Standard Error Estimation in PROC ARIMA

[sasalex2024]

Dear SAS Community,

For proc arima estimation, I couldn’t find detailed information about how standard errors (and thus t-values) are computed. The manual states: 

The t-values reported in the table of parameter estimates are approximations whose accuracy depends on the validity of the model, the nature of the model, and the length of the observed series...

Could you please clarify what method SAS uses to estimate standard errors for ARMA model parameters? For example, does it rely on the Outer Product of Gradients (OPG), the Observed Information Matrix (OIM), Huber/White/sandwich estimator, or something similar when method=ml is selected?

Additionally, is there any way in proc arima to specify alternative methods for standard error estimation, or are the defaults considered standard for ARMA models?

Thank you very much.

[sbxkoenk · Accepted solution]

I never wondered and my quick search yields only this :

• The standard error estimates are based on large sample theory. Thus, the standard errors are labeled as approximate, and the standard errors and t values might not be reliable in small samples.
• Cautions during Estimation
  The ARIMA procedure uses a general nonlinear least squares estimation method that can yield problematic results if your data do not fit the model. Output should be examined carefully. The GRID option can be used to ensure the validity and quality of the results. Problems you might encounter include the following:
  • ... ... ... ... ...
  • If the data are differenced and a moving-average model is fit, the parameter estimates might try to converge exactly on the invertibility boundary. In this case, the standard error estimates that are based on derivatives might be inaccurate.

[Edit] ... I think PROC ARIMA relies on OIM for "calculation" of standard errors (of parameter estimates).
OIM is the inverse of the negative matrix of second derivatives.
OIM is not so computer-time-friendly.

Koen

[sasalex2024]

Thank you, Koen, for your prompt and insightful reply. I managed to run a simple regression with ARMA errors and input series in both SAS and Stata. Stata has a default vce(opg) option that uses the sum of the outer product of the gradient (OPG) vectors. It reports results as z-values and P>|z|, which are extremely different from the SAS output. In fact, for my data SAS indicates that all the coefficients are significant, whereas Stata suggests that some are not significant at all.

However, Stata offers two other options:

1. vce(robust) uses the robust or sandwich estimator of variance.
2. vce(oim) uses the observed information matrix (OIM).

i saw that vce(oim) produces z-values that are fairly close to the t-values reported in SAS. This suggests that your guess—that PROC ARIMA relies on OIM for calculating the standard errors of parameter estimates—is correct. Did I understand you correctly? Thank you.

[sbxkoenk]

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@sasalex2024 wrote:

Did I understand you correctly? Thank you.

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It seems you did understand me correctly.

Thanks for feedback !

Koen