Learning outcomes

After this module you should be able to:

1. Derive the Dickey–Fuller regression from an AR(1) and the augmented form from an AR($p$), derive the limiting distribution of $T(\hat\rho-1)$ and of the DF $t$-statistic under a unit root, and explain why the critical values are not Student-$t$.
2. Show analytically and by simulation why regressions of one random walk on another produce large $t$-statistics and $R^2$, and state what this implies for "pairs" chosen by correlation.
3. Define cointegration, prove the Granger representation for a cointegrated VAR(1), run the Engle–Granger two-step test with the correct critical values, and derive Johansen's trace and maximum-eigenvalue statistics from the reduced-rank regression.
4. Estimate an Ornstein–Uhlenbeck process exactly from its AR(1) form, quantify the bias and imprecision of the estimated speed of mean reversion, and compute half-lives with honest uncertainty.
5. Derive the expected first-passage times of an OU process and Bertram's expected cycle length, and compute the entry and exit bands that maximise expected profit per unit time after costs.
6. Compare OLS, reverse-OLS and total-least-squares hedge ratios, and derive the Kalman filter for a random-walk hedge ratio, including its likelihood and steady-state gain.
7. Derive and apply the Chow test, explain why choosing the break date from the data invalidates it, and use sup-F, CUSUM and Bai–Perron appropriately.
8. Test Indian pairs (HDFC Bank/ICICI Bank, Nifty/Bank Nifty, MCX gold against COMEX parity, gold/silver) properly, and price the frictions of an Indian stat-arb book: costs, the SLB market, futures as the short leg, and lot sizes.