Guide · Quantitative methods

Cointegration and pairs trading on the NSE

Statistical arbitrage sells itself on a clean idea: find two stocks that belong together, trade the gap when it opens, and collect as it closes, indifferent to whether the market rises or falls. The idea is sound. The way most retail treats it is not, because it starts from correlation, and correlation is the wrong instrument. This guide sets out the distinction that decides everything, the tests that separate a tradeable spread from a coincidence, the mechanics of the trade itself, and the NSE-specific frictions and the one risk that ends these strategies: the relationship breaking.

Correlation is not cointegration

Correlation is a statement about returns. It asks whether, day to day, two series tend to move up and down together, and it is scale-free and short-memoried. Two stocks can post a correlation of returns near one and still see the gap between their price levels widen without limit, because nothing in the correlation coefficient constrains the levels. A stock compounding faster than another can march away from it while every daily wiggle stays synchronised. Correlation is high; the spread is a runaway.

Cointegration is a statement about levels. Two price series are each non-stationary: a random-walk-like series with no fixed mean, wandering wherever the last shock left it. Cointegration says that although each series wanders, there exists a fixed combination of them, price of A minus a constant times price of B, that does not wander. That combination, the spread, is stationary: it has a constant mean and variance and it reverts. The two prices are tied by an invisible spring. That spring, and only that spring, is what a pairs trade harvests.

Testing for cointegration: the Engle-Granger two-step

For a single pair, the workhorse is the Engle-Granger two-step procedure, introduced by Robert Engle and Clive Granger in 1987. Step one estimates the long-run relationship by ordinary least squares, regressing the price of one stock on the other. The slope is the hedge ratio, the number of units of B that offset one unit of A, and the regression residual is the spread. Step two tests that residual for stationarity with an Augmented Dickey-Fuller (ADF) test, whose null hypothesis is that the residual has a unit root, meaning it is non-stationary. Reject the null and the spread is stationary: the pair is cointegrated.

import numpy as np
import pandas as pd
import statsmodels.api as sm
from statsmodels.tsa.stattools import adfuller, coint

# a and b are pandas Series of daily closing prices for two same-sector large-caps,
# aligned on a common trading calendar. Supply your own historical data.
prices = pd.concat([a, b], axis=1).dropna()
a, b = prices.iloc[:, 0], prices.iloc[:, 1]

# STEP 1  ---  OLS: regress price A on price B to get the hedge ratio and spread
X = sm.add_constant(b)                 # intercept + slope
ols = sm.OLS(a, X).fit()
hedge_ratio = ols.params.iloc[1]       # units of B per 1 unit of A
spread = a - hedge_ratio * b - ols.params.iloc[0]   # the regression residual

# STEP 2  ---  ADF test on the spread: is the residual stationary?
adf_stat, adf_p, *_ = adfuller(spread, autolag="AIC")
print(f"hedge ratio = {hedge_ratio:.3f}")
print(f"ADF p-value on spread = {adf_p:.4f}")   # small p-value => reject unit root => stationary

There is a subtlety worth stating plainly, because it is the kind of detail generic articles miss. Running adfuller on a residual you obtained from your own regression does not use the right critical values. The residual was chosen to look as stationary as possible, so the standard ADF distribution is too lenient. The correct test uses Engle-Granger critical values (MacKinnon's), which statsmodels provides directly through coint. Prefer it for the verdict, and keep the manual two-step for the hedge ratio and the spread series:

# Correct Engle-Granger test in one call (uses the proper critical values)
eg_stat, eg_p, eg_crit = coint(a, b)
print(f"Engle-Granger p-value = {eg_p:.4f}")   # NOT the same as the naive adfuller p-value
# A common decision rule: treat the pair as cointegrated only if eg_p < 0.05.

The order matters too. Because the regression is not symmetric, regressing A on B and B on A gives slightly different hedge ratios and residuals, so the verdict can wobble near the threshold depending on which series you place on the left. That asymmetry is one reason the multivariate test below exists.

The Johansen test, for baskets

When you want cointegration among three or more series at once, say a basket of same-sector names, the Engle-Granger two-step no longer fits cleanly, because it hinges on an arbitrary choice of dependent variable and can only find one relationship. The Johansen test handles the general case. It estimates the cointegration rank, how many independent stationary combinations exist among the series, using a trace statistic and a maximum-eigenvalue statistic, and it returns the cointegrating vectors directly rather than assuming one.

from statsmodels.tsa.vector_ar.vecm import coint_johansen

# basket: a DataFrame whose columns are price levels of the candidate names
result = coint_johansen(basket, det_order=0, k_ar_diff=1)
# compare each trace statistic against its 90/95/99% critical value
for i, (trace, crit) in enumerate(zip(result.lr1, result.cvt)):
    print(f"rank <= {i}:  trace = {trace:.2f}   95% crit = {crit[1]:.2f}")
# the first i where trace < the 95% critical value estimates the cointegration rank

For a two-stock pair, Engle-Granger is simpler and usually enough. Johansen earns its keep on baskets and when you need the relationship itself, not just a yes or no.

The trade: spread, z-score, and thresholds

A cointegration verdict gives you a stationary spread. The trade standardises it. Subtract a rolling mean and divide by a rolling standard deviation to get a z-score, a spread expressed in units of its own deviation, oscillating around zero. A z-score of +2 says the spread sits two deviations above its recent normal: unusually wide. The rule then trades the extremes back toward the middle. When the z-score stretches past a positive threshold, the spread is rich, so you short the spread, selling the leg that has run up and buying the leg that has lagged, in the hedge ratio. When it stretches past the negative threshold you do the mirror. You exit as the z-score reverts toward zero.

The half-life of mean reversion

A stationary spread is not enough; you need to know how fast it reverts, because that sets your holding period and decides whether the edge survives costs. Model the spread as an Ornstein-Uhlenbeck process, the continuous-time analogue of a first-order autoregression. Its speed is captured by fitting an AR(1): regress the daily change in the spread on the lagged level of the spread. The slope, call it lambda, is negative for a mean-reverting series, and the half-life, the expected time to close half the distance to the mean, is minus the natural logarithm of two divided by lambda.

# Ornstein-Uhlenbeck / AR(1) half-life of the spread
lag = spread.shift(1)
delta = (spread - lag).dropna()
lag = lag.loc[delta.index]

reg = sm.OLS(delta, sm.add_constant(lag)).fit()
lam = reg.params.iloc[1]                 # slope on the lagged level; negative if mean-reverting
half_life = -np.log(2) / lam
print(f"lambda = {lam:.4f}   half-life = {half_life:.1f} periods")

Read the number with judgement. A very short half-life, a day or two, usually means you are fitting microstructure noise rather than a real equilibrium, and costs will dominate. A very long half-life ties capital up for months against a small per-trade edge, and gives a slow-moving relationship more time to break. There is no universal sweet spot to quote, and quoting one would be false precision; the point is that the half-life, the threshold, the transaction and borrow costs, and the capital tied up have to be weighed together before a spread is worth trading. The arithmetic below is illustrative, with round numbers chosen only to show the shape of the trade, not a claim about any real pair.

NSE specifics: the short leg and its frictions

A pairs trade is market-neutral only because it is long one leg and short the other. The short leg is where Indian-market reality bites. Regulation prohibits naked short selling: you cannot sell stock for delivery that you have not arranged to deliver. To hold a genuine cash-segment short you must borrow the stock through the Securities Lending and Borrowing (SLB) framework, an exchange-run, clearing-corporation-secured mechanism under the SEBI short-selling rules. Borrow is not free and not always available: the lending pool for a given name can be thin, the fee varies, and a hard-to-borrow stock can make one leg of an otherwise valid pair impractical.

For this reason many pairs are constructed in the derivatives segment, taking the short leg in single-stock futures where the name is available there, which sidesteps the stock-borrow step but introduces its own frictions: fixed lot sizes that will not match your computed hedge ratio exactly, so you round and accept a small residual imbalance; the roll from one expiry to the next; and, for positions carried into expiry, physical settlement obligations on the equity that underlies the future. None of this changes the statistics. It changes which cointegrated pairs are actually tradeable, and at what cost, which is a different and equally binding filter. If you want the mechanics of the short leg itself, see the companion guide on short selling in India.

The failure modes: where pairs trades die

The central risk is not that the spread wiggles. It is that the spring breaks. Cointegration is an empirical relationship that holds only while the two businesses keep sharing the drivers that pinned their prices together. A merger, a demerger, a regulatory change, a broken supply chain, a balance-sheet shock, a shift in the competitive order, any of these can end the equilibrium. The spread then stops reverting and starts trending away, and the trade is now long a diverging spread with no mean to return to. Because you are short one leg, the loss on that side has no natural ceiling. This regime break, not ordinary noise, is what ends most pairs strategies, and it is why a hard stop on divergence is not optional.

Two quieter failure modes compound the first. The per-trade edge in a stationary spread is typically small, so transaction costs, the borrow fee on the short leg, the bid-ask on both legs, and slippage can quietly eat it; a spread that looks profitable gross is often flat or worse net. And when you scan many pairs looking for the ones that test as cointegrated, you invite multiple-testing bias: run enough tests on the same history and a handful will pass by chance, look wonderful in-sample on the very window you searched, then fail out of sample. This is the same trap that undoes naive backtests, and the discipline that avoids it, honest out-of-sample testing and a costs-first mindset, is covered in the guides on backtesting with pandas and the backtesting mistakes retail makes.

Where this fits

Cointegration pairs trading is a genuine, statistically grounded style, but it is not a shortcut. Its edge is thin, its short leg is constrained by Indian rules, and its defining risk is a relationship quietly ceasing to hold. The value is in the discipline the method demands: testing the right thing, sizing the horizon by the half-life, respecting costs, and stopping on divergence rather than hoping. That upstream statistical judgement, telling a real equilibrium from a coincidence and a survivable trade from an overfit one, is the harder and more durable half of the skill, and it is where careful quantitative work in India actually lives. The same mean-reversion-and-arbitrage family also shows up in ETF arbitrage on the NSE, where the equilibrium is enforced by a creation-redemption mechanism rather than a statistical one.

Frequently asked questions

Where the facts come from

• Engle and Granger (1987). The original two-step representation, estimation and testing of co-integration, which defines the regress-then-test-the-residual procedure used above. Engle, R. F. and Granger, C. W. J., "Co-integration and error correction," Econometrica 55(2).
• statsmodels documentation. The coint function implements the Engle-Granger test with the correct (MacKinnon) critical values, distinct from a naive adfuller on the residual; coint_johansen in the vecm module returns the trace and maximum-eigenvalue statistics for the multivariate case. statsmodels.org
• NSE Securities Lending and Borrowing. The exchange-run, clearing-corporation-secured SLB scheme is the route to borrow a stock for a cash-segment short leg. nseindia.com
• SEBI short-selling framework. The regulator's framework permits short selling only through disclosed, covered routes and prohibits naked short selling, which is why the short leg of a pairs trade needs SLB or a futures position. sebi.gov.in

Related guides

• Backtesting Indian equities with pandas: building an honest spread and signal backtest.
• The backtesting mistakes retail makes: lookahead, survivorship, and overfitting a scan.
• ETF arbitrage on the NSE: an equilibrium enforced by creation-redemption, not statistics.