panelmed

panelmed implements the double/debiased machine learning estimator for causal mediation analysis in panel data of Wu and Xiao (2026), arXiv:2610.21339, in Python. You give it a panel with an outcome, a mediator, a treatment and covariates; it returns the direct, indirect and total effects of the treatment with standard errors, tables ready for a manuscript, and figures.

• Documentation: https://merwanroudane.github.io/panelmed/
• PyPI: https://pypi.org/project/panelmed/
• Author: Dr Merwan Roudane · merwanroudane920@gmail.com · https://github.com/merwanroudane

Table of contents

1. What this package does
2. Start from absolutely nothing
3. Installation
4. Your first analysis in 60 seconds
5. Complete example with real data
6. Full syntax reference
7. Interpreting the output
8. Using your own Excel or CSV file
9. Exporting tables and figures
10. The method
11. Practical guidance and pitfalls
12. Limitations
13. References
14. Citation
15. Author and licence

1. What this package does

Many questions ask not only whether a treatment changes an outcome but through which channel. Employment raises local income: how much of that goes through wages? Mediation analysis splits the total effect into a direct effect and an indirect effect that passes through a mediator. With panel data the difficulty is that unobserved, time-invariant characteristics of each unit can drive treatment, mediator and outcome at once, and that observed confounders may act nonlinearly.

The estimator of Wu and Xiao (2026) handles both. It allows unit and period fixed effects in all three equations, lets unknown nonlinear functions of the covariates absorb observed confounding, and uses cross-fitting so that the machine-learning step does not bias the estimates.

What you get:

• Estimator: DML-GMM for the outcome and mediator equations (Algorithm 1 of the paper).
• Effects: controlled direct, natural direct, natural indirect and total effects, as level contrasts between two treatment histories and as derivatives.
• Inference: analytical standard errors from influence functions, joint Wald tests, and a cross-sectional bootstrap that repeats the whole procedure.
• Comparisons: the naive benchmark of the paper, and a paired test of whether two specifications give different decompositions.
• Beyond the paper: two-step GMM with the Hansen J test, an analytical standard error for the mediated share, and a moment-diagnostic figure.
• Output: text, LaTeX, Markdown and HTML tables; six figure types.
• Data: the U.S. county panel used in the paper's application, and the paper's Monte Carlo generator.

2. Start from absolutely nothing

If you have never used Python, do these four steps once. Fifteen minutes.

Step 1 - install Python

Download Anaconda (recommended for researchers: it includes Python, Jupyter and Spyder in one installer) from https://www.anaconda.com/download, or plain Python from https://www.python.org/downloads/. On the python.org installer, tick "Add Python to PATH" on the first screen.

Step 2 - open a terminal

• Windows: press Win, type Anaconda Prompt (or cmd), press Enter.
• macOS: press Cmd+Space, type Terminal, press Enter.

Check it works. Type this and press Enter:

python --version

You should see something like Python 3.11.5.

Step 3 - install this package

pip install panelmed

Step 4 - run your first command

Type python, press Enter, then paste:

import panelmed
print(panelmed.__version__)

Type exit() to leave.

Where do I write my code?

Any of these; pick one and stay with it.

3. Installation

pip install panelmed

With interactive 3-D surfaces:

pip install "panelmed[plotly]"

Development version:

pip install git+https://github.com/merwanroudane/panelmed.git

Requirements: Python >= 3.10, numpy, scipy, pandas, matplotlib, scikit-learn, joblib (installed automatically).

4. Your first analysis in 60 seconds

from panelmed import PanelMediation, simulate_panel

# a simulated panel with known true effects: 1,600 units, 10 periods
df = simulate_panel(n=1600, t=10, seed=2026, as_frame=True)

model = PanelMediation(
    "y", "m", "d", ["x1", "x2", "x3", "x4", "x5"],      # outcome, mediator, treatment, covariates
    data=df, unit="unit", time="time",
    outcome=["d", "d**2", "L1_d", "d*x3", "m", "L1_m"],  # what enters the outcome equation
    mediator=["d", "d**2", "L1_d"],                      # what enters the mediator equation
)
res = model.fit(n_folds=10, seed=2026)
print(res.effects(d=1, d_star=0).summary())

======================================================================================
                  Panel mediation, level effects  (Wu & Xiao (2026))
======================================================================================
Outcome            : y                                 Units (N)        :         1600
Mediator           : m                                 Periods (T)      :           10
Treatment          : d                                 Folds (K)        :           10
Covariates         : 5                                 Instrument lags  :            2
Basis              : polynomial deg. 3 (55 terms)      Moments          :           83
Estimator          : DML-GMM                           Weighting        :     identity
--------------------------------------------------------------------------------------
Estimand        Coef.  Std.Err.        z   P>|z|      [95% Conf. Int.]
--------------------------------------------------------------------------------------
Effects
CDE            4.5065    0.0737   61.184   0.000     4.3621     4.6508 ***
NDE            4.5065    0.0737   61.184   0.000     4.3621     4.6508 ***
NIE            2.7899    0.0942   29.630   0.000     2.6053     2.9744 ***
TE             7.2964    0.1072   68.033   0.000     7.0862     7.5066 ***
--------------------------------------------------------------------------------------
Mediated share
NIE/TE         0.3824    0.0095                      0.3637     0.4010
======================================================================================
Significance: *** p<0.01, ** p<0.05, * p<0.10
Standard errors: estimated influence functions (Eq. 21, Theorem 3).
Treatment history d versus d*; mediator fixed at m for the CDE.
Total = natural direct + natural indirect. No test against zero is reported for the mediated share.

The true values in this simulated design are CDE = NDE = 4.5, NIE = 3 and TE = 7.5.

5. Complete example with real data

The package ships the panel used in the empirical application of Wu and Xiao (2026, Section 6): 3,058 U.S. counties observed every year from 2000 to 2019, assembled by Cerqua, Letta and Pinto (2026). The question: when the workplace employment rate of a county rises, personal income per capita rises too. How much of that goes through the average wage?

load_us_counties() returns the panel prepared as in the paper: years 2001-2019 (the year 2000 only supplies the lagged covariates), every variable winsorised at its 1% and 99% quantiles and standardised. Effects are therefore in standard deviations of the outcome per standard deviation of the treatment.

The whole example is the script examples/panelmed_empirical.py; the output below is copied from a run of that script.

Step 1 - load the data

from panelmed import PanelMediation, compare_effects, load_us_counties
from panelmed.datasets import US_COUNTIES_COVARIATES as X

df = load_us_counties()
print(df.head(3).round(3).to_string(index=False))

 county_id        county_name  year      y      m      d  log_pop  ui_share  share_under18  share_over65  births  deaths  mig_international  mig_domestic
      1001 Autauga County, AL  2001 -1.054 -1.189 -0.929    0.297    -0.769          1.507        -1.392  -2.104  -2.174             -0.581         0.353
      1001 Autauga County, AL  2002 -0.991 -1.079 -1.036    0.310    -0.636          1.425        -1.368   0.798  -0.457             -0.568         0.714
      1001 Autauga County, AL  2003 -0.795 -1.029 -1.013    0.326    -0.450          1.473        -1.346   0.471  -0.427             -0.960         1.156

Counties: 3058, years: 2001-2019

Step 2 - Specification I: the same effect in every county

common = dict(data=df, unit="county_id", time="year", basis_degree=3, basis_max_vars=2)

spec1 = PanelMediation("y", "m", "d", X, outcome=["d", "m"], mediator=["d"], **common)
res1 = spec1.fit(n_folds=5, max_lags=2, seed=2026)
print(res1.summary())

======================================================================================
              Panel mediation, structural parameters  (Wu & Xiao (2026))
======================================================================================
Outcome            : y                                 Units (N)        :         3058
Mediator           : m                                 Periods (T)      :           19
Treatment          : d                                 Folds (K)        :            5
Covariates         : 8                                 Instrument lags  :            2
Basis              : polynomial deg. 3 (108 terms)     Moments          :           99
Estimator          : DML-GMM                           Weighting        :     identity
--------------------------------------------------------------------------------------
Parameter       Coef.  Std.Err.        z   P>|z|      [95% Conf. Int.]
--------------------------------------------------------------------------------------
Outcome equation (theta)
theta[d]       0.2089    0.0875    2.388   0.017     0.0375     0.3803 **
theta[m]       0.6593    0.0943    6.991   0.000     0.4744     0.8441 ***
--------------------------------------------------------------------------------------
Mediator equation (gamma)
gamma[d]       0.6063    0.0780    7.775   0.000     0.4535     0.7592 ***
======================================================================================
Significance: *** p<0.01, ** p<0.05, * p<0.10
Sandwich standard errors from the stacked individual scores (Eq. A.15); N -> infinity, T fixed.
Moments: 66 outcome + 33 mediator; Jacobian condition number 2.0.

Step 3 - Specification II: effects that vary with county characteristics

inter = [f"d*{c}" for c in X]          # d*log_pop, d*ui_share, ...
spec2 = PanelMediation("y", "m", "d", X, outcome=["d", "m"] + inter,
                       mediator=["d"] + inter, **common)
res2 = spec2.fit(n_folds=5, max_lags=2, seed=2026)
print(res2.summary())

==================================================================================================
                    Panel mediation, structural parameters  (Wu & Xiao (2026))
==================================================================================================
Outcome            : y                                             Units (N)        :         3058
Mediator           : m                                             Periods (T)      :           19
Treatment          : d                                             Folds (K)        :            5
Covariates         : 8                                             Instrument lags  :            2
Basis              : polynomial deg. 3 (108 terms)                 Moments          :          627
Estimator          : DML-GMM                                       Weighting        :     identity
--------------------------------------------------------------------------------------------------
Parameter                       Coef.  Std.Err.        z   P>|z|      [95% Conf. Int.]
--------------------------------------------------------------------------------------------------
Outcome equation (theta)
theta[d]                       0.3657    0.1176    3.109   0.002     0.1352     0.5962 ***
theta[m]                       0.5264    0.1110    4.743   0.000     0.3088     0.7439 ***
theta[d*log_pop]              -0.0058    0.0613   -0.095   0.924    -0.1259     0.1142
theta[d*ui_share]              0.0045    0.0102    0.439   0.661    -0.0155     0.0245
theta[d*share_under18]         0.1907    0.0748    2.550   0.011     0.0441     0.3373 **
theta[d*share_over65]          0.2314    0.0648    3.571   0.000     0.1044     0.3583 ***
theta[d*births]               -0.0206    0.0110   -1.873   0.061    -0.0421     0.0010 *
theta[d*deaths]                0.0089    0.0132    0.669   0.503    -0.0171     0.0348
theta[d*mig_international]     0.0344    0.0116    2.966   0.003     0.0117     0.0572 ***
theta[d*mig_domestic]         -0.0055    0.0079   -0.698   0.485    -0.0210     0.0100
--------------------------------------------------------------------------------------------------
Mediator equation (gamma)
gamma[d]                       0.4715    0.0905    5.210   0.000     0.2942     0.6489 ***
gamma[d*log_pop]              -0.0247    0.0560   -0.440   0.660    -0.1345     0.0851
gamma[d*ui_share]             -0.0217    0.0074   -2.923   0.003    -0.0362    -0.0071 ***
gamma[d*share_under18]        -0.0331    0.0279   -1.189   0.234    -0.0877     0.0215
gamma[d*share_over65]         -0.0038    0.0346   -0.109   0.913    -0.0715     0.0640
gamma[d*births]               -0.0083    0.0067   -1.232   0.218    -0.0214     0.0049
gamma[d*deaths]                0.0125    0.0092    1.359   0.174    -0.0055     0.0306
gamma[d*mig_international]     0.0006    0.0062    0.102   0.919    -0.0115     0.0127
gamma[d*mig_domestic]         -0.0012    0.0054   -0.222   0.824    -0.0118     0.0094
==================================================================================================
Significance: *** p<0.01, ** p<0.05, * p<0.10
Sandwich standard errors from the stacked individual scores (Eq. A.15); N -> infinity, T fixed.
Moments: 330 outcome + 297 mediator; Jacobian condition number 23.6.

Step 4 - direct, indirect and total effects, averaged over 2001-2019

eff1 = res1.derivatives(d=0, period="average")
eff2 = res2.derivatives(d=0, period="average")
print(eff1.summary())
print(eff2.summary())

======================================================================================
               Panel mediation, derivative effects  (Wu & Xiao (2026))
======================================================================================
Outcome            : y                                 Units (N)        :         3058
Mediator           : m                                 Periods (T)      :           19
Treatment          : d                                 Folds (K)        :            5
Covariates         : 8                                 Instrument lags  :            2
Basis              : polynomial deg. 3 (108 terms)     Moments          :           99
Estimator          : DML-GMM                           Weighting        :     identity
--------------------------------------------------------------------------------------
Estimand        Coef.  Std.Err.        z   P>|z|      [95% Conf. Int.]
--------------------------------------------------------------------------------------
Effects
CDD            0.2089    0.0875    2.388   0.017     0.0375     0.3803 **
NDD            0.2089    0.0875    2.388   0.017     0.0375     0.3803 **
NID            0.3997    0.0703    5.689   0.000     0.2620     0.5375 ***
TD             0.6086    0.0786    7.746   0.000     0.4546     0.7626 ***
--------------------------------------------------------------------------------------
Mediated share
NID/TD         0.6568    0.1199                      0.4218     0.8918
======================================================================================
Significance: *** p<0.01, ** p<0.05, * p<0.10
Standard errors: estimated influence functions (Eq. 21, Theorem 3).
Derivative with respect to the treatment of the effect's own period.
Total = natural direct + natural indirect. No test against zero is reported for the mediated share.

======================================================================================
               Panel mediation, derivative effects  (Wu & Xiao (2026))
======================================================================================
Outcome            : y                                 Units (N)        :         3058
Mediator           : m                                 Periods (T)      :           19
Treatment          : d                                 Folds (K)        :            5
Covariates         : 8                                 Instrument lags  :            2
Basis              : polynomial deg. 3 (108 terms)     Moments          :          627
Estimator          : DML-GMM                           Weighting        :     identity
--------------------------------------------------------------------------------------
Estimand        Coef.  Std.Err.        z   P>|z|      [95% Conf. Int.]
--------------------------------------------------------------------------------------
Effects
CDD            0.3657    0.1177    3.107   0.002     0.1350     0.5964 ***
NDD            0.3657    0.1177    3.107   0.002     0.1350     0.5964 ***
NID            0.2482    0.0478    5.193   0.000     0.1545     0.3419 ***
TD             0.6139    0.1051    5.843   0.000     0.4080     0.8198 ***
--------------------------------------------------------------------------------------
Mediated share
NID/TD         0.4043    0.1068                      0.1949     0.6137
======================================================================================
Significance: *** p<0.01, ** p<0.05, * p<0.10
Standard errors: estimated influence functions (Eq. 21, Theorem 3).
Derivative with respect to the treatment of the effect's own period.
Total = natural direct + natural indirect. No test against zero is reported for the mediated share.

Compared with Table 2 of the paper:

The small differences come from the random fold split, which the paper does not publish. The paper reports bootstrap standard errors (500 replications); the tables above use the analytical ones. For bootstrap standard errors run python panelmed_empirical.py 200 (slow: every replication repeats the whole procedure).

Step 5 - does the decomposition change between the two specifications?

print(compare_effects(eff1, eff2))

======================================================================================
         Paired difference between specifications (B - A)  (Wu & Xiao (2026))
======================================================================================
Outcome            : y                                 Units (N)        :         3058
Mediator           : m                                 Periods (T)      :           19
Treatment          : d                                 Folds (K)        :            5
Covariates         : 8                                 Instrument lags  :            2
Basis              : polynomial deg. 3 (108 terms)     Moments          :          627
Estimator          : DML-GMM                           Weighting        :     identity
--------------------------------------------------------------------------------------
Estimand        Coef.  Std.Err.        z   P>|z|      [95% Conf. Int.]
--------------------------------------------------------------------------------------
CDD            0.1568    0.0762    2.056   0.040     0.0074     0.3062 **
NDD            0.1568    0.0762    2.056   0.040     0.0074     0.3062 **
NID           -0.1515    0.0510   -2.972   0.003    -0.2515    -0.0516 ***
TD             0.0053    0.0698    0.075   0.940    -0.1315     0.1421
NID/TD        -0.2525    0.0813   -3.105   0.002    -0.4119    -0.0931 ***
======================================================================================
Significance: *** p<0.01, ** p<0.05, * p<0.10
Standard errors: difference of the unit-level influence functions.

The total effect is the same in both specifications, but the share going through wages drops from about two thirds to about 40%. This is the paper's main empirical message: a stable total effect does not validate its direct/indirect split.

Step 6 - figures

fig, ax = eff2.plot()
fig.savefig("decomposition.pdf")
res2.derivatives(d=0, period="all").plot_profile()
res2.plot_dose_response()
res2.plot_surface()
res1.plot_moments()

Direct, indirect and total effect under Specification II, with 95% intervals.

Year-by-year effects under Specification II. All three profiles are mildly U-shaped, as in Figure A.1 of the paper.

Effects of moving the employment rate from 0 (its mean) to each value d, in standard deviations.

Total effect for every pair of treatment histories (d, d*).

Standardised sample moments of Specification I at the estimate.

6. Full syntax reference

PanelMediation(y, m, d, x=None, *, outcome, mediator, ...)

Sets up the model of Eq. (4) of the paper.

How to write a dictionary. Each term is a Python expression using d (treatment), m (mediator), your covariate names, and the lag prefix L1_, L2_, ...

Do not include terms that depend on covariates only; the nuisance functions absorb them.

model.fit(n_folds=5, max_lags=2, penalty="plugin", method="dml", weighting="identity", seed=None)

Returns a PanelMediationResults object.

PanelMediationResults

Arguments of effects and derivatives:

EffectsResult

Other functions

7. Interpreting the output

The coefficient table (res.summary()). theta[...] are the coefficients of the outcome equation and gamma[...] those of the mediator equation. theta[m] is the effect of the mediator on the outcome; gamma[d] is the effect of the treatment on the mediator. Their product is the core of the indirect effect. The last line reports the number of moment conditions and the condition number of the moment Jacobian: a very large condition number (above about 1,000) signals weak identification.

The effects table (res.effects(...).summary()).

CDE and NDE coincide when the outcome dictionary has no treatment-mediator interaction. The derivative table reads the same way for an infinitesimal increase in one period's treatment (CDD, NDD, NID, TD).

Stars mark p-values below 0.01, 0.05 and 0.10 for the null of a zero effect. Standard errors are valid as the number of units grows with the number of periods fixed, and allow any serial dependence within a unit.

Figures.

• Decomposition (eff.plot()): one row per effect with its confidence interval.
• Profile (eff.plot_profile()): effect in each period with a confidence band. When the coefficients are constant, movements over time come from changes in the covariates that interact with the treatment.
• Dose-response (res.plot_dose_response()): direct, indirect and total effect of moving the treatment from d* to each value d.
• Surface (res.plot_surface()): an effect for every pair (d, d*); the white line marks zero.
• Moment diagnostic (res.plot_moments()): each bar is a standardised sample moment. Many bars far outside the dashed lines suggest a misspecified dictionary or invalid instruments.

8. Using your own Excel or CSV file

Your file must be in long format: one row per unit and period, with a column identifying the unit and a column identifying the period.

import pandas as pd
from panelmed import PanelMediation

df = pd.read_excel("C:/Users/you/Documents/mydata.xlsx")   # or pd.read_csv("mydata.csv")
df = df.dropna(subset=["income", "wage", "employment", "pop", "age"])

model = PanelMediation(
    "income", "wage", "employment", ["pop", "age"],
    data=df, unit="county", time="year",
    outcome=["employment", "wage"], mediator=["employment"],
)
res = model.fit(seed=1)
print(res.summary())
print(res.effects(d=1, d_star=0).summary())

Notes:

• On Windows use forward slashes C:/... or double backslashes C:\\....
• The panel must be balanced: every unit observed in every period, no missing values.
• Covariate column names must be plain identifiers (letters, digits, underscore) and cannot be d or m; rename with df = df.rename(columns={"old name": "new_name"}).
• Standardising the variables ((x - x.mean()) / x.std()) makes d=1 versus d_star=0 a one-standard-deviation change.

9. Exporting tables and figures

print(res.to_latex())                           # paste into your manuscript
res.to_frame().to_excel("coefficients.xlsx")    # open in Excel or Word
eff = res.effects(d=1, d_star=0)
eff.to_frame().to_excel("effects.xlsx")

fig, ax = eff.plot()
fig.savefig("figure1.pdf")                      # vector, for the journal
fig.savefig("figure1.png", dpi=300)             # raster, for slides

The LaTeX output uses booktabs and the \sym command; add \newcommand{\sym}[1]{\ensuremath{^{#1}}} to your preamble.

10. The method

The model for unit i and period t is

Y_it = tau_Y(D, M, X)' theta + f0(X) + alpha^Y_i + lambda^Y_t + eps^Y_it
M_it = tau_M(D, X)'   gamma + g0(X) + alpha^M_i + lambda^M_t + eps^M_it

where tau_Y and tau_M are your dictionaries, f0 and g0 are unknown functions of the covariates, alpha are unit effects and lambda period effects. Identification rests on sequential conditional independence given the observed history and the unit effects (Assumption 4 of the paper), so any time-invariant confounder is allowed.

Estimation, in the order of the code:

The full map from every function to its equation, and the choices made where the paper's text leaves room, are in docs/methods.md.

Check against the paper. examples/monte_carlo_table1.py reruns the design of the paper's Table 1 (N = 1600, T = 10). With 300 replications:

The naive benchmark fails in the same way as in the paper: gamma[d] has mean 0.7996 and coverage 0.000 (paper: 0.8230 and 0.000), and the total effect has coverage 0.003 (paper: 0.012). The paper uses 1,000 replications, so coverage differences of about 0.02 are within Monte Carlo error.

11. Practical guidance and pitfalls

• Sample size. The theory is for many units and few periods. Use at least a few hundred units; with fewer, reduce n_folds to 3-5 and keep the dictionaries small.
• Periods. You need T >= 3. Two periods are consumed by differencing and by the lagged instruments.
• Folds. n_folds=5 is a sensible default; 10 uses more data for the nuisance functions when N is large. Results depend slightly on the random split, so always set seed and check that conclusions hold for a second seed.
• Dictionaries. Start simple (["d", "m"] and ["d"]), then add lags, squares and interactions. The paper's application shows that the total effect can be stable while the direct/indirect split changes with the specification; report both.
• Identification error. If fit raises IdentificationError, the instruments cannot separate the coefficients: increase max_lags, simplify the dictionary, or check that the treatment varies within units over time.
• Standardise. Put variables on comparable scales before fitting; the polynomial library is numerically better behaved and effects are easier to read.
• Bootstrap. Each replication repeats the full procedure, so it is slow on large panels; use n_jobs=-2 to run in parallel. Analytical standard errors are instant.
• A common mistake: putting the mediator in the mediator dictionary, or leaving it out of the outcome dictionary. The package stops with a clear message in both cases.

12. Limitations

• One scalar, continuous mediator. Vector mediators (Appendix A.1 of the paper) are not implemented.
• Balanced panels only, as in the paper.
• The nuisance functions are fitted by a penalised regression on a basis of the covariates; other machine-learning learners are not offered, because under sequential exogeneity the paper only specifies penalised IV/GMM first stages.
• The paper does not prove bootstrap consistency; it uses the bootstrap in its application, and so does this package.
• The causal interpretation requires Assumption 4 of the paper (sequential conditional independence given the unit effects). The package cannot test it.
• No public code of the original authors exists for this estimator. Agreement was checked against the paper's reported Monte Carlo and empirical tables, not against another implementation.

13. References

• Wu, P. and Xiao, Z. (2026). "Causal Mediation Analysis for Panel Data with Double/Debiased Machine Learning". arXiv:2610.21339. https://doi.org/10.48550/arXiv.2610.21339
• Wu, P. and Xiao, Z. (2026). "Double/Debiased Machine Learning for Treatment Effects in Dynamic Panels". Journal of the American Statistical Association. https://doi.org/10.1080/01621459.2026.2696607
• Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W. and Robins, J. (2018). "Double/debiased machine learning for treatment and structural parameters". The Econometrics Journal, 21(1), C1-C68. https://doi.org/10.1111/ectj.12097
• Kapetanios, G. (2008). "A bootstrap procedure for panel data sets with many cross-sectional units". The Econometrics Journal, 11(2), 377-395. https://doi.org/10.1111/j.1368-423X.2008.00243.x
• Hansen, L. P. (1982). "Large sample properties of generalized method of moments estimators". Econometrica, 50(4), 1029-1054. https://doi.org/10.2307/1912775
• Cerqua, A., Letta, M. and Pinto, G. (2026). "On the (mis)use of machine learning with panel data". Oxford Bulletin of Economics and Statistics, 88(3), 506-518. https://doi.org/10.1111/obes.70019

(Every DOI above was checked on https://doi.org/.)

14. Citation

@software{roudane_panelmed,
  author  = {Roudane, Merwan},
  title   = {{panelmed}: causal mediation analysis for panel data with double/debiased machine learning},
  year    = {2026},
  url     = {https://github.com/merwanroudane/panelmed},
  version = {0.1.1}
}

Please also cite the original paper: Wu, P. and Xiao, Z. (2026), "Causal Mediation Analysis for Panel Data with Double/Debiased Machine Learning", arXiv:2610.21339, https://doi.org/10.48550/arXiv.2610.21339.

15. Author and licence

Dr Merwan Roudane merwanroudane920@gmail.com https://github.com/merwanroudane

Released under the MIT Licence; see LICENSE.