Large language models are increasingly used in social sciences, but their training data can introduce lookahead bias and training leakage. A good chronologically consistent language model requires efficient use of training data to maintain accuracy despite time-restricted data. Here, we overcome this challenge by training a suite of chronologically consistent large language models, ChronoBERT and ChronoGPT, which incorporate only the text data that would have been available at each point in time. Despite this strict temporal constraint, our models achieve strong performance on natural language processing benchmarks, outperforming or matching widely used models (e.g., BERT), and remain competitive with larger open-weight models. Lookahead bias is model and application-specific because even if a chronologically consistent language model has poorer language comprehension, a regression or prediction model applied on top of the language model can compensate. In an asset pricing application predicting next-day stock returns from financial news, we find that ChronoBERT's real-time outputs achieve a Sharpe ratio comparable to state-of-the-art models, indicating that lookahead bias is modest. Our results demonstrate a scalable, practical framework to mitigate training leakage, ensuring more credible backtests and predictions across finance and other social science domains.