## reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

Occasional Contributor
Posts: 6

# reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

Hi all,

We are using PROC AUTOREG to fit our models, and we need confidence interval/limit for our forecast. There is a formular for that SAS uses for calculation of variable "v" in Section "Predicting Future Series Realizations" in

http://support.sas.com/documentation/cdl/en/etsug/63939/HTML/default/viewer.htm#etsug_autoreg_sect02...

I sent a request to SAS support for the reference/proof, but here is the response I got from them:

"I have checked with PROC AUTOREG developer who indicated that first, he does not really have a copy of more detailed derivations than what is currently provided in the documentation. Secondly, there are proprietary reasons that we may not be able to share the derivations with our customers. You may just cite our SAS/ETS documentation as reference for this formula."

The point is that there is no reference in the webpage or SAS online documentation for it. I wonder if any smart/knowledgable guys among you might know the reference/proof, such that I can be sure of what I am using, instead of being given a formula and being told to just use it. Thanks in advance.

Best,

Jonathan

Occasional Contributor
Posts: 6

## Re: reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

anyone has any idea? thanks.

Super User
Posts: 23,787

## Re: reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

There are two formula's for V, which method were you using?

The explanation under the computational methods didn't seem too unclear to me, but I'd have to verify the math specifically.

SAS/ETS(R) 9.3 User's Guide

Occasional Contributor
Posts: 6

## Re: reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

thank you for your reply Reeza. the one I am mentioning is the formula for regression with auto-regessive errors:

v=SQRT( z'Vz + s^2 r)

which is in Section "Predicting future series realizations" in

http://support.sas.com/documentation/cdl/en/etsug/63939/HTML/default/viewer.htm#etsug_autoreg_sect02...

Also here is what I heard back from SAS:

> I heard back from the development and here are the references he was

> able to locate that was used to derive at the results:

>

> Baillie, R., 1979, The Asymptotic Mean Squared Error of Multistep

> Prediction from the Regression Model with Autoregressive Errors,

> Journal of the American Statistical Association.  Vol. 74, No. 365,

> Mar., 1979,

> 175-184

>

> Baillie, R., and Bollerslev, T., 1992, Prediction in dynamic models

> with time-dependent conditional variances, Journal of Econometrics,

> 52, 91-113.

Those two papers are

are available on the author's webpage:

https://www.msu.edu/user/baillie/JASA_1979.pdf

https://www.msu.edu/user/baillie/J_Econometrics.1992.pdf

These two papers are helpful, but it's still beyond my capabilities to give a proof of the formula. Hope there could be some experts or smart people that would give a proof for it. thank you again!!

Super User
Posts: 23,787

## Re: reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

I don't have the time or the skills anymore to go through and prove that.

I will say one thing though. For a 'linear regression' the variance of the predictor is a similar formula: http://jackman.stanford.edu/classes/350B/07/predictionforWeb.pdf (see Slide 10).

Which would be enough for me to go with it, but obviously you need to be as well.

Good Luck and hopefully someone else can help more.

Occasional Contributor
Posts: 6

## Re: reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

Thank you for your quick response Reeza. that formula in your link is for simple regressions without auto-regressive error. the story is quite different if the regression has auto-regressive errors. Thank you again for your effort Reeza, and hope someone else could figure it out...

SAS Super FREQ
Posts: 4,247

## Re: reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

By "proof," do you mean that you want someone to supply a derivation of the formula based on the model assumptions, the properties of the expected value operator, asymptotic properties of distributions, and so forth?  Or do you want an explanation of what this formula means (intuitively)?

Occasional Contributor
Posts: 6

## Re: reference/proof for a formula used in confidence interval/limit calculation in AUTOREG PROC

Hi Rick,

First I want thank you for many great posts in your blog on IML which I, as well as many others, benefit a lot from. We use that formula for the confidence interval of predictions in our model, and we are asked to provide a reference where the formula comes from, or a deriviation if no such reference exists. We actually use regression with auto-regressive errors in many of our models, so we do want to be assured of the correctness of the formula.

thanks,

Jonathan

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