9 Panel VARs
Panel VARs have the same structure as VAR models, in the sense that all variables are assumed to be endogenous and interdependent, but a cross sectional dimension is added to the representation. There are \(N\) units indexed by \(i\in\{1,...,N\}\). The index \(i\) is generic and could indicate countries, sectors, markets… Then a panel VAR is \[ y_{it}=c_i+\Phi_i(L) y_{t-1}+\varepsilon_{it}.\] where \(y_t\) is the stacked version of \(y_{it}\) and \(\varepsilon_t\) is i.i.d., with variance-covariance matrix \(\Omega\). Vector \(c_i\) and the lag polynomial \(\Phi_i(L)\) may depend on the unit. Canova and Ciccarelli (2013) provide a survey of panel estimation methods.
Contrary to standard VARs, panel VARs may help study * similarities/differences in the transmission of shocks; * Spillovers, contagion.
But panel VARs are subject tothe curse of dimensionality. Indeed, they can be characterized by * Dynamic interdependence: potentially, the lags of all endogenous variables of all units can enter the model for unit \(i\). * Static interdependence: \(\varepsilon_{it}\) are generally correlated across \(i\). * Cross sectional heterogeneity: the intercept, the slope and the variance of the shocks may be unit-specific.
9.1 Without Dynamic interdependence
A panel VAR, assuming no dynamic interdependence, is of the form: \[ y_{it}=c_i+\Phi_i(L) \color{red}{y_{it-1}}+\varepsilon_{it}.\]
As a comparison, consider micro panel data, in the univariate cae (AR(1) case): \[y_{it}=c_i+{\color{red}\phi} y_{it-1}+\varepsilon_{it}.\] In that kind of context, we usually have no cross-sectional heterogeneity as \(\phi_i=\phi\) for all \(i\). Typically, we have a large cross-sectional dimension \(N\), and a small time dimension \(T\). If one uses a ``Fixed-effect’’ regression: \[y_{it}-\frac{1}{T}\sum_{s=1}^Ty_{is}=\phi(y_{it-1}-\frac{1}{T}\sum_{s=1}^Ty_{is})+\varepsilon_{it}-\frac{1}{T}\sum_{s=1}^T\color{red}{\varepsilon_{is}},\] then one faces the Nickell bias: with a lagged dependent variable, the estimator is biased, with a bias of size \(\sim 1/T\). One can then use GMM regressions (Arellano and Bond (1991)) so as to get unbiased estimates.
Macro panel data have a different structure, with typically a moderate cross-sectional dimension \(N\) and a large time dimension \(T\), so that the Nickell bias is negligible (\(\rightarrow 0\) as \(T\rightarrow\infty\)).
9.1.1 Mean Group Estimator
When we etimate \[ y_{it}=c_i+\Phi_i(L) y_{it-1}+\varepsilon_{it},\] we need to take into account the cross-sectional heterogeneity in the coefficients, i.e., different \(\Phi_i\)’s across \(i\)’s. Pooled estimators (assuming cross-sectional homogeneity, i.e. identical \(\Phi_i\)’s across \(i\)s) are not consistent (biased) if the underlying dynamics are actually heterogeneous. By contrast, the Mean Group (MG) estimator, which consists in estimating \(N\) separate regressions and calculating the coefficient means, is consistent.
9.2 With Dynamic Interdependencies
A panel VAR that accommodates dynamic interdependence is of the form: \[ y_{it}=c_i+\Phi_i(L) y_{t-1}+\varepsilon_{it}.\]
We face a serious curse of dimensionality here: there are \(NGp+1\) coefficients to estimate in each equation.
A solution is to select some eligible dynamic links (See for instance Negro (2011)). Another alternative is to use a factor model. This consists in capturing the dynamic interdependencies by a set of unobservable factors (See Canova and Ciccarelli (2004) and Canova and Ciccarelli (2009)). See Section 10 for more details on FAVAR models.