Out-of-sample checks for linear growth fits

A scored growth line

A least-squares line is a first-order fit through ordered observations that minimizes squared residuals and yields a slope treated as a constant growth rate. The archive then scored later intervals after a stated condition, using a batting average rather than leaving the slope as a description of the fitted window.

Hit rates and expected frequency

A score built from every monthly close of a broad equity index from 1947 through mid-1991, with no indicator filter, still produced more later gains than a chance model at a stated 0.999 confidence. In that hit-rate design, a random result was described as a batting average near 0.5 and confidence below 0.9.

When a market has risen 53 percent of intervals and fallen 47 percent, those shares become the expected frequency against which an indicator batting average is tested. The chance counts are the unfiltered rise and fall shares of the same series.

Taking the natural logarithm of each period close was proposed as a bias correction. The reply held that logarithms would not remove the extra time spent rising.

What the next interval must print

Comparison series were set with earnings of 1.00 in 1980 and 2.00 in 1985. Least-squares lines then produced in-sample fitted values and a calculated 1986 figure, the first year after the observed window. That calculated print is the out-of-sample extension: continue the fitted line one defined sampling interval past the last observed point to see what print would preserve the in-sample slope.

An idealized path rose in five equal 0.20 steps. Holding a 20 percent slope into 1986 implied a 2.20 value. The rates used were regression slopes rather than an average-slope statistic.

Extending the least-squares window through 1980-86, only one comparison series could keep an overall 20 percent slope after a 1986 decline. The others required a 1986 print above every figure in the prior six years.

Window selection and short spans

Window selection decides which consecutive observations enter the fit. A short favorable slice can reverse which series appears stronger.

For a span of five or six years a first-order equation was judged sufficient. Uncertainty was said to grow rapidly beyond the last actual year. Six years were treated as a comfortable base for a two-year extension only if the entire record is used.