Using a roof box will reduce fuel economy but probably not by as much as you would expect, usually a few MPG. Fuel economy when using a roof box will be better than if you are carrying bikes on the roof, or on a high-mount tailgate carrier, or using some roof bars on their own, so it is very important that you remove the bars when you are not using them. The exception to this is wing-shaped aluminium aero bars. Not only are the aluminium bars much quieter than the square bars but they create so little drag that most can be left in place without any significant effect on fuel consumption.
More aerodynamic boxes will give improved efficiency – look out for wind tunnel testing and aerodynamic features that diffuse the air around the box and bars.
Roof boxes