The battery light may illuminate on the instrument panel because the alternator wire harness connector is damaged. A new connector kit is available and should be installed to correct this concern.
Problems for specific Mercury Cougar years:
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Most reported 2001 Mercury Cougar problems
The Check Engine Light can illuminate with codes due to the lack of EGR (exhaust gas recirculation) flow caused by the intake manifold passage plugging up with carbon and oil. The EGR valve has to be removed and the passage cleaned thoroughly. Partial restriction of this passage can cause Smog test failures for the NOx emissions without causing a drivability symptom.
If the vehicle hesitates or idles roughly after being parked for twenty to thirty minutes, with the engine still warmed to normal operating temperature, you may need to reprogram the PCM (powertrain control module) with new calibration.