Comparing Apples to Apples: “Synthetic Real‑Time” Estimates of R‑Star

Sophia Cho and John C. Williams

Estimates of the natural rate of interest, commonly called “r-star,” garner a great deal of attention among economists, central bankers, and financial market participants. The natural interest rate is the real (inflation-adjusted) interest rate expected to prevail when supply and demand in the economy are in balance and inflation is stable. The natural rate cannot be measured directly but must be inferred from other data. When assessing estimates of r-star, it is important to distinguish between real-time estimates and retrospective estimates. Real-time estimates answer the question: “What is the value of r-star based on the information available at the time?” Meanwhile, retrospective estimates answer the question: “What was r-star at some point in the past, based on the information available today?” Although the latter question may be of historical interest, the former question is typically more relevant in practice, whether in financial markets or central banks. Thus, given their different nature, comparing real-time and retrospective estimates is like comparing apples to oranges. In this Liberty Street Economics post, we address this issue by creating new “synthetic real-time” estimates of r-star in the U.S. for the Laubach-Williams (2003) and Holston-Laubach-Williams (2017) models, using vintage datasets. These estimates enable apples-to-apples comparisons of the behavior of real-time r-star estimates over the past quarter century.

Some History

One advantage of the Laubach-Williams (LW) and Holston-Laubach-Williams (HLW) models is their long-standing use. Real-time estimates for the LW model are available on the New York Fed’s website from the first quarter of 2005 onward, with a reporting gap from the third quarter of 2020 to the third quarter of 2022 due to the COVID-19 pandemic. During this period, publication of estimates was paused as the extreme volatility in economic data was at odds with the underlying structure of the models (Holston, Laubach, and Williams 2023). Real-time estimates for the HLW model are available from the fourth quarter of 2015 onward, with the same pandemic-related reporting gap.

The specifications and estimation methodologies of these models, which were unchanged during pre-pandemic years, were modified to take into account the effects of the COVID-19 pandemic. Specifically, for estimates covering the first two quarters of 2020, both models were modified to incorporate the effects of pandemic-related closures (HLW 2020). When publication of estimates resumed with data for the fourth quarter of 2022, modified versions of the models were used that accounted for pandemic-related closures and the extreme volatility in economic data (HLW 2023). Since then, the specifications and methodologies of both models have remained unchanged.

Creating Synthetic Real-Time Estimates

To extend real-time estimates back to the mid-1990s and to fill in the pandemic-related reporting gap, we constructed “synthetic real-time” estimates for the HLW and LW models, using real-time vintages of U.S. data as closely as possible. We refer to these estimates as “synthetic real-time” because we apply estimation methods that did not yet exist to past real-time data. This contrasts with fully real-time estimates, which were produced at the same time as the real-time data.

For the HLW model, we use real-time data for real GDP and the price index for personal consumption expenditures excluding food and energy (“core inflation”) available on the Philadelphia Fed’s real-time data website. When these data are unavailable, we rely on the St. Louis Fed’s ALFRED real-time database. To be consistent with the usual practice of publishing real-time estimates for the HLW model, we use data from the second release of the National Income and Product Accounts from the Bureau of Economic Analysis. Real-time data are available starting from the fourth quarter of 1995. Since federal funds rate data is not revised retroactively, we use currently available data for this series.

For the LW model, we use the same data sources for real GDP and core inflation as the HLW model. Additionally, the LW model incorporates data on imported oil prices and non-oil imported goods prices, which are not provided on public real-time databases. For these series, we rely instead on internal archives of vintage data where available. We have only a few vintages of non-oil imported goods prices before 2005, with the earliest dating back to mid-2001. In cases where real-time data is unavailable, we use the closest available vintage. The resulting estimates of r-star are not very sensitive to changes in the specific vintages of import prices used.

To construct synthetic real-time estimates before 2020, we use the standard pre-pandemic versions of the HLW and LW models. For estimates covering the pandemic-related reporting gap, we use the modified versions of these models. In all cases, we use the model code available on the New York Fed’s website. Note that the two versions of each model yield nearly identical estimates of r-star for the pre-pandemic period (HLW 2023).

Properties of HLW Real-Time Estimates of R-Star

As illustrated in the chart below, synthetic real-time estimates from the HLW model reveal two episodes of large movements in r-star: a 1-3/4 percentage point rise and subsequent reversal during the productivity boom of the late 1990s and early 2000s and a sharp 1-1/2 percentage point decline following the global financial crisis and Great Recession. In between these two episodes, real-time estimates of r-star are remarkably stable, at around 2-1/4 percent during the mid-2000s and ½ percent during the 2010s. The blue line in the chart presents a consistent history of HLW real-time estimates of r-star by combining the existing published real-time and synthetic real-time estimates. The red line depicts the corresponding four-quarter moving average of these combined estimates, smoothing quarter-to-quarter variation.

During the COVID-19 pandemic and its aftermath, HLW real-time estimates of r-star rose to a little above 1 percent before falling back to about ¾ percent, with a net increase of only a quarter of a percentage point from 2019 to 2024. Excluding the temporary bulge from late 2020 to early 2022, when the economy was subject to extreme pandemic-related volatility, these real-time estimates of r-star have remained below 1 percent since 2010.

In the HLW and LW models, r-star is determined by the trend growth rate of GDP as well as a second, unobserved factor, called “z,” which encapsulates the effects of all other determinants of r-star beyond trend growth. Although not specified in these models, the determinants of z may include demographics, the relative demand for safe assets like Treasury securities, and government indebtedness, all of which affect the global demand and supply for savings.

Changes in real-time estimates of trend growth explain most of the persistent movements in HLW real-time estimates of r-star from the mid-1990s to 2011. The chart below shows HLW real-time estimates of trend growth, which closely tracked estimates of r-star over this period.

Since 2012, estimates of trend growth have gradually increased, but other determinants have exerted increasing downward pressure on r-star. The chart below shows HLW real-time estimates of z, which held steady from the mid-1990s to mid-2000s.

From 2012 to 2019, estimates of z trended lower, offsetting the rise in estimates of trend growth. This pattern of rising estimates of trend growth and declining estimates of z has continued since the onset of the pandemic. The net effect is that the most recent estimate of r-star is slightly above estimates from 2019.

Properties of LW Real-Time Estimates of R-Star

Real-time estimates of r-star from the LW model display patterns similar to those from the HLW model. The chart below presents the combined published real-time and synthetic real-time estimates of r-star from the LW model.

LW estimates from 2024 are about half a percentage point greater than LW estimates from 2019. These estimates exhibit the same pattern as HLW estimates of a temporary rise following the pandemic, which is partially reversed.

Conclusion

In this post, we constructed synthetic real-time estimates of r-star in the U.S. dating back to the mid-1990s, using the HLW and LW models. Combined with the published real-time estimates, these synthetic real-time estimates provide a more comprehensive history of apples-to-apples comparisons across time and across other real-time measures of r-star.

Note: Estimates of r-star and related variables from the LW and HLW models are published quarterly. The most recent estimates for 2024:Q4 were released on February 28 for the LW model; estimates for the HLW model post today. Visit Measuring the Natural Rate of Interest for more information and additional release dates.

How to cite this post:
Sophia Cho and John C. Williams, “Comparing Apples to Apples: “Synthetic Real‑Time” Estimates of R‑Star,” Federal Reserve Bank of New York Liberty Street Economics, March 3, 2025, https://libertystreeteconomics.newyorkfed.org/2025/03/comparing-apples-to-apples-synthetic-real-time-estimates-of-r-star/.