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I want to fit the following model to a time series:

$$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1}+\alpha_{2}y_{t-1}^{2}+\lambda h_{t}+\varepsilon_{t} $$

$$ h_{t}=\beta_{0}+\beta_{1}\varepsilon_{t-1}^{2}+\beta_{2}h_{t-1} $$

How can I do this with R or with any other statistical software?


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Sorry ... I missed the "non" in "non-linear" and the square ... I will delete my answer, it is not an answer to your question. – Richard Mar 8 '13 at 14:54

I would fit an AR(1)-GARCH(1,1) with arch in mean effect and with the square of the serie at lag 1 as explanatory variable in the mean process :

The AR(1)-GARCH(1,1) component for :

$$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1}+\varepsilon_{t} $$ $$ h_{t}=\beta_{0}+\beta_{1}\varepsilon_{t-1}^{2}+\beta_{2}h_{t-1} $$ + the arch in mean effect ($\lambda h_{t}$ ): $$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1} +\lambda h_{t}+\varepsilon_{t} $$ $$ h_{t}=\beta_{0}+\beta_{1}\varepsilon_{t-1}^{2}+\beta_{2}h_{t-1} $$ + the explanatory variable in the mean process ($ y_{t-1}^{2} $):

$$ y_{t}=\alpha_{0}+\alpha_{1}y_{t-1}+\alpha_{2}y_{t-1}^{2}+\lambda h_{t}+\varepsilon_{t} $$ $$ h_{t}=\beta_{0}+\beta_{1}\varepsilon_{t-1}^{2}+\beta_{2}h_{t-1} $$ All econometrical packages allowing to add 1)an arch in mean effect and 2) an explanatory variable can be used (the G@rch package (Ox) for example).

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