Panel VAR Theory --- Dynamic Multivariate Analysis¶

Key Takeaway

Panel VAR models extend vector autoregression to panel data, allowing researchers to study dynamic interdependencies between multiple variables across entities. GMM-based estimation handles individual heterogeneity, while orthogonalized impulse response functions and variance decompositions reveal causal transmission mechanisms.

Motivation¶

Standard panel regression models capture the effect of one variable on another, but many economic systems involve simultaneous feedback between multiple variables. For example:

Investment and profitability: Does investment drive profits, or do profits drive investment?
Trade and growth: Do trade openness and GDP growth reinforce each other?
Government spending and output: What are the dynamic fiscal multiplier effects?

Panel VAR models address these questions by:

Treating all variables as jointly endogenous
Capturing dynamic interdependencies through lagged values
Exploiting the panel dimension for more efficient estimation
Providing tools for structural analysis (IRF, FEVD, Granger causality)

Model Specification¶

The Panel VAR(p) Model¶

For entity \(i\) at time \(t\), the Panel VAR of order \(p\) is:

\[ Y_{it} = \sum_{j=1}^{p} A_j Y_{i,t-j} + \mu_i + \varepsilon_{it} \]

where:

\(Y_{it}\) is a \(K \times 1\) vector of endogenous variables
\(A_j\) are \(K \times K\) coefficient matrices for lag \(j\)
\(\mu_i\) is a \(K \times 1\) vector of entity-specific fixed effects
\(\varepsilon_{it}\) is a \(K \times 1\) innovation vector with \(E[\varepsilon_{it}] = 0\) and \(E[\varepsilon_{it}\varepsilon_{it}'] = \Sigma\)

Stationarity Conditions¶

The system is stable if the eigenvalues of the companion matrix lie inside the unit circle:

\[ \det\left(I_{Kp} - A_1 z - A_2 z^2 - \cdots - A_p z^p\right) \neq 0 \quad \text{for } |z| \leq 1 \]

Estimation¶

Forward Orthogonal Deviations (Helmert Transformation)¶

To eliminate fixed effects \(\mu_i\), PanelBox uses the Helmert transformation (forward orthogonal deviations) instead of first-differencing:

\[ \tilde{y}_{it} = \sqrt{\frac{T-t}{T-t+1}} \left( y_{it} - \frac{1}{T-t} \sum_{s=t+1}^{T} y_{is} \right) \]

This transformation:

Removes individual effects while preserving orthogonality of errors
Avoids the serial correlation introduced by first-differencing
Allows use of lagged levels as valid GMM instruments

GMM Estimation¶

The transformed model is estimated by GMM using lagged levels as instruments:

\[ \hat{A} = \left(\tilde{X}' Z W^{-1} Z' \tilde{X}\right)^{-1} \tilde{X}' Z W^{-1} Z' \tilde{Y} \]

where \(Z\) is the instrument matrix constructed from appropriate lags.

Structural Analysis¶

Impulse Response Functions (IRF)¶

Orthogonalized IRFs use a Cholesky decomposition of the residual covariance matrix \(\Sigma = PP'\) to identify structural shocks:

\[ \Theta_h = \Phi_h P \]

where \(\Phi_h\) are the moving average coefficients at horizon \(h\), computed recursively:

\[ \Phi_h = \sum_{j=1}^{h} \Phi_{h-j} A_j, \quad \Phi_0 = I_K \]

Ordering Sensitivity

Cholesky-based identification assumes a recursive causal ordering. The first variable in the system is assumed to be contemporaneously exogenous to all others. Results can be sensitive to variable ordering --- always test robustness.

Forecast Error Variance Decomposition (FEVD)¶

FEVD measures the proportion of the \(h\)-step forecast error variance of variable \(i\) attributable to shock \(j\):

\[ \text{FEVD}_{ij}(h) = \frac{\sum_{s=0}^{h-1} (\Theta_s)_{ij}^2}{\sum_{s=0}^{h-1} \sum_{k=1}^{K} (\Theta_s)_{ik}^2} \]

Granger Causality¶

Variable \(j\) Granger-causes variable \(i\) if lagged values of \(j\) significantly predict \(i\), conditional on lagged values of all other variables. This is tested via a Wald test:

\[ H_0: (A_1)_{ij} = (A_2)_{ij} = \cdots = (A_p)_{ij} = 0 \]

Panel VECM¶

When variables are cointegrated (non-stationary but with stable long-run relationships), the Panel VECM representation is:

\[ \Delta Y_{it} = \Pi Y_{i,t-1} + \sum_{j=1}^{p-1} \Gamma_j \Delta Y_{i,t-j} + \mu_i + \varepsilon_{it} \]

where:

\(\Pi = \alpha \beta'\) is the long-run impact matrix with rank \(r\) (number of cointegrating relationships)
\(\alpha\) is the \(K \times r\) matrix of adjustment speeds
\(\beta\) is the \(K \times r\) matrix of cointegrating vectors
\(\Gamma_j\) are short-run dynamics matrices

Confidence Intervals¶

PanelBox computes confidence bands for IRFs and FEVDs using Monte Carlo simulation from the estimated coefficient distribution, providing reliable inference on dynamic responses.

References¶

Abrigo, M. R. M. & Love, I. (2016). Estimation of panel vector autoregression in Stata. The Stata Journal, 16(3), 778--804.
Holtz-Eakin, D., Newey, W. & Rosen, H. S. (1988). Estimating vector autoregressions with panel data. Econometrica, 56(6), 1371--1395.
Love, I. & Zicchino, L. (2006). Financial development and dynamic investment behavior. The Quarterly Review of Economics and Finance, 46(2), 190--210.