Multivariate STAR Model (STVAR)
Posted: Tue Apr 21, 2015 12:02 pm
This is an example of a bivariate smooth transition regression, or STVAR. The example is from Martin, Hurn and Harris, Econometric Modelling with Time Series, Application 19.9.2, from pp 745-748, and is based upon Anderson, Anthansopoulos, and Vahid(2007), "Nonlinear autoregressive leading indicator models of output in G-7 countries", Journal of Applied Econometrics, vol 22, 63–87.
Program file
The base model is a VAR on the GDP growth rate and an interest rate spread. The threshold variables are different lags of the growth rate: lag one in GDP equation and lag two in the spread equation. In this case, all four branches (two regimes x two equations) use the standard two lag VAR explanatory variables, though that isn't required.
Because the threshold variables (and transition points) are different in the two equations, the equations have only a minimal link statistically---the joint estimates will be only slightly different from the single equation estimates. You could also use the exact same threshold variable and a common transition point, which would be more similar to the (hard) threshold VAR literature---that would require adding a constraint PARMSET that adds CG=CS.
As I've told a number of people who have asked about this, it's a straightforward extension of the STAR model to a multivariate setting. In fact, this uses the univariate regressions (done with NLLS) to get guess values for the multivariate regression (done with NLSYSTEM). The main reason there's a relatively thin literature with actual data is that it can be hard to get it to work properly. In this case, for instance, the spread equation fits better with a "sharp" rather than "smooth" transition, which means that the optimal value for the gamma (the scale in the logistic) is infinity and the center point can't be estimated well using non-linear least squares since the sum of squares isn't differentiable.
Program file
The base model is a VAR on the GDP growth rate and an interest rate spread. The threshold variables are different lags of the growth rate: lag one in GDP equation and lag two in the spread equation. In this case, all four branches (two regimes x two equations) use the standard two lag VAR explanatory variables, though that isn't required.
Because the threshold variables (and transition points) are different in the two equations, the equations have only a minimal link statistically---the joint estimates will be only slightly different from the single equation estimates. You could also use the exact same threshold variable and a common transition point, which would be more similar to the (hard) threshold VAR literature---that would require adding a constraint PARMSET that adds CG=CS.
As I've told a number of people who have asked about this, it's a straightforward extension of the STAR model to a multivariate setting. In fact, this uses the univariate regressions (done with NLLS) to get guess values for the multivariate regression (done with NLSYSTEM). The main reason there's a relatively thin literature with actual data is that it can be hard to get it to work properly. In this case, for instance, the spread equation fits better with a "sharp" rather than "smooth" transition, which means that the optimal value for the gamma (the scale in the logistic) is infinity and the center point can't be estimated well using non-linear least squares since the sum of squares isn't differentiable.