Normal Distribution
nothing is called 95% until it has been counted
Field

When the observations repeat each other

Every standard error on this site divides by √n, which claims the observations carry independent information. In time order they usually do not: at a lag-one correlation of 0.8 a fifty-point series is worth about six independent observations, its 95% interval covers 47%, and two series that wander are called related three times out of four.
Twenty series with a lag-one correlation of 0.8. Every series has a true mean of zero and 60 observations. The marks on the right are the twenty sample means. The variance of that mean is 8.3 times what 60 independent observations would give, so the series is worth about 7 of them.

The observations that repeat each other

Almost every standard error divides by √n, which claims the observations carry independent information. At a lag-one correlation of 0.8 a fifty-point series is worth about six independent observations, and its 95% interval covers 47%.

 Two independent random walks, 100 steps. Nothing connects these two series: each is generated from its own independent draws. Regressing one on the other gives a slope with t = -10.9, R² = 0.55 and p = 0.0e+0 — a result that would be reported as a finding by any standard output.

Two walks and a finding

Regress one random walk on another, independently generated, and the slope is significant 76.7% of the time with a median R² of 0.17. Nothing connects the two series, nothing in the output says so, and more data makes it worse.

 What differencing fixes, and what it costs, 100 steps. The first pair is the false-positive rate for two independent random walks: 77% on the levels, 4.9% on the differences. The second pair is how much of a real relationship survives: R² falls from 0.91 to 0.33. The same operation does both.

What differencing costs

Differencing takes the false-positive rate between two unrelated walks from 76.7% to 4.9%, and takes a genuine relationship's R² from 0.91 to 0.33. Applied to a series that did not need it, it doubles the variance and installs a correlation of −0.5 that the data never had.

 An AR(1) at φ = 0.5, 200 observations. The bars are the measured correlations; the curve is φᵏ, which is what an AR(1) must have. The band is ±1.96/√n, where an independent series would stay. The first bar is 0.53 against a band of ±0.14.

The check before the standard error

One number decides whether every interval in an analysis is trustworthy, and the check for it flags a lag-one correlation of 0.5 nine times in ten — and one of 0.2 only one time in five, where the interval already covers 88.6% instead of 95%.

 The damage and the warning, against the same dial. Two readings at each persistence. In the darker colour, how often a regression between two independent series of 200 steps is called significant at 5%: 4.9% at φ = 0, 34.2% at 0.8, 52.4% at 0.9, 83.4% at a unit root. In the lighter, how often the standard unit-root test refuses a unit root on one of those series — the chance the analyst is told the series is stationary and may be regressed: 87.2% at φ = 0.9 and 31.9% at 0.95. At φ = 0.9 both are high at once, which is a correct diagnostic licensing a regression that is wrong half the time.

The cliff that is a slope

A regression between two independent series is called significant 4.9% of the time at no persistence, 52.4% at a lag-one correlation of 0.9, and 83.4% at a unit root. The rule the field offers asks whether the last of those holds, and at 0.9 the unit-root test correctly refuses one 87.2% of the time.

 Six cells, and 5% is the right answer in all of them. How often a regression between two independently generated series is called significant at the 5% level, for two worlds and three treatments, at 200 observations. Every pair is independent by construction, so 5% is correct everywhere and every other reading is a failure. Untreated: 82.9% and 100.0%. With a fitted line removed: 74.2% and 33.5%. Differenced: 5.0% and 5.2%. The treatment that controls the rate in both worlds is the one that discards the level and the trend, which is the quantity a study of trending series was about.

The repair that keeps the question

A regression between two independent trending series is significant 82.9% of the time on random walks and 100.0% on trend-stationary ones. Subtracting a fitted line leaves 74.2% and 33.5%; differencing leaves 5.0% and 5.2% and throws away the trend the study was about.

 How fast a gap has to close before a sample can see it close. The power of the test against the half-life of a disagreement, at 100, 200, 400 observations, each read against its own simulated critical value. Every pair in every reading is genuinely tied together, so a non-rejection is a miss. At 200 observations a gap that halves in 3 steps is found 99.9% of the time and one that halves in 12 steps is found 15.3% of the time — and by 35 steps the reading is 6.1%, which is the test's own size. Beyond that the curves are flat because there is nothing left to detect with.

How slow a return a sample can see

At two hundred observations the test finds a gap that halves in five steps four times in five, one that halves in eight 37.3% of the time, and one that halves in fifty 4.95% of the time — which is the rate at which it finds pairs with no mechanism at all. The boundary moves with the sample, not with its square root.

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