Estimating the Interest Rate Trend in a Shadow Rate Term Structure Model

Wed, 01 Jan 2025 00:00:00 +0000

This paper proposes a shadow rate no-arbitrage dynamic term structure model (SDTSM) with drifting trends to estimate the long-run trend of the real interest rate using yield curve data from the U.S., U.K., and Germany from January 1972 to April/March 2022. The model combines the shadow rate approach of Wu and Xia (2016) to handle the zero lower bound with the shifting endpoint of Bauer and Rudebusch (2020) to capture low-frequency movements. Interest rate trends in all three countries have declined since the 1990s, with strong co-movement among them. The model provides better yield forecasts than existing models. Term premium estimates from the model are stationary and positively correlated with inflation uncertainty measures, corroborating Wright (2011). Under the convention that all permanent shocks to real interest rates are derived from real shocks, the model’s trend estimate also serves as a measure of the natural rate of real interest.

Summary based on a working paper version, AI-assisted and human-reviewed. See the linked published article for the authoritative version.

In depth

Q1. What are the two key modeling innovations?

The model combines two innovations: (1) a shadow rate approach following Wu and Xia (2016) to handle the zero lower bound (ZLB)—defining the policy rate as max(shadow rate, lower bound) so that the model remains valid when rates are near zero; and (2) a drifting trend (shifting endpoint) following Bauer and Rudebusch (2020) to capture the slow downward movement of the interest rate trend since the 1990s. Combining these two features is the paper’s key contribution: existing shadow rate models (Wu-Xia) do not model the low-frequency trend; existing shifting-endpoint models (Bauer-Rudebusch) do not account for the ZLB. The combination produces better-identified trend estimates because the shadow rate summarizes financial conditions including the effects of unconventional monetary policy.

Q2. Why use the full yield curve rather than a few selected maturities?

Using the full yield curve with no-arbitrage restrictions allows the model to exploit all information in the Treasury bond market and impose internally consistent restrictions on how maturities are related, improving estimation efficiency relative to models that select a few yields and do not impose no-arbitrage restrictions (e.g., Del Negro et al. 2017; Johannsen and Mertens 2021). The failure of the pure expectations hypothesis implies that a model handling term premiums coherently and flexibly is necessary to correctly extract interest rate trends from long-term yields; the no-arbitrage DTSM provides this structure while also being free of the liquidity premium complications in TIPS-based models.

Q3. What are the main empirical findings about the interest rate trend?

Interest rate trends in the U.S., U.K., and Germany have all declined since the 1990s, with strong co-movement among them; under the convention that all permanent shocks to real interest rates are derived from real shocks, the paper’s trend estimate can be interpreted as a trend estimate of the natural rate of real interest. The strong international co-movement is consistent with global factors—such as declining trend output growth, rising savings, and global safe asset demand—driving the secular decline in real interest rates rather than purely country-specific factors.

Q4. What is the relationship between term premiums and inflation uncertainty?

Term premium estimates from the model are stationary (rather than trending downward as in some models where the trend and the term premium are not well separated) and are positively correlated with inflation uncertainty measures, corroborating Wright (2011)’s finding that term premiums are driven partly by inflation risk. The stationarity of term premiums is a desirable property that results from properly separating the trend component (modeled via the shifting endpoint) from the cyclical component; models that do not include a shifting endpoint may attribute some of the trend to the term premium, producing non-stationary term premium estimates.

Key concepts

shadow rate dynamic term structure model (SDTSM) : a term structure model in which the policy rate is defined as the maximum of a latent shadow rate and the effective lower bound, following Wu and Xia (2016); allows the model to be estimated without modification when short-term rates are near zero. drifting trend (shifting endpoint) : a slow-moving unconditional mean of interest rates that evolves over time, following Bauer and Rudebusch (2020); captures the secular decline in interest rates since the 1990s and separates trend from cyclical variation and term premiums. natural rate of real interest : the long-run equilibrium real interest rate consistent with stable inflation and output at potential; under the assumption that all permanent shocks to real rates are real shocks, the paper’s trend estimate provides a measure of this rate. Beveridge-Nelson trend : the long-run forecast of the shadow rate derived from the model; used here as the operational definition of the interest rate trend; transforms the information in the entire yield curve into a single macroeconomic equilibrium measure.