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What is the mathematical form of an ARIMA forecast?

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What is the mathematical form of an ARIMA forecast?

I created an ARIMA model in SAS, but I need to implement the formula
for the forecasts in Excel. Below is the SAS code for the model. The model is
AR(5,10,15,20) MA(1,5) with differencing of 5 periods.

proc arima data=test;
identify var=calls (5) nlag=49;
estimate p=(5 10 15 20) q=(1 5);
run;

I tried using Y(t)-Y(t-5)=Mu+MA/AR, but the forecasts on the spreadsheet are still different from the SAS output. This notation is based on a document from SAS Support here: http://support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/viewer.htm#etsug_tffordet_sect0...

Can you please help me with translating the ARIMA forecast to Excel? Thank you.
Valentin
SAS Employee
Posts: 416

Re: What is the mathematical form of an ARIMA forecast?

Hello -
This is not the answer you are looking for, but in my opinion you will have a hard time replicating ARIMA models created in SAS using Excel. The implementation of these kinds of models is more sophisticated than you might expect - and as such difficult to translate to simple Excel statements (in a straightforward manner that is).

Wouldn't it be more appropriate to leave the calculation of the ARIMA model in SAS and export the results to Excel only?

Alternatively you might want to export the ARIMA model parameters to Excel and take it from there; see: http://www.mathkb.com/Uwe/Forum.aspx/sas/38646/ARIMA-MODEL

Thanks,
Udo
New Contributor
Posts: 3

Re: What is the mathematical form of an ARIMA forecast?

i have tried replicating the SAS ARIMA (AR model) model in excel and have been successful for the most part... below is the equation that i am using.. 

 

below is the equation for p =(1,7,14) model with no d and q terms

Yt = Constant Estimate + ar1*lag1 +ar7*lag7 +ar14*lag14

 

if the model has other input variables apart from these 3 lags, below is the equation (p =(1,7,14), with input=Temp, with no d and q terms):

Yt =  Constant Estimate + ar1*lag1 +ar7*lag7 +ar14*lag14+ num1*Temp-ar1*num1*temp1-ar7*num1*temp7-ar14*num1*temp14

 

hope this helps.

thanks 

New Contributor
Posts: 3

Re: What is the mathematical form of an ARIMA forecast?

Thanks for the Logic, I am struggling with the notations num1. Can you please clarify?

Thanks in advance!

New Contributor
Posts: 3

Re: What is the mathematical form of an ARIMA forecast?

num1,num2 etc. are coefficients for other terms you may include in your arima model.. proc arima lets you add other variables (like regressive) into your model using the followin statement:

 

  estimate input=XYZ;

referrence: http://support.sas.com/documentation/cdl/en/etsug/63348/HTML/default/viewer.htm#etsug_arima_sect012....

 

hope that helps... if you just have AR term you don't have to worry about the num1,num2 etc.

New Contributor
Posts: 3

Re: What is the mathematical form of an ARIMA forecast?

[ Edited ]

Thanks for the quick response. It helped me move 1 step ahead, I could replicate ARMA (1,1) with 2 input variables. Now I am trying extend it to a general form of more than 1 AR term (AR 1,2 ) with 2 or more input variables. The below mentioned form does not conform with SAS output.

 Yt =  Constant Estimate + ar1*lag1 +ar2*lag2 + num1*Temp-ar1*num1*Temp1-ar2*num1*Temp2    {changed the subscripts}

 Yt =  Constant Estimate + ar1*lag1 +ar2*lag2 + num1*(Temp- ar1*Temp1- ar2*Temp2)                    {brought num1 forward}

 Yt =  Constant Estimate + ar1*lag1 +ar2*lag2 + num1*(Temp- (ar1*Temp1+ ar2*Temp2))                 {Regression - AR contribution to get the marginal contribution for temp}

 

I believe the above form suggests that whenever you have AR with input variables, we need to take out the AR interaction out of the input variable to see the marginal contribution of the input variable (Temp in our case). This form does not match with SAS

 

So i think, there should be a term to take care of the effect of interaction between Ar1 and Ar2 something like A1*Ar2

 Yt =  Constant Estimate + ar1*lag1 +ar2*lag2 + num1*(Temp-(ar1*temp1 + ar2*temp2 - ar1*temp1*ar2*temp2)

 

Unfortunately this does not help either.

Nav1, any guidance would be helpful. Thanks a ton!

 

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