Methods and systems are provided for carrying out on-board diagnostics of a plurality of components of an exhaust heat exchange system. In one example, degradation of one or more of a heat exchanger and a coolant system fluidically coupled to the heat exchanger may be detected based on a first temperature estimated upstream of the heat exchanger, a second temperature sensor estimated downstream of the heat exchanger, a coolant temperature, and a pressure estimated upstream of the heat exchanger. Also, degradation of a diverter valve of the heat exchange system may be detected based on inputs of a position sensor coupled to the diverter valve.