The engine may idle roughly or stall when it's cold because the oil pump pressure relief plunger is sticking. Failure is usually due to wear and it causes excessive oil pressure to pump up the hydraulic valve lifters keeping the valves from closing completely, which results in a loss of compression. The oil pump will need to be replaced in this case.
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Repairpal verified 1992 Mazda Miata MX-5 problems
A loud ticking noise (hydraulic valve lifter noise) may be heard from the top of the engine, particularly when the engine is still cold. The noise may occur more frequently on higher mileage vehicles; Hydraulic valve lash adjusters may need to be replaced to correct this issue. Following the maintenance schedule for oil changes will also help prevent a recurrence of the noise.