perform econometric panel data regression on a simple dataset

I have a panel dataset (year, state, unemployment rates, jobs vacancy rates) and I need to perform some econometric regressions to replicate a research paper analysis (see attached - only need to replicate Table 2, 3 and 4). The job doesn't sound too time-consuming. This job will need to be done on R and the output will be an R code file (fully executable) and a one-page summary.