I am having a problem regarding the measurement equation in estimation. Can you give me some advice please?

For example with the nominal interest rate:

Quarterly Nominal Interest rate is computed by:
Robs=FFR/400;
Let “consr” be the steady-state implied by the model, so the measurement for interest rate is:
Robs=Rhat + consr (Rhat is the deviation of R from steady state)

But as I learned from Smets and Wouter, it can be done by:
Robs=FFR/4
then the measurement equation is:
Robs=Rhat+consr*100 (consr" is the steady-state implied by the model)

When I use the second way, my model can be estimated successfully and the Hessian matrix at the mode is positive definite.
But if I employ the first way, the Hessian matrix is not positive definite.

It depends on your scaling in the rest of the model. Dividing by 400, you are saying that 0.01 is a 1% quarterly net interest rate. Dividing by 4, you are saying that 1 is a 1% quarterly net interest rate. Due to linearity, you can do this, but this will require you to scale up also the other variables by a factor of 100 to maintain consistency. A good check typically is to simulate the observed model variables with sensible parameters and compare them to the actually observed data. They should look similar. See also sites.google.com/site/pfeiferecon/Pfeifer_2013_Observation_Equations.pdf?attredirects=0

I tried to estimate a linear model using two alternative methods of treating the data, but one has positive definite Hessian matrix, the other doesnt. Do you know why it happens? I guess I have to change the prior of standard deviation of shocks, right? the standard deviation of shocks should be smaller when I divide data by 100 (and 400 for annual data like interest rate)

If I want to estimate a non-linear model (from 2nd-order perturbation) using particle filter, what kind of data transformation should I use? What did you do in your paper called “Policy uncertainty and Business Cycles”?

There are many reasons why this could happen. The prior has to be consistent with your data specification and should be smaller for the standard deviations if you take 0.01 to be 1%.

It depends. My experience is that you should opt for a transformation that keeps the mean in the data (for at least 1 variable). If you cannot do that, use first differences.What we did is