In Herbst and Schorfheide "Bayesian Estimation of DSGE Models", in the appendix named "Data Sources", they show several formulas relating the empirical data which should be used for the model.
Here's a snapshot from the book.
I've taken the data from the FRED. One can notice that the observations were taken monthly for some variables, and for others were taken quarterly. Some begin in the 40's, while others begin in the 50's.
However, the formulas in the book assume the same time period $t$. Hence, what I did was to discover the intersection of the dates of all observations, i.e. quarterly data beginning from the 50's, and cleaned the data accordingly.
Now, I'm using the formulas in the book to create those new variables, namely per capita real output, annualized inflation, and federal funds rate.
My doubt resides on whether my procedure was correct.