Constructing detrended tests for calendar seasonality

A date is not a regime

Seasonality is a repeating calendar or other chosen cycle. It becomes discussable only after one period's average differs significantly from another, not when a calendar story is widely repeated.

Seasonal-analysis is a market-regime read that places one calendar idea against multi-month prices, residual trend, and month-bucket dispersion rather than treating the date as self-justifying.

Three matrices, one screen

The workbench is three matrices: raw monthly levels, each month as a percent of that year's average, and month-to-month percent change.

Hypothesis-testing is a predeclared comparison of period means with a baseline on ordered prices, a stated lookback, and a p-value, not a ranking of column totals. Each matrix is scored with a two-sample-t of column mean versus matrix mean. A two-sample-t is a standardized gap between a month-column mean and the full-matrix mean that uses both column and matrix dispersion and counts. A fixed polynomial then maps that T into a p-value.

Weekly buckets need a matrix about four times wider. A single global trend line can raise residual variance when the series is range-bound for years and then steepens.

Raw levels inherit the secular rise

In a 30-year monthly matrix of a broad equity index from 1961 to 1990, no month's average level differed significantly from any other month, because a secular rise left large standard deviations.

Detrending shrinks a long-run rise so month columns are not compared on raw levels that inherit the secular drift.

Year-average-normalization still misses the cutoff

Year-average-normalization expresses each month as a percentage of that calendar year's twelve-month mean. Dividing each month by that year's 12-month average and expressing the ratio as a percentage shrinks monthly standard deviations, but every comparison with the year average still stays above the significance-threshold.

Comparing only December with January of the same calendar year can manufacture a significant gap from leftover upward drift even when the rest of the year-normalized matrix shows no seasonal pattern.

Month-to-month changes keep an upward step

Month-to-month percentage changes still carry an overall 0.55% rise. The December-to-January step has a p-value of 0.08, and May is the only month with a negative average change, which remains insignificant.

A failed window is abstention

A November-to-April equity window is not established as a regime in the 1961-1990 sample. November exceeded April of the same year in 18 of 30 years, and several multi-year intervals ran against the window, so a failed screen is treated as abstention rather than an entry.

Seasonal-trading is a single testable procedure that turns a calendar window into an entry, an exit, or an abstention instead of a folklore holding period.

Short slices measure how long a contrast lasts

The construction asks for at least 10 years, or an equally long column, before treating a pattern as validated. Pattern-duration is the observation that a genuine seasonal contrast can fade, so a short sample may hint at how long it lasts while remaining too small to validate it. Three- or five-year slices may hint at how long a contrast lasts but are usually too small for the significance screen to accept a pattern.