The first vector autoregressive panel model (PVAR) was introduced by Holtz-Eakin et al. (1988). Binder et al. (2005) extend their equation-by-equation estimator for a PVAR model with only endogenous variables that are lagged by one period. We further improve this model in Sigmund and Ferstl (2021) to allow for \(p\) lags of \(m\) endogenous variables, \(k\) predetermined variables and \(n\) strictly exogenous variables.
Therefore, we consider the following stationary PVAR with fixed effects.
yi,t = μi + ∑l=1pAlyi,t-l + Bxi,t + Csi,t + εi,t
Let yi,t ∈ ℜm be an m×1 vector of endogenous variables for the ith cross-sectional unit at time t. Let yi,t-l ∈ ℜm be an m×1 vector of lagged endogenous variables. Let xi,t ∈ ℜk be an k×1 vector of predetermined variables that are potentially correlated with past errors. Let si,t ∈ ℜn be an n×1 vector of strictly exogenous variables that neither depend on εi,t nor on εi,t-s for s = 1,…,T. The idiosyncratic error vector εi,t ∈ ℜm is assumed to be well-behaved and independent from both the regressors xi,t and si,t and the individual error component μi. Stationarity requires that all unit roots of the PVAR model fall inside the unit circle, which therefore places some constraints on the fixed effect μi. The cross section i and the time section t are defined as follows: i = 1,…,N and t = 1,…T. In this specification we assume parameter homogeneity for Al (m×m), B (m×k) and C (m×n) for all i.
A PVAR model is hence a combination of a single equation dynamic panel model (DPM) and a vector autoregressive model (VAR).
First difference and system GMM estimators for single equation dynamic panel data models have been implemented in the STATA package xtabond2 by Roodman (2009) and some of the features are also available in the R package plm.
For more technical details on the estimation, please refer to our paper Sigmund and Ferstl (2021).
There we define the first difference moment conditions (see Holtz-Eakin et al., 1988; Arellano and Bond, 1991), formalize the ideas to reduce the number of moment conditions by linear transformations of the instrument matrix and define the one- and two-step GMM estimator. Furthermore, we setup the system moment conditions as defined in Blundell and Bond (1998) and present the extended GMM estimator. In addition to the GMM-estimators we contribute to the literature by providing specification tests (Hansen overidentification test, lag selection criterion and stability test of the PVAR polynomial) and classical structural analysis for PVAR models such as orthogonal and generalized impulse response functions, bootstrapped confidence intervals for impulse response analysis and forecast error variance decompositions. Finally, we implement the first difference and the forward orthogonal transformation to remove the fixed effects.