Panel VAR Estimation

Quick Reference

Class: panelbox.var.model.PanelVAR Data: panelbox.var.data.PanelVARData Import: from panelbox.var import PanelVARData, PanelVAR Stata equivalent: pvar (community-contributed) R equivalent: panelvar::pvargmm()

Overview

Panel Vector Autoregression (Panel VAR) extends the classical VAR framework to panel data settings where multiple entities (countries, firms, individuals) are observed over time. Each endogenous variable is modeled as a function of its own lagged values and the lagged values of all other endogenous variables in the system, while controlling for entity-specific fixed effects.

The Panel VAR(p) model with \(K\) endogenous variables, \(N\) entities, and \(p\) lags is:

\[ Y_{it} = A_1 Y_{i,t-1} + A_2 Y_{i,t-2} + \ldots + A_p Y_{i,t-p} + \alpha_i + \varepsilon_{it} \]

where \(Y_{it}\) is the \(K \times 1\) vector of endogenous variables for entity \(i\) at time \(t\), \(A_1, \ldots, A_p\) are \(K \times K\) coefficient matrices, \(\alpha_i\) are entity-specific fixed effects, and \(\varepsilon_{it} \sim (0, \Sigma)\) is the error term.

PanelBox estimates Panel VAR using OLS equation-by-equation with within transformation (entity demeaning) to remove fixed effects. This approach follows the methodology of Holtz-Eakin, Newey, and Rosen (1988) and the implementation strategy of Abrigo and Love (2016).

Quick Example

import pandas as pd
from panelbox.var import PanelVARData, PanelVAR

# Step 1: Prepare data container
var_data = PanelVARData(
    data=df,
    endog_vars=["gdp", "inflation", "unemployment"],
    entity_col="country",
    time_col="year",
    lags=2,
    trend="constant",
)

# Step 2: Create and estimate model
model = PanelVAR(data=var_data)
results = model.fit(method="ols", cov_type="clustered")

# Step 3: View results
print(results.summary())
print(f"Stable: {results.is_stable()}")
print(f"AIC: {results.aic:.4f}, BIC: {results.bic:.4f}")

When to Use

• Macroeconomic dynamics: Studying interactions between GDP, inflation, interest rates, and unemployment across countries
• Financial contagion: Analyzing how shocks propagate across markets or firms
• Policy analysis: Evaluating the dynamic effects of monetary or fiscal policy
• Firm-level dynamics: Modeling investment, employment, and output interactions across firms
• No strong prior on causality: When the causal ordering among variables is unknown

Key Assumptions

• Stationarity: All endogenous variables must be stationary (or cointegrated -- see VECM)
• No cross-entity contamination: Lags are constructed within each entity separately
• Homogeneous slope coefficients: The \(A_l\) matrices are assumed identical across entities
• Continuous time series: No internal gaps allowed within any entity's time series
• Sufficient time periods: \(T > K \times p + 1\) for each entity after lag construction

Detailed Guide

Data Preparation

The PanelVARData class handles all data preparation, including lag construction, gap detection, and missing value handling.

from panelbox.var import PanelVARData

var_data = PanelVARData(
    data=df,                                    # pandas DataFrame in long format
    endog_vars=["gdp", "inflation", "rate"],    # Endogenous variables (K)
    entity_col="country",                       # Entity identifier column
    time_col="year",                            # Time identifier column
    exog_vars=None,                             # Optional exogenous variables
    lags=2,                                     # Number of lags (p)
    trend="constant",                           # Deterministic terms
    dropna="any",                               # Missing value strategy
)

# Inspect data properties
print(f"Variables (K): {var_data.K}")
print(f"Lags (p): {var_data.p}")
print(f"Entities (N): {var_data.N}")
print(f"Observations: {var_data.n_obs}")
print(f"Balanced: {var_data.is_balanced}")
print(f"T range: [{var_data.T_min}, {var_data.T_max}]")

Parameters:

Parameter	Type	Default	Description
data	pd.DataFrame	required	Panel data in long format
endog_vars	list[str]	required	Names of endogenous variables
entity_col	str	required	Entity identifier column name
time_col	str	required	Time identifier column name
exog_vars	list[str]	None	Optional exogenous variable names
lags	int	1	Number of lags (\(p\))
trend	str	"constant"	"none", "constant", "trend", or "both"
dropna	str	"any"	"any" (drop if any variable missing) or "equation"

Critical Safety Feature

Lags are constructed using .groupby(entity).shift() to ensure that lag \(t\) of entity A never contains an observation from entity B. A cross-contamination verification check runs automatically after lag construction.

Estimation

The PanelVAR class performs OLS equation-by-equation estimation with within transformation (entity demeaning) to remove fixed effects.

from panelbox.var import PanelVAR

model = PanelVAR(data=var_data)
results = model.fit(
    method="ols",              # Estimation method
    cov_type="clustered",      # Covariance estimator
)

Within Transformation: Before OLS estimation, each variable is demeaned within each entity:

\[ \tilde{y}_{it} = y_{it} - \bar{y}_i \]

This removes the entity-specific fixed effects \(\alpha_i\) and avoids the incidental parameters problem.

Estimation proceeds equation by equation: For each equation \(k = 1, \ldots, K\):

1. Apply within transformation to \(y_k\) and \(X\)
2. Estimate \(\hat{\beta}_k = (X'X)^{-1} X' y_k\)
3. Compute residuals \(\hat{\varepsilon}_k = y_k - X \hat{\beta}_k\)
4. Compute covariance matrix using the specified method

Covariance Types

cov_type	Description	When to Use
"clustered"	Cluster-robust by entity	Default (recommended). Accounts for within-entity correlation
"driscoll_kraay"	Driscoll-Kraay HAC	Cross-sectional dependence suspected
"hc1"	Heteroskedasticity-robust (HC1)	Heteroskedastic errors, no clustering
"nonrobust"	Classical OLS	Homoskedastic, no clustering (rarely appropriate)
"sur"	Seemingly Unrelated Regressions	Exploit cross-equation correlation

# Cluster-robust standard errors (recommended)
results = model.fit(cov_type="clustered")

# Driscoll-Kraay for cross-sectional dependence
results = model.fit(cov_type="driscoll_kraay", max_lags=3)

# SUR covariance (exploits cross-equation correlation)
results = model.fit(cov_type="sur")

Lag Selection

Choosing the optimal lag order is crucial. Too few lags can lead to omitted variable bias; too many waste degrees of freedom and reduce efficiency.

# Automatic lag selection
lag_result = model.select_lag_order(max_lags=8, cov_type="clustered")
print(lag_result.summary())

# Access optimal lag by criterion
optimal_bic = lag_result.selected["BIC"]
optimal_aic = lag_result.selected["AIC"]

# Visualize information criteria
fig = lag_result.plot(backend="plotly")

The select_lag_order method tests \(p = 1, 2, \ldots, \text{max\_lags}\) and computes four information criteria:

Criterion	Formula	Properties
AIC	$\log	\hat{\Sigma}
BIC	$\log	\hat{\Sigma}
HQIC	$\log	\hat{\Sigma}
MBIC	$\log	\hat{\Sigma}

The LagOrderResult object contains:

• criteria_df: DataFrame with all criteria values for each lag
• selected: Dictionary mapping criterion name to optimal lag
• summary(): Formatted summary table
• plot(): Visual comparison of criteria

Practical Advice

BIC is generally recommended for Panel VAR because it penalizes complexity more heavily and is consistent (selects the true lag order as \(N, T \to \infty\)). Start with max_lags=8 and reduce if you get warnings about insufficient observations.

Interpreting Results

The PanelVARResult object provides comprehensive access to estimation results.

# Coefficient matrices A_1, A_2, ..., A_p
for lag in range(1, results.p + 1):
    print(f"\nA_{lag}:")
    print(results.coef_matrix(lag))  # Returns labeled DataFrame

# Residual covariance matrix
print(f"\nSigma (residual covariance):\n{results.Sigma}")

# Information criteria
print(f"AIC: {results.aic:.6f}")
print(f"BIC: {results.bic:.6f}")
print(f"HQIC: {results.hqic:.6f}")
print(f"Log-likelihood: {results.loglik:.2f}")

# Per-equation summary
for k in range(results.K):
    print(results.equation_summary(k))

# System-level summary (compact)
print(results.summary_system())

# Full summary with coefficient tables
print(results.summary())

Key Result Attributes:

Attribute	Type	Description
params_by_eq	list[np.ndarray]	Coefficient vectors per equation
std_errors_by_eq	list[np.ndarray]	Standard errors per equation
A_matrices	list[np.ndarray]	\(K \times K\) coefficient matrices \([A_1, \ldots, A_p]\)
Sigma	np.ndarray	Residual covariance matrix (\(K \times K\))
aic, bic, hqic	float	Information criteria
loglik	float	Log-likelihood
K, p, N, n_obs	int	Dimensions

Stability Analysis

A Panel VAR is stable (stationary) if all eigenvalues of the companion matrix have modulus strictly less than 1. Stability is essential for meaningful impulse response functions and forecasts.

# Check stability
print(f"Stable: {results.is_stable()}")
print(f"Max eigenvalue modulus: {results.max_eigenvalue_modulus:.6f}")
print(f"Stability margin: {results.stability_margin:.6f}")

# Eigenvalues of companion matrix
eigenvalues = results.eigenvalues
print(f"Eigenvalues: {eigenvalues}")

# Companion matrix (Kp x Kp)
F = results.companion_matrix()

# Visual stability check
results.plot_stability(backend="matplotlib")

The companion matrix \(\mathbf{F}\) reformulates VAR(p) as VAR(1):

\[ \mathbf{F} = \begin{bmatrix} A_1 & A_2 & \cdots & A_p \\ I_K & 0 & \cdots & 0 \\ 0 & I_K & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & I_K & 0 \end{bmatrix} \]

The plot_stability() method visualizes eigenvalues on the complex plane with the unit circle, making it easy to identify whether any eigenvalues lie outside the circle (unstable system).

Unstable VAR

If is_stable() returns False, the system is explosive. IRFs will diverge rather than converge to zero, and forecasts will blow up. Consider:

• Differencing non-stationary variables
• Using VECM for cointegrated variables (see VECM)
• Reducing the lag order
• Checking for data issues

Export Results

# LaTeX table
latex = results.to_latex()
with open("var_results.tex", "w") as f:
    f.write(latex)

# HTML table
html = results.to_html()

# Single equation export
latex_eq1 = results.to_latex(equation=0)

Configuration Options

PanelVAR.fit() Parameters

Parameter	Type	Default	Description
method	str	"ols"	Estimation method (currently "ols" only)
cov_type	str	"clustered"	Covariance estimator type
**cov_kwds			Additional keyword arguments for covariance (e.g., max_lags for Driscoll-Kraay)

PanelVAR.select_lag_order() Parameters

Parameter	Type	Default	Description
max_lags	int	8	Maximum number of lags to test
criteria	list[str]	["AIC", "BIC", "HQIC", "MBIC"]	Information criteria to compute
cov_type	str	"clustered"	Covariance type for each estimation

Complete Workflow Example

import pandas as pd
from panelbox.var import PanelVARData, PanelVAR

# Load macroeconomic panel data
# df has columns: country, year, gdp_growth, inflation, unemployment
df = pd.read_csv("macro_panel.csv")

# Step 1: Create data container
var_data = PanelVARData(
    data=df,
    endog_vars=["gdp_growth", "inflation", "unemployment"],
    entity_col="country",
    time_col="year",
    lags=2,  # Start with initial guess
    trend="constant",
)

print(f"Panel: N={var_data.N}, T_avg={var_data.T_avg:.1f}, "
      f"K={var_data.K}, balanced={var_data.is_balanced}")

# Step 2: Select optimal lag order
model = PanelVAR(data=var_data)
lag_result = model.select_lag_order(max_lags=6)
print(lag_result.summary())
optimal_p = lag_result.selected["BIC"]
print(f"\nOptimal lag (BIC): {optimal_p}")

# Step 3: Re-estimate with optimal lag
var_data_opt = PanelVARData(
    data=df,
    endog_vars=["gdp_growth", "inflation", "unemployment"],
    entity_col="country",
    time_col="year",
    lags=optimal_p,
)
model_opt = PanelVAR(data=var_data_opt)
results = model_opt.fit(method="ols", cov_type="clustered")

# Step 4: Check stability
print(f"\nStable: {results.is_stable()}")
print(f"Max eigenvalue modulus: {results.max_eigenvalue_modulus:.4f}")

# Step 5: Full summary
print(results.summary())

# Step 6: Proceed to analysis
# IRF, FEVD, Granger causality, forecasting
irf = results.irf(periods=10, method="cholesky")
gc = results.granger_causality(cause="inflation", effect="gdp_growth")
print(gc.summary())

Tutorials

Tutorial	Description	Link
Panel VAR Notebook	Full estimation workflow

See Also

• Impulse Response Functions -- Dynamic effects of shocks
• Forecast Error Variance Decomposition -- Relative importance of shocks
• Granger Causality -- Testing predictive relationships
• VECM -- For cointegrated (non-stationary) variables
• Forecasting -- Multi-step ahead predictions

References

• Holtz-Eakin, D., Newey, W., & Rosen, H. S. (1988). Estimating vector autoregressions with panel data. Econometrica, 56(6), 1371-1395.
• Abrigo, M. R., & Love, I. (2016). Estimation of panel vector autoregression in Stata. The Stata Journal, 16(3), 778-804.
• Andrews, D. W. K., & Lu, B. (2001). Consistent model and moment selection procedures for GMM estimation with application to dynamic panel data models. Journal of Econometrics, 101(1), 123-164.
• Luetkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer-Verlag.