Most countries in the world are not at the technological frontier, yet their economies grow and fluctuate. This paper sets up a quantitative model of endogenous growth with business cycle fluctuations to analyze the medium frequency fluctuations in non-frontier countries. The growth mechanism is a Schumpeterian creative destruction framework embedded into a real business cycle dynamic stochastic general equilibrium model, with standard and non-standard features. We allow multinational firms to enter the economy and challenge existing incumbents, which permits us to study the tension between their direct positive productivity contribution and their indirect negative contribution through the expected obsolescence of domestic innovators. We estimate the model using a full-information approach and show that multinationals' entry is both boon and bane for non-frontier economies.