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## Question Regarding Vector Autoregressive Bayesian Priors; SAS ETS: Proc VARMAX

Hello All,

I was hoping someone could shed some light on the difference between the "Theta" and "Lambda" Bayesian prior options within Proc VARMAX of SAS ETS. The both say that they "specify the prior standard deviation of the AR coefficient parameter matrices" but their doesn't seem to be any explanation of how the two are different or what each really does. I was also hoping to find out if their was any connection to these and the "Minnesota Priors" introduced by Robert Litterman in the academic literature.

They are defined in SAS help as follows:

LAMBDA=value

specifies the prior standard deviation of the AR coefficient parameter matrices. It should be a positive number. The default is LAMBDA=1. As the value of the LAMBDA= option is increased, the BVAR() model      becomes closer to a VAR() model.

THETA=value

specifies the prior standard deviation of the AR coefficient parameter matrices. The value is in the interval (0,1). The default is THETA=0.1. As the value of the THETA= option approaches 1, the specified BVAR()      model approaches a VAR() model.

Thank you very much for your time. Any help is greatly appreciated.

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## Re: Question Regarding Vector Autoregressive Bayesian Priors; SAS ETS: Proc VARMAX

I think this is a case of the documentation missing something.  If you go to the Bayesian VAR and VARX Modeling part of the documentation, it gives Litterman's definition of the variance, with lambda as prior standard deviation of the diagonal elements, and theta as a constant used to fit the ratio of the variances.  I hope this helps.  If not, on the Forecasting and Time Series forum is an excellent resource.

Steve Denham

2 REPLIES 2

## Re: Question Regarding Vector Autoregressive Bayesian Priors; SAS ETS: Proc VARMAX

I think this is a case of the documentation missing something.  If you go to the Bayesian VAR and VARX Modeling part of the documentation, it gives Litterman's definition of the variance, with lambda as prior standard deviation of the diagonal elements, and theta as a constant used to fit the ratio of the variances.  I hope this helps.  If not, on the Forecasting and Time Series forum is an excellent resource.

Steve Denham

Calcite | Level 5

## Re: Question Regarding Vector Autoregressive Bayesian Priors; SAS ETS: Proc VARMAX

Thanks a lot for taking the time to reply to my question. That definitely helps clarify things, I appreciate the help!

All the best,

Chris

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