Nonlinear VAR: Some Theory and an Application to US GNP and Unemployment

A generalization of the endogenous threshold model is developed by extending this class to a multivariate framework and to cases where the feedback acts at multiple lags. The feedback is specified, following Beaudry and Koop, by a variable which measures the depth of recessions. We give conditions for the ergodicity of the model and prove strong consistency of the maximum likelihood estimator, although the objective function is discontinuous in the threshold parameter. The model is applied to a bivariate VAR of output growth and changes in the unemployment rate for the US economy. The nonlinearity is found to be statistically significant only in the unemployment equation and it transmits to GNP through the cross-correlation between the series. We also find that, owing to the nonlinear structure, shocks hitting the economy in downturns have lower persistence than those occurring during expansions. Since this dampening effect is stronger for negative than for positive shocks, the feedback from recessions is found to contribute positively to the long-run growth of the economy and we estimate this contribution to be about 1/6 of the total growth over the sample period. We interpret this result as an empirical validation of those economic theories that model recessions as cleansing times. Finally, we suggest that the state-dependence in persistence is a possible key to interpret the divergence in the measures of persistence existing in the literature.