Traders love calendar patterns: “Buy Monday, sell Friday.” “Crypto dumps on weekends.”

Bitcoin trades 24/7, including weekends when stock markets are closed, which makes it a natural place to look for day-of-week effects.

Using 2,921 daily returns (2018–2025):

Do Bitcoin’s average daily returns depend on the day of the week?

And the deeper question: if we find a “significant” day, should we believe it?

A p-value answers: if there were truly no effect, how likely is data at least this extreme?

The trap: we will test seven days, not one.

• Each single test has a 5% chance of a false alarm.
• Seven tests give seven chances to be fooled.
• Finding one “significant” day is expected even if no pattern exists.

This is the multiple testing problem, one of the most common ways people (and trading strategies) fool themselves with p-values.

Daily log return on day \(t\), from closing price \(P_t\):

\[ r_t = 100 \cdot \ln\!\left(\frac{P_t}{P_{t-1}}\right) \quad (\%) \]

For each weekday \(d\), compare its mean return \(\mu_d\) to the mean of all other days \(\mu_{-d}\):

\[ H_0:\ \mu_d = \mu_{-d} \qquad \text{vs.} \qquad H_1:\ \mu_d \neq \mu_{-d} \]

Welch’s test statistic and two-sided p-value:

\[ t = \frac{\bar r_d - \bar r_{-d}}{\sqrt{s_d^2/n_d + s_{-d}^2/n_{-d}}}, \qquad p = P\big(|T| \ge |t_{\text{obs}}| \mid H_0\big) \]

Daily returns have a standard deviation of about 3.4%. Any day-of-week signal has to be detected through that noise.

Every interval crosses or comes close to zero. Visually, the “pattern” is weak.

day_labels <- c("Mon", "Tue", "Wed", "Thu", "Fri", "Sat", "Sun")

results <- lapply(day_labels, function(d) {
  t <- t.test(btc$ret[btc$day == d], btc$ret[btc$day != d])  # Welch
  data.frame(day = d, mean_ret = mean(btc$ret[btc$day == d]),
             t = unname(t$statistic), p = t$p.value)
}) |> dplyr::bind_rows()

# Correct for running 7 tests at once
results$p_bonf <- p.adjust(results$p, method = "bonferroni")

One test per weekday, then a Bonferroni correction for running seven tests.

• Smallest raw p-value: Thu (p = 0.0451)
• Days significant before correction: 1; after Bonferroni: 0

If each of \(m\) independent tests has false-positive rate \(\alpha\), the chance of at least one false alarm (the family-wise error rate) is

\[ \text{FWER} = 1 - (1 - \alpha)^m \]

With \(m = 7\) and \(\alpha = 0.05\):

\[ 1 - 0.95^7 \approx 0.302 \]

That’s about a 30% chance of finding a “significant” day even if no effect exists.

Bonferroni fix: test each day at \(\alpha / m = 0.05/7 \approx 0.0071\), or equivalently use \(p_{\text{adj}} = \min(1,\ m \cdot p)\).

We scrambled the weekday labels so no real pattern can exist, yet “winners” keep appearing. Real-data p-value for Thu: 0.0451.

Drag to rotate. A real effect would show a ridge running across every year. Instead, the “best day” moves around from year to year.

• Before correction, 1 weekday(s) looked significant; after accounting for 7 tests, 0 remained. There is no reliable evidence of a day-of-week effect in Bitcoin returns.
• Testing many patterns guarantees some “discoveries.” With 7 tests, there’s a 30% chance of a false alarm.
• The permutation experiment showed that pure noise produces “significant” weekdays regularly.
• Takeaway for finance: a backtested pattern with p < 0.05 means little until you ask how many patterns were tried. Corrections, out-of-sample checks, and stability over time matter as much as the p-value itself.

Data: BTC-USD daily closing prices, Yahoo Finance via the quantmod R package, 2018–2025.