### Chan-McAleer(2003), STAR-STGARCH model

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**Thu Aug 11, 2016 2:37 pm**This is a replication file for Felix Chan & Michael McAleer, 2003. "Estimating smooth transition autoregressive models with GARCH errors in the presence of extreme observations and outliers," Applied Financial Economics, vol. 13, no 8, 581-592. This demonstrates estimation of STAR-STGARCH models (STAR for the mean with a Smooth Transition GARCH process for the variance).

While we have had quite a few requests for this technique, we have tried to steer people away from it. If you read the paper, and the comments in our program, we hope you will think hard before deciding to pursue this. While STAR models have some practical uses, they rarely, if ever, work in the presence of outliers. If the data have GARCH properties you, almost by definition, have some decided outliers. In this paper, the model is applied to daily returns of the SP500 over a period which includes the wild swings in the stock market in 1987. A basic (non GARCH) STAR model really can't be fit to the data set because the STAR, with four added parameters (two in the second branch of the AR(1) and the two transition parameters) can typically figure out some combination which somehow "explains" one or two extreme values. The STAR-GARCH (STAR for the mean with a fixed GARCH(1,1) process for the variance) might have some potential for improvement since the "GARCH" part will often downweight some of the extreme values. However, the authors find (and we can confirm) that the likelihood is extremely flat in the STAR processâ€”apparently very different parameters for the STAR part yield almost identical likelihoods and almost identical parameters for the GARCH parameters.

Adding the ST to the GARCH process tends to magnify the difficulties. The same types of problems with outliers affect the ability to get sensible results as you can end up with GARCH processes which (for most data points) aren't even stableâ€”it's only the fact that the GARCH transition is on the lagged residual (thus making the GARCH process a very non-linear function of the lagged data) that allows the STGARCH to generate even reasonable looking results for the variance process.

Particularly with the STGARCH part, we would strongly recommend fixing the center of the transition at the obvious value of 0. It's the ability to adjust that location (combined with the other four parameters added in moving to STGARCH from GARCH) that gives rise to the numerical difficulties due to the parameters trying to "fix" outliers.

While we have had quite a few requests for this technique, we have tried to steer people away from it. If you read the paper, and the comments in our program, we hope you will think hard before deciding to pursue this. While STAR models have some practical uses, they rarely, if ever, work in the presence of outliers. If the data have GARCH properties you, almost by definition, have some decided outliers. In this paper, the model is applied to daily returns of the SP500 over a period which includes the wild swings in the stock market in 1987. A basic (non GARCH) STAR model really can't be fit to the data set because the STAR, with four added parameters (two in the second branch of the AR(1) and the two transition parameters) can typically figure out some combination which somehow "explains" one or two extreme values. The STAR-GARCH (STAR for the mean with a fixed GARCH(1,1) process for the variance) might have some potential for improvement since the "GARCH" part will often downweight some of the extreme values. However, the authors find (and we can confirm) that the likelihood is extremely flat in the STAR processâ€”apparently very different parameters for the STAR part yield almost identical likelihoods and almost identical parameters for the GARCH parameters.

Adding the ST to the GARCH process tends to magnify the difficulties. The same types of problems with outliers affect the ability to get sensible results as you can end up with GARCH processes which (for most data points) aren't even stableâ€”it's only the fact that the GARCH transition is on the lagged residual (thus making the GARCH process a very non-linear function of the lagged data) that allows the STGARCH to generate even reasonable looking results for the variance process.

Particularly with the STGARCH part, we would strongly recommend fixing the center of the transition at the obvious value of 0. It's the ability to adjust that location (combined with the other four parameters added in moving to STGARCH from GARCH) that gives rise to the numerical difficulties due to the parameters trying to "fix" outliers.