Examples of Cointegration in Time Series: 3 Practical Examples You’ll Actually Use

When people talk about examples of cointegration in time series: 3 practical examples, they almost always start with stock pairs. And for good reason: equity markets give some of the cleanest, most intuitive real examples of cointegration.

Imagine two large U.S. bank stocks, say JPMorgan Chase (JPM) and Bank of America (BAC). Both prices are non‑stationary: they trend, react to news, and exhibit volatility clustering. If you plot them separately, each looks like a random walk with drift.

But now look at the spread:

[
S_t = \log(P^{JPM}_t) - \beta \log(P^{BAC}_t)
]

If these two banks share similar risk exposures, funding costs, and regulatory environments, the spread can be surprisingly stable over long periods. The log prices wander, but the linear combination of the two (the spread) behaves like a stationary process. That is the hallmark of cointegration.

Traders use this relationship in pairs trading:

• They estimate a long‑run relationship between the two stock prices (often via linear regression on historical log prices).
• They test the residuals for stationarity using something like the Engle–Granger approach.
• If the residuals look mean‑reverting, they treat deviations as temporary mispricings and trade the spread.

This is not just theory. Academic work going back to the 1990s shows that cointegration‑based pairs trading can generate statistically significant profits in some markets, even after transaction costs, although profitability has declined as strategies became more crowded.

More stock market examples include sector and ETF pairs

Beyond big bank stocks, other examples of cointegration in time series in equity markets include:

• Dual‑listed shares: A company listed on both the NYSE and LSE, for instance, will have prices in different currencies that move closely together once you adjust for FX. The two price series are natural candidates for cointegration because they represent claims on the same underlying cash flows.
• Parent–subsidiary relationships: A holding company and a publicly traded subsidiary sometimes show cointegrated prices, reflecting the parent’s stake in the subsidiary.
• ETF vs. index futures: The SPDR S&P 500 ETF (SPY) and S&P 500 futures prices often form a cointegrated pair after accounting for cost‑of‑carry. Arbitrageurs enforce this relationship.

If you’re building a practical toolkit, the best examples to start with are:

• A pair of large, liquid stocks in the same sector (e.g., Coca‑Cola vs. PepsiCo in beverages, ExxonMobil vs. Chevron in energy).
• An index ETF vs. the corresponding index futures contract.

In both cases, you’ll often find that although each series is non‑stationary, a stable spread emerges. That’s cointegration in action.

For a deeper theoretical foundation, the Federal Reserve’s materials on time series and cointegration in macro models are a solid starting point: federalreserve.gov.

Cointegration between interest rates across maturities

Another classic example of cointegration in time series is the relationship between interest rates of different maturities on the same yield curve.

Think about the U.S. Treasury yield curve: 3‑month, 2‑year, 10‑year, and 30‑year yields. Each yield series is usually modeled as an integrated process, influenced by monetary policy, inflation expectations, and risk sentiment. Yet these yields are tied together by no‑arbitrage conditions and expectations about future short‑term rates.

Empirically, researchers often find that:

• Individual yields look non‑stationary.
• Certain linear combinations of yields (for example, level, slope, and curvature factors) are stationary.

That means the yields are cointegrated. If the 10‑year yield drifts too far from what is implied by the path of short‑term rates and term premia, arbitrage and policy expectations tend to pull it back.

Real examples: policy shifts and the post‑pandemic period

The last few years provided vivid real examples of how cointegration plays out:

• Pandemic shock (2020): Short‑term yields collapsed as the Federal Reserve cut the federal funds rate to near zero. Long‑term yields also fell, but not as much, as markets priced in future recovery and inflation.
• Tightening cycle (2022–2024): As the Fed raised rates aggressively to combat inflation, short‑term yields surged. Long‑term yields moved higher too, but with different dynamics as markets debated the long‑run neutral rate.

Across these episodes, yields moved a lot, but the long‑run relationships between different maturities remained relatively stable. Studies using Johansen cointegration tests often continue to find one or more cointegrating vectors among Treasury yields even in the post‑COVID period.

For background on yield curve modeling and term structure, the Federal Reserve Bank of New York has accessible research and data: newyorkfed.org.

Cross‑country interest rate cointegration

Cointegration is not limited to a single country’s curve. Another example of cointegration in time series is cross‑country interest rates in tightly linked economies.

Consider:

• U.S. and Canadian government bond yields.
• German Bunds and French OATs within the euro area.

These markets are highly integrated through trade, capital flows, and monetary policy coordination (especially inside the euro area). While each country’s yields follow its own path, long‑run spreads between them often behave like stationary series. That is, the level of the spread may shift after regime changes, but within a regime, the spread tends to revert.

Researchers at central banks and institutions like the Bank for International Settlements (BIS) routinely test for cointegration among international yields to understand financial integration and contagion.

Cointegration in commodity and energy markets: one of the best examples

If you want some of the best examples of cointegration in time series: 3 practical examples, commodity and energy markets are hard to beat. Physical constraints, transportation costs, and storage economics all generate long‑run pricing relationships.

Spot and futures prices for the same commodity

Take crude oil. The spot price and the front‑month futures price both follow noisy, non‑stationary paths driven by global demand, supply shocks, OPEC decisions, and geopolitical risk. Yet storage and arbitrage conditions tie them together.

Under the cost‑of‑carry model, the futures price should roughly equal the spot price plus storage costs and financing, minus any convenience yield. In practice, this relationship is imperfect, but over time the spot and futures prices tend to move together in a way that often looks cointegrated.

Empirical studies in the 2010s and 2020s have repeatedly found cointegration between spot and futures prices for commodities like:

• Crude oil (WTI, Brent)
• Natural gas
• Gold and silver
• Agricultural products (corn, wheat, soybeans)

The residuals from a regression of futures prices on spot prices often behave like a mean‑reverting series, making this one of the clearest examples of cointegration in time series used in risk management and hedging.

The U.S. Energy Information Administration (EIA) provides long historical series for energy prices that you can use to test these relationships yourself: eia.gov.

Regional price spreads in energy markets

Another powerful example of cointegration in time series involves regional price spreads that are constrained by transportation and infrastructure.

Consider:

• Natural gas prices at Henry Hub vs. other U.S. hubs: Pipeline capacity and transport costs limit how far regional prices can diverge for long.
• Electricity prices in neighboring ISO/RTO markets: Markets like PJM and NYISO are interconnected. While local congestion and weather create short‑term differences, long‑run spreads tend to be bounded.

In both cases, price series at different locations often appear non‑stationary on their own, but their differences (after adjusting for typical transport costs) frequently look stationary. That’s cointegration driven by physical flows.

Beyond the big three: more real examples of cointegration in time series

So far we’ve focused on three headline cases—equity pairs, interest rates, and commodities—that anchor most discussions of examples of cointegration in time series: 3 practical examples. But cointegration shows up in plenty of other applied settings.

Here are several more real examples that analysts use in practice.

Exchange rates and purchasing power parity

Exchange rates are notoriously hard to model, but long‑run parity conditions sometimes create cointegrated relationships.

A classic example of cointegration in time series is the test of purchasing power parity (PPP). In its simplest form, PPP says that the nominal exchange rate between two countries should adjust so that the price of a basket of goods is comparable in both currencies.

Empirically:

• The nominal exchange rate (e.g., USD/EUR) is non‑stationary.
• Relative price levels (e.g., U.S. vs. euro area CPI) are also non‑stationary.
• But the combination implied by PPP can sometimes behave like a stationary process over long horizons.

Researchers often test for cointegration between exchange rates and relative price indices to evaluate whether PPP holds in the long run. Major data providers include the Federal Reserve Economic Data (FRED) system at the St. Louis Fed: fred.stlouisfed.org.

Macroeconomic aggregates: income, consumption, and money

Macroeconomists regularly use cointegration to capture long‑run equilibrium relationships among economic aggregates. Common examples include:

• Consumption and income: Permanent income and life‑cycle theories suggest that consumption and disposable income should move together in the long run. Both series are typically non‑stationary, but many studies find cointegration.
• Money supply, prices, and output: In monetarist frameworks, the quantity of money, price level, and real output may form a cointegrated system reflecting long‑run monetary neutrality.
• GDP and CO₂ emissions: In environmental economics, researchers test for cointegration between output and emissions to understand decoupling and the environmental Kuznets curve.

These relationships are not just academic curiosities. Central banks and policy institutions build vector error correction models (VECMs) around cointegrated macro variables to forecast inflation, output, and interest rates.

Climate and environmental data

In climate science, cointegration can help distinguish between shared trends and independent variation across environmental indicators.

For instance, researchers have examined cointegration between:

• Global mean surface temperature and greenhouse gas concentrations (e.g., atmospheric CO₂).
• Sea level and temperature over multi‑decade horizons.

Both variables in each pair usually show strong upward trends over the past century. Testing for cointegration helps assess whether they share a common stochastic trend or just happen to move in the same direction over a finite sample.

Agencies like NASA and NOAA provide open climate time series suitable for this kind of analysis: climate.nasa.gov and noaa.gov.

How to work with these examples of cointegration in practice

It’s one thing to list examples of cointegration in time series: 3 practical examples and a handful of others. It’s another to actually work with them. In applied settings, the workflow usually looks like this:

First, you identify candidate series. You look for pairs or groups of variables that:

• Are non‑stationary on their own (fail standard unit root tests like ADF or KPSS in the stationary direction).
• Have a strong economic, physical, or institutional reason to be linked in the long run (no‑arbitrage, regulations, physical constraints, accounting identities).

Second, you test for cointegration. Common approaches include:

• Engle–Granger two‑step method for pairs: regress one series on the other, then test the residuals for stationarity.
• Johansen test for systems of three or more variables: estimate a vector autoregression (VAR) and test for the rank of the cointegration space.

Third, you build models that respect those long‑run relationships:

• Error correction models (ECMs) add a term that pulls the system back toward the cointegrating equilibrium whenever it drifts away.
• VECMs generalize this to multivariate systems.

In finance, this might mean a pairs trading strategy that bets on mean reversion in the spread between two cointegrated stocks. In macro, it might mean a forecasting model where inflation, output, and interest rates share one or more cointegrating vectors.

FAQ: common questions about real examples of cointegration

Q1. What are some of the best real examples of cointegration in time series?
Some of the best examples are:

• Pairs of related stocks (same sector, dual‑listed shares, ETF vs. futures).
• Interest rates across maturities on the same yield curve.
• Spot and futures prices for the same commodity.
• Regional energy prices linked by transport infrastructure.
• Exchange rates and relative price levels (PPP tests).
• Macroeconomic aggregates like consumption and income.

Q2. Can you give a simple example of cointegration that a beginner can test?
A straightforward example of cointegration to test is the relationship between an index ETF (like SPY) and its corresponding index futures contract. Both series are non‑stationary, but their spread is often stationary. You can download daily data, regress ETF prices on futures prices, and run a unit root test on the residuals.

Q3. Are cointegrated series always good candidates for trading strategies?
Not always. Cointegration is a statistical property, not a guarantee of profit. Transaction costs, changing regimes, structural breaks, and execution risk can destroy the edge. Many pairs that looked promising in backtests underperform in live trading. Cointegration is a starting point, not a trading signal by itself.

Q4. How has the relevance of cointegration changed around 2024–2025?
The concept is as relevant as ever, but the context has shifted. Post‑pandemic macro volatility, rapid interest‑rate moves, and the energy transition have created new structural breaks. Many analysts now re‑test cointegration relationships more frequently and use rolling or regime‑switching models. In energy markets, for example, the rise of renewables and shifting gas flows after geopolitical shocks have altered some historical cointegrating relationships.

Q5. Where can I find data to explore more examples of cointegration in time series?
For U.S. and international data, good sources include:

• Federal Reserve Economic Data (FRED): fred.stlouisfed.org
• U.S. Energy Information Administration (EIA): eia.gov
• NASA and NOAA for climate series: climate.nasa.gov, noaa.gov
  These sources cover many of the examples of cointegration in time series discussed above, from interest rates and macro aggregates to energy and climate indicators.