Mathematical representation of a particular state space model

[rrremedio]

Hello,

I am working with the following state space model and would like to confirm my understanding of how EViews interprets the error structure.

Here is the model specified in the sspace object:

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depvar1 = SV1 + [ENAME=E1, VAR=192*EXP(C(1))] depvar2 = C(3)*depvar2(-1) + E1 @STATE SV1= SV1(-1) + [VAR=EXP(C(1))]

Using the notation from the EViews User Guide II on State Space Models, my interpretation is as follows:

Signal Error Covariance (2x2, H matrix): Since the same named error E1 is used in both signal equations, the resulting H matrix is singular, with all of its elements being equal to 192*EXP(C(1)). Is this correct?

State Error Covariance (2x2, Q matrix): Since SV1 is a random walk with variance EXP(C(1)) and the implicit state for the coefficient C(3) is deterministic, the Q matrix for the state disturbances is effectively [EXP(C(1))] for the SV1 state, and zero for any other states. Is this logic correct?

Cross-Covariance (G matrix): This is my main question. Because the same parameter, C(1), is used to define the variance of a state disturbance (SV1) and the observation disturbance (E1), does EViews interpret this as a correlation between the two? In other words, does this specific syntax result in a non-zero cross-covariance matrix G (between the state and signal errors)?

[EViews Glenn]

That's an interesting model. Let me see if I can take a crack at this.

First off, in the DEPVAR1 equation you define the error E1 with a variance specification and reuse the error E1 in the DEPVAR2 equation. I must admit that while I think your text specification is fine, I would probably write the system so that the E1 error is defined separately and inserted into the individual signal equations:

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depvar1 = sv1 + [ename=e1] depvar2 = c(3)*depvar2(-1) + [ename=e1] @evar var(e1) = 192*exp(c(1)) @state sv1= sv1(-1) + [var=exp(c(1))]

While I do think your method of specifying will work, I tend to be paranoid about separating out the definition from the use of a shared error as it makes it a tad clearer that the E1 in the DEPVAR2 equation is the same as the E1 in the DEPVAR1 equation. I think you'll see why in my last example below.

That said, your interpretation of the H matrix is correct, H will be singular = 192*exp(c(1)) times a 2 x 2 matrix of 1s.

As to the state error covariance, I am not sure why you say that it is 2 x 2. Perhaps you left off an equation, but to my reading, there is a single state SV1 in the system with variance exp(c(1)), and signal equation coefficient matrix [1, 0]'. I personally think of the c(2)*depvar2(-1) as a fancy expression for a predetermined intercept in the equation, and not an implicit state.

Note that as written, your state specification indicates that there is an error term in the state equation with the variance exp(c(1)), and that this error term is unrelated to the error term in the signal equations. The best way to think about this is that since the error is not named explicitly, there is an unnamed error defined for this equation. The variance of this second error is restricted to be 1/192 times the variance of the E1 variance, but the errors themselves are independent, and the cross covariance matrix G=0 accordingly.

So the verbose method of writing your system is

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depvar1 = sv1 + [ename=e1] depvar2 = c(3)*depvar2(-1) + [ename=e1] @state sv1= sv1(-1) + [ename=e2] @evar var(e1) = 192*exp(c(1)) @evar var(e2) = exp(c(1))

If you do want to allow for cross-covariance, you will have to name the error term in the state equation as in the latter example, and define the covariance between the named E1 and a named E2 explicitly.