No, this would not be a common approach to unemployment forecasting. The common approaches would be something like VAR, i.e. time series modeling. The field which deals with this stuff is called macroeconomic forecasting.
The reason why particularly unemployment would be unsuitable for what you called causal forecasting is the endogeneity. Unemployment itself is such an important variable, that its state impacts the state of other variables. So, if you make model where you forecast macroeconomic variables, then use them to forecast unemployment, your model will be nonsensical, in my opinion. Unemployment, inflation and GDP must be in your core model in some form. It may not necessarily be unemployment rate, it could employment rate or some other form labor capacity utilization.
Having said that there are academic models which do not have unemployment in them, see e.g. the DSGE model: F. Smets and R. Wouters (2002), An Estimated Stochastic Dynamic General Equilibrium Model of the Euro Area, European Central Bank, Working Paper Series, No. 171. Also in Journal of the European Economic Association, Vol. 1, No. 5, 2003, pp. 1123-1175.
This MATLAB example will give you an idea of how typical forecasting models may look like. Note, they call their model DSGE, ignore it. It's not a DSGE model at all, this is a simple VAR model.