Course Material
This section is the mathematical reference behind StochX. It is deliberately separate from the Python API: equations, definitions, hypotheses and decision rules are explained here first; implementation details live in the API and User Guide sections.
Track A — Time Series and Econometrics
This is the primary applied track for users coming from EViews or a university time-series course.
Workflow
observed series
→ descriptive analysis
→ transformations
→ stationarity / deterministic specification
→ ACF / PACF
→ model identification
→ estimation
→ residual validation
→ selection
→ forecast
The software workflow mirrors this sequence through Workfile, expressions, ADF/DF/KPSS/PP tests, correlograms, AR/MA/ARMA/ARIMA/SARIMA estimators, diagnostics, Box–Jenkins helpers and forecast objects.
The Time Series User Guide is the practical entry point; Time Series contains the longer course-oriented workflow notes.
Course topics to expand here
- weak stationarity and unit roots;
- deterministic components and trend specifications;
- lag operators and differencing;
- ACF/PACF interpretation;
- AR, MA and ARMA equations;
- ARIMA and seasonal ARIMA construction;
- information criteria: AIC, SC/BIC and HQ;
- residual whiteness and model adequacy;
- prediction and interval forecasting;
- linear state-space representation;
- Kalman filtering, smoothing and innovations.
Track B — Stochastic Processes
Chapter 1 — Discrete-Time Markov Chains
Transition matrices, n-step transition probabilities, Chapman–Kolmogorov, accessibility, communication classes, recurrence and transience, periodicity, irreducibility, stationary distributions, limiting behavior, hitting/return times and simulation.
Chapter 1 — DTMC · Markov-chain guide
Chapter 2 — Poisson Processes
Counting processes, independent and stationary increments, exponential inter-arrival times, event-count probabilities, conditional properties and non-homogeneous intensities.
Chapter 2 — Poisson processes · Poisson guide
Chapter 3 — Continuous-Time Markov Chains
Generator matrices, holding times, transition semigroups, matrix exponentials, Chapman–Kolmogorov in continuous time, trajectories and the birth-death specialization.
Chapter 4 — Conditional Expectation
Finite probability spaces, random variables, partitions, sigma-algebra intuition, conditional expectation on partition cells and the tower-property viewpoint.
Chapter 4 — Conditional expectation · Probability objects guide
Chapter 5 — Discrete-Time Martingales
Filtrations, adapted processes, martingales, integrability, stopping times, stopped processes and finite-sample checks.
Chapter 5 — Martingales · Martingale guide
Track C — Reproducible simulation
Simulation is the computational bridge between theory and testing. Every generator should be paired with theoretical moments, path properties or deterministic fixtures where possible.
Documentation philosophy
The course section answers why the mathematics works. The User Guide answers when and how to use the method. The API Reference answers what the Python object accepts and returns. Worked examples show the complete workflow.