13A.12 Indian applications, tested

All statistics in this section are computed for this course (6 Nov 2026) from Yahoo Finance adjusted daily closes (tickers HDFCBANK.NS, ICICIBANK.NS, ^NSEI, ^NSEBANK), 4 Jan 2010 to 4 Nov 2026, on log prices, using statsmodels (coint with automatic lag selection; coint_johansen with det_order=0, one lagged difference). Yahoo's adjusted series can contain errors; a production study should use exchange data with its own corporate-action adjustments.

13A.12.1HDFC Bank and ICICI Bank#

Sample	$n$	OLS $\hat b$	EG stat	EG $p$	Residual AR(1) half-life	Johansen trace $r=0$ (5% cv 15.49)
2010–2026	4,135	0.919	−1.44	0.785	468 days	8.79
2010–2014	1,221	1.018	−1.82	0.622	104 days	5.82
2015–2019	1,222	1.125	−3.10	0.089	69 days	10.25
2020–2026	1,692	0.381	−1.87	0.596	70 days	11.08
2023–2026	946	0.329	−1.50	0.762	107 days	4.27

In rolling two-year windows (504 trading days, stepped by 63 days, 58 windows) the EG test rejected at 5% in 12% of windows, and the OLS hedge ratio ranged from −0.25 to 1.40.

Reading There is no sample in which the two are convincingly cointegrated. The best case, 2015–19, rejects only at 10%, and that is one of five samples tried. The 12% rolling rejection rate is barely above the 5% expected under the null, and overlapping windows make it weaker evidence still. The hedge ratio is unstable to the point of changing sign. "Half-lives" of 70–100 days are computed on residuals that are not demonstrably stationary and should not be quoted. The two banks are strongly correlated in returns (0.54) and not tied in levels.

13A.12.2Nifty Bank and Nifty 50#

Sample	$\hat b$	EG stat	EG $p$	Half-life	Johansen trace $r=0$	max-eig $r=0$ (5% cv 14.26)
2010–2026	1.102	−2.97	0.116	193 days	10.87	10.81
2010–2014	1.181	−2.45	0.299	82 days	6.39	6.11
2015–2019	1.458	−4.64	0.001	21 days	22.09	21.92
2020–2026	1.029	−3.65	0.021	65 days	29.05	23.11
2023–2026	0.885	−1.84	0.612	109 days	7.22	4.25

The log ratio $\ln(\text{Nifty Bank}/\text{Nifty})$ itself (the known vector $(1,-1)$, so DF critical values apply) has an ADF statistic with constant of −2.80 ($p = 0.059$); the ratio moved from 1.74 to 2.33 over the sample. Rolling two-year EG windows rejected at 5% in 22% of cases; the hedge ratio ranged from 0.63 to 1.64.

Reading This pair has an economic tie (the banks are a large part of Nifty) and the evidence is correspondingly stronger: both EG and Johansen reject in 2015–19 and 2020–26. But the hedge ratio differs between those two samples (1.46 against 1.03), the full sample and the most recent three years do not reject, and the ratio drifted up by a third. A partial accounting link produces cointegration in windows with a drifting vector. A trader using the 2015–19 hedge ratio in 2020–26 would have held a different spread from the one the data then supported. This is the setting where the Kalman filter of 13A.9 earns its keep, with the trap noted there.

13A.12.3MCX gold against COMEX-implied parity#

M10 (10.15) gives the duty-paid rupee price of gold. In logs, for a futures contract,

with $F^{\text{COMEX}}$ in USD per troy ounce for the matching month, $X^{\text{fwd}}$ the USDINR forward to the same date, $d_t$ the customs duty (M10 · 10.5.2) and the last term converting a fine troy ounce into 10 g of 995 gold. The cointegrating vector is known: $(1,-1,-1)$ on the three log prices, with $d_t$ a known step function. So test $s_t$ with the DF/ADF critical values (constant case), not Engle–Granger ones; nothing is estimated.

What $s_t$ measures is the local premium or discount $q$ of M10 plus basis and timing noise. Three cautions.

• Duty changes are known-date level shifts in $s_t$ if $d_t$ is omitted or mis-dated: 24 Jul 2024 (15% to 6%) and 13 May 2026 (6% to 15%) (M10 · 10.5.2). Include $d_t$ or add step dummies at those dates. A break test that "discovers" these dates teaches nothing.
• Who arbitrages. Closing the spread physically requires importing, paying duty and delivering on MCX, which a trader cannot do. The band within which $s_t$ wanders is set by importers' costs, not by a trader's. Mean reversion is enforced, but slowly and with a wide band.
• Weak exogeneity. In the VECM for $(\ln F^{\text{MCX}}, \ln F^{\text{COMEX}}, \ln X)$ one expects the COMEX and USDINR loadings to be near zero: MCX adjusts to the world, not the reverse. That is a testable restriction on $\alpha$ (13A.4.3), and it decides which leg an arbitrageur expects to move.

M10's worked example of 2 Nov 2026 found the crude parity within 0.034% of the MCX December price, with purity and carry corrections nearly cancelling. A daily $s_t$ series with these corrections applied is a good first project for the methods of this module; MCX historical data are on the exchange's website.

13A.12.4The gold/silver ratio, revisited#

M10 · 10.7.2 tested $\ln(\text{Gold}/\text{Silver})$ with the known vector $(1,-1)$: ADF $p = 0.055$ on 2001–2026 month-end data, not rejecting in either sub-sample, half-life about 13.6 months before bias correction. Two additions from this module.

• Known versus estimated vector. Imposing $(1,-1)$ is what allows DF critical values. If you estimate the hedge ratio instead, you need EG critical values (−3.34 at 5% for a pair), which are harder to beat: M10's statistic of −2.83 would be further from rejection. Imposing a correct known vector buys power; imposing a wrong one destroys the test.
• Economics of trading it. Suppose (illustratively) monthly residual s.d. 0.05 around M10's $\hat\varphi = 0.9503$. Then $\kappa = 0.612$ per year and the stationary s.d. is 0.161. By (13A.28), a cycle between $\pm1$ stationary s.d. takes 9.8 years on average, and between $\pm0.5$ s.d. 4.3 years (Problem 13A.16). Even if the ratio is stationary, it offers a handful of round trips per career, each estimated from a sample containing only a few half-lives.