If the Fed is feeling any inflation pressure, it certainly isn't coming from the labor market.

A closely watched gauge of what's happening with worker salaries showed only muted progress as 2016 came to a close.

Compensation costs for the private sector rose just 0.5 percent for the final quarter of the year, which didn't even match the relatively muted Wall Street expectations of 0.6 percent, according to employment cost index data the Bureau of Labor Statistics released Tuesday.

For the full year, the increase translated to just 2.2 percent, a shade ahead of the 2 percent growth rate from the previous year.

Moreover, much of the fuel for the compensation increase came from benefit costs, which rose 2.1 percent during the year, owing largely to a 2.7 percent surge in health-care benefit expenses.

Wages and salaries increased 2.3 percent for the year, compared with 2.1 percent for the previous period.

"The latest data suggest that inflation might even be slowing rather than rising, indicating that there is no reason whatsoever for the Fed to weaken the labor market and slow job growth with further rate hikes," the Center for Economic Policy and Research said in a post on its website. "In other words, the Fed is shooting at phantom inflation."