The characteristics of an output of an exhaust gas sensor (5) with respect to an air-fuel ratio of an engine (1) are expressed according to a quadratic function, and the parameters of the quadratic function are sequentially identified according to a sequential identifying algorithm using data of the output of the exhaust gas sensor and data of an output of an air-fuel sensor (4) which detects the air-fuel ratio of the engine. Deterioration evaluating parameters whose values change as the deterioration of a catalytic converter processes are sequentially determined from the identified values of the parameters of the quadratic function, and the deteriorated state of the catalytic converter (3) is evaluated based on the determined deterioration evaluating parameters.