^^ what he said. Most events will not allow you to have the top down. Some organizations would let me drive with the top down in my s2000.
Also most will not even allow convertibles unless you have a roll bar/hoops. S2k and Boxter have factory roll hoops for example. Either way the soft top offers hardly any protection so I always thought it silly to require the top to be up.
As for being faster with the top down. It depends on the track. On a tight course like Streets of Willow I was faster with the top down. On higher speed courses I would be faster with the top up. Aerodynamics and lower center of gravity being the differences.
Top down track days are a blast
