Help please! prior distributions for measurement equation for consumer sentiment and I am about to submit thesis, thank you!

Estimation ML/Bayesian estimation
1

Dear All,
Thank you for those people who helped me previously especially big thanks to Johannes and who take time to look at my posts. I am grateful.
My current DSGE model involved using consumer sentiment data as one of observable variable, and use sentiment shock, sentiment shock is the ratio of expected future stock bubble divided by current stock bubble.

Define consumer sentiment observable variable as CSI, steady state value is CSIbar,
linearised state variable of output is y, linearised sentiment shock is Vs,
My measurement equation for consumer sentiment observable variable is as follows:
CSI=CSIbar+beta1*(y-y(-1))+beta2*Vs
higher consumer sentiment is intuitively associated with higher output growth and higher sentiment (growth in stock bubbles),
therefore I set both beta1 and beta2 follow gamma distributions (positive distributions),
however, i have questions about the prior mean for gamma distributions,
I set beta1’s prior mean as 1 and beta2’s prior mean also as 1,

I chose these prior means this because I used a dogmatic gamma prior for them, estimate the model, then I use ols to run a regression of consumer sentiment index on state variable (y-y(-1)) and shock Vs, both of the regression coefficients are close to 1. Is this method of getting prior mean ok or is double counting data?
What other prior means can I use?

My question is kind of urgent, because I am going to submit my thesis soon and I need verification of this question.

Any suggestions will be helpful!
Thank you very much!
Best regards,
Jesse

2
  1. This way of getting the prior mean is not OK. There is the obvious issue of using the data twice. But even if you accept this as an “endogenous” prior, there is the issue that your OLS regression is generally misspecified. The variables you put in on the right-hand-side are not exogenous, i.e. there will be an endogeneity issue. Generally, you cannot estimate single equations from your model based on the equilibrium behavior of your model.
  2. If you do not know how to set your prior distribution, why do you not go for a flat prior, i.e. a uniform distribution?