A ticking engine is usually an indication of a problem with the valvetrain. The actual issue isn't always obvious and could be mechanical problems, an incorrect filter or a failing oil pump.
A majority of automobiles use hydraulic valve lifters that create a tight clearance between the rocker arm and camshaft with oil pressure. Whenever the oil pressure is off, the engine may start ticking because there is play between the valvetrain and the lifters. There's a chance that cold oil at startup can help with the lifters, but the oil gradually starts to thin out as the engine starts to warm up. If the oil is too thin, it's unable to work the lifters. The ticking noise gradually fades as the oil starts to thicken and as the automobile travels a few thousand miles.
It's best to use filters supplied by the automobile manufacture in order to lower the chances of the engine ticking. Low oil pressure is also attributed to an incorrect flow rate through a generic filter. The worst possibility is that the oil pump could be failing and needs to be replaced. The vehicle should be taken in for a professional inspection as soon as possible to avoid further problems.