Tools for Forcasting Unemployment Rate
I think one might use either vector auto-regression or perhaps factor-augmented vector auto-regression to make such predictions. The later will allow an economist to condition on hundreds of time series to predict future unemployment values without encountering the degrees-of-freedom issues that plague VAR whenever VAR attempts to condition on larger information sets. 

Here is a paper detailing Kalman filtering and VAR for predicting unemployment: 
https://www.degruyter.com/downloadpdf/j/ngoe.2015.61.issue-3/ngoe-2015-0009/ngoe-2015-0009.pdf

TLDR: VAR does better than does KF. This is for Romania's economy, FYI. 

Here is another paper detailing the use of VAR to forecast unemployment: 
https://www.brookings.edu/wp-content/uploads/2012/09/2012b_barnichon.pdf

Here is the seminal paper that introduced FAVAR:
http://faculty.wcas.northwestern.edu/~lchrist/finc520/QJE.pdf 

Wiki for VAR: 
https://en.wikipedia.org/wiki/Vector_autoregression

within the wiki, you will find a link to the seminal paper by Sims that introduces the VAR methodology to economics. Fair warning: this paper is dense:
http://www.ekonometria.wne.uw.edu.pl/uploads/Main/macroeconomics_and_reality.pdf