I have a dataset which contains daily NO2 and ozone levels of the 50 largest cities in Spain for one year (so 50 x 365 observations). The data are retrieved from http://www.eea.europa.eu/data-and-maps/data/airbase-the-european-air-quality-database-7. When regressing ozone on NO2, I find a significantly negative coefficient for contemporaneous NO2 (-0.35, standard error 0.027), the first lag (-0.05, standard error 0.014), insignificant coefficients on NO2-levels on the second and third lag (-0.0014 and 0.014 respectively), and a positive coefficient on the level of NO2 five days ago (0.50, standard error 0.013). I am also controlling for time-invariant local characteristics (fixed effects) and (lagged) weather conditions (temperature, sunshine, cloud cover, precipitation, wind).
I'm an economist so I don't know much about these things. I guess this could be explained by the complex mechanics of ozone formation? It would be great if you have some reference or idea why such a relation can be observed.
Edit: Added some more hopefully useful details.