  We prove a twist formula for the epsilon factor of a constructible sheaf on a
projective smooth variety over a finite field in terms of characteristic class
of the sheaf. This formula is a modified version of the formula conjectured by
Kato and Saito in [Ann. Math., 168 (2008):33-96, Conjecture 4.3.11].
  We give two applications of the twist formula. Firstly, we prove that the
characteristic classes of constructible \'etale sheaves on projective smooth
varieties over a finite field are compatible with proper push-forward.
Secondly, we show that the two Swan classes in the literature are the same on
proper smooth surfaces over a finite field.
