97 Jetta electric powers on with no key
 

My 97 Jetta GLS, 81,000 miles, has developed this problem: when no key is in the ignition, the car will power on intermittently. The dash warning lights and gauges come on, and if the alarm is armed, the alarm goes off. I've sat in the car with no key, and watched the dash and seen the car get electricity, stay on for about 15 seconds, go off, and then repeat. Only the electric accessories power on, the car doesn't attempt to start (thankfully!)

The car starts fine and when the engine is running, nothing ever cuts off (yet). Ignition switch, short circuit somewhere else, or where to start looking?