zhang, K., and Zhao, L. (2013). “Valid post-selection inference.” Annals of Statistics, 41(2):802-837.