DynamicPanelModels.jl

Panel data with a lagged dependent variable ($y_{it} = \alpha y_{i,t-1} + x_{it}'\beta + \eta_i + \varepsilon_{it}$) cannot be consistently estimated with OLS or the standard fixed-effects (within) estimator — removing the individual effect $\eta_i$ by differencing or demeaning mechanically correlates the transformed lagged-dependent-variable regressor with the transformed error (Nickell, 1981). DynamicPanelModels.jl provides the standard GMM fix for this: the Arellano-Bond (Difference GMM), Blundell-Bond (System GMM), and Anderson-Hsiao estimators, with correct instrument construction, robust and Windmeijer-corrected inference, and the diagnostic tests needed to check whether the identifying assumptions actually hold before trusting the results.

This package covers linear dynamic panel models (a continuous dependent variable with a lagged-dependent-variable regressor). Nonlinear dynamic panel models — binary choice (logit/probit) or count data (Poisson/negative binomial) with a lagged dependent variable — are a distinct econometric problem (the fixed effect cannot be differenced away, and consistent estimators such as Honoré-Kyriazidou (2000) have no widely-used reference implementation to validate against) and are out of scope here.

The package targets performance on large-$N$ datasets through sparse matrix operations, correct inference via robust and Windmeijer-corrected standard errors, and a suite of diagnostic tools and visualizations for checking model validity.

Features

Difference GMM (Arellano-Bond, 1991) removes fixed effects by first-differencing and supports one-step and two-step estimation, robust standard errors, and instrument collapsing for large instrument sets. System GMM (Blundell-Bond, 1998) jointly estimates the differenced and level equations (the latter with its own intercept, _cons, as in xtabond2), offering efficiency gains for persistent series. Anderson-Hsiao (1981) is provided as a simple IV baseline for comparison.

Standard errors are one-step or two-step, cluster-robust, with the Windmeijer (2005) finite-sample correction available for the two-step case. Regressors declared via exog are used as their own ("IV-style") GMM instruments, following standard dynamic panel practice (Roodman, 2009), which improves identification and efficiency for strictly exogenous covariates.

Diagnostics include the Sargan/Hansen J-test for overidentifying restrictions, Arellano-Bond AR(1)/AR(2) tests for serial correlation, an instrument proliferation check, a Wald test for linear restrictions, a Jarque-Bera normality test, a pseudo-$R^2$, and a difference-in-Hansen (C) test for comparing nested instrument sets. Diagnostic plots (residuals, fitted vs. actual, Q-Q, and a combined dashboard) are available via RecipesBase.

The package integrates with the wider ecosystem: any Tables.jl-compatible input is accepted (not only DataFrames), formulas can be given as strings or as StatsModels @formulas, and fitted models implement the standard StatsAPI accessors (coef, vcov, stderror, residuals, fitted, predict, confint, nobs, dof, formula).

The formula syntax supports whitelisted transforms (log, log10, log2, exp, sqrt, abs) on either side of the formula and nested in lags (log(y) ~ lag(log(y)) + log(x)), explicit and ranged lags (lag(x, 2), lag(x, 0:1)), and automatic period dummies via time_effects=true. Instrument sets can be controlled with forward orthogonal deviations (transform=:fod, better for unbalanced panels), xtabond2-style lag limits (min_lag/max_lag), instrument collapsing (collapse), and automatic dropping of collinear instruments.

Validation

The estimators are checked against published and reference-implementation results, not just internal consistency. On the Arellano-Bond (1991) EmplUK data (examples/example.jl, and test/test_replication.jl with a bundled copy), the package reproduces Table 4 of the paper to the published precision: two-step $n_{t-1} = 0.629$, $n_{t-2} = -0.065$, $w = -0.526$, $k = 0.278$, $ys = 0.592$, Hansen $J = 31.4$ (25 df); one-step $n_{t-1} = 0.686$ (0.145), Sargan 65.8. Coefficients, robust and Windmeijer-corrected standard errors, the J-statistics and the AR(1)/AR(2) statistics all match R's plm::pgmm, plm::vcovHC and plm::mtest to three or four decimals, and Anderson-Hsiao matches a hand-computed 2SLS to machine precision.

Installation

The package is registered in the Julia General registry and requires Julia 1.12 or higher:

using Pkg
Pkg.add("DynamicPanelModels")

For the development version, Pkg.add(url="https://github.com/MichalS16/DynamicPanelModels.jl").

Quick start

using DynamicPanelModels
using DataFrames
using Plots

# df must be in long format with :id and :time columns, e.g.
# | id: 1 | time: 2020 | y: 5.2 | x1: 0.5 |

# x1 is declared exogenous via `exog`, so it is additionally used as its own
# ("IV-style") instrument, following standard GMM dynamic panel practice.
model = fit(DifferenceGMM(robust=true), df;
            formula = "y ~ lag(y) + x1",
            id_col = :id,
            time_col = :time,
            exog = ["x1"])

println(model)
diagnose(model)
plot(model)

See the Getting Started guide for a full walkthrough.

Key functions

Component	Description
fit(Estimator, df; ...)	Fits the specified GMM estimator to the provided panel data. Key arguments: formula, id_col, time_col, exog.
exog = ["x1", ...]	Names of RHS regressors (as they appear in formula) that are strictly exogenous; each is additionally used as its own GMM instrument. Defaults to none — non-lagged regressors are otherwise only weakly identified through incidental correlation with the y-lag instruments.
time_effects = true	Adds automatically-generated exogenous period dummies (one per period identified in the transformed sample) to absorb common time shocks.
DifferenceGMM(robust=true)	Arellano-Bond (1991). Settings: robust, steps, windmeijer.
SystemGMM(robust=true)	Blundell-Bond (1998), for persistent series ($\rho \approx 1$). Settings: robust, steps, windmeijer.
AndersonHsiao()	Anderson-Hsiao (1981) IV baseline (no settings): one lagged level $y_{i,t-k-1}$ per lag(y, k) regressor.
diagnose(model)	Runs Sargan, AR, and normality tests together.
sargan_test(model)	Tests instrument validity (overidentifying restrictions).
ar_test(model, order)	Arellano-Bond test for serial correlation of order 1, 2, …; cached after diagnose.
check_proliferation(model)	Checks whether instrument count exceeds group count.
wald_test(model, R, r)	Wald test for linear restrictions $R\beta = r$.
diff_hansen_test(restricted, unrestricted)	Difference-in-Hansen test for a nested instrument subset (e.g. the same model with and without exog).
plot(model) / plot(model, :residuals) / :fitted / :qq / :histogram	Diagnostic dashboard or individual plots.
coef, vcov, stderror, confint, residuals, fitted, nobs, ngroups, ninstruments	Standard result accessors.

Current limitations

First differences (:fd) and forward orthogonal deviations (:fod, Arellano-Bover) are supported for Difference GMM; System GMM currently supports first differences only. Unbalanced panels are generally supported, but highly sparse datasets may need manual preprocessing to avoid estimation errors or excessive data loss. Estimation is restricted to linear dynamic panel models with continuous dependent variables (see the scope note above). The Blundell-Bond level-equation instruments are valid only under the mean-stationarity assumption (the deviation of the initial condition from its long-run mean is uncorrelated with that long-run mean); this is not automatically tested, so inspect the Sargan/Hansen diagnostics before trusting System GMM results on persistent series.

Contributing

Contributions are welcome. See CONTRIBUTING.md for the development workflow and the Code of Conduct.

References

Nickell, S. (1981). Biases in dynamic models with fixed effects. Econometrica, 49(6), 1417-1426.

Arellano, M., & Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. The Review of Economic Studies, 58(2), 277-297.

Blundell, R., & Bond, S. (1998). Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics, 87(1), 115-143.

Windmeijer, F. (2005). A finite sample correction for the variance of linear efficient two-step GMM estimators. Journal of Econometrics, 126(1), 25-51.

Roodman, D. (2009). How to do xtabond2: An introduction to difference and system GMM in Stata. The Stata Journal, 9(1), 86-136.

Anderson, T. W., & Hsiao, C. (1981). Estimation of dynamic models with error components. Journal of the American Statistical Association, 76(375), 598-606.

Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press.