Federal Reserve Chair Janet Yellen and her colleagues have tightened monetary policy by a lot more than their benchmark interest rate increase would suggest. It's just that they've done it in the shadows.

The so-called shadow federal funds rate rose about 300 basis points from the middle of 2014 through the end of last year as the central bank tapered, then ended its asset purchase program and prepared the way for its single rate increase in December.  The shadow rate's increase helped fuel a rise of the dollar that depressed U.S. exports and economic growth, according to James Hamilton, an economics professor at the University of California at San Diego. 

"The Fed's rate hike in December was not the beginning of the tightening cycle," said Peter Schiff, president of Euro Pacific Capital Inc. in Westport, Connecticut. "If you look at the shadow rate, the tightening cycle began two years before that."The shadow rate serves as a way to measure the economic impact of Fed policy makers' more unconventional actions, such as quantitative easing and forward guidance, after they cut the actual funds rate to near zero.It's calculated based on the historical relationship of the federal funds rate to the rest of the yield curve, especially at the short end. The shadow rate slipped deeper and deeper into negative territory from 2009 through the mid-2014 as the Fed's policies pushed Treasury bill and other short-term rates down toward zero.

It rose subsequently as those rates climbed with the end of the Fed's quantitative easing program and in anticipation of liftoff at the end of last year.

The "shadow rate is extracted from the whole yield curve," Jing Cynthia Wu, an associate professor at the University of Chicago's Booth School of Business, said in an e-mail. "It's the hypothetical short-end of the yield curve, if there is no zero lower bound" on the federal funds rate.

It's not clear how much stock Fed policy makers put in the concept, though calculations of the rate by Wu and fellow economist Fan Dora Xia are carried on the Atlanta Fed's website. And St. Louis Fed President James Bullard referenced separate computations of the rate by Leo Krippner of the Reserve Bank of New Zealand in a 2012 speech.

While the shadow rate may indicate tighter policy, Yellen and her colleagues contend that the sheer size of the Fed's balance sheet — currently  a record $4.5 trillion — provides the economy with support even though the central bank is no longer adding to it with asset purchases. And longer-term interest rates, such as the yield on the Treasury's 10-year note, are lower than they were two years ago when the shadow rate began to rise.

What is clear though is that the dollar began advancing in the middle of 2014 just as the shadow rate began to climb. And the currency's appreciation had a noticeable impact on the economy, reducing growth last year by about one percentage point, according to Michael Feroli, chief U.S. economist at JPMorgan Chase & Co. in New York. Gross domestic product expanded 2 percent in 2015 on a fourth quarter-to-fourth quarter basis, down from 2.5 percent in 2014.

"While the Fed has so far hiked rates only once, and by 25 basis points, the shift away from massive balance sheet expansion and forward guidance may have been the unconventional equivalent to as much as 300 basis points" of tightening, Feroli said in a May 6 note to clients.