The matrix AT has dimension Tx(T-1)k and UT is a (T-l)kxl vector of
parameter disturbances. The stochastic specification of VT is
E(VT) = 0 (6a)
E(VTVT) = a T + AT(I T-1Q)AT = T, (6b)
where a 2 denotes the variance of the individual structural disturbances
E
contained in Et.
If IT is known estimation of the parameter vector at time T is
given by
T = (XT XT) XT2T YT (7)
2
Of course, a is rarely known and cannot be estimated given BT since
T is conditioned by a 2. Cooper has suggested an iterative procedure
to calculate a2 which can be efficiently implemented. The method starts
with an initial estimate of 2 and then derives T and BT. Then a new
estimate of a2 is given by
^2 A ), l 1
= (YT-XT T)T1 (8)
and the procedure is repeated until convergence is achieved.
The forecasting and updating problem for the varying parameter
model may be solved by using the appropriate Kalman filter recursions.
The relationship between the Kalman filter model and generalized least
squares models has been developed by Duncan and Horn and Sant (1977).
The one step ahead prediction is given by the parameter time update
aT+1/T = T (9)