WASHINGTON (Reuters) -The Chicago Federal Reserve said on Tuesday it had combined government and private data into a new estimate of the unemployment rate that would give policymakers faster-moving, twice monthly updates of a statistic central to the central bank's current interest rate debate.

The new "real-time" set of labor market statistics, relying on the government's Current Population Survey along with data from sources including payroll processor ADP, online job site Indeed, and Google, projects the unemployment rate most likely stayed stable in September at 4.3%.

The hiring rate for unemployed workers increased slightly, a bank press release said, but that was offset by a rise in layoffs and other job separations. 

The estimate comes nearly two weeks before the next employment report on October 3, covering payroll job growth and the jobless rate for September.

While Fed officials study an array of indicators to gauge the health of the labor market, the unemployment rate is paramount. Under the current data release timetable, Fed officials would receive the September official unemployment rate only before their next policy meeting at the end of October. The Chicago data would now supplement that with an early estimate of the October jobless rate, and possibly show whether the labor market was weakening or had stayed stable, a central issue in any decision to further cut interest rates.

"One of the hardest things a central banker has to do is get the timing right in moments of transition, and that's why real-time data can be so important," said Austan Goolsbee, president and CEO of the Chicago Fed.

The Fed cut the benchmark policy rate by a quarter of a percentage point at its meeting last week, and Fed chair Jerome Powell speaks later today in comments that may shed more light on the debate over whether rates should fall again at the October meeting.

Since the pandemic and through the early months of the new Trump administration, when government policies shifted fast, Fed officials have looked for reliable outside indicators that could show how the economy is shifting at a tempo faster than the government's often lagged statistical reports.

Outside data can also help validate or serve as a check against official sources at a time when the government has canceled some data reporting and moved to fire officials at statistical and scientific agencies.

(Editing by Chizu Nomiyama)