The Efficient-market hypothesis: the BTCUSDT example

Introduction

Succinctly put, the Efficient Market Hypothesis (EMH) asserts that asset prices fully reflect all available information. A direct implication is that it is impossible to consistently "beat the market" on a risk-adjusted basis.

In practical terms, this means no trading strategy—after costs—can systematically outperform a simple buy-and-hold approach.

In the next section, I’ll present a more formal definition of the EMH, followed by an overview of the tests we use on our Market Efficiency Tests page. Feel free to skip ahead if you're already familiar with the theory.

This blog will inevitably be more academic than some of my previous posts, but I hope it remains accessible to beginners and engaging for experienced investors alike.

The Efficient Market Hypothesis (EMH)

The term “efficient capital market” was first introduced by Fama, Fisher, Jensen, and Roll in 1969 (Eugene F. Fama, Lawrence Fisher, Michael C. Jensen, and Richard Roll, 1969, "The Adjustment of Stock Prices to New Information,"International Economic Review, Vol. 10, No. 1, pp. 1–21). They defined an efficient market as one that quickly adjusts to new information.

In 1970, Fama refined this concept (Eugene F. Fama, 1970, "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance, Vol. 25, No. 2, pp. 383–417), defining an efficient market as one where prices reflect all available information. According to him, three sufficient conditions for efficiency are: (i) no transaction costs, (ii) free access to all relevant information, and (iii) homogeneous expectations—meaning all participants interpret information the same way.

Fama also proposed three forms of market efficiency, each corresponding to a different information set:
1. Weak-form efficiency – prices reflect all past trading data.
2. Semi-strong form efficiency – prices reflect all publicly available information.
3. Strong-form efficiency – prices reflect all public and private information.

While these conditions are sufficient, they are not strictly necessary. A market may still be considered efficient as long as no investor can consistently exploit deviations to earn excess returns.

Excess returns, often referred to as alpha, represent the additional return generated beyond what would be expected given the investment’s risk profile and prevailing market conditions. In practical terms, we can think of alpha as the return achieved beyond that of a passive buy-and-hold strategy, adjusted for risk.

In 1991, Fama added a practical nuance to his view (Eugene F. Fama, 1991, "Efficient Capital Markets: II," Journal of Finance, Vol. 46, No. 5, pp. 1575–1617), suggesting that prices reflect information only up to the point where the marginal benefit of processing new data equals its marginal cost.

In summary, the central message of EMH is this: no investor can consistently achieve superior risk-adjusted returns using a given information set. In other words, market inefficiency can only be demonstrated if a repeatable method exists that allows for the consistent generation of alpha.

EMH Tests

In this section, I will use the Binance BTCUSDT spot price data from August 17, 2017, to May 23, 2025, as an example.

Normality of returns

In its more traditional form, the EMH assumes that returns are normally distributed. However, in my view, normality is not a necessary condition for the EMH to hold. The essential requirement is that returns are i.i.d. (independent and identically distributed), regardless of the specific distribution.

The following figure compares BTCUSDT returns with a normal distribution:

There are several possible normality tests, but, at the time of writing this blog, TradingShepherd uses the Anderson-Darling and D’Agostino-Pearson tests to assess normality.

Both the Anderson-Darling and D’Agostino-Pearson tests strongly reject the null hypothesis of normality for BTCUSDT returns, with p-values effectively equal to zero. This indicates a significant deviation from the normal distribution.

Autocorrelation and partial autocorrelation functions

According to the Efficient Market Hypothesis (EMH), the sequence of price changes (the returns) has no memory.

The autocorrelation and partial autocorrelation functions test whether the returns series exhibits memory or serial dependence.

The Autocorrelation Function (ACF), measures the linear relationship between a time series and its lagged values. The ACF at lag k tells you how correlated the series is with itself k periods ago. The ACF can be constructed using the following equation:

$\rho_k = \frac{\mathrm{Cov}(r_t, r_{t-k})}{\mathrm{Var}(r_t)} = \frac{\mathbb{E}[(r_t - \mu)(r_{t-k} - \mu)]}{\sigma^2}$

The Partial Autocorrelation Function (PACF), measures the correlation between a time series and its lagged values, after removing the effect of all shorter lags. The PACF at lag k shows the direct effect of lag k on the series, controlling for lags 1 to k–1.

Consider the following linear regression:

$r_t = \phi_{k1} r_{t-1} + \phi_{k2} r_{t-2} + \dots + \phi_{kk} r_{t-k} + \varepsilon_t$

In this setup, the partial autocorrelation at lag k corresponds to the k-th regression coefficient:

$\text{PACF}(k) = \phi_{kk}$

No autocorrelation coefficient should be significantly different from zero. However, for instance, at a 5% significance level, it is expected that 1 in 20 (or 5 in 100) autocorrelation coefficients will differ from zero purely by chance, without this implying rejection of the hypothesis that returns are uncorrelated. One possible way to address this issue is to combine the autocorrelation coefficients into a single test statistic, such as the Ljung-Box or Box-Pierce test.

Typically, the ACF and PACF show similar patterns. This is also true for BTCUSDT. The next figure show the ACF.

The next table shows the Ljung-Box test for the first 10 lags.

The Ljung-Box test shows strong and consistent evidence of autocorrelation across lags, using a p-value threshold = 2.50%.

This result suggests that the return series is not fully memoryless, which challenges the strict form of the EMH. However, statistical significance doesn't necessarily imply economic significance, and further analysis is required to assess whether these patterns are exploitable after costs.

Tests of Non-Random Structure

Brock, Dechert and Scheinkman (BDS) test

The Brock, Dechert and Scheinkman (BDS) test was originally proposed in 1987 as a way of detecting non-linear dependencies in time series. For those interested the paper to read is Brock, Dechert, Scheinkman and LeBaron, "A test for independence based on the correlation dimension" 1986 paper.

The BDS statistic tests the null hypothesis that the sample data are drawn from an iid random process, against an unspecified alternative, using non-parametric techniques. Rejection of H₀ indicates that there is some form of dependence among the data — a dependence that may stem from a linear stochastic system, a nonlinear stochastic system, or a nonlinear deterministic system.

The BDS has been shown to be able to detect deviations from dependence in time-series data quite reliably (see Christopher Baum, Stan Hurn and Kenneth Lindsay, 2021, "The BDS test of independence", The Stata Journal, 21, Number 2, pp. 279–294).

Runs test

The runs test is a non-parametric test introduced by Wald and Wolfowitz in 1940 (A. Wald and J. Wolfowitz, 1940, "On a Test Whether Two Samples are from the Same Population", Annals of Mathematical Statistics, 11(2), 147–162). The test checks for randomness in a binary sequence, like successive price changes (returns), looking for positive/negative return clusters.

Runs are sequences of identical values. For example, a run could be the price of BTCUSDT going up in 3 consecutive days (UUU) or a down a day followed by 2 up days (DUU). If returns are random, the observed number of runs should conform to the expected frequency under randomness.

Variance ratio

The modern version of this test was introduced por Andrew Lo and Craig Mackinlay in 1987 (see, for instance: Andrew Lo and Craig Mackinlay, 1987, "Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test, Rodney L. White Center for Financial Research, The Wharton School, University of Pennsylvania).

The random walk property states that variance must change linearly with the time interval over which it is calculated. Consider, for instance, the variance of daily and the monthly returns. If the variance scales linearly then 30 times the daily variance must be approximately equal to the monthly variance.

Hurst exponent

Originally developed by hydrologist Harold Hurst during his efforts to determine optimal dam sizing for the Nile River in Egypt, the Hurst exponent was later popularized by Benoît Mandelbrot (see, for instance, Mandelbrot, 1983, The Fractal Geometry of Nature, W. H. Freeman and Company, 249-255).

The Hurst exponent (H) is a statistical measure used to evaluate the long-term memory of time series data. It quantifies the tendency of a time series to either regress to the mean or to cluster in a particular direction over time. In the context of financial markets, it's often used to assess the degree of market efficiency.

Interpretation of H typically follows this scale:

• H > 0.5: The time series is persistent, meaning that trends are likely to continue. If the market has gone up in the past, it is more likely to continue going up — indicative of momentum effects.
• H = 0.5: The time series behaves like a random walk, where past values have no predictive power. This is consistent with the weak form of the EMH.
• H < 0.5: The time series is mean-reverting, implying that extreme values tend to be followed by reversals. This can be associated with overreaction and correction dynamics.

In EMH testing, Hurst exponent values that significantly differ from 0.5 may suggest potential predictability in returns, and therefore possible inefficiencies. However, interpreting H requires caution, as noisy data, non-stationarity, and structural breaks can all influence the estimate.

Non-random structure tests summary

The next table summarizes the non-random structure tests results.

The non-random structure tests present strong and consistent evidence against ramdomness, using a p-value threshold = 2.50%.

Excess returns: the ultimate test

Determining whether it is possible to "beat the market" consistently on a risk-adjusted basis is the ultimate test of the Efficient Market Hypothesis (EMH). In our previous blog posts,

• "Trading Bitcoin using a fast Shepherd's Momentum signal",
• "Trading Ethereum using a fast Shepherd's Momentum binary signal",
• "Trading Ripple (XRP) using a fast Shepherd's Momentum signal",
• "Trading Solana using a fast Shepherd's Momentum binary signal",

we showed that, in the absence of costs, our fast Shepherd's Momentum strategy does generate excess returns (i.e., alpha).

Of course, we still don't know if these results would hold once trading costs are taken into account.

Final remarks

• The hypothesis of normally distributed BTCUSDT returns is strongly rejected.
• The ACF shows strong autocorrelation. However, as noted earlier, statistical significance does not necessarily imply economic significance.
• Our tests for non-random structure show strong evidence against randomness.
• Excess returns may be possible, and certainly are before trading costs. It is poetic that while, according to Fama, zero transaction costs is one of the conditions for market efficiency, it is often precisely these transaction costs that prevent market inefficiencies from being exploited.
• This blog is not an invitation to trade, nor does it constitute trading advice. Trading involves risk, and readers are solely responsible for their decisions.

Curious about how efficient your favorite asset is? Explore our Market Efficiency Tests page to see the results and run your own analysis.

At TradingShepherd, we’re building tools to help you understand market behavior—beyond the noise. Stay tuned.