proc autoreg direction of AR1 estimate

[Taliah]

Hello, in the results of proc autoreg we get a negetive AR1 estimate. Based on the data an on buisness logic we know that the higher AR1 the higher the dependent veriable and it's prediction. Does that make sense and does that mean that the direction AR1 influences the dependent veriable is oposite to the estimate's direction we get from SAS? Thank you

[sbxkoenk]

Your question makes me think about the one you have posted on 30-OCT last year.

Autogerression direction of the AR estimate
https://communities.sas.com/t5/SAS-Forecasting-and-Econometrics/Autogerression-direction-of-the-AR-e...

Do you have additional info since then?

Koen

[Taliah]

 I was searching for that question and answer and couldn't find it

Thank you!

[SASCom1]

Hi @Taliah 

I think this is probably what you are confused about:

If you look at the regression model with AR error process discussed in PROC AUTOREG documentation:

SAS Help Center: Autoregressive Error Model

you can see that there is a negative sign in front of the AR parameter φ1 in PROC AUTOREG specification, if you write the complete model with AR(1) error:

yt=xt′β+νt

νt=ϵt−φ1*νt−1

since vt-1 = yt-1 - xt-1*β, this implies that 

yt = xt*β - φ1*(y_t-1 - x_t-1*β) + ϵt        (1)

Note the negative sign in front of φ1 in the above equation. This has opposite sign for the AR parameter than that specified in the usual ARIMA model expression, as in PROC ARIMA:

yt = xt′β +   ϵt/ϕ(B) , 

where ϕ(B)=1−ϕ1B for AR(1) case.

If you multiply both sides of the above equation by (1−ϕ1B), then you get the following:

yt - ϕ1*y_t-1  = xt′β - ϕ1*x_t-1*β + ϵt 

this implies

yt = ϕ1*y_t-1  +  xt′β - ϕ1*x_t-1*β + ϵt        (2)

If you compare (1) and (2), you can see that the two specifications have opposite signs on the AR parameter ϕ1. 

I hope this helps.