If you want to study both long and short run relationship between GDP and budget deficit, the most appropriate framework is vector error correction model.
There are several techniques that you can use in this case.
1. Co-integration test (to examine the long run relationship between the two variables)
2. VAR or VECM (if there is no co-integrating relationship you can use VAR and and if there is such relationship, you need to use VECM.
3. More over using the results of VAR or VECM, you can examine the granger-cauaslity or pair-wise granger causality to see the nature of relationship (uni-directional or bi-directional) between the two variables.
I've one more concern about nominal GDP and budget deficit, it's better to take all these at constant prices. It will make your analysis more accurate.
There are several techniques that you can use in this case.
First of all check for unit Root and depending on the attainment of stationarity of the variables , you can decide what to do. i.e. which Method to use.