Dynamic Models API Reference¶

Module

Import: from panelbox.models.dynamic import LSDVC, LSDVCResults, AndersonHsiao Source: panelbox/models/dynamic/

Overview¶

Dynamic panel models include lagged dependent variables as regressors. The LSDVC estimator corrects the Nickell (1981) bias in Fixed Effects estimation using analytical bias formulas from Kiviet (1995), while Anderson-Hsiao provides a consistent IV estimator often used as an initial step.

Estimator	Class	Reference	Use Case
LSDVC	`LSDVC`	Kiviet (1995), Bruno (2005)	Bias-corrected FE for dynamic panels
Anderson-Hsiao	`AndersonHsiao`	Anderson & Hsiao (1982)	Consistent IV estimator; initial estimator for LSDVC

LSDVC vs GMM

LSDVC is preferred over GMM when N is small to moderate (N < 100) and T is not too large. GMM requires large N for its asymptotic properties, while LSDVC works well even with small N. See the decision guide for details.

Classes¶

LSDVC¶

Least Squares Dummy Variable Corrected estimator. Estimates a dynamic panel model with fixed effects and corrects for Nickell bias using analytical bias formulas.

The model is:

\[ y_{it} = \rho \, y_{i,t-1} + X_{it}'\beta + \alpha_i + \varepsilon_{it} \]

LSDVC(
    formula: str,
    data: pd.DataFrame,
    entity_col: str,
    time_col: str,
    lags: int = 1,
    initial_estimator: str = "ab",
    bias_order: int = 2,
    max_iter: int = 50,
    tol: float = 1e-6,
)

Parameter	Type	Default	Description
`formula`	`str`	required	R-style formula, e.g. `"y ~ x1 + x2"`. The lagged dependent variable is added automatically.
`data`	`pd.DataFrame`	required	Panel data in long format
`entity_col`	`str`	required	Column identifying entities (individuals/firms)
`time_col`	`str`	required	Column identifying time periods
`lags`	`int`	`1`	Number of AR lags (currently only 1 supported)
`initial_estimator`	`str`	`"ab"`	Initial consistent estimator: `"ah"` (Anderson-Hsiao), `"ab"` (Arellano-Bond), `"bb"` (Blundell-Bond)
`bias_order`	`int`	`2`	Order of Kiviet bias approximation (1, 2, or 3)
`max_iter`	`int`	`50`	Maximum iterations for iterated bias correction
`tol`	`float`	`1e-6`	Convergence tolerance for iterated bias correction

Initial Estimator Options¶

Value	Method	When to Use
`"ah"`	Anderson-Hsiao IV	Fast; good default for small panels
`"ab"`	Arellano-Bond (one-step)	More efficient; requires N > T
`"bb"`	Blundell-Bond (one-step)	Best for persistent processes (high \(\rho\))

Bias Order

Higher orders provide more accurate bias correction but may increase variance. Order 2 (default) is recommended for most applications. Order 3 adds the Bun & Kiviet (2003) refinement but gains are typically small.

`.fit()` Method¶

model.fit(
    n_bootstrap: int = 500,
    ci_level: float = 0.95,
    seed: int | None = None,
) -> LSDVCResults

Parameter	Type	Default	Description
`n_bootstrap`	`int`	`500`	Number of bootstrap replications for inference. Set to 0 to skip bootstrap.
`ci_level`	`float`	`0.95`	Confidence level for bootstrap percentile intervals
`seed`	`int \\| None`	`None`	Random seed for reproducibility

Returns: LSDVCResults

Pipeline:

Build lag matrices from panel data
LSDV (Fixed Effects with lagged dependent variable) estimation
Initial consistent estimate of \(\rho\) via the chosen initial estimator
Iterative Kiviet bias correction
Parametric bootstrap for standard errors and confidence intervals

Example¶

from panelbox.models.dynamic import LSDVC
from panelbox import load_grunfeld

data = load_grunfeld()

# Fit LSDVC with Arellano-Bond initial estimator, order-2 bias correction
model = LSDVC(
    "invest ~ value + capital", data, "firm", "year",
    initial_estimator="ab", bias_order=2,
)
results = model.fit(n_bootstrap=500, seed=42)
print(results.summary())

# Compare initial estimators
for init in ["ah", "ab", "bb"]:
    model = LSDVC("invest ~ value + capital", data, "firm", "year",
                   initial_estimator=init)
    res = model.fit(n_bootstrap=200, seed=42)
    print(f"{init.upper()}: rho = {res.params.iloc[0]:.4f}")

LSDVCResults¶

Results container for LSDVC estimation. Inherits from PanelResults and adds LSDVC-specific attributes for bias correction diagnostics and bootstrap inference.

Attributes¶

All standard PanelResults attributes (params, std_errors, cov_params, resid, fittedvalues, nobs, rsquared, etc.) plus:

Attribute	Type	Description
`bias_correction`	`pd.Series`	Bias correction vector applied to each parameter
`lsdv_params`	`pd.Series`	Uncorrected LSDV parameters (before bias correction)
`initial_estimator`	`str`	Method used for initial consistent estimate (`'ah'`, `'ab'`, `'bb'`)
`initial_rho`	`float`	Initial consistent estimate of \(\rho\)
`n_iterations`	`int`	Number of iterations until convergence
`converged`	`bool`	Whether the iterated bias correction converged
`bias_order`	`int`	Order of the Kiviet bias approximation used
`sigma2`	`float`	Error variance estimate from LSDV
`bootstrap_params`	`np.ndarray \\| None`	Bootstrap parameter distribution (B × K matrix), or None
`bootstrap_ci`	`tuple \\| None`	`(ci_lower, ci_upper)` bootstrap percentile confidence intervals

Methods¶

`.summary(title=None)`¶

Generate formatted summary of LSDVC estimation results. Includes:

Model information (formula, initial estimator, bias order, convergence)
Sample size details
Coefficient table with bootstrap standard errors, z-statistics, p-values, and confidence intervals
Bias correction comparison table (LSDV vs LSDVC estimates)

results = model.fit(n_bootstrap=500)
print(results.summary())

`.bias_summary()`¶

Focused table showing the magnitude of bias correction per parameter, including the initial estimator details and convergence information.

print(results.bias_summary())

`.plot_bootstrap_distribution(variable=None)`¶

Plot histogram of the bootstrap parameter distribution for a given variable.

Parameter	Type	Default	Description
`variable`	`str \\| None`	`None`	Variable name to plot. If None, plots the first parameter (lagged dependent variable).

Returns: matplotlib.figure.Figure

fig = results.plot_bootstrap_distribution("L.invest")
fig.savefig("bootstrap_rho.png")

Inspecting bootstrap results

Access the raw bootstrap distribution via results.bootstrap_params (a B × K numpy array) for custom analysis beyond the built-in plots.

AndersonHsiao¶

Anderson-Hsiao (1982) IV estimator for dynamic panel models. Provides consistent (but inefficient) estimation using instrumental variables on the first-differenced equation.

The model estimated is:

\[ \Delta y_{it} = \rho \, \Delta y_{i,t-1} + \Delta X_{it}'\beta + \Delta \varepsilon_{it} \]

using \(y_{i,t-2}\) (levels) or \(\Delta y_{i,t-2}\) (differences) as an instrument for the endogenous \(\Delta y_{i,t-1}\).

AndersonHsiao(
    data: pd.DataFrame,
    dep_var: str,
    entity_col: str,
    time_col: str,
    lags: int = 1,
    exog_vars: list[str] | None = None,
    instrument: str = "levels",
)

Parameter	Type	Default	Description
`data`	`pd.DataFrame`	required	Panel data in long format
`dep_var`	`str`	required	Name of the dependent variable
`entity_col`	`str`	required	Column identifying entities
`time_col`	`str`	required	Column identifying time periods
`lags`	`int`	`1`	Number of AR lags (currently only 1 supported)
`exog_vars`	`list[str] \\| None`	`None`	Names of exogenous regressors
`instrument`	`str`	`"levels"`	Instrument type: `"levels"` (\(y_{t-2}\), more efficient) or `"differences"` (\(\Delta y_{t-2}\))

`.fit()` Method¶

ah.fit() -> dict

Returns a dictionary with:

Key	Type	Description
`"rho"`	`float`	AR(1) coefficient estimate
`"beta"`	`np.ndarray \\| None`	Exogenous variable coefficients
`"params"`	`np.ndarray`	All parameters combined
`"resid"`	`np.ndarray`	Residuals
`"param_names"`	`list[str]`	Parameter labels
`"nobs"`	`int`	Number of observations used

Example¶

from panelbox.models.dynamic import AndersonHsiao
from panelbox import load_grunfeld

data = load_grunfeld()
ah = AndersonHsiao(data, "invest", "firm", "year",
                   exog_vars=["value", "capital"])
result = ah.fit()
print(f"rho = {result['rho']:.4f}")
print(f"N = {result['nobs']}")

Primary use case

Anderson-Hsiao is primarily used as an initial consistent estimator for LSDVC (initial_estimator="ah"). For standalone dynamic panel estimation, consider LSDVC or GMM estimators.

Comparison Example¶

from panelbox.models.dynamic import LSDVC
from panelbox import FixedEffects, load_grunfeld

data = load_grunfeld()

# Standard FE (biased with lagged dependent variable)
fe = FixedEffects("invest ~ value + capital", data, "firm", "year")
fe_result = fe.fit()

# LSDVC (bias-corrected) with different initial estimators
for init in ["ah", "ab", "bb"]:
    model = LSDVC("invest ~ value + capital", data, "firm", "year",
                   initial_estimator=init)
    result = model.fit(n_bootstrap=200, seed=42)
    rho = result.params.iloc[0]
    rho_lsdv = result.lsdv_params.iloc[0]
    print(f"LSDVC ({init.upper()}): rho={rho:.4f}  (LSDV={rho_lsdv:.4f}, "
          f"bias correction={result.bias_correction.iloc[0]:.4f})")

Dynamic Models API Reference¶

Overview¶

Classes¶

LSDVC¶

Initial Estimator Options¶

.fit() Method¶

Example¶

LSDVCResults¶

Attributes¶

Methods¶

.summary(title=None)¶

.bias_summary()¶

.plot_bootstrap_distribution(variable=None)¶

AndersonHsiao¶

.fit() Method¶

Example¶

Comparison Example¶

See Also¶

`.fit()` Method¶

`.summary(title=None)`¶

`.bias_summary()`¶

`.plot_bootstrap_distribution(variable=None)`¶

`.fit()` Method¶