ARIMAX with a specified nonlinear model using the arima function in R

I am interested in fitting an ARIMAX model using R. As known, ARIMAX can be understood as a composition of ARIMA models and regression models with exogenous (independent) variables. I have a time series $Y_i$, and want to estimate the ARIMA and nonlinear coefficients. The nonlinear model is the following:

$y_i=β_0+β_1t_i+β_2d+β_3 sin(2πt_i/β_4 )+β_5 (-1^{t_i})+ε_i$, nonlinear regression with an exogenous variable. Where $t_i$ =1, 2…, 60 and

d = dummy variable with 20 0's and 40 number 1's

d=c(rep(0,20),rep(1,40))

And an ARIMA model (1,1,1) for $Y_i$. Therefore, I want to estimate simultaneously the $β_i$ and the ARIMA coefficients in order to avoid the confusion between the exogenous coefficients and ARIMA coefficients. I know that $arima()$ can deal with this formulation but, how do the nonlinear model can be set within function function?. It seems that the xreg term only deals with linear parameters.

• As parameterized, your model isn't identifiable (choose some value for $\beta_3/\beta_4$, and then note that doubling both $\beta_3$ and $\beta_4$ leaves that ratio unchanged) - as the model stands, you can only estimate $\beta_3/\beta_4$. You have to fix that problem with your model before even considering trying to estimate it. – Glen_b Oct 15 '14 at 0:37

Do the parameters $\beta_3$ and $\beta_4$ have any particular interpretation or meaning in the context of your model? You can consider reparameterizing the model merging $\beta_3 t_i /\beta_4$ as $\theta\,t_i$. In this way the model becomes linear in the parameters and you can pass the regressors $t_i$ and $d_t$ through argument xreg in function arima.
You should include $t_i$ only once in the reparameterized model. In your model, the regressor $t_i$ is proportional to the regressor $2\pi t_i$. With this type of collinearity among the regressors the optimization algorithm won't be able to invert the system matrix.
• Thanks for your comments @Glen_b and javlacalle . I have edited my post and the model has varied a little. With the new model, $beta_3$ and $beta_4$ have a particular interpretation, period and amplitude of a sinus wave. Is there any way to estimate the ARMA parameters and this model parameters in only a single stage? (not estimating first the NLM and then the ARMA). – Hector Oct 15 '14 at 8:47