Three real time-series cases

Three end-to-end exercises that integrate everything we've covered. Each case is a workflow a desk quant might run on a Monday morning. We won't reproduce the code in detail — that's for the Python and R courses — but we'll articulate the analytical steps, the diagnostics, and the conclusions.

Case 1 — GARCH vol model on NSE-20 daily returns

1. Pull 5 years of daily NSE-20 closing prices. Compute log returns. Plot the time series — look for breaks, gaps, regime changes (a multi-week trading halt or a dramatic policy shift).
2. Test for stationarity on returns: ADF should reject unit root easily. KPSS should not reject stationarity.
3. Inspect ACF of returns (should be ≈ zero) and ACF of squared returns (should be strongly positive — evidence of vol clustering).
4. Fit GARCH(1,1) with Gaussian innovations. Get α̂, β̂, ω̂. Check α + β ≈ 0.95-0.99 (typical), and the implied unconditional vol.
5. Refit with Student-t innovations. Compare log-likelihood and AIC.
6. Compute conditional one-day VaR_95 and VaR_99 using the fitted GARCH. Compare to historical-simulation VaR over the same period.
7. Back-test: count how often realised loss exceeds predicted VaR. Run Kupiec and Christoffersen tests. A correctly-calibrated 99% VaR has ~1% violation rate.

Case 2 — Cointegration test, Safaricom vs the NSE-20

1. Pull 3 years of daily prices for Safaricom (SAFCOM) and the NSE-20 index. Work in log levels.
2. ADF on each series in levels: both should fail to reject the unit-root null (both are I(1)).
3. ADF on first differences (log returns): both should reject (returns are I(0)). Confirms both are I(1).
4. Estimate the cointegrating regression: log SAFCOM_t = α + β log NSE20_t + u_t. Note β̂ ≈ 1 in many empirical setups — a 1% move in the index corresponds to ~1% move in Safaricom.
5. Engle-Granger ADF on residuals. If you reject the unit root (using Engle-Granger critical values, not standard ADF), the two are cointegrated.
6. Estimate the speed of adjustment: fit AR(1) to the residuals. The AR coefficient ρ gives half-life log(0.5)/log(ρ).
7. Translate into a trading rule: when residual exceeds 2σ_u, take the spread position; exit when it returns to zero.
8. Back-test on out-of-sample data. Compute the Sharpe of the rule. Stress-test on subperiods.

Case 3 — Kalman filter on a noisy commodity price

1. Take daily Brent crude futures settlements. Construct a 'noisy proxy' by adding observation noise (or use a high-frequency tick series).
2. Specify state-space: latent log-price follows random walk, observation = latent + noise.
3. Estimate Q (process variance) and R (observation variance) by MLE through the Kalman filter likelihood.
4. Run the filter: get the smoothed log-price estimate and its uncertainty band at each time.
5. Use the filtered series as input to a downstream model (e.g., a forecasting AR or GARCH). Compare performance to the raw series.
6. Extension: allow a slowly-drifting state mean by adding a level-and-slope local-linear-trend model. Compare BIC.

Where to go next

• Stochastic Calculus course: continuous-time analogues of these models — Brownian motion, Ornstein-Uhlenbeck for mean-reverting series, Heston for stochastic vol.
• Optimization course: portfolio construction takes the covariance estimated here and turns it into a portfolio.
• Numerical Methods course: putting these algorithms into production code with the right variance reduction and numerical stability.
• Python and R courses: the implementations of every model in this course.