Time Series Forecasting System shows wrong estimates

[niam]

I have come across this strange case in SAS Time Series Forecasting System and it seems that SAS is displaying wrong estimates.

Here is a simple example. 

When I fit a quadratic trend to my dataset using TSFS, I get the following estimates:

Model Parameter	Estimate	Std. Error	T	Prob>|T|
Intercept	3.85465	0.3471	11.1041	<.0001
Linear Trend	-0.03373	0.0089	-3.7788	0.0003
Quadratic Trend	-0.00117	0.000353	-3.3153	0.0013
Model Variance (sigma squared)	5.08917	.	.	.

Now, when I run a simple regression (proc reg) on the same dataset, using proc reg, I get totally different results

Parameter Estimates
Variable	DF	Parameter Estimate	Standard Error	t Value	Pr> |t|
Intercept	1	2.52733	0.71316	3.54	0.0006
t	1	0.08578	0.03465	2.48	0.0152
t2	1	-0.00117	0.00035340	-3.32	0.0013

Interestingly, the oredicted values shown by TSFS system seem to be following the estimates shown by the PROC REG. 

Here is the predicted values of TSFS for the first three observations

Obs	PERIOD	ACTUAL	Predicted value for INCIDENTS	Upper 95% Confidence Limit	Lower 95% Confidence Limit	Prediction error for INCIDENTS	Prediction standard error	Normalized prediction error for INCIDENTS	Linear Trend	Quadratic Trend
1	NOV2009	1.0000	2.6119	7.0335	-1.8096	-1.6119	2.2559	-0.7145	-50	2500
2	DEC2009	2.0000	2.6942	7.1157	-1.7273	-0.6942	2.2559	-0.3077	-49	2401
3	JAN2010	4.0000	2.7741	7.1956	-1.6474	1.2259	2.2559	0.5434	-48	2304

The values in the fourth column (Predicted value for INCIDENTS) are based the parameter estimates form PROC REG and not the estimates from TSFS. for example, for first observations:

2.52733+ 0.08578*1 -0.00117*(1^2)=2.6119

I noticed that this discrepancy between the estimates of proc OLS and TSFS happens only when there are high order terms included in the model (such as quadratic or cubic). 

Why is this happening and how can I get correct estimates using TSFS. 

[dw_sas · Accepted solution]

Hi,

The differences you observed in the parameter estimates between the TSFS results and PROC REG are due to a difference in the definition of the linear and quadratic time trend terms.  For the quadratic time trend curve, the TSFS defines the _LINEAR_ regressor as  t - c, and the _QUAD_ regressor as ( t - c ) **2, whereas your PROC REG step uses t and t**2, respectively.  Details on the definitions of the time trend terms used by the TSFS can be found in the following section of the documentation:

http://support.sas.com/documentation/cdl/en/etsug/68148/HTML/default/viewer.htm#etsug_tffordet_sect0...

For your data (and most of the other series I tested), c=50.  You can determine the value of "c" used by the TSFS by looking at the Forecast Table in the Model Viewer.  The last two columns of this table contain the values of the _LINEAR_ and _QUAD_ variables.

To reproduce the parameter estimates observed in the TSFS by using PROC REG, you will need to define your time trend variables in the same manner as was used in the TSFS.  Please note that the predicted values will be the same between the TSFS and PROC REG, regardless of which definition of the time trend terms is used.

I hope this helps!

DW

[niam]

Thank you very much for your response! It was really helpful.