In Discovering Statistics Using SPSS 4e, Andy Field writes on p835 that:
SPSS gives you the choice of two methods for estimating the parameters in the analysis: maximum likelihood (ML), which we have encountered before, and restricted maximum likelihood (REML). The conventional wisdom seems to be that ML produces more accurate estimates of fixed regression parameters, whereas REML produces more accurate estimates of random variances (Twisk, 2006). As such, the choice of estimation procedure depends on whether your hypotheses are focused on the fixed regression parameters or on estimating variances of the random effects. However, in many situations the choice of ML or REML will make only a small difference to parameter estimates. Also, if you want to compare models you must use ML.
I'm most interested in the last sentence. Why is it that to compare models you must use ML and not REML?