FEDS Notes
September 05, 2018
An Estimate of the LongTerm Neutral Rate of Interest
An estimate of the economy's equilibrium interest rate can be a helpful guide for a central bank's setting of interest rates. Woodford (2003), for example, argues that offsetting highfrequency movements in the equilibrium interest rate should be a key consideration of monetary policy, to be supplemented by deviations from the equilibrium interest rate to achieve the central bank's inflation and employment stabilization goals. This note proposes a new measure of the highfrequency equilibrium interest rate, one that falls naturally out of a common textbook notion of the economy's equilibrium interest rateand which is rooted in one particularly simple and wellknown model.
Consistent with the theory underlying it, the new measure is an estimate of the equilibrium value of the longterm interest ratein particular, the tenyear Treasury yield. That is in contrast to many other estimates that are focused on the equilibrium value of shortterm interest rates, such as the federal funds rate, an overnight rate. (For example, Laubach and Williams (2003), Holston, Laubach, and Williams (2017), Edge, Kiley, and Laforte (2008), and Barsky, Justiniano, and Melosi (2014).)
There are several reasons to focus on longterm interest rates. First, in many macroeconomic models, longterm interest rates are more important for spending decisions than are shortterm interest rates. Second, while monetary policy makers traditionally use shortterm interest rates as their policy instrument, in the wake of the financial crisis, policy makers were constrained by limits on how much they could reduce shortterm interest rates, and so central banks focused on policies that affect longerterm interest rates, such as forward guidance and longerterm asset purchases. As well, the experience at the effective lower bound may have altered the relationship between shortterm interest rates and the overall economy.
The estimate of the neutral interest rate proposed here pertains to longterm interest rate, and, in particular, it is an estimate of shortrun longterm equilibrium interest rate. As emphasized by Woodford (2003), the shortrun equilibrium rate can be useful in helping determine where policymakers may want to lead interest rates in achieving maximum sustainable employment in the short run. The shortrun focus is in contrast to measures such as those discussed in Laubach and Williams (2003), Kiley (2017), and Holston, Laubach, and Williams (2017), which emphasize the economy's longrun equilibrium interest rate. The key distinction is that a shortrun equilibrium rate is one that stabilizes the economy period by period, whereas the longrun equilibrium is the value of the interest rate that will stabilize the economy down the road; in the long run. Estimates of the shortrun equilibrium interest rate are often computed using DSGE models; see, for example, Edge, Kiley, and Laforte (2008), Barsky, Justiniano, and Melosi (2014) and Curdia et al. (2015). The estimates from DSGE models, however, are typically for the shortrun, shortterm or overnight equilibrium interest rate, whereas the estimates here are of the shortrun, longterm (tenyear) equilibrium rate. Also, as will be clear shortly, the estimates developed in this note are based on a model that is much simpler than the typical DSGE model.
The table below summarizes the relationship between the measure of the equilibrium interest rate developed in this paper and other approaches. To reiterate, the expressions "shortterm" and "longterm" refer to the maturity of the debt instrument: The focus of much of the literature has been on the estimation of neutral values of the federal funds rate, a shortterm interest rate, whereas the focus here will be on the neutral value of the tenyear Treasury yield, a longterm interest rate. The expressions "shortrun" and "longrun" refer to the horizon over which the neutral rate stabilizes output. The DSGE literature has emphasized periodbyperiod stabilization and thus is focused on shortrun stabilization whereas the interest of Laubach and Williams and subsequent papers has been on the value of the interest rate that would eventually stabilize the economy, and thus is focused on longrun stabilization.^{1}
Relationship among estimates of the neutral interest rate
Shortterm (e.g., Federal funds rate) 
Longterm (e.g., Tenyear Treasury) 


Shortrun (stabilizes period by period)  DSGE models  This paper (main focus) 
Longrun  Laubach and Williams (2003)  This paper (final section); Del Negro et al. (2017) 
The Model
The point of departure for the analysis is the textbook IS curve, which states that, other things equal, higher longterm interest rates should imply weaker economic activity. A simple model that captures this idea is:
where xgap is the output gap, R_{t} is the real longterm interest rate, and R_{t}^{n} is the neutral rate of interest. Equation 1 says that when the longterm interest rate rises, the output gap will fall. The equation allows for some lag in the effect of interest rates on the output gap. The neutral rate of interest is one type of equilibrium interest rate; it is defined to be the value of R_{t} that will lead to GDP growing at potential (and thus to no change in the output gap) provided the output gap is initially zero. An implication of equation 1 is that when (real longterm) interest rates are higher than their neutral level, the output gap will be negative.
Although equation 1 is much simpler than the typical empirical DSGE model or the Board staff's FRB/US model, it is broadly consistent with their properties. Indeed, an equation very much like equation 1 governs consumer spending in most DSGE models.^{2} In particular, equation 1 has only two parameters, one reflecting the sensitivity of the output gap to interest rates and the other, the speed at which interestrate changes are reflected in spending.
To derive an estimate of the neutral rate of interest, Equation 1 can be rearranged as,
To use equation 2 to infer the neutral rate of interest, the requirements are: (a) data on the output gap and the real longterm interest rate and (b) assumptions about the two model parameters. These choices are discussed in the next section.
Data and Calibration
For most of the analysis, the output gap will be the measure that is part of the Federal Reserve's FRB/US model.^{3} An advantage of the FRB/US output gap is that it attempts to abstract from measurement error. The CBO's output gap measure will be considered as an alternative. The longterm real interest rate will be the difference between the tenyear Treasury yield and an estimate of longterm inflation expectations.^{4}
The parameters of Equation 1 are chosen to match the properties of some common macroeconomic models. The persistence parameter, η, is set equal to 0.75. This value is consistent with many estimates from the DSGE literature. Several values of σ are considered, informed by the effects of interest rates on output in a variety of macroeconomic models. In particular, Chung (2015) compares the properties of three macroeconomic models, two from the Board of Governors staffthe FRB/US and EDO modelsplus the wellknown SmetsWouters (2007) model. As Figure 1 of Chung (2015) shows, DSGE models such as EDO and SmetsWouters are considerably more interest sensitive than is FRB/US, with the peak GDP effect of an interestrate shock about two to twoandahalf times bigger in the DSGE models, suggesting it will be worthwhile to consider a range of values for σ.^{5} With η = 0.75 and σ = 3.5, the model comes close to matching the impulse responses from the two DSGE models. The lower interest sensitivity of the FRB/US model is captured with a value of σ equal to 0.75.^{6}
Results
Figure 1 shows the estimate with the FRB/US output gap, η = 0.75, and σ = 0.75. Variation in the shortrun neutral rate is fairly wide, suggesting that stabilization of the output gap would require considerable adjustment in longterm interest rates. For example, over the period 1990 to 2007a period of relatively stable macroeconomic conditionsthe shortrun neutral real tenyear Treasury yield varied between 1 percent and 6 percent, reaching its peak value during the hightech boom around 19992000 and its lowest values in the aftermath of the recessions of the early 1990s and 2001.
According to this model, the neutral tenyear Treasury yield just prior to the 200809 financial crisis was around 3 1/2 percent in real terms. During the crisis, the neutral tenyear Treasury yield dropped as low as 3 percent, in early 2009. By 2010, R_{t}^{n} had rebounded to around 1/2 percent. Longterm inflation expectations remained stable at around 2 percent during the crisis period, so a real interest rate of 1/2 percent would have been feasible, for example, if the nominal tenyear Treasury yield had dropped to 1 1/2 percent, which could potentially have been achieved through sufficient asset purchases and forward guidance about future shortterm interest rates. In 2012, near the peak of the euro crisis, R_{t}^{n} dipped again, this time to around 1 3/4 percent. Again, such a real tenyear Treasury yield is potentially feasible, because it is lower, in absolute value, than longerterm inflation expectations. By late 2013, R_{t}^{n} had recovered, and it remained between 1/2 and 1 percent through the end of 2014.
In the couple of years at the end of the sample, the estimate of R_{t}^{n} has moved up, from around 0 percent in 2016, to just below 1 percent in the last period shown, 2017:Q4. While 1 percent matches the high point since the financial crisis, it is still well below the values that prevailed in the years before the crisis. Of course, the estimates of the neutral rate are only as good as the data underlying them, and this estimate of the output gap, as are all others, is estimated with considerable uncertainty, especially at the end of the sample.
Figure 2 compares the baseline estimate of R_{t}^{n} with an alternative based on the CBO output gap. The broad contour is similar. In particular, the neutral rate dropped sharply during the crisis and remained depressed relative to precrisis levels for many years thereafter. At the end of the sample, there is also a recovery, although using the CBO output gap, R_{t}^{n} is only about 1/2 percent in 2017:Q4.^{7}
Figure 3 compares the baseline estimate of R_{t}^{n}, which uses the FRB/US output gap, η = 0.75, and σ = 0.75 with an alternative that assumes η = 0.75, and σ = 3.5. As discussed earlier, with this setting of σ, the interest sensitivity of output is closer to that of DSGE models. Under this alternative, the decline in the real tenyear Treasury yield needed to stabilize the output gap during the financial crisis was considerably smaller than in the baseline case: A real tenyear Treasury yield of about 0 percent would have been sufficient to stabilize the output gap in early 2009, compared with 3 percent under the baseline parameter setting. Under the alternative, the trough of R_{t}^{n} occurred during the euro crisis, in 2012, when R_{t}^{n} fell as low as 3/4 percent. Near the end of the sample, both measures show a recovery in R_{t}^{n}, although the alternative reaches a somewhat smaller value, of around 1/2 percent.
There is considerable quartertoquarter variation in the estimates of R_{t}^{n}. One possible source of highfrequency variation is measurement error in the output gap. That possibility is minimized in the preferred estimates, however, because the FRB/US output gap excludes an estimate of measurement error. Another possibility is that the dynamics of the economy are more complex than in equation 1, and this specification error could be reflected in excess highfrequency movements in R_{t}^{n}.
To smooth through highfrequency variation, Figure 4 plots the fourquarter moving average of R_{t}^{n}. The same broad pattern emerges with, notably, the increase near the end of the sample marking the third period of upward movement since the financial crisis.
The longrun, longterm neutral rate
As has been discussed in a variety of papers, including Kiley (2015), Holston, Laubach, and Williams (2017), and Del Negro et al (2017), the extended period of very low interest rates has raised the prospect that the longrun value of R_{t}^{n} may have fallen. The estimates of R_{t}^{n} presented so far have made no assumption about the persistence of the movements in the neutral rate of interest; it is simply the result of applying Equation 2 to the data. Laubach and Williams (2003), and subsequent papers, use the Kalman filter to extract a persistent component of the equilibrium interest rate.
To derive an estimate of the longrun component of the longterm neutral interest rate, we can assume the following statespace model of R_{t}^{n}:
R_{t}^{n} = R_{t}^{n} + cyc_{t} ,
R_{t}^{n} = R_{t1}^{n} + ε_{t} ,
cyc_{t} = α cyc_{t1} + v_{t}
As in Laubach and Williams (2003) and other papers in this literature, the longrun component of R_{t}^{n}, R_{t}^{n}, follows a random walk. The stationary component, cyc_{t}, follows an AR(1) process. The residual error terms ε_{t} and v_{t} are assumed to follow white noise error processes.
Figure 5 plots the baseline estimate of R_{t}^{n} along with a model estimate of its longrun component.^{8} In the first twenty years of the sample, the model estimate of the longrun neutral real tenyear Treasury yield fluctuated in a narrow range around 3 3/4 percent. It edged down slightly over the 1990s and more steeply from 1999 to 2011. Over the 2012to2017 period, the longrun neutral rate has averaged 3/4 percent, with little variation. This estimate of the longrun neutral interest rate is thus broadly consistent with the results of Holston, Laubach, and Williams (2017), Kiley (2015), and Del Marco et al. (2017), who also find that the longrun neutral interest rate has been lower in recent years than in the decades before the financial crisis.
The gap between the shortrun estimate of R_{t}^{n} and its longrun value, R_{t}^{n}, can be taken as a measure of the relative aggregate demand facing the economy: When R_{t}^{n} < R_{t}^{n}, the level of the real tenyear Treasury yield needed to maintain the output gap at zero is below its longrun value. Such a situation can be characterized as the economy facing "headwinds."^{9} According to the estimates shown in Figure 5, during the financial crisis, in 200809, considerable headwinds developed, as R_{t}^{n} fell well below R_{t}^{n}. As emphasized by Yellen (2015), these headwinds proved to be persistent, and R_{t}^{n} remained below R_{t}^{n} for many years. R_{t}^{n} briefly exceed R_{t}^{n} in 2014 but fell back in 201516. At the end of the sample, R_{t}^{n} had once again reached the model estimate of its longrun value. If R_{t}^{n} were to rise further, so that it exceeded R_{t}^{n} on a sustained basis, the economy could be said to be benefitting from "tailwinds."^{10} Of course, as with any econometric estimate, there is considerable uncertainty around the specific results.
References
Barsky, Robert, Alejandro Justiniano, and Leonardo Melosi (2014). "The Natural Rate of Interest and Its Usefulness for Monetary Policy," American Economic Review 104(5), pp. 3743.
Board of Governors (2017). FRB/US Model Packages, https://www.federalreserve.gov/econres/usmodelspackage.htm.
Brainard, Lael (2018). "Navigating Monetary Policy as Headwinds Shift to Tailwinds," speech presented at the Money Marketeers of New York University, New York, New York, March 6, https://www.federalreserve.gov/newsevents/speech/brainard20180306a.htm.
Congressional Budget Office (CBO, 2018). Budget and Economic Outlook: 2018 to 2028, April, https://www.cbo.gov/publication/53651.
Chung, Hess (2015). "The Effects of Forward Guidance in Three Macro Models, FEDS Note 20150226, https://doi.org/10.17016/23807172.1488
Curdia, Vasco, Andrea Ferrero, Ging Cee Ng, and Andrea Tambalotti (2015). "Has U.S. Monetary Policy Tracked the Efficient Interest Rate?" Journal of Monetary Economics 70, pp. 7283.
Del Negro, Marco, Domenico Giannone, Marc P. Giannoni, and Andrea Tambalotti (2017). "Safety, liquidity, and the natural rate of interest," Brookings Papers on Economic Activity, Spring 2017, pp. 235294, https://www.brookings.edu/wpcontent/uploads/2017/08/delnegrotextsp17bpea.pdf.
Edge, Rochelle M., Michael T. Kiley, and JeanPhilippe Laforte (2008). "Natural Rate Measures in an Estimated DSGE Model of the U.S. Economy," Journal of Economic Dynamics and Control 32, pp. 25122535.
Fleischman, Charles, and John M. Roberts (2011). "From Many Series, One Cycle: Improved Estimates of the Business Cycle from a Multivariate Unobserved Components Model, Finance and Economics Discussion Series 201146, Board of Governors of the Federal Reserve System, https://www.federalreserve.gov/pubs/feds/2011/201146/201146abs.html.
Holston, Kathryn, Thomas Laubach, and John C. Williams (2017). "Measuring the Natural Rate of Interest: International Trends and Determinants," Journal of International Economics 108, May 2017, S59S75, https://www.frbsf.org/economicresearch/publications/workingpapers/2016/wp201611.pdf
Kiley, Michael T. (2015). "What Can the Data Tell Us About the Equilibrium Real Interest Rate?" Finance and Economics Discussion Series 2015077. Board of Governors of the Federal Reserve System (U.S.).
Laubach, Thomas, and John C. Williams (2003). "Measuring the Natural Rate of Interest," Review of Economics and Statistics, 85(4), November, pp. 10631070.
Powell, Jay (2018). "Statement on the Semiannual Monetary Policy Report to the Congress," testimony before the Committee on Financial Services, U.S. House of Representatives, Washington, D.C., February 27, https://www.federalreserve.gov/newsevents/testimony/powell20180226a.htm.
Roberts, John M. (2014). "Estimates of Latent Variables for the FRB/US Model," available at the Federal Reserve Board's Public FRB/US website, https://www.federalreserve.gov/econres/usmodelspackage.htm.
Smets, Frank, and Rafael Wouters (2007). "Shocks and Frictions in U.S. Business Cycles: A Bayesian DSGE Approach," American Economic Review 97, pp. 586606, https://www.aeaweb.org/articles?id=10.1257/aer.97.3.586.
Woodford, Michael (2003). Interest and Prices, Princeton University Press.
Yellen, Janet (2015). "Normalizing Monetary Policy: Prospects and Perspectives," speech presented at "The New Normal Monetary Policy," a research conference sponsored by the Federal Reserve Bank of San Francisco, San Francisco, California, March 27, https://www.federalreserve.gov/newsevents/speech/yellen20150327a.htm.
1. Using reducedform methods similar to those of Laubach and Williams (2003), Del Negro et al. (2017) derive an estimate of the longrun, longterm neutral interest rate. Return to text
2. The textbook consumption Euler equation with habits can be written as:
(a) x_{t} − η x_{t1} = E_{t} (x_{t+1} − η x_{t} ) + ς (r_{t}^{n} − r_{t} ),
where x_{t} is consumption and r_{t} is the shortterm real interest rate. This equation can be solved forward to yield:
(b) x_{t} = η x_{t1} − ς (R_{t} − R_{t}^{n} ),
where R_{t} is the longterm real interest rate. In the empirical implementation, we will use the real tenyear Treasury yield as the proxy for the longterm real interest ratea valid assumption if shortterm interest rates are close to their longrun values after ten years. In that case, σ = 10ς, assuming that the tenyear Treasury is a moving average of expected future short rates over the next 40 quarters, and adjusting for the fact that the shortterm interest rate in the theoretical model is typically not annualized. Return to text
3. The model underlying the FRB/US output gap is described in Roberts (2014). See also Fleischman and Roberts (2011). Return to text
4. The estimate of longterm inflation expectations is also taken from the FRB/US model's database (mnemonic PTR). Starting in the late 1970s, PTR is based on surveys of professional forecasters. Before that, it is based on a simple timeseries model. See the online documentation of the FRB/US model (Board of Governors, 2017) for further details. Return to text
5. The value of η needed to match FRB/US impulse responses in Chung (2015) is larger than thisperhaps 0.9 or higher. However, in many of its spending equations, FRB/US forces lagged adjustment onto "intrinsic" sources of persistence, such as habit persistence, and does not allow for "extrinsic" persistence, such as persistent shocks to R_{t}^{n}. Because FRB/US estimation does not allow for many of the channels of extrinsic persistence that are present in DSGE models, it is likely that the intrinsic persistence in FRB/US is too high. Return to text
6. The parameterizations were chosen to match the responses to a monetarypolicy shock in a simple New Keynesian DSGE model comprising Equation 1; the inertial Taylor rule used in Chung (2015); an assumption that shortterm and longterm interest rates are linked via the expectations hypothesis; and a New Keynesian Phillips curve. Return to text
7. The estimate of the CBO output gap is based on the CBO estimate of potential output published April 2018, with the GDP estimates from the BEA as the numerator. Return to text
8. In the estimated model underlying this estimate, the standard deviation of ε_{t} is 0.7; α is 0.82; and the standard deviation of v_{t} is 0.85. The value of the standard deviation of ε_{t} is constrained; the other two parameters are statistically significant at the 1 percent level. The estimation sample period is 1970 to 2017; the model was estimated using statespace methods under maximum likelihood. Return to text
9. See Yellen (2015), Powell (2018), and Brainard (2018) for examples of the use of the term "headwinds" to characterize adverse aggregate demand conditions. Return to text
10. See Powell (2018) and Brainard (2018). Return to text
Roberts, John M. (2018). "An Estimate of the LongTerm Neutral Rate of Interest," FEDS Notes. Washington: Board of Governors of the Federal Reserve System, September 5, 2018, https://doi.org/10.17016/23807172.2227.
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