Dynamic Spatial Panel¶

Quick Reference

Class: panelbox.models.spatial.DynamicSpatialPanel Import: from panelbox.models.spatial import DynamicSpatialPanel Stata equivalent: No direct equivalent (custom GMM) R equivalent: SDPDmod::SDPDm()

Overview¶

The Dynamic Spatial Panel model combines temporal dynamics (a lagged dependent variable) with spatial dependence (spatial lag of the dependent variable). This is the most realistic specification for many applied settings where both time persistence and spatial spillovers are present.

The model is specified as:

\[y_{it} = \gamma y_{i,t-1} + \rho W y_{it} + X_{it}\beta + \alpha_i + \varepsilon_{it}\]

where:

\(\gamma\) is the temporal autoregressive parameter (persistence)
\(\rho\) is the spatial autoregressive parameter (spillovers across units)
\(y_{i,t-1}\) is the lagged dependent variable
\(Wy_{it}\) is the contemporaneous spatial lag
\(X_{it}\) are exogenous covariates
\(\alpha_i\) are individual fixed effects
\(\varepsilon_{it}\) is the idiosyncratic error

Both \(y_{i,t-1}\) and \(Wy_{it}\) are endogenous: the lagged dependent variable is correlated with \(\alpha_i\), and the spatial lag creates simultaneous feedback. This double endogeneity requires instrumental variable or GMM estimation.

Quick Example¶

from panelbox.models.spatial import DynamicSpatialPanel, SpatialWeights

# Create weight matrix
W = SpatialWeights.from_contiguity(gdf, criterion='queen')

# Fit dynamic spatial panel with GMM
model = DynamicSpatialPanel("y ~ x1 + x2", data, "region", "year", W=W.matrix)
results = model.fit(effects='fixed', method='gmm', lags=1, spatial_lags=1)

# View results
print(results.summary())
print(f"Temporal persistence (gamma): {results.params['y_lag1']:.4f}")
print(f"Spatial spillover (rho):      {results.rho:.4f}")

# Impulse response: how a shock to entity 0 propagates through space and time
irf = results.compute_impulse_response(shock_entity=0, periods=10)

When to Use¶

Regional economics: GDP growth persists over time AND spills across regions
Epidemiology: disease prevalence has temporal persistence AND geographic contagion
Housing markets: prices have momentum AND are influenced by neighboring areas
Technology diffusion: adoption has path dependence AND network effects
Fiscal policy: public spending has inertia AND fiscal competition between jurisdictions

In general, use the dynamic spatial model when:

You suspect both temporal autocorrelation (persistence, momentum)
AND spatial autocorrelation (spillovers, contagion, competition)
Standard spatial models (SAR, SEM, SDM) ignore the lagged dependent variable
Standard dynamic models (Arellano-Bond) ignore spatial dependence

Key Assumptions

Balanced panel: required for the spatial structure
Stationarity: \(|\gamma| < 1\) and \(|\rho| < 1\) for a stable process
Sufficient time periods: at least \(T \geq 4\) to construct valid instruments (after differencing and lagging)
Exogeneity of \(X\): covariates must be strictly exogenous
Fixed \(W\): the spatial weight matrix is time-invariant

Detailed Guide¶

Data Requirements¶

The dynamic spatial model uses first-differencing and lagging, which consumes time periods:

# Minimum T depends on lag structure
# With lags=1, spatial_lags=1, time_lags=2:
# Need at least T >= 4 (1 for lag + 2 for instruments + 1 for differencing)

T_actual = data.groupby("region").size().iloc[0]
print(f"Available time periods: {T_actual}")
print(f"Minimum required: 4")

Estimation¶

The model is estimated via two-step efficient GMM:

Step 1 (First Stage): 2SLS with instruments consisting of:

Deeper lags of \(y\): \(y_{i,t-2}, y_{i,t-3}, \ldots\)
Spatial lags of \(X\): \(WX_{it}, W^2X_{it}\)
Interactions: \(Wy_{i,t-2}\), etc.

Step 2 (Second Stage): Optimal GMM with the efficient weighting matrix computed from first-stage residuals.

results = model.fit(
    effects='fixed',
    method='gmm',
    lags=1,           # Number of temporal lags of y
    spatial_lags=1,   # Number of spatial lags in instruments
    time_lags=2,      # Depth of temporal instruments
    maxiter=1000,
    tol=1e-6,
    verbose=False
)

Interpreting Results¶

Parameters¶

Parameter	Symbol	Interpretation
`y_lag1`	\(\gamma\)	Temporal persistence: how much of last period's outcome carries forward
`rho`	\(\rho\)	Spatial spillover: how much neighbors' current outcomes affect own outcome
`x1`, `x2`, ...	\(\beta\)	Direct effects of covariates

Short-Run vs Long-Run Effects¶

The model produces different multiplier effects over time:

Short-run (contemporaneous): effect within the same period, through spatial multiplier \((I - \rho W)^{-1}\)
Long-run (steady state): cumulative effect including temporal persistence, through \((I - \gamma - \rho W)^{-1}\) (when it converges)

The long-run spatial multiplier is larger because it includes the compounding of temporal persistence with spatial feedback.

Impulse Response Function¶

The impulse response function (IRF) traces how a shock to one entity propagates through space and time:

# Shock to entity 0, tracked for 10 periods
irf = results.compute_impulse_response(shock_entity=0, periods=10)
# Returns array of shape (periods, N)
# irf[t, i] = response of entity i at time t to unit shock at entity 0

# Plot the response
import matplotlib.pyplot as plt

# Response of the shocked entity over time
plt.plot(range(10), irf[:, 0], label='Shocked entity')

# Response of neighbors
neighbor_indices = [1, 2, 3]  # indices of neighboring entities
for idx in neighbor_indices:
    plt.plot(range(10), irf[:, idx], alpha=0.5)

plt.xlabel('Periods after shock')
plt.ylabel('Response')
plt.title('Spatio-Temporal Impulse Response')
plt.legend()
plt.show()

The IRF combines two propagation channels:

Temporal propagation: \(\gamma\) causes the shock to persist over time (exponential decay if \(|\gamma| < 1\))
Spatial propagation: \(\rho\) causes the shock to spread to neighbors, who then spread it further

Prediction¶

# One-step-ahead prediction using last observed values
y_pred = results.predict(steps=1)

# Multi-step prediction with future covariates
y_pred_5 = results.predict(steps=5, X_future=X_future_df)

Multi-step prediction combines the spatial structure (applying \(W\) at each step) with temporal dynamics (feeding predicted values back as lagged values).

Configuration Options¶

`fit()` Parameters¶

Parameter	Type	Default	Description
`effects`	`str`	`'fixed'`	`'fixed'` or `'random'`
`method`	`str`	`'gmm'`	`'gmm'` (only available method)
`lags`	`int`	`1`	Number of temporal lags of \(y\)
`spatial_lags`	`int`	`1`	Number of spatial lags in instruments
`time_lags`	`int`	`2`	Depth of temporal instruments
`initial_values`	`np.ndarray`	`None`	Starting values for optimization
`maxiter`	`int`	`1000`	Maximum GMM iterations
`tol`	`float`	`1e-6`	Convergence tolerance
`verbose`	`bool`	`False`	Print optimization progress

Results Attributes¶

Attribute	Type	Description
`params`	`pd.Series`	All parameters (\(\gamma\), \(\rho\), \(\beta\))
`rho`	`float`	Spatial autoregressive parameter
`bse`	`pd.Series`	Standard errors (sandwich formula)
`tvalues`	`pd.Series`	t-statistics
`pvalues`	`pd.Series`	p-values
`llf`	`float`	Quasi-log-likelihood
`aic`	`float`	AIC
`bic`	`float`	BIC
`resid`	`np.ndarray`	GMM residuals

Results Methods¶

Method	Parameters	Description
`compute_impulse_response()`	`shock_entity`: int, `periods`: int (default 10)	Spatio-temporal IRF
`predict()`	`steps`: int (default 1), `X_future`: array	Multi-step prediction
`summary()`	—	Formatted results table

Diagnostics¶

Hansen J-Test¶

The GMM estimator provides an overidentification test:

# Hansen J-test is computed during estimation
# Access via the results (if available in the output)
print(results.summary())  # J-test reported in summary

The Hansen J statistic tests whether the instruments are valid (orthogonal to the error term). Under \(H_0\) (instruments are valid), \(J \sim \chi^2(L - K)\) where \(L\) is the number of instruments and \(K\) is the number of parameters.

Stability Check¶

Verify that the estimated parameters imply a stable process:

gamma = results.params['y_lag1']
rho = results.rho

print(f"|gamma| = {abs(gamma):.4f} < 1: {abs(gamma) < 1}")
print(f"|rho|   = {abs(rho):.4f} < 1: {abs(rho) < 1}")

# The characteristic roots of the system should be inside the unit circle
# A necessary (but not sufficient) condition: |gamma| + |rho| < 1
print(f"|gamma| + |rho| = {abs(gamma) + abs(rho):.4f}")

Limitations

QML estimation is not yet implemented for the dynamic spatial model; only GMM is available
The model requires a balanced panel
Computational cost grows quickly with \(N\) and the instrument count
For very large \(N\) with many instruments, consider reducing time_lags or spatial_lags

Tutorials¶

Tutorial	Description	Links
Spatial Econometrics	Includes dynamic spatial panel example

References¶

Yu, J., de Jong, R., and Lee, L.F. (2008). Quasi-maximum likelihood estimators for spatial dynamic panel data with fixed effects when both \(n\) and \(T\) are large. Journal of Econometrics, 146(1), 118-134.
Lee, L.F. and Yu, J. (2010). Estimation of spatial autoregressive panel data models with fixed effects. Journal of Econometrics, 154(2), 165-185.
Elhorst, J.P. (2014). Spatial Econometrics: From Cross-Sectional Data to Spatial Panels. Springer.
Anselin, L., Le Gallo, J., and Jayet, H. (2008). Spatial panel econometrics. In Matyas, L. and Sevestre, P. (eds), The Econometrics of Panel Data, 625-660.