Estimates of both the human impact on climate as well as the economic impacts of climate change are crucial to inform policy decisions. Econometric modelling allows us to quantify these impacts and their uncertainties, but models have to be consistent with the underlying physics and the time series properties of the data. Here, I show that energy-balance models of climate are equivalent to an econometric cointegrated system and can be estimated in discrete time. This equivalence provides a basis for the use of cointegration methods to estimate climate responses and test their feedback. Further, it is possible to use the estimated parameters to quantify uncertainties in integrated assessment models of the economic impacts of climate change. In an application I estimate a system of temperatures, ocean heat content, and radiative forcing including greenhouse gases and find statistical support for the cointegrated energy balance model. Accounting for structural breaks from volcanic eruptions highlights large parameter uncertainties and shows that previous empirical estimates of the temperature response to increased CO2 concentrations may be misleadingly low due to model-misspecification.