ABSTRACT
Our discussions up to now have focused on the model having p × 1 response vectors
y(t) = H(t)x(t) + e(t)
corresponding to q × 1 state vectors
x(t + 1) = F (t)x(t) + u(t),
where H(t), F (t), t = 1, . . ., n, are known matrices, u(0), . . ., u(n − 1) are zero mean, random q-vectors that are uncorrelated with each other and with the uncorrelated, zero mean, random p-vector e(1), . . ., e(n). The covariance structure of the model is then determined from the specifications
Var (u(t)) = Q(t), t = 0, . . ., n − 1,
Var (e(t)) = W (t), t = 1, . . ., n,
for known matrices Q(t−1), W (t), t = 1, . . ., n, and the condition that the initial state vector has zero mean and
the KF we want to expand this model somewhat to allow for more general applications. Perhaps the first thing to realize is that our previous
restriction that the y(·) and x(·) vectors all have fixed dimensions p and q, respectively, has never been necessary. Indeed, all the recursions we have developed up to now will still work perfectly well with dimensions that change with the t index. Thus, from this point on we can proceed as if y(t) and x(t) are, respectively, pt × 1 and qt × 1 with H(t), F (t), W (t), Q(t) now representing pt × qt, q(t+1) × qt, pt × pt and qt × qt matrices, respectively. The next step is to broaden our original model formu-
lation by allowing for nonzero means. To accomplish this we will use a response equation of the form
y(t) = AY (t)β + H(t)x(t) + e(t) (8.1)
for t = 1, . . ., n, coupled with the state equation
x(t + 1) = AX (t)β + F (t)x(t) + u(t), (8.2)
for t = 0, . . ., n−1, where AY (t), AX (t−1), t = 1, . . ., n, are, respectively, pt×r and qt×r known matrices, β is a r-vector of parameters, the e(·) and u(·) processes are as before and x(0) = 0 in the sense that
x(1) = AX (0)β + u(0). (8.3)
As a result of (8.3) the mean vector for x(1) is E[x(1)] = AX (0)β which has the consequence that
E[y(1)] = AY (1)β + H(1)AX (0)β.
