Before 2008, U.S. monetary policy was primarily conducted in terms of a target set by the Federal Reserve for the fed funds rate, which is the interest rate a bank pays to borrow funds overnight from other banks. A large academic literature used the fed funds rate as a summary of monetary policy, looking at its correlations in dynamic regressions with other variables of macroeconomic interest. But the fed funds rate has been stuck near zero for the last 5 years, and will likely be replaced by an alternative policy focus even once we exit the zero lower bound. Economic researchers face not just the difficulty of summarizing what the Fed has been doing in the current and future environment, but also the practical challenge of how to update their historical regressions to try to describe the full set of historical data along with the new experience in a coherent way. Here I describe a new research paper that suggests one solution to these problems.

One approach that some researchers have been using to model the dynamics of interest rates when the short end of the yield curve is stuck near zero is based on the shadow rate model first proposed by Black (1995). This hypothesizes the existence of a “shadow” short-term interest rate that might in some circumstances be quite negative. In normal times (when the observed short-term interest rate is sufficiently high), this shadow rate is positive and coincides with the observed short rate, and the dynamic relations among interest rates of various maturities are described using familiar tools. The hypothesis is that when the shadow rate falls below some bound, we could continue to calculate an implied negative shadow rate as if it followed the usual historical dynamics as well as calculate forecasts for that shadow rate. To predict the actual short rate at any future horizon, we just compute the expectation of the maximum of the future shadow rate and the lower bound. We can then get predicted values for all other yields from the assumption that risk gets priced in a consistent way across assets.

A recent paper by Dora Xia, a UCSD graduate student who expects to complete her Ph.D. this spring, and Cynthia Wu, a former UCSD student who is now an assistant professor at the University of Chicago, makes several contributions to this literature. First, most previous applications of the shadow rate model have involved arduous numerical simulations to calculate its full predictions. By contrast, Wu and Xia develop a very simple closed-form expression that gives a very good approximation to the predictions of the model for the yield of any maturity. Here is a graph showing the estimate of the shadow rate that comes out of their approach. Up until 2009, this basically coincides with the observed fed funds rate, but since then, the implied shadow rate has been quite negative.

Their model tracks the recent behavior of the yields on Treasury securities of various maturities quite closely. For example, the red dots in the graph below describe the average one-month forward rates in 2012 associated with 3-month, 6-month, 1-year, 2-year, 5-year, 7-year, and 10-year Treasury securities, while the blue curve gives the model’s predictions for what those rates should have been. It appears to be an approach that is quite accurate and easy to use.

*n*-year one-month forward rates as a function of horizon

*n*(horizontal axis, in years). Source: Wu and Xia (2013).

Of particular interest is Wu and Xia’s observation that their series for the shadow rate exhibits similar correlations with other macro variables since 2009 as the fed funds rate did in data up until the end of 2007. Wu and Xia took a popular model that had been estimated before the Great Recession in which the fed funds rate was used as the summary of monetary policy, and just replaced the fed funds rate with the Wu-Xia shadow rate to get a data set that continues after 2009. Although this device could not fully account for all that happened to interest rates and unemployment during the Great Recession over 2007-2009, Wu and Xia found the evidence to be consistent with the hypothesis that data since 2009 could be described using the spliced series as the monetary policy indicator and using the same model that described pre-2007 data. In other words, the shadow rate displays the same sort of correlation with lagged macro variables since 2009 as the fed funds rate did in earlier data, and likewise the value of macro variables that one would predict using the new shadow rate series is close to the value one would have predicted in earlier data using the fed funds rate instead.

The suggestion is that we then might use the shadow rate series as a way of summarizing what the Fed has been doing with its unconventional policy measures such as large-scale asset purchases and forward guidance. If the Wu-Xia framework is correct, these unconventional policies can all be summarized in terms of what effect they had on the shadow short rate. By comparing the shadow rate with the value that traditional models would have predicted for the fed funds rate, Wu and Xia get a measure of the shocks to monetary policy. Wu and Xia find that monetary policy has recently been a bit more expansionary than usual (pushing the shadow rate about 0.6% more negative over 2011-2013 than the traditional monetary policy rule would imply), as a result of which their estimates imply that the current unemployment rate is 0.23% lower than it otherwise would have been.

The Wu-Xia series for the shadow rate is available online. I recommend this as a practical patch for researchers who want to update analysis that made use of historical data on the fed funds rate.

And while I’m on the topic of useful new research coming out of UCSD, let me also call attention toanother paper from Dora’s Ph.D. dissertation, which develops a practical way to get term-structure models that actually generate decent forecasts.

*This piece is cross-posted from Econbrowser with permission.*