1990issue C041-6
Chi-square evaluation of lagged breadth correlations
Weekly breadth observations from 1964 through 1988 were differenced, then checked for shared movement and self-echo. Only lags whose correlation still cleared a chi-square cutoff of 3.84 were kept.
- Replace each breadth-accumulator level with a detrended-change before pairing, so association is read from movement rather than from shared trend.
- Use correlation-analysis or an autocorrelation-test as the first gate, then keep a lag only when its chi-square-test statistic still exceeds the 95 percent cutoff of 3.84.
- Zero-lag pairings were the strongest, a one-week shift cut the coefficients sharply while remaining above the cutoff, and longer trading-window coefficients were small but still passed at a shared cluster of lags.
- Self-lag coefficients stayed between 0.06 and 0.1, and the three series tracked one another closely enough to treat as interchangeable for lag screening.
What the archive evaluated
The historical workflow used weekly observations from 1964 through 1988. Each series was differenced into a detrended-change, the period-to-period difference used in place of the raw level, so association was measured on movement, not trend. Those differenced observations were then paired for cross-series correlation-analysis and for an autocorrelation-test of each series against its own earlier changes. A chi-square-test was computed for every coefficient.
Three interchangeable breadth-accumulator series
The three series were built as a running advance-minus-decline issue sum, a running advance-minus-decline volume sum, and a cumulative index updated from successive Quotron increments. Each is a breadth-accumulator: a running sum of advancing versus declining issues, advancing versus declining volume, or successive index increments. The three tracked one another closely enough that the evaluation treated them as interchangeable for lag screening.
First gate: shared sign
Correlation-analysis tested whether contemporaneous or lagged changes in two market series shared the same sign more often than chance. The correlation coefficient came from four sign-compartment-tally cells, a four-cell count of same-direction and opposite-direction change pairs. A hand approximation used the same-sign count minus the opposite-sign count over their sum. The same-sign-rate, the share of paired changes that moved in the same direction, was the readable stand-in for association strength.
Second gate: the chi-square screen
The chi-square-test is a significance screen that converts a correlation coefficient and its sample size into a statistic compared with a 95 percent cutoff. In this workflow the statistic equaled sample size times the squared coefficient. A chi-square value of 3.84 marked the 95 percent confidence level.
One worked pairing produced 736 same-up cells, 465 same-down cells, 55 and 31 mixed-sign cells, an approximate coefficient of 0.86, and a chi-square of 954, which exceeded 3.84.
Contemporaneous pairing, a one-week fade, and longer lags
Zero-lag pairings produced the strongest associations. A one-week shift cut the coefficients sharply, though they remained above the 3.84 cutoff. Across the tested pairs, one series' change matched the next change in another series at an average same-sign-rate between 0.61 and 0.66.
At longer lags that still cleared the 95 percent screen, coefficients ranged from 0.045 to 0.149. Nearly every series showed the same cluster of lags: 236, 154, 130, and 41 weeks backward and 194, 158, 56, and 23 weeks forward. Each of those long serial-dependence lags was treated as a trading-window, a candidate horizon held only after the significance screen was applied.
What the self-lag test kept
The autocorrelation-test asked whether an indicator's current change was associated with its own earlier changes at a chosen lag. Self-lag coefficients were low, from 0.06 to 0.1, with same-sign agreement of 53 to 54 percent. The shared echo lengths clustered near 194, 157, 70, 48, and 23 weeks.
Editorial: redundancy, fade, and echo
Editorial interpretation: close tracking among the three breadth-accumulator series is redundancy, which is why the archive treated them as interchangeable for lag screening. The one-week shift is a fade. Association drops sharply but still clears 3.84. The small coefficients at the shared longer lags are a non-random long-horizon echo, not a contemporaneous substitute. Students can list those echo lengths as a trading-window only after both gates have been passed.
All readings on this track · 15 readings
- 1987Evaluating weekly price cycles with autocorrelation cutoffs
- 1988Serial correlation is not an independence test
- 1988Event-triggered chi-square tests for jittered serial links
- 1988Reverse the rule before you trust the system
- 1990Chi-square evaluation of lagged breadth correlations
- 1990Two-gate out-of-sample tests for trading rules
- 1990Sign-cell correlation, chi-square checks, and lag shifts
- 1998Test serial dependence before you optimize a mechanical system
- 2003How a one-rank rotation is constructed for an offset autocorrelation check
- 2004Constructing first-order residual autocorrelation tests
- 2004A two-stage Durbin-Watson residual screen
- 2011Constructing trend labels from increment correlation atoms
- 2012Overnight versus daytime return evaluation
- 2012Constructing a multi-horizon folded-correlation reversal filter
- 2016Tuning indicators to measured market cycles