SAS ARIMA forecast estimate statement

Question

Im trying to forecast a timeseries (daily intervals) but I am unsure of the syntax of the estimate statement. I know p is for autoregression and q is for moving averages, but what do the (2)(12) and (1 3) mean, and what values should they be if the interval = day?

proc arima data=daily;
 i var=SUM_of_Total_Revenue2;
run;
e p=(2)(12) q=(1 3);
 f lead=365
    out=daily2
    id=Order_date
    interval=day
    noprint;

Answer 1

SAS has extensive documentation on all their procedures. Please see here for the syntax for PROC ARIMA.

As you noted p is for autoregressive and q is for moving average in an arima model. In proc arima if p or q is separated by a bracket then it means that there is a seasonal autoregressive model.For your example, what p = (2)(12) means is that current day of SUM_of_Total_Revenue2 is related to 2 days (nonseasonal ar) before and also related to 12 days (seasonal ar) before the current day.

Mathematically using back-shift notation this could be represented as follows. I'm following the notation used in Makridakis et al:

$(1-\phi_1B^2)(1-\Phi_1B^{12}) Y_t = (1-\theta_1B-\theta_1B^3)e_t$

Left hand is non seasonal and seasonal ar(p) and right hand side is the nonseasonal ma(q).

Hope this helps.

Answer 2

My experience tells me that the analysis of daily data would never lead to the ARIMA model that you propose. How did you go about identifying such a model for daily data (7 days in a week) ? Are you analysing daily data that has 6 readings per week ? A more appropriate model might include daily fixed effects, weekly or monthly effects , holiday effects and perhaps an ARIMA structure or order 1 or 7 or perhaps 6 if the data is 6 periods per week . Care should be taken to deal with and identify and incorporate pulses, level shifts and perhaps trends.