# DLM regression with parameter restriction

Good afternoon,

I am attempting to fit a state space regression model of the form:

$$Y_{t} = \beta_{1}Y_{t-1} + (1-\beta_{1})[i^* + \beta_{2}X_{t}] + \epsilon_{1,t}$$

$$i^* = i^*_{t-1} + \epsilon_{2,t}$$

Should I estimate $$\beta_{1}$$ and $$\beta_{2}$$ trough ML imposing that restriction and then run KF to get the $$i^*$$ latent variable? Is there a best way to proceed?

• Hi: is there an error term in the observation equation ( the equation for $Y_{T}$ ) or is $i^{*}$ the only stochastic variable ? Since you're estimating $\beta_1$ and $\beta_2$, then I would think that the first equation would have its own error term also ? – mlofton Apr 28 at 21:15
• Yes, i've already corrected. There is an error term in the first equation too. – Lucas Queiroz Apr 28 at 22:25
• so, in that case, it's a kalman filter setup but the problem is that the observation equation is non-linear. In that case, I think you have to either linearize the first equation (somehow ) or use a KF that deals with non-linear observation equations. I have no experience with non-linear KF's so I shouldn't say more but that's what you have because of the $\beta_1 \times \beta_2$ term. – mlofton Apr 29 at 17:11
• @mlofton It is a linear model so the Kalman filter can be used to obtain the likelihood. What is needed is to impose the restriction on the parameters in the optimization (or MCMC). – hejseb May 3 at 9:20
• @hejseb: so you are saying that you don't run the KF recursions ( don't view as KF ) but instead use optimization or mcmc ? if that's the case, I learned something. if not, could you provide slightly more details. thanks. – mlofton May 4 at 12:12