Build a chi-square stationarity screen before you forecast

What stationarity is asking

A price, volume, or rate chart is a time series. A stationarity check asks whether observations in the first half of a chosen lookback still measure the same process as observations in the second half.

Stationarity is the condition in which the same governing influences operate across a sampled series so that early observations remain comparable to later ones. The lookback is the historical window that is split and compared before a later forecast is trusted.

How the two halves are built

The construction converts the raw series into ordered changes, splits the lookback into two equal halves, and assigns each change to one of a fixed set of mutually exclusive bins so that two frequency tables can be compared. A bin is a mutually exclusive value interval that receives each observation once when a frequency table is built.

The chi-square test is a goodness-of-fit comparison of two binned frequency tables from successive halves of the same ordered series. In the Treasury-rate illustration, an 11-bin frequency table of daily changes over two 12-year halves was used to display that later observations did not share the earlier distribution after slope and volatility shifted.

Why a chart is not the baseline

Charts can show a break, but they lack the precision of a chi-square comparison of the two binned halves. That comparison is the quantitative baseline used to decide whether the series is statistically consistent.

Declare the sampling interval first

The same construction is sensitive to sampling interval. Daily and weekly versions of the same rate series can produce different visual and statistical impressions of stability, so the horizon must be declared before the test is read as a forecast screen.

Sampling interval is the chosen spacing, such as daily or weekly changes, used to form the observations that enter the bins.

When the screen finds mixed data

Nonstationarity is a material change in the distribution of later observations relative to earlier ones in the same series. Methods that assume a stationary, homogeneous series, including uses of regression beyond simple description, can give misleading out-of-sample readings when the chi-square screen finds the two halves do not match.

If the test flags nonstationarity, the construction can be revised by lengthening or shortening the lookback or by changing the lag used to form differences, then rerun so that the revised baseline is compared with the later sample.