Time-Series, Monte Carlo & Intro ML-for-Finance

Learning objectives

By the end you can:

1. Explain, in mechanism, why time-ordered data breaks the independence assumption behind ordinary statistics, and define the autocorrelation function (ACF) and partial autocorrelation function (PACF) that measure exactly how much memory a series carries.
2. Write down an AR(1) and an MA(1) process from first principles, state their stationarity/invertibility conditions, derive each one's theoretical ACF/PACF signature, and use the Box-Jenkins identification grid to tell an AR process from an MA process by eye.
3. Fit an AR(1) and an MA(1) to a real (synthetic, India ₹ and US $) return series in Python, reading off the fitted coefficient, its standard error and significance, the implied half-life of a shock, and the model's R², and interpret every number correctly.
4. Define stationarity precisely, explain why a random walk is the canonical non-stationary process, and run and correctly interpret an Augmented Dickey-Fuller test, including why its critical values are not the ordinary t-table.
5. Explain, and demonstrate numerically, why regressing one non-stationary series on another produces a spurious regression (high R², significant-looking t-stats, near-zero Durbin-Watson) with no real relationship behind it, and state the cointegration exception: when two non-stationary series share a genuine stable long-run relationship, and how the Engle-Granger two-step test detects it correctly (including why the test needs its own, stricter critical values).
6. Build a Monte Carlo simulation from scratch, specify a data-generating process, draw correlated random variates via a Cholesky decomposition, simulate paths, and compute a distribution of outcomes, and compute a Monte Carlo Value-at-Risk and Expected Shortfall on a two-asset India/US book, cross-checked against a closed-form parametric estimate.
7. State the Monte Carlo standard-error law (∝ 1⁄√N), explain why 10× more simulations buys only ≈3.16× more precision, and correctly judge when a variance-reduction technique (antithetic variates) helps a lot versus only a little.
8. Frame supervised versus unsupervised learning correctly, state the train/validation/test discipline and k-fold cross-validation, define overfitting as a bias-variance tradeoff and demonstrate it numerically on a fitted-polynomial example, and name the finance-specific trap (look-ahead bias from a non-chronological split) that breaks all of the above in a time-series setting.
9. **(Productivity objective: R10 duality, the QM/AA angle: copilot-but-you-interpret.)** Use an AI assistant to accelerate order-selection, scenario narration, and concept-tutoring on this module's tasks, while never letting the tool's suggestion replace your own diagnostic read of the ACF/PACF, the test statistic, or the convergence table.
10. State the binding C-3 boundary from memory: this module teaches the concepts, the math, and the failure modes; the quant-code region (DA/QD) teaches every one of these ideas in code (implementation, validation, pipelines, evaluation) and you can name exactly which module to go to for each build.

The duality, stated once (R10). Every module in this Ring carries two objectives at once. The understanding objective (items 1–8 and 10) is what the mastery gate rewards. The productivity objective (item 9) is the payoff you keep: knowing how to let a copilot draft the order-selection narrative or the scenario write-up while you remain the one who reads the diagnostic. A tool can suggest an ARIMA order; only you can look at the ACF and PACF and decide whether that suggestion is defensible.

Prerequisites & connections

Builds on. QM1.01 (probability, distributions, the CLT, and point/interval estimation, the Monte Carlo section below is the law of large numbers put to work, and its standard-error arithmetic is a direct descendant of QM1.01's sampling-distribution machinery) and QM1.02 (hypothesis testing and regression, every AR/MA fit below is an OLS or maximum-likelihood regression, and the diagnostic instincts QM1.02 built (read the standard error, read the t-stat, don't trust R² alone) are exactly what catches a spurious regression here). EX2.03 (Financial & Statistical Functions) already taught you LINEST, FORECAST.LINEAR/TREND, CORREL/RSQ, and the four ways an Excel regression lies (causation, small samples, non-stationarity, extrapolation), this module is the inferential/econometric layer under that Excel skill: EX2.03 told you non-stationarity was one of four traps; this module gives you the actual test that catches it and the actual math that explains why it's a trap. Nothing here re-derives LINEST. M2.07 (Shenanigans II & Quantitative Screens) used the Beneish M-Score, the Sloan accruals ratio, Altman's Z, and the Piotroski F-Score, every one of them a statistically estimated model built on the same sampling and estimation logic QM1.01/1.02 supplied; this module doesn't re-teach those screens, but it is worth noticing in passing that Altman's Z-score is literally a linear discriminant classifier, an early, hand-built cousin of the supervised-learning classifiers framed in §9 below.