5. Panel VAR Modeling

(Panel reduced-form VAR – the modern factory is prfvar_model, which extends the reduced-form VAR object to a cross-section of units.)

A prfvar_model object models a panel of (possibly Markov-switching) reduced-form VARs: the same set of variables observed for several cross-sectional units (countries, sectors, …), stacked into one system whose coefficients are linked across units according to a chosen homogeneity assumption. It extends the reduced-form VAR object, so data handling, estimation, identification, forecasting and the various decompositions are exactly as in Reduced-form VAR Modeling; this page covers what is specific to the panel case.

5.1. The model

Stacking the \(n\) units gives

\[\begin{split}\left[ \begin{array}{c} y_{1t} \\ y_{2t} \\ \vdots \\ y_{nt} \end{array} \right] = C(r_{t}) \left[ \begin{array}{c} x_{1t} \\ x_{2t} \\ \vdots \\ x_{nt} \end{array} \right] + B_{1}(r_{t}) \left[ \begin{array}{c} y_{1t-1} \\ y_{2t-1} \\ \vdots \\ y_{nt-1} \end{array} \right] + \cdots + B_{p}(r_{t}) \left[ \begin{array}{c} y_{1t-p} \\ y_{2t-p} \\ \vdots \\ y_{nt-p} \end{array} \right] + u_{t}\end{split}\]

with \(r_{t} = 1, 2, \dots, h\) and transition probabilities \(p_{r_{t}, r_{t+1}}(I_{t})\). The blocks \(B_{1}, \dots, B_{p}\) (dynamic / lag coefficients, named b1, …, bp – b(row, col), with the lag index omitted, refers to all lags) and \(C\) (deterministic / exogenous coefficients, named c) carry, on and off the diagonal, both within-unit and cross-unit dynamics. How much of this is shared across units is governed by the homogeneity assumption:

  • 'pooled' – all units share the same coefficients;

  • 'meanGroup' – mean-group estimator (average across units);

  • 'static' – the deterministic (constant / exogenous) coefficients are common, the dynamics are unit-specific;

  • 'dynamic' – the lag coefficients are common, the constants are unit-specific;

  • 'independent' – nothing in common (a separate VAR per unit).

5.2. Creating a panel VAR

The first argument is a panel struct describing the cross-section; the rest of the signature is the rfvar_model one:

panel             = struct();
panel.members     = {'US','CA','MX','BR'};
panel.homogeneity = 'dynamic';

endog = {'GROWTH','PAI','R'};

mdl = prfvar_model(panel, endog, ...
    lag_length    = 4, ...
    constant_term = true);

You declare the variables once (endog) and the units once (panel.members); internally RISE expands the variable list by appending the unit names – GROWTH_US, GROWTH_CA, …, PAI_US, … – so the time-series database you pass to estimate should carry those per-unit series. RISE’s panel data handling takes care of the stacking; translate_panel_output gives the results back unit by unit. The list of units and the homogeneity of an existing object are available through members and homogeneity.

5.3. Estimation, identification, IRFs, decompositions, forecasting

These are called exactly as for a reduced-form VAR – estimate (with an optional prior built from var_priors.minnesota(...) and any linear restrictions on the b / c coefficients), identification (and then structural_shocks / irf with the resulting Rfunc), variance_decomposition, historical_decomposition, forecast, bootstrap. Restrictions are written on the expanded parameter names (b1(GROWTH_US, R_CA) = 0 for a cross-unit exclusion, etc.), but the homogeneity assumption already imposes the bulk of the cross-unit structure. See Reduced-form VAR Modeling for the call patterns and the plotting helpers (quick_irfs, plot_fanchart, plot_decomp).

5.4. Adding regime switching

Pass a Markov-chain structure through the markov_chains keyword and list the parameters it controls (on the expanded names). Time-varying transition probabilities are specified exactly as in Reduced-form VAR Modeling, and the switching parameters are given priors through estim_priors.