An important transition in the economic history of countries occurs when they move from a regime of low prosperity, high child mortality and high fertility to a state of high prosperity, low child mortality and low fertility. Researchers have proposed various theories to explain this demographic transition and its relation to economic growth. In this article, we test the validity of some of these theories by fitting a non-linear dynamic model for the available cross-country data. Our approach fills the gap between the micro-level models that discuss causative mechanisms but do not consider if alternative models may fit the data well, and models from growth econometrics that show the impact of different factors on economic growth but do not include non-linearities and complex interactions. In our model, mortality and fertility decline and economic growth are endogenized by considering a simultaneous system of equations in the change variables. The model shows that the transition is best described in terms of a development cycle involving child mortality, fertility and GDP per capita. Fertility rate decreases when child mortality is low, and is weakly dependent on GDP. As fertility rates fall, GDP increases, and as GDP increases, child mortality falls. We further test the hypothesis that female education drives down fertility rates rather than child mortality, but find only weak evidence for it. The Bayesian methodology we use ensures robust models and we identify non-linear interactions between indicators to capture real-world non-linearities. Hence, our models can be used in policymaking to predict short-term evolutions in the indicator variables. We also discuss how our approach can be used to evaluate policy initiatives such as the Millennium Development Goals or the Sustainable Development Goals and set more accurate, country-specific development targets.