On page 349 of gretl's user guide the state space model is defined as

Later, when presenting the state-space GUI (p. 369) we read  that it supports models of the form

My only question is about the time indices on the state vector in the transition equation. 

In the first model (p. 349), the starting values for the state variables (alpha) should be for time t, and will also affect the first time point of the measurement equation. In the second formulation (p. 369), these starting values should be for time (t-1), they should lead to alpha values for time t through the transition equation alone, which will then affect the first time point of the measurement equation.

Which of the two does gretl do?

-- 
Alecos Papadopoulos PhD
Affiliate Researcher
Dpt of Economics, Athens University of Economics and Business
Foundation for Economic and Industrial Research (IOBE)
web: alecospapadopoulos.wordpress.com/
ORCID:0000-0003-2441-4550