Objective crossover tests of Fibonacci wave ratios

Ratios treated as expected fractions

Fibonacci retracement uses Fibonacci-sequence proportions, especially the limits near 0.618 and 1.618, as expected fractions of a prior price swing. After the first four Fibonacci-sequence pairs, successive ratios cluster near 0.618 and inverse pairs cluster near 1.618.

Elliott wave analysis is a self-similar count of five-wave impulses and three-wave corrections applied at more than one time scale. A recursively subdivided five-up and three-down wave count would be expected to produce ratios near 0.618 if that self-similar structure described prices.

The golden ratio is that limiting Fibonacci value near 0.618, with inverse near 1.618, treated as a testable mean for wave ratios rather than a chart decoration.

A cited split near the limit

One cited split of 8143 bull-market days against 4972 bear-market days equals 0.611, near 0.618, while a single cited primary-swing ratio of 1.621 was not matched by the other eight bull markets in that set.

Editorial reading: one day-count split near 0.618, and one primary-swing ratio near 1.618, do not show that those constants are general wave-ratio means.

Waves locked by crossovers

A moving-average crossover is a trend-turn rule that labels an upwave or downwave from the local extreme between index crossings of a moving average. Upwaves and downwaves were defined from local extremes between moving-average crossovers so trend turns did not depend on discretionary labeling.

A wave ratio is a measured comparison inside a defined wave, such as advancing days to declining days or total advance to total decline. Wave statistics were computed on Dow Jones Industrial Average prices through January 30, 1986 across moving-average lengths of 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, and 100 days.

Tests of a 1.618 mean

For 10-day crossovers, a two-tailed test rejected a mean of 1.618 at the 1% level for total-advance-to-net-advance and advancing-to-declining days, and at the 5% level for total advances versus total declines.

For 20- and 30-day crossovers the same 1.618 mean generally could not be rejected at the 1% level, except that the 20-day total-advance-to-net-advance ratio could be rejected at the 10% level.

None of the crossover-defined wave-ratio means equaled 1.618, and every moving-average length showed large dispersion with standard deviations on the order of the respective means.

Subwaves in two 1985 intervals

Inside the September 18, 1985 to January 7, 1986 advance, 10-day subwave advance/decline ratios were 2.534, 2.982, 5.903, and 2.758, with 50 advancing versus 26 declining days and a total-advance-to-net-advance ratio of 1.338.

In the June 20, 1985 to September 18, 1985 interval there were 35 advancing and 26 declining days, and the average advance of 4.606 points versus the average decline of 6.108 gave a ratio of 0.754.