Evaluating stationarity, randomness, and dependence in an index series

What each test is asking

A chi-square test is a goodness-of-fit comparison of observed frequencies with an explicit baseline. In this historical workflow it was used for stationarity, randomness, zero-point symmetry, and transition counts.

A transition matrix is a table of observed moves among categorized change states, judged against the product of the states' separate probabilities. An autocorrelation test is a serial-correlation screen for one linear form of dependence. It is treated here as incomplete when used alone on jittered raw prices.

Stationarity means the generating rules stay stable across subperiods, a precondition for tools that assume an unchanging process. Randomness means each realized change is equally likely inside a fixed probability density. Independence means one period's change does not influence a later period's change. Those three questions are not interchangeable.

Stationarity and randomness came first

On monthly index observations from 1945 through 1985, a stationarity chi-square of 14.84 was not statistically significant, so the generating process was treated as unchanged over that sample.

The same monthly series produced a randomness chi-square of 10.79 that was not significant, consistent with individual changes being equally likely inside the observed density. That result concerns draws inside a fixed density. It is not the same as independence across time.

Independence is a later gate

A single-frequency histogram of changes, with tails cut at plus or minus 5.0, yielded a chi-square of 181.59 against symmetry around zero, a highly significant rejection of independence. Repeating the symmetry test with tails cut at plus or minus 2.5 still produced a highly significant chi-square of 154.68.

A digram transition matrix on the categorized series had a chi-square of 30.60, and a trigram matrix had 94.73. Both were highly significant versus an independence baseline.

Tertile-coding without raw-price jitter

After coding each change into one of three histogram tertiles, waiting-time histograms triggered on a large rise, a large fall, or a mid-range change were visually uneven across months, including at a tenth subsequent match. Tertile-coding maps each change into one of three histogram bins so sequential state patterns can be counted without raw-price jitter dominating the test.

Calendar-window histograms that start from the first or last month of a trading year did not place the first subsequent tertile class equally across the twelve months.

Repeated-state triplets such as 1-1-1, 2-2-2, and 3-3-3 occurred more often than an independence product of state probabilities would imply, while mixed sequences such as 1-2-3 and 1-3-2 occurred less often.

Why serial screens are incomplete

Serial and autocorrelation procedures, and runs tests, are described as incomplete screens because they capture only some dependence forms and can be blurred by jitter in uncategorized prices. Editorial reading: those screens can sit beside a chi-square or transition-matrix check, but they do not replace the independence gate.

The same trigger-histogram and chi-square comparison applied to large money-supply increases versus later large index moves produced uneven monthly counts and statistically dissimilar fifth-lag histograms for large declines versus large advances.