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I've obtained a Transfer Function model to predict the value of $y_t$ which is:

$$ y_t - \mu = \frac{0.0034 + 0.0024B^9}{1- 0.9B}x_{t-9} + \frac{1}{1+0.6B} a_t$$

I obtained this model with SAS, and each parameter is estimated as a T-Student distribution and the standard deviation for each parameter also is available. The $x_t$ is the input parameter and $a_t$ is the white noise.

I obtained predictions of $y_t$ with assigned values for different scenarios. Also, I need the confidence interval for the predicted values, but I do not know how I can calculated them. Please guide me how I can calculate them according to the obtained formula or with SAS commands.

kjetil b halvorsen
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Afshin Oroojlooy
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    SAS automatically provides confidence interval. Adding forecast statement will provide you prediction interval – forecaster Sep 06 '15 at 22:24
  • @forecaster I wanna obtain the forecast and also the prediction interval for the assigned value of $x_t$. How I can do that with SAS? – Afshin Oroojlooy Sep 06 '15 at 22:27
  • I don't believe that SAS allows that which is why I added that selfsame feature to some software I wrote. Additionally the confidence limits that SAS provides naively premise that the estimated parameters are the population parameters as only the Psi Weights are used which do not reflect the uncertainties in the estimated transfer function's coefficients.You might want to see http://stats.stackexchange.com/questions/122660/diagnose-arima-seasonality-model-residual-auto-and-partial-correlation-plots which has a discussion about robust (realistic) confidence limits – IrishStat Sep 06 '15 at 22:44
  • @IrishStat I also obtained the scenario forecast with excel. Can you please explain how I can obtain the confidence interval? I do not know the formula or the procedure that I can obtain them. – Afshin Oroojlooy Sep 06 '15 at 22:48
  • If you obtained scenario (i.e.forecasts for the x variable) from excel , you must also obtain the variance of the individual forecasts extending into the future. On the premise that your transfer function has a set of errors that are independent of your excel model errors you can simply add the variance for the x variable to the variance from the TF model and then take the square root. Again all of this is flawed because nowhere in this recipe is there concern/recognition for the uncertainties in the estimated coefficients.or the possibility of pulse outliers arising in the future. – IrishStat Sep 06 '15 at 22:56
  • Apparently SAS has caught up in this regard as their documentation suggests that do include the uncertainty in the predictors ... see http://www.stat.cmu.edu/~hseltman/618/LNTS10.pdf – IrishStat Sep 06 '15 at 23:16
  • Yes, SAS allows prediction interval, as noted simply use [forecast](http://support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/viewer.htm#etsug_arima_sect024.htm) statement which alpha value providing prediction interval range example 95% or provide a reproducible example for me to show how to do this. – forecaster Sep 07 '15 at 00:18
  • @forecaster SAS does not do it as I want. When I feed the values of $x_t$ of a scenario, it use them to change the prediction model and change the value of the parameters and RMSE, ... – Afshin Oroojlooy Sep 07 '15 at 21:06
  • @forecaster Do you know how I can tell SAS to not use the value of $x_t$ from a given period? – Afshin Oroojlooy Sep 07 '15 at 21:07
  • @IrishStat I did not understand the procedure of obtaining the confidence interval which you mentioned. Could you please explain it in more detail? – Afshin Oroojlooy Sep 07 '15 at 21:09
  • Simply don't supply the value for the period. As I said before, please provide sample data for replication. – forecaster Sep 07 '15 at 21:16
  • @forecaster But without the value of $x_t$, only it only gives me the forecast of the next 9 periods, and I need more period's forecast. I have to feed it with the values of $x_t$ in the scenario to get a longer forecast. Do you have any suggestion? – Afshin Oroojlooy Sep 07 '15 at 21:23

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