In my model, I have a monopolistic labor market and sticky wage. Should I have to use wage or employment data in the estimation for a better identification of the labor market? If these data are not available, would this make the results not so good?
it is always better to include observable variables in the model as the likelihood then cannot imply implausible moments for these data. The best example is the relative price of investment, whose importance for business cycle contributions to hugely overestimated, if it is not included as an observable into DSGE models, see Justiniano et al. 2010 faculty.wcas.northwestern.edu/~gep575/ISRP10_gt.pdf.
However, if the data is not available, the best you can do is estimate your model with data available.