# Research

**Wu-Xia Shadow Federal Funds Rate**

Note: When the shadow fed funds rate is at least 25 basis points, this model's short-term interest rate is identical to the shadow fed funds rate and has been highly correlated with other short-term market interest rates such as the effective federal funds rate. Consequently, we will not provide regular updates of the shadow federal funds rate as long as the target range for the federal funds rate is at or above 25 to 50 basis points.

From December 16, 2008, to December 15, 2015, the effective federal funds rate was in the 0 to 1/4 percent range targeted by the Federal Open Market Committee. In this "zero lower bound" environment, a number of researchers used shadow rate models to characterize the term structure of interest rates (Kim and Singleton (2012) and Bauer and Rudebusch (2013)) or quantify the stance of monetary policy (Bullard (2012) and Krippner (2013)). This web page hosts estimates from the Wu and Xia (2015) model of the shadow rate.

**Latest update — January 6, 2016**

In the Wu and Xia model, the model's short-term interest rate is the maximum of the shadow federal funds rate and a lower bound calibrated to be 25 basis points, since that was the rate paid on both required and excess reserve balances during the December 16, 2008, to December 15, 2015, period when the Federal Open Market Committee (FOMC) set the target range for the federal funds rate at 0 to 25 basis points. On December 16, 2015, the FOMC raised the rate paid on reserve balances to 50 basis points and the target range for the federal funds rate to 25 to 50 basis points. When the shadow fed funds rate is at least 25 basis points, the model's short-term interest rate is identical to this rate and has been highly correlated with other short-term market interest rates such as the effective federal funds rate. As noted above, we will not provide regular updates of the shadow federal funds rate as long as the target range for the federal funds rate is at or above 25 to 50 basis points. We'll continue to keep historical estimates of the shadow fed funds rate through November 2015 on this site.

### Model Description

Unlike the observed short-term interest rate, the shadow rate—first introduced by Fischer Black (1995)—is

*not*bounded below by 0 percent. Whenever the Wu-Xia shadow rate is above 1/4 percent, it is exactly equal to the model implied one-month interest rate by construction.The input data for the Cynthia Wu and Fan Dora Xia model are one-month forward rates beginning

*n*years hence. Wu and Xia use forward rates corresponding to*n*= 1/4, 1/2, 1, 2, 5, 7, and 10 years. These forward rates are constructed with end-of-month Nelson-Siegel-Svensson yield curve parameters from the Gurkaynak, Sack, and Wright (2006) dataset. The full details of the Wu and Xia model are described in their accompanying working paper. In short, the shadow rate is assumed to be a linear function of three latent variables called factors, which follow a VAR(1) process. The latent factors and the shadow rate are estimated with the extended Kalman filter.### Related Resources

- The Shadow Knows (the Fed Funds Rate) (macroblog)
- What Is the Stance of Monetary Policy? (macroblog)
- Shadow Interest Rates and the Stance of U.S. Monetary Policy (St. Louis Fed)
- Summarizing Monetary Policy (Econbrowser)