I estimated three models, 2021-2025M08, resulting in adjusted R² of between 0.54 (based on ADP data) to 0.71 (based on ADP data). Here’s what I got for forecasts of September and October private NFP as reported by BLS:

For the ARIMA(1,1,1) in the preliminary benchmark BLS series.

For an ECM in the ADP ratio of total private NFP to employment in firms 20-49:

For an ECM of private NFP to firms 1-499:

A comparison:

Obviously, without much difference between adjusted R²‘s, there’s not a lot of basis to choose one over the other, although the adjusted R² for an ECT involving employment in firms 1-499 employees vs. total private has the highest (at 0.71).  Using the highest R² specification, change in employment is -36K and -199K in September and October (although increases are entirely possible given the large prediction intervals).

One interesting point is that I can reject the null hypothesis that the ratio of firm employment 1-499 to total private Granger causes the log difference in BLS private NFP (at 10% msl), but not the reverse.