I have a response time series(Y) & Input time series Xt & Zt. My only objective is to identify functional form Yt=f(Yt-1,Xt,Zt) where f(Yt-1,Xt,Zt) contains only lags of Yt , Xt & Zt as variables because I want to use this functional form as a constraint in an optimization model.

I used ARIMAX/Dynamic regression to do the same. But it gives a function form which have ARMA(p,q) of residuals also in f(Yt-1,Xt,Zt).

Having ARMA(p,q) of residuals in the functional form will not serve my purpose because my objective is to write f(Yt-1,Xt,Zt) in terms of lags of Yt, Xt & Zt only. And I assume i cannot just ignore ARMA(p,q) of residuals in the model when writing f(Yt-1,Xt,Zt). Please let me know if I can just drop it.

I tried linear regression but since this is time series data, residuals is having autocorrelation.

Other than ARIMAX & linear regression is there any model which will help me achieve my objective.



Your specification (Yt=f(Yt-1,Xt,Zt) where f(Yt-1,Xt,Zt) contains only lags of Yt , Xt & Zt as variables) is evidently inadequate as the errors have auto-regressive structure. If you form an ARMAX model correctly incorporating any contemporary or lagged structure AND further validate that there are no pulses/level/shifts/seasonal pulses/local time trends AND that the error variance is constant over time AND that the model parameters are constant over time you should be good to go as it may not be possible with your flawed specification.

Modified to help explain PDL/ADL. Presented here is an excerpt of a file called RHSIDE.TXT produced by AUTOBOX to help explain the ARMAX equation .

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  • $\begingroup$ ARMAX model will always estimate ARMA(p,q) for residual right?( If p=0,d=0,q=0 then it becomes equivalent of linear regression.)So are you suggesting that ignore ARMA(p,q) from output of ARIMAX and write functional form without ARMA(p,q) of residual? Please explain. $\endgroup$ Jun 12 '15 at 15:25
  • $\begingroup$ you raise an interesting question here. No I would not ignore the arima structure but rather I would take that structures complement (the arma component) and multiply it by Y(t).X(t) and Z(t) essentially expressing Y as a PDL//ADL. I have written such procedures which take the parsimonious ARMAX model and presents it to AUTOBOX users as a pure-right-hand-side model. It helps the user to understand the equation and it's implications/narrative. $\endgroup$
    – IrishStat
    Jun 12 '15 at 15:55
  • $\begingroup$ This is very interesting. Can you explain how i can take error structure complement mathematically? Can you please explain what do you mean "express Y as PDL (ADL)". Please forgive my lack of knowledge. $\endgroup$ Jun 12 '15 at 16:03
  • $\begingroup$ If the ar structure is 1-.5B and the ma structure is 1+.3B12 then compute [1=.3b12]/[1-.5B] and use that polynomial as a multiplier on equation $\endgroup$
    – IrishStat
    Jun 12 '15 at 20:22

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