Inspired by the reservoir computing paradigm in machine learning, we introduce a new class of observation-driven time series models in which the time-varying parameters are reconstructed from a high-dimensional basis of randomly generated states, called reservoirs. Compared with existing observation-driven specifications, the proposed class offers several advantages: the invertibility condition is tractable and invariant across model specifications, while the gradient and Hessian of the likelihood function are available in closed form, making estimation computationally efficient also in multivariate applications. We establish conditions for consistency and asymptotic normality of the quasi-maximum likelihood estimator. Furthermore, building on universality and random-feature approximation results for reservoir computing, we show how these approximation properties carry over to the observation-driven setting and yield bounds for the quasi-maximum likelihood estimated readouts. Simulated and empirical results show that the method outperforms alternative observation-driven specifications in recovering highly nonlinear parameter dynamics, including jumps, structural breaks, and chaotic behavior.