Hi All I have an expression for the H step ahead forecast of a model but I want to check if it's correct. Suppose I have the following model.

Y_(T+30,T+31) = rho* Y_(T,T+30) +  beta * (1-rho) X_T,(T+1)

Explanation of above:

beta, rho are  estimated parameters. The subscripting just denotes value of the respective variable over that time period. So, X_T,T+1 denotes the value of X over the period from T to T + 1, etc.

So, now I want to calculate the H step ahead forecast for Y when one is sitting at t = T+30  so Y_(T+30,T+30+ H) . Note that I am sitting at T+30 when I want the forecast so any X's with subscripts at T+30 or earlier are known exactly.

Also, one should assume that "New X's" are zero over the period being forecasted are ( basically it can be  assumed  that the X_t's  from time T+30 to T+30 + H are random shocks with N(0,1) distribution ). Thank you very much for any help.