I have a disagreement with friend over if an observable that depend on lagged observable, should be essentially thrown out when doing Kalman filter, since the lagged observable are observed at time t and present no innovation.

I have the following system:

xt=u+z(1,t-1)+x(t-1)+d·[y(t-1)-x(t-1)]+e(x,t)

yt=u+z(2,t-1)+y(t-1)-d·[y(t-1)-x(t-1)]+e(y,t)

z1 and z2 are AR(1) processes

z(1,t)=p1z(1,t-1)+e(1,t)

z(2,t)=p2z(2,t-1)+e(2,t)

The observations are xt and yt

I say the state variables are z1 and z2, but my friend say I am wrong in that the state variable must also include x(t-1) and y(t-1)

Since I think that x(t-1) and y(t-1) are observed and known at time t, can I claim that they are known and thus can be filtered out

And as far as the Kalman filter gain and the variance of estimate P , as well as their steady state values are concerned, I can derive K and P and their steady state values by pretending the system is the following, which will yield the exact same K and P:

xt=u+z(1,t-1)+x(t-1)+e(x,t)

yt=u+z(2,t-1)+y(t-1)+e(y,t)

z1 and z2 are AR(1) processes

z(1,t)=p1z(1,t-1)+e(1,t)

z(2,t)=p2z(2,t-1)+e(2,t)

Am I right that the 2nd system will yield the same K and P and their respective steady state value as the original system?

But my friend say I am wrong because x(t-1) and y(t-1) are not known at time 0, they must be state variables, and thus xt and yt are both observables variables and state variables.

I have the following system:

xt=u+z(1,t-1)+x(t-1)+d·[y(t-1)-x(t-1)]+e(x,t)

yt=u+z(2,t-1)+y(t-1)-d·[y(t-1)-x(t-1)]+e(y,t)

z1 and z2 are AR(1) processes

z(1,t)=p1z(1,t-1)+e(1,t)

z(2,t)=p2z(2,t-1)+e(2,t)

The observations are xt and yt

I say the state variables are z1 and z2, but my friend say I am wrong in that the state variable must also include x(t-1) and y(t-1)

Since I think that x(t-1) and y(t-1) are observed and known at time t, can I claim that they are known and thus can be filtered out

And as far as the Kalman filter gain and the variance of estimate P , as well as their steady state values are concerned, I can derive K and P and their steady state values by pretending the system is the following, which will yield the exact same K and P:

xt=u+z(1,t-1)+x(t-1)+e(x,t)

yt=u+z(2,t-1)+y(t-1)+e(y,t)

z1 and z2 are AR(1) processes

z(1,t)=p1z(1,t-1)+e(1,t)

z(2,t)=p2z(2,t-1)+e(2,t)

Am I right that the 2nd system will yield the same K and P and their respective steady state value as the original system?

But my friend say I am wrong because x(t-1) and y(t-1) are not known at time 0, they must be state variables, and thus xt and yt are both observables variables and state variables.

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