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I estimated three fashions, 2021-2025M08, leading to adjusted R2 of between 0.54 (based mostly on ADP knowledge) to 0.71 (based mostly on ADP knowledge). Right here’s what I bought for forecasts of September and October personal NFP as reported by BLS:
For the ARIMA(1,1,1) within the preliminary benchmark BLS collection.

For an ECM within the ADP ratio of complete personal NFP to employment in companies 20-49:

For an ECM of personal NFP to companies 1-499:

A comparability:

Clearly, with out a lot distinction between adjusted R2‘s, there’s not a number of foundation to decide on one over the opposite, though the adjusted R2 for an ECT involving employment in companies 1-499 workers vs. complete personal has the best (at 0.71). Utilizing the best R2 specification, change in employment is -36K and -199K in September and October (though will increase are completely potential given the massive prediction intervals).
One attention-grabbing level is that I can reject the null speculation that the ratio of agency employment 1-499 to complete personal Granger causes the log distinction in BLS personal NFP (at 10% msl), however not the reverse.
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