Assessing and testing for threshold nonlinearity in stock returns

This paper proposes a test for threshold nonlinearity in a time series with generalized autoregressive conditional heteroscedasticity ( GARCH) volatility dynamics. This test is used to examine whether financial returns on market indices exhibit asymmetric mean and volatility around a threshold value, using a double-threshold GARCH model. The test adopts the reversible-jump Markov chain Monte Carlo idea of Green, proposed in 1995, to calculate the posterior probabilities for a conventional GARCH model and a double-threshold GARCH model. Posterior evidence favouring the threshold GARCH model indicates threshold nonlinearity with asymmetric behaviour of the mean and volatility. Simulation experiments demonstrate that the test works very well in distinguishing between the conventional GARCH and the double-threshold GARCH models. In an application to eight international financial market indices, including the G-7 countries, clear evidence supporting the hypothesis of threshold nonlinearity is discovered, simultaneously indicating an uneven mean-reverting pattern and volatility asymmetry around a threshold return value.