A lag-regression audit of stochastic forecast skill

What the archive tested

A stochastic-oscillator is a 0-to-100 %K series built from a short ordered-price lookback and used as the independent variable when testing whether oversold or overbought readings precede later price changes. The archive workflow computed daily %K from the prior five days of prices, kept it bounded between 0 and 100, and set it against daily changes in a broad exchange composite. The sample covered roughly 400 observations from September 1989 through March 1991.

Same-day fit is not a lead

Linear-regression is a two-variable fit of later price changes on a contemporaneous or lagged indicator, summarized by r-squared, a slope, and a fitted line that can be compared across leads. R-squared is a 0-to-1 measure of how much of the price-change variation a single linear equation attributes to the indicator at a chosen lag. A same-day linear-regression of those price changes on %K produced an r-squared of 0.376, read as a moderately good coincident association rather than a lead. The same-day fitted line was written as price change equals -2.5851 plus 0.04561 times the %K reading.

A lead-lag-sweep is a sequence of otherwise identical regressions in which the indicator is shifted one sampling interval at a time so coincident association can be separated from forecast skill. Lagging %K by one day dropped r-squared to 0.0008. The same lead scan through 10 days stayed near zero, so the oscillator behaved as a coincident series in that sample.

The slope still needs a t-test

Hypothesis-testing is a t-test that the indicator slope is not zero, formed by dividing the estimated coefficient by its standard error and comparing the result with sample-size critical values. For regressions with more than 120 observations, t-statistics above 1, 1.98, and 2.62 were presented as 67 percent, 95 percent, and 99 percent confidence that the true slope is not zero. The coincident slope t-statistic was 15.34, while the one-day-lead t-statistic was 0.57, so only the contemporaneous coefficient cleared the 99 percent threshold cited for large samples.

A monthly contrast that peaked at a lead

A contrast monthly lead sweep of a stock-price index on net changes in consumer installment debt from January 1967 through October 1987 reached its highest r-squared at a 13-month lead. Editorial reading: that contrast is included only to show that a lead-lag-sweep can retain a lag when r-squared actually peaks away from coincidence.

When the linear setup is unreliable

The linear setup was treated as unreliable if the true map is nonlinear, if forecasts are extrapolated outside the observed indicator range, or if the relationship itself changes over time.