Tsay JASA 1998

Tsay(1998) includes two examples of multivariate threshold models (TVAR's). The paper extends his "arranged autoregression" test for thresholds (@TSAYTEST) to a multivariate model and uses a search over grid(s) in the chosen threshold variable to estimate the model. Unfortunately, the results in the paper aren't really reproducible, due to an identifiable error in the interest rates example and some less obvious error in the river flow example. The interest rates example is discussed in detail in the Structural Breaks and Switching Models e-course. Another (more applied) example of a threshold VAR is Balke(2000).

iceland_river.rpf

This does a bivariate model on river flow data from two Icelandic rivers, including two exogenous variables (rainfall and temperature) and using temperature as the threshold variable. (This dataset has been used in other examples of non-linear time series models, but previously a bivariate one). Not surprisingly, the optimal threshold value is near the freezing mark. It uses a full model to select the threshold, then does a separate analysis treating that as given to select lag lengths for the variables.

usrates.rpf

This does a bivariate threshold model on yields on two US treasury instruments, using a (lagged) smoothed moving average of the spread between the yields as the threshold variable. (Because the spread is endogenous, the current value can't be used as the threshold without creating a simultaneity problem). This estimates both a one-break and a two-break model using a grid search across possible threshold values (nested grid search for two-break case). Due to a computational error, the published results strongly favor the two-break model, while, when done correctly, the evidence in favor of two- vs one-breaks is weak.