Proc ARIMA to Fit ARIMA(p,d,q) model

Characteristics of Time Series Data:

1. Trend.
Trend is a long-term, consistent change in the time series values from beginning to end. It often represents a steady increase or a steady decrease in the values of time series, although other trend patterns are possible, e.g., quadratic or cubic trends in the data.

2. Seasonal Components.
Seasonal components involve a regular change in the data values that occurs at the same time in a given time period, e.g., Turkey sales are highest in November of every year.

3. Cyclical Components.
Cyclical components are long-term patterns of rising and falling data values in the time series. Many time series follow the business cycle of growth and recession in the economy.

4. Random Fluctuations.
Random fluctuations are what is 'left over' in a time series after the other three components have been accounted fro. They can represent measurement error or other unpredictable facors that affect the behaviour of a time series.

Check ACF, IACF, and PACF plots to identify p,q.
1. All three plots tail off exponentially for ARIMA(p,d,q) process.
2. For the ACF, r=max(p-1,q) beginning values followed by the behavior characteristic of an AR(p), that is, tailing off.
3. For the IACF and PACF, there are r=max(p-1,q) beginning values followed by the behavior characteristic of an MA(q), that is, tailing off.

Also, check seasonal pattern of time series as follow:
1. For seasonal AR process, the seasonal lags, s, 2s, ..., Ns, die off exponentially in ACF plot. In the IACF and PACF plots, there are spikes at each seasonal lag up to lag Ps, where there are P seasonal auto-regressive parameters necessary to model the data.
2. For seasonal MA process, the seasonal lags, s, 2s, ..., Ns, die off exponentially in IACF and PACF plot. In the ACF, there are spikes at each seasonal lag up to lag Qs, where Q is the number of seasonal moving average parameters necessary to model the data.
3. For seasonal ARMA process, the seasonal lags in the ACF tail off exponentially after lag Qs, where Q is the number of seasonal MA parameters necessary to model the data. The seasonal lags in the PACF and IACF tail of exponentially after lag Ps, where P is the number of seasonal AR parameters necessary to model the data.

proc arima data=predicted_series ;
identify var=severity(1) nlag=24 stationarity=(ADF=(0,1));
estimate p=(1,2) (12) q=(1,2) (12) method=ml;
forecast lead=60 interval=month id=failure_date out=outlead;
run;

axis1 width=1 offset=(1 pct) label=(a=90 r=0 'Severity');
axis2 width=1 offset=(1 pct) label=('Failure Date') value=(h=1.25);
symbol1 v=star ci=red height=1 cells interpol=join l=1 w=2;
symbol2 v=dot ci=green height=1 cells interpol=join l=1 w=2;
legend1 label=('') value=('Actual Claim Severity' 'Predicted Claim Severity') across=2 mode=protect position=(top center inside);
title 'Severity Prediction';
proc gplot data=outlead;
format failure_date monyy. severity dollar.;
plot (severity forecast)*failure_date/ overlay
caxis = BLACK
ctext = BLACK
vaxis = axis1
haxis = axis2
legend= legend1
grid
hminor=0
;
run;
quit;