Deseasonalize data AND deflate with CPI? I have property return variables and economic variables that I am using in a VECM/VAR to generate Impulse Response Functions. I have deflated my data with CPI, but do I also have to deseasonalize the data? My results are much better when the data is non seasonally adjusted but deflated with the Consumer Price Index (CPI) to adjust for inflation, than when it is deseasonalized and deflated. 
Is there sufficient theory to support the use of non seasonally adjusted data where it is deflated with CPI?
 A: CPI itself can be seasonally adjusted, for instance the CPIAUCSL series are already adjusted. Just make sure that when mixing unadjusted data you don't introduce the artefacts of the phase and frequency differences. 
Imagine, your series A and B are both annual seasonality but one is lowest in July while the other is in January. If you divide A/B you can get weird seasonality effects that are result of the transformation itself. For instance, this may show up like 6 or month seasonality. In this case it's best to deseasonalize series before transformation.
In your case if the series that you're deflating have annual seasonality and CPI is not adjusted, then if you first deseasonalize the series then divide by CPI, your resulting series will acquire seasonality from CPI. That's why simple deflation may seem to work better
A: Deflating the data does not exclude the need for seasonal adjustment. If you are using month over month return, I recommend seasonal adjustment. For YoY or 12mo moving average, there is no need. 
If you use R, you can check seasonal::seas.
