These can be produced by
REML to assess the contributions of individual terms in the fixed model. In a balanced design, the Wald statistic corresponds to the treatment sum of squares divided by the stratum mean square. The statistic would have an exact chi-square distribution if the variance parameters were known but, as they must be estimated, it is only asymptotically distributed as chi-square.
In practical terms, the chi-square values will be reliable if the residual degrees of freedom for the fixed term is large compared to its own degrees of freedom. In a balanced design, the number of residual degrees for a fixed (or treatment) term is simply the number of residual degrees of freedom for the stratum where the term is estimated. Also, if the design is balanced, the Wald statistic divided by its degrees of freedom will be distributed as Fm,n, where m is the number of degrees of freedom of the fixed effect, and n is the number of residual degrees of freedom for the fixed effect.
For unbalanced designs the F distribution is only approximate and, in any case, it may be difficult to deduce the appropriate residual numbers of degrees of freedom. The important point to remember, though, is that use of the chi-square distribution tends to give significant results rather too frequently, so you need to be careful especially if the value is close to a critical value.