I'm trying to fit linear mixed models to 3 different DV (so three models). I understand that REML gives less biased variance estimates. As im more interested in the fixed effects, I use ML for the initial stepwise model reduction based on AIC-values, and use REML to fit my final (reduced) models.

However, if I got that right, REML ignores the fixed part for fitting the model, right? And since 2 of my 3 models have only very little random variance, I'm confused whether I should completely stick to ML-estimates? What is your opinion on this? Am I right in my understanding of REML vs. ML?