Data in modern economic and financial applications often arrive as a stream, requiring models and inference to be updated in real time -- yet most semiparametric methods remain batch-based and computationally impractical in large-scale streaming settings. We develop an online learning framework for semiparametric monotone index models with an unknown monotone link function. Our approach uses a two-phase learning paradigm. In a warm-start phase, we introduce a new online algorithm for the finite-dimensional parameter that is globally stable, yielding consistent estimation from arbitrary initialization. In a subsequent rate-optimal phase, we update the finite-dimensional parameter using an orthogonalized score while learning the unknown link via an online sieve method; this phase achieves optimal convergence rates for both components. The procedure processes only the most recent data batch, making it suitable when data cannot be stored (e.g., memory, privacy, or security constraints), and its resulting parameter trajectories enable online inference such as confidence regions--on parameters including policy-effect analysis with negligible additional computation. Monte Carlo experiments on both simulated and real data show adequate performance especially relative to full sample methods.
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