ArticleOct 13, 202612 min read

Correlation & Cointegration

BTC and ETH move together almost every day, yet they are not cointegrated. What the two concepts really mean, why correlation is everywhere in crypto while cointegration is almost nowhere, and what that means for pairs trading.

Naively, since BTC and ETH are highly correlated, I thought they would be cointegrated. Unfortunately, they are not. Of course: correlation isn't cointegration.

So can we find other pairs in the crypto market that are cointegrated, or a way to use cointegration to build a trading strategy? That's what we'll see.

1. Definitions

Let's start with a few definitions to understand both concepts: correlation and cointegration.

Correlation

Correlation between two assets answers one question: how consistently do their returns move in the same direction at the same time?

DefinitionCorrelation

Let $P_i(t)$ be the price of asset $i$ at time $t$ , and $r_i(t)$ its log return:

r_i(t) = \ln P_i(t) - \ln P_i(t-1)

The (Pearson) correlation between assets $A$ and $B$ over $N$ periods is:

\rho_{AB} = \frac{\operatorname{Cov}(r_A, r_B)}{\sigma_A \, \sigma_B}

with

\operatorname{Cov}(r_A, r_B) = \frac{1}{N-1} \sum_t \big(r_A(t) - \bar r_A\big)\big(r_B(t) - \bar r_B\big)

where $\sigma_i$ is the standard deviation of $r_i$ and $\bar r_i$ its mean.

$-1 \le \rho_{AB} \le +1$ : $+1$ means always together, $0$ no linear relation, $-1$ always opposite.
$\rho$ measures how consistently they co-move, not by how much. The "how much" is the beta:

\beta = \frac{\operatorname{Cov}(r_A, r_B)}{\sigma_B^2} = \rho_{AB} \, \frac{\sigma_A}{\sigma_B}

The key point: correlation is computed on returns, not on prices. Two highly correlated series move together day to day, yet the gap between their prices can keep growing over time.

Two simulated assets with a 0.92 correlation of daily returns: they move together every day, but asset A ends up 2.9 times higher than asset B — Simulated data: one common market factor, small independent noise, opposite long-run drifts.

Cointegration

To introduce this concept, Murray (1994) uses a simple image. A drunk man walks with his dog. His path is a random walk, and so is his dog's. But when they get too far apart, he calls the dog, and the distance between them shrinks again.

Animation of two random walks in 2D, a drunk and his dog: each path wanders, but the distance between them keeps coming back down — Each path is a random walk on its own. The leash, the distance between them, is not: it keeps coming back.

Each walker on his own is unpredictable. The distance between them is not: it keeps coming back. That's cointegration.

DefinitionStationarity and cointegration

A series is stationary, written $I(0)$ , if its mean and variance are constant over time: it keeps coming back to its mean.

A series is integrated of order 1, written $I(1)$ , if it is not stationary but its changes are. The typical example is the random walk:

X(t) = X(t-1) + \varepsilon(t) \qquad\Rightarrow\qquad X \sim I(1), \quad \Delta X(t) = \varepsilon(t) \sim I(0)

Log prices are (almost always) $I(1)$ ; log returns are $I(0)$ .

Two $I(1)$ series $\ln P_A$ and $\ln P_B$ are cointegrated if there exist $\mu$ and $\beta$ such that the spread

s(t) = \ln P_A(t) - \mu - \beta \ln P_B(t)

is stationary, $I(0)$ . Each price wanders on its own, but this specific combination does not: it keeps coming back to 0. $(1, -\beta)$ is called the cointegrating vector.

Why does the spread come back? If it is stationary, at least one of the two prices must react to the gap: when the spread is too wide, $A$ and/or $B$ must move in the direction that closes it. So today's returns depend on yesterday's spread. That is the error correction model of Engle & Granger (1987).

DefinitionError correction model (VECM)

The Granger representation theorem states that $\ln P_A$ and $\ln P_B$ are cointegrated if and only if their log returns $r_A(t) = \Delta \ln P_A(t)$ and $r_B(t) = \Delta \ln P_B(t)$ can be written as

\begin{aligned} r_A(t) &= \alpha_A \, s(t-1) + \sum_{i=1}^{p} \big( \Gamma_{AA,i} \, r_A(t-i) + \Gamma_{AB,i} \, r_B(t-i) \big) + \varepsilon_A(t) \\ r_B(t) &= \alpha_B \, s(t-1) + \sum_{i=1}^{p} \big( \Gamma_{BA,i} \, r_A(t-i) + \Gamma_{BB,i} \, r_B(t-i) \big) + \varepsilon_B(t) \end{aligned}

with at least one $\alpha \neq 0$ . This system is called a VECM, Vector Error Correction Model. It has two parts:

Long run: $\alpha \, s(t-1)$ depends on the level of the gap between the prices. It is what pulls them back together.
Short run: the sums are the past returns of both assets over the last $p$ periods ( $p$ chosen by AIC). $\Gamma_{XY,i}$ is how much the return of $Y$ at $t-i$ feeds into the return of $X$ at $t$ : $\Gamma_{AA,i}$ is $A$ 's own momentum, $\Gamma_{AB,i}$ is $B$ leading $A$ . They capture dynamics unrelated to the gap, such as momentum or one asset leading the other by a few hours. Without them, these effects would end up in $\varepsilon$ and bias the estimate of $\alpha$ , the same role the lagged differences play in the ADF test below. They can be ignored for the intuition.

$\alpha_A$ and $\alpha_B$ are the adjustment coefficients ("speed of adjustment"): how strongly each price reacts to the gap. They are estimated by OLS, regressing each asset's return on the previous spread and the past returns.

Signs: $\alpha_A < 0$ and/or $\alpha_B > 0$ pull the spread back to 0.
Who does the work: if $\alpha_B = 0$ , only $A$ corrects. $B$ wanders freely and $A$ follows it: the dog comes back, the drunk doesn't care.

Combining the two equations, the spread itself returns to its mean at speed $\lambda = \alpha_A - \beta\,\alpha_B$ , which gives its half-life (defined below, with DOGE/SOL). The VECM also tells us who closes the gap: $A$ , $B$ or both.

A quick example. Say $s(t-1) = +0.10$ ( $A$ is 10% too expensive relative to $B$ ), $\alpha_A = -0.05$ and $\alpha_B = +0.03$ :

expected return of $A$ : $-0.05 \times 0.10 = -0.5\%$ , so $A$ goes down;
expected return of $B$ : $+0.03 \times 0.10 = +0.3\%$ , so $B$ goes up.

Both moves shrink the spread, and the wider the gap, the stronger the pull. This pulling force is the dog's leash.

This is exactly what we look for in trading. If BTC and ETH were cointegrated, their spread would have this mean-reverting property, and we could trade it: sell the spread when it is too wide, buy it back when it closes. Unfortunately, things don't go as expected.

2. Correlation in practice

Let's start with correlation again, and first look at how it evolves over time.

Prices of BTC, ETH, XRP and LINK, and the 30, 90 and 365-day rolling correlations of daily returns for BTC/ETH, BTC/XRP and ETH/LINK, with bull and bear markets shaded — Daily closes (USDT), 2019-01-17 to 2026-10-11. Green = bull market, red = bear market (BTC cycle tops and bottoms).

Correlation is high, but not really stable over time, and it's even less stable for altcoins. Worse: correlation goes up in bear markets and down in bull markets. Diversification helps you in bull markets, but much less in bear markets, exactly when you need it most.

Pair	Bull	Bear
BTC/ETH	0.81	0.89
BTC/XRP	0.57	0.80
ETH/LINK	0.71	0.82

Correlation of daily returns in bull vs bear markets.

Here is the correlation matrix for a few cryptos. As you can see, Bitcoin explains a large part of these correlations. That's why, on the right, I removed the Bitcoin factor from the log returns ( $r - \beta\, r_{BTC}$ ), to focus on what is left.

Two correlation matrices of daily returns for 13 cryptos: raw returns with an average altcoin correlation of 0.69, and after removing the Bitcoin factor with an average of 0.41 — Daily log returns, 2024-03-25 to 2026-10-11. Once the Bitcoin factor is removed, the average correlation between altcoins drops from 0.69 to 0.41.

3. Cointegration in practice

The tests

Before showing you some charts, I need to introduce the test we'll use: the ADF test (Dickey & Fuller, 1979), and its two-asset version, the Engle-Granger test.

DefinitionADF test (Augmented Dickey-Fuller)

On a series $s(t)$ , we estimate by OLS:

\Delta s(t) = c + \gamma \, s(t-1) + \sum_{i=1}^{k} \varphi_i \, \Delta s(t-i) + \varepsilon(t)

The lagged differences $\Delta s(t-i)$ absorb short-term autocorrelation; $k$ is chosen by AIC.

$H_0: \gamma = 0$ , a unit root: $s$ is a random walk, not stationary.
$H_1: \gamma < 0$ : $s$ is pulled back to its mean, stationary.

The test statistic is the t-stat of $\hat\gamma$ : the more negative, the stronger the pull. We reject $H_0$ (so $s$ is stationary) if it is below the critical value: with a constant, $-2.86$ at 5% and $-3.43$ at 1%.

The p-value is the probability of getting a statistic at least this negative if $s$ really were a random walk. If $p < 5\%$ , we reject the random walk: the spread is stationary.

BTC and ETH: correlated, not cointegrated

So, are BTC and ETH cointegrated?

Engle-Granger cointegration test between ETH and BTC: log prices, rolling and expanding beta, p-value over time, and the spread — Daily closes (USDT), 2017-08-18 to 2026-10-11, ETH regressed on BTC. The top and bottom panels use μ and β estimated on the whole period: fine to visualise, but it contains future information.

And the answer is no. First, the beta of the regression isn't stable over time, which is a big problem: our starting hypothesis, $\ln(\text{ETH}) = \mu + \beta \ln(\text{BTC}) + \text{noise}$ with a constant $\beta$ , simply doesn't hold.

Of course, the ADF test fails over the full period. Interestingly, with a rolling 365-day window, there are some phases where the test passes ( $p < 5\%$ about 6% of the time). Keep in mind that pure chance would already give about 5%: if two assets had no relation at all, the test would still pass 5% of the time. So 6% is no better than noise.

DOGE and SOL: too good to be true

Even though BTC and ETH fail, I found a pair with better results:

Engle-Granger cointegration test between DOGE and SOL: the expanding beta is stable around 0.42 and the p-value stays below 5% most of the time after the warm-up — Daily closes (USDT), 2021-06-01 to 2026-10-11. Expanding window starting with 365 days; β and p-values recomputed every 7 days.

For DOGE/SOL, the good news is that after the warm-up period, beta is relatively stable, so our hypothesis $\ln(\text{DOGE}) = \mu + \beta \ln(\text{SOL}) + \text{noise}$ is more realistic. The ADF test passes most of the time after the warm-up period, so it seems we could have an edge with this pair.

But it's not that simple:

Selection bias. I only kept the pair that worked, and that could be pure luck. Spoiler: it is. When I start the dataset in 2022 instead, the pair is cointegrated only 3% of the time, and beta isn't stable anymore.
No out-of-sample data. I studied the pair on data from 2021 to 2026, so I can't backtest the strategy on that same period. A real backtest must run on data that was never used in the analysis; otherwise the result is biased, because I already know how the pair behaved up to 2026.
The half-life is too long. The half-life is the average time it takes for a deviation to shrink by half. If it's too long, there aren't enough trades. Even if we backtested between 2022 and 2026 (which we shouldn't), we would get something like 5 trades, which isn't enough.

DefinitionHalf-life of mean reversion

We model the spread as a mean-reverting process (Ornstein-Uhlenbeck, discrete form) and estimate $\lambda$ by OLS:

\Delta s(t) = \lambda \big(s(t-1) - \bar s\big) + \varepsilon(t) \qquad\Rightarrow\qquad \text{half-life} = -\frac{\ln 2}{\lambda}

with $\lambda < 0$ if the spread is mean-reverting. It is the average time for a deviation from the mean to shrink by half. If $\lambda \ge 0$ there is no pull back to the mean, and the half-life is infinite. It only makes sense if the spread is stationary.

Link with the error correction model: since $\Delta s(t) = \Delta \ln P_A(t) - \beta\,\Delta \ln P_B(t)$ , this $\lambda$ is exactly $\alpha_A - \beta\,\alpha_B$ . Same number, two ways to get it: directly on the spread (what we use here), or from the two prices (which also tells who corrects).

So here are our problems:

beta isn't constant over time;
the assets aren't cointegrated over the full period;
the half-life is too long.

Conclusion

Correlation is everywhere in crypto, while cointegration is almost nowhere: tokens move together day to day, but nothing ties their price levels together in the long run. When we thought we had found a cointegrated pair like DOGE/SOL, it was most likely just selection bias.

Could a rolling window fix this? I tried it on hourly data, and nothing held up out-of-sample. It also makes the model more and more complicated, and cointegration is already hard enough.

Real cointegration does exist when a mechanism forces it: 1 stETH can always be redeemed for 1 ETH, so their spread comes back within about an hour. But there, the gaps are smaller than the trading fees.

References

Murray, M. P. (1994). A Drunk and Her Dog: An Illustration of Cointegration and Error Correction. The American Statistician, 48(1), 37–39.
Engle, R. F., & Granger, C. W. J. (1987). Co-Integration and Error Correction: Representation, Estimation, and Testing. Econometrica, 55(2), 251–276.
Dickey, D. A., & Fuller, W. A. (1979). Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association, 74(366), 427–431.
MacKinnon, J. G. (2010). Critical Values for Cointegration Tests. Queen's Economics Department Working Paper No. 1227.

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