This period marks the foundation of modern time series analysis. Researchers began studying random processes, seasonal patterns and long term trends in a systematic way. Early smoothing and decomposition techniques emerged, such as moving averages and basic exponential smoothing. These developments created the mathematical groundwork that later statistical models and forecasting methods would build upon.

The introduction of ARIMA models by Box and Jenkins marked a major shift toward systematic statistical forecasting. Their framework unified autoregression, differencing and moving averages into a single model class capable of capturing trends, seasonality and autocorrelation. The method standardized model identification, diagnostics and validation, establishing a rigorous workflow that shaped time series analysis for decades.

State space representations brought a more flexible way to model dynamic systems, allowing hidden states, measurement noise and time varying structures. The Kalman filter provided an efficient algorithm to update estimates as new data arrived, making real time forecasting and control applications possible. These models became essential in economics, engineering and signal processing due to their adaptability and probabilistic foundation.

The ETS framework formalized exponential smoothing as a family of statistical models with clear error, trend and seasonality components. This brought theoretical grounding to methods that were previously heuristic. ETS models are fast to compute, robust to noise and well suited for business forecasting, especially when data show gradual trends and stable seasonal patterns. Their simplicity and reliability contributed to widespread adoption.

The rise of machine learning introduced new ways to forecast by extracting patterns from engineered features rather than modeling the time series directly. Gradient boosting, random forests and support vector methods became popular for handling nonlinear relationships, external variables and large datasets. These algorithms often delivered competitive accuracy, particularly when domain specific features were incorporated into the training process.

Neural sequence models such as RNNs, LSTMs and CNN based architectures enabled direct learning from raw time series. They captured complex temporal dependencies, long range patterns and nonlinear interactions without the need for manual feature engineering. As computational power increased, deep learning became a prominent approach for high dimensional and high frequency time series across domains such as finance, energy and sensors.

Transformers introduced attention mechanisms that allowed models to focus on relevant parts of a sequence and handle long range dependencies more effectively than recurrent networks. Their scalability and parallelization made them attractive for large forecasting tasks. Recent adaptations, such as Informer, FEDformer and TimeGPT, extended transformers specifically for time series, offering strong performance on diverse datasets and paving the way for foundation models in forecasting.