  Vertical federated learning (VFL) is a promising area for time series
forecasting in industrial applications, such as predictive maintenance and
machine control. Critical challenges to address in manufacturing include data
privacy and over-fitting on small and noisy datasets during both training and
inference. Additionally, to increase industry adaptability, such forecasting
models must scale well with the number of parties while ensuring strong
convergence and low-tuning complexity. We address those challenges and propose
'Secret-shared Time Series Forecasting with VFL' (STV), a novel framework that
exhibits the following key features: i) a privacy-preserving algorithm for
forecasting with SARIMAX and autoregressive trees on vertically partitioned
data; ii) serverless forecasting using secret sharing and multi-party
computation; iii) novel N-party algorithms for matrix multiplication and
inverse operations for direct parameter optimization, giving strong convergence
with minimal hyperparameter tuning complexity. We conduct evaluations on six
representative datasets from public and industry-specific contexts. Our results
demonstrate that STV's forecasting accuracy is comparable to those of
centralized approaches. They also show that our direct optimization can
outperform centralized methods, which include state-of-the-art diffusion models
and long-short-term memory, by 23.81% on forecasting accuracy. We also conduct
a scalability analysis by examining the communication costs of direct and
iterative optimization to navigate the choice between the two. Code and
appendix are available: https://github.com/adis98/STV
