Just to clarify my understanding of the generation of fitted values for levels of Yt

do the estimated coefficients transpose directly into the specified equation

Yt=a1 Yt-1+ b1Xt + d +ut   (which begs the question of recovery of d )

or if the GUI difference estimator is the default estimator , are the fitted values derived through expansion of the differenced form of the equation with terms collected such that Yt=(1+a1)Yt-1.....etc.?..