## 2. Correlation between Yield Curve Factors and Macroeconomic Variables

## Source details

**Canonical URL:** [2. Correlation between Yield Curve Factors and Macroeconomic Variables](https://www.imf.org/-/media/files/publications/wp/2018/wp18140.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2018/wp18140.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2018/wp18140.pdf.json)

---

### Introduction and motivation
- Term premium defined: TP_t(τ) = y_t(τ) − (1/τ) ∫_{t}^{t+τ} E_t^P(r^f_s) ds.
- Recent behavior:
  - Term premium described as "relatively small or even negative" in recent years.
  - Some approaches give a negative term premium during 2011-2014, while another suggests a positive risk premium (of up to 175 basis points).
- Policy relevance:
  - Yield curve informs market expectations of future monetary policy and tests central bank communication including forward guidance.
  - Risk: further monetary tightening could compress short- vs long-term spreads and invert the yield curve, a reliable indicator of recessions in the United States.
- Modeling challenge:
  - Both expected future short-term rates and term premium are unobservable; different modeling choices yield wide range of term premium estimates.
  - Finance literature commonly uses affine Gaussian term structure models; growing practice to add macroeconomic variables and survey data.

### Data and empirical facts
- Sample and maturities:
  - Sample covers June 1962 until March 2018.
  - Maturities used (months): 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and 120.
  - Yields are month-end data; unemployment and core PCE inflation matched from previous month; Federal Reserve long-run inflation expectations observed in month of survey.
- Macro variables:
  - Unemployment gap = U3 unemployment less CBO NAIRU.
  - Inflation gap = core PCE inflation minus long-run inflation expectations.
- Selected descriptive statistics (from Table 1; maturities in months):
  - Level (120): Mean 6.26, St. dev. 2.73, Min 1.50, Max 14.89, AC(1) 0.99, AC(12) 0.90, AC(36) 0.75.
  - Slope: Mean -1.33, St. dev. 1.47, Min -4.32, Max 4.02, AC(1) 0.93, AC(12) 0.51, AC(36) -0.15.
  - Curvature: Mean -0.13, St. dev. 0.88, Min -3.34, Max 3.16, AC(1) 0.85, AC(12) 0.47, AC(36) 0.27.
  - Unemployment gap: Mean 0.50, St. dev. 1.56, Min -2.44, Max 4.98, AC(1) 0.99, AC(12) 0.77, AC(36) 0.22.
  - Inflation: Mean 3.31, St. dev. 2.18, Min 0.95, Max 10.23, AC(1) 1.00, AC(12) 0.87, AC(36) 0.66.
- Empirical correlations (Table 2):
  - Level vs Level: 1
  - Level vs Slope: 0.170
  - Level vs Curvature: 0.485
  - Level vs Unemployment gap: 0.038
  - Level vs Inflation: 0.741
  - Slope vs Slope: 1
  - Slope vs Curvature: -0.166
  - Slope vs Unemployment gap: -0.583
  - Slope vs Inflation: 0.345
  - Curvature vs Curvature: 1
  - Curvature vs Unemployment gap: -0.123
  - Curvature vs Inflation: 0.260
  - Unemployment gap vs Unemployment gap: 1
  - Unemployment gap vs Inflation: 0.029
  - Inflation vs Inflation: 1
- Time-series properties:
  - Dickey-Fuller and Phillips-Peron tests: inflation and level of yield curve exhibit unit roots; slope and curvature and unemployment rate appear mean reverting.
  - Engel-Granger stationarity test rejects that level of yields minus inflation is stationary; Johansen test indicates two independent unit roots drive series.

### Model structure and identification
- Modeling strategy:
  - State space model estimated with Kalman Filter.
  - Uses Nelson-Siegel-Svensson curve fitted to off-the-run U.S. Treasury bonds (GSW dataset).
  - Splits level term L_t into long-run trend L_t^T and cyclical component L_t^C.
  - Transition dynamics composed of two blocks:
    - Cyclical block: stationary series including L_t^C (cyclical level), slope, curvature, and cyclical components of inflation and unemployment.
    - Trend block: unit-root processes for long-run expected inflation and long-run real yield (two unit roots).
  - Imposes shrinkage and parsimony to improve robustness and reduce overfitting.
  - Diagonal covariance matrix for state innovations to prevent direct instantaneous interactions between cyclical and trend blocks.
- Observation and measurement choices:
  - Observed unemployment and inflation each decomposed into cyclical component + trend + observation error with standard deviation of 2 basis points.
  - Macro trends proxies:
    - NAIRU series from CBO and long-run inflation expectations from the Fed (PTR series from FRB/US staff).
    - Measurement error fixed at 10 basis points for NAIRU and 25 basis points for inflation expectations.
  - The model relaxes assumption that macro variables are observed without error; unemployment and inflation lagged by 1 month to mimic real-time information and allow conservative uncertainty.
- Parameter notes and identification constraints:
  - Estimated parameter controlling curvature peak loading: 0.0423 (from Kalman Filter estimation in standard model).
  - Trend equations (examples):
    - L_t^T = Π_t^* + Ψ_t^T
    - Π_t^* = Π_{t−1}^* + ε_t
    - Ψ_t^T = Ψ_{t−1}^T + ε_t
  - NAIRU modeled as unit root: NAIRU_t = NAIRU_{t−1} + ε_t.
  - L_t^C assumed solely from observed interest rates.
  - Means of L_t^C, U_t^C, and Π_t^C constrained to zero to aid long-run properties.

### Empirical findings and term premium estimates
- Long-run trend vs cyclical components:
  - Allowing trend level of interest rates to vary over time produces more stable long-run expectations.
  - Temporary (cyclical) component to level of yields influences yields but overall level is a martingale.
- Comparison with other methods:
  - Four methodologies’ 10-year term premia compared: ACM (authors’ model), Kim-Wright, Christensen–Rudebusch, Kopp-Williams.
  - In early phase of current expansion, most approaches estimate 10-year term premia around 2 percent and then compress; authors’ estimate is less prone to short-lived spikes and reacts over cycles as expected.
- Average term premia across maturities:
  - Average term premia display a concave shape from 1 to 10 years.
  - Implied ex ante Sharpe ratios decline as maturities increase (implied Sharpe ratio calculated as avg. term premium divided by product of st.dev. of changes in term premium and bond duration).
- Model behavior:
  - Medium-term dynamics in this macro-focused model show risk-free rates often over- or undershooting long-run values (driven by unemployment and inflation gaps), unlike other models where risk-neutral yields asymptotically approach long-horizon forward.
- Robustness:
  - Short-run term premium forecasts from the model are very similar to those estimated from models targeting high short-term precision.
  - Long-run risk-free rate estimates are smoother and more stable than other approaches.

### Macroeconomic implications and projections (conditional on information as of end-2017)
- Survey evidence:
  - Survey of Professional Forecasters and Fed/SMP/PDS surveys indicate slow revision of long-run interest rate expectations after the crisis; markets slow to revise equilibrium fed funds rate.
- Model projections (conditional on end-2017 information):
  - Unemployment rate projected to drop to around 3.5 percent by 2020, then start rising again.
  - Risk-free rate projected to approach 3 percent by 2021.
  - 10-year rate projected to reach 3.8 percent by 2020-2021 (noted as in line with IMF WEO projections for the United States).
- Interpretation:
  - Including macroeconomic trends and cyclical gaps supports more realistic medium- to long-run forecasts where structural expectations adjust slowly.
  - Medium-term forecasts are driven importantly by interactions of unemployment and inflation gaps with trend dynamics.

### Conclusions and policy-relevant takeaways
- A macroeconomically informed state-space term structure model:
  - Exploits both cross-sectional and temporal information in the term structure.
  - Adds cyclical dynamics and separable trend components to support realistic medium- and long-run forecasting.
  - Produces smoother and more stable estimates of expected long-term risk-free rates, consistent with slow-moving structural beliefs (inflation expectations, potential growth).
- Practical implications:
  - Model suggests that term premia estimates broadly align with other studies while delivering more stable long-run interest rate forecasts—a key input for macroeconomic forecasting and policy-making.
  - Forecasts of short- and long-term interest rates and cyclical gaps are close to those from large-scale macroeconomic models used by CBO, IMF, and private forecasters.

*Source: IMF Working Paper "2. Correlation between Yield Curve Factors and Macroeconomic Variables" (wp18140), content from the supplied PDF chapter.*

### REFERENCES _____________________________________________________________________________________ 20

### REFERENCES _____________________________________________________________________________________ 20

### FIGURES

- 1. Term Structure of Interest Rates ____________________________________________________ 4
- 2. Term Premium Estimates ____________________________________________________________ 5
- 3. Unit Roots and Mean-Reverting Variables ________________________________________ 10
- 4. Macroeconomic Outputs __________________________________________________________ 12
- 5. Trends in Interest Rates ___________________________________________________________ 14
- 6. Cyclical components of interest rates _____________________________________________ 14
- 7. Comparison of 10-year Term Premium Estimates _________________________________ 16
- 8. Average Term Premia _____________________________________________________________ 16
- 9. Survey-based Long Horizon Interest Rate Forecasts ______________________________ 17
- 10. Expected Risk-Free Rates in the Longer-Term ___________________________________ 18
- 11. Survey-Based Long Horizon Interest Rate Forecasts _____________________________ 19

### TABLES

- 1: Destriptive Statistics of the Input Data ______________________________________________ 8

*wp18140 - REFERENCES _____________________________________________________________________________________ 20 — https://www.imf.org/-/media/files/publications/wp/2018/wp18140.pdf*

### 2. Correlation between Yield Curve Factors and Macroeconomic Variables ___________ 9

### 2. Correlation between Yield Curve Factors and Macroeconomic Variables

### Introduction and motivation
- Term premium defined: TP_t(τ) = y_t(τ) − (1/τ) ∫_{t}^{t+τ} E_t^P(r^f_s) ds.
- Recent behavior:
  - Term premium "relatively small or even negative" in recent years.
  - Some approaches give a negative term premium during 2011-2014, while another suggests a positive risk premium (of up to 175 basis points).
- Policy relevance:
  - Yield curve informs market expectations of future monetary policy and tests central bank communication including forward guidance.
  - Risk: further monetary tightening could compress short- vs long-term spreads and invert the yield curve, a reliable indicator of recessions in the United States.
- Modeling challenge:
  - Both expected future short-term rates and term premium are unobservable; different modeling choices yield wide range of term premium estimates.
  - Finance literature commonly uses affine Gaussian term structure models; growing practice to add macroeconomic variables and survey data.

### Data and empirical facts
- Data sample and maturities:
  - Sample covers June 1962 until March 2018.
  - Maturities used (months): 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and 120.
  - Yields are month-end data; unemployment and core PCE inflation matched from previous month; Federal Reserve long-run inflation expectations observed in month of survey.
- Variables:
  - Macro variables include unemployment gap (U3 unemployment less CBO NAIRU) and inflation gap (core PCE inflation minus long-run inflation expectations).
- Descriptive statistics (selected from Table 1; maturities in months):
  - Level (120): Mean 6.26, St. dev. 2.73, Min 1.50, Max 14.89, AC(1) 0.99, AC(12) 0.90, AC(36) 0.75.
  - Slope: Mean -1.33, St. dev. 1.47, Min -4.32, Max 4.02, AC(1) 0.93, AC(12) 0.51, AC(36) -0.15.
  - Curvature: Mean -0.13, St. dev. 0.88, Min -3.34, Max 3.16, AC(1) 0.85, AC(12) 0.47, AC(36) 0.27.
  - Unemployment gap: Mean 0.50, St. dev. 1.56, Min -2.44, Max 4.98, AC(1) 0.99, AC(12) 0.77, AC(36) 0.22.
  - Inflation: Mean 3.31, St. dev. 2.18, Min 0.95, Max 10.23, AC(1) 1.00, AC(12) 0.87, AC(36) 0.66.
- Empirical correlations (Table 2):
  - Level vs Level: 1
  - Level vs Slope: 0.170
  - Level vs Curvature: 0.485
  - Level vs Unemployment gap: 0.038
  - Level vs Inflation: 0.741
  - Slope vs Slope: 1
  - Slope vs Curvature: -0.166
  - Slope vs Unemployment gap: -0.583
  - Slope vs Inflation: 0.345
  - Curvature vs Curvature: 1
  - Curvature vs Unemployment gap: -0.123
  - Curvature vs Inflation: 0.260
  - Unemployment gap vs Unemployment gap: 1
  - Unemployment gap vs Inflation: 0.029
  - Inflation vs Inflation: 1
- Time-series properties:
  - Dickey-Fuller and Phillips-Peron tests: inflation and level of yield curve exhibit unit roots; slope and curvature and unemployment rate appear mean reverting.
  - Engel-Granger stationarity test rejects that level of yields minus inflation is stationary; Johansen test indicates two independent unit roots drive series.

### Model structure and identification
- Modeling strategy:
  - State space model estimated with Kalman Filter.
  - Uses Nelson-Siegel-Svensson curve fitted to off-the-run U.S. Treasury bonds (GSW dataset).
  - Splits level term L_t into long-run trend L_t^T and cyclical component L_t^C.
  - Transition dynamics composed of two blocks:
    - Cyclical block: stationary series including L_t^C (cyclical level), slope, curvature, and cyclical components of inflation and unemployment.
    - Trend block: unit-root processes for long-run expected inflation and long-run real yield (two unit roots).
  - Imposes shrinkage and parsimony to improve robustness and reduce overfitting.
  - Diagonal covariance matrix for state innovations to prevent direct instantaneous interactions between cyclical and trend blocks.
- Observation and measurement choices:
  - Observed unemployment and inflation each decomposed into cyclical component + trend + observation error with standard deviation of 2 basis points.
  - Macro trends proxies:
    - NAIRU series from CBO and long-run inflation expectations from the Fed (PTR series from FRB/US staff).
    - Measurement error fixed at 10 basis points for NAIRU and 25 basis points for inflation expectations.
  - The model relaxes assumption that macro variables are observed without error; unemployment and inflation lagged by 1 month to mimic real-time information and allow conservative uncertainty.
- Parameter notes:
  - Estimated parameter controlling curvature peak loading: 0.0423 (from Kalman Filter estimation in standard model).
  - Trend equations (examples):
    - L_t^T = Π_t^* + Ψ_t^T
    - Π_t^* = Π_{t−1}^* + ε_t
    - Ψ_t^T = Ψ_{t−1}^T + ε_t
  - NAIRU modeled as unit root: NAIRU_t = NAIRU_{t−1} + ε_t.
- Identification constraints:
  - L_t^C assumed solely from observed interest rates.
  - Means of L_t^C, U_t^C, and Π_t^C constrained to zero to aid long-run properties.

### Empirical findings and term premium estimates
- Long-run trend vs cyclical components:
  - Allowing trend level of interest rates to vary over time produces more stable long-run expectations.
  - Temporary (cyclical) component to level of yields influences yields but overall level is a martingale.
- Comparison with other methods:
  - Figure 7 (described) shows four methodologies’ 10-year term premia: ACM (authors’ model), Kim-Wright, Christensen–Rudebusch, Kopp-Williams.
  - In early phase of current expansion, most approaches estimate 10-year term premia around 2 percent and then compress; authors’ estimate is less prone to short-lived spikes and reacts over cycles as expected.
- Average term premia across maturities:
  - Average term premia display a concave shape from 1 to 10 years.
  - Implied ex ante Sharpe ratios decline as maturities increase (implied Sharpe ratio calculated as avg. term premium divided by product of st.dev. of changes in term premium and bond duration).
- Model behavior:
  - Medium-term dynamics in this macro-focused model show risk-free rates often over- or undershooting long-run values (driven by unemployment and inflation gaps), unlike other models where risk-neutral yields asymptotically approach long-horizon forward.
- Robustness:
  - Short-run term premium forecasts from the model are very similar to those estimated from models targeting high short-term precision.
  - Long-run risk-free rate estimates are smoother and more stable than other approaches.

### Macroeconomic implications and projections (conditional on information as of end-2017)
- Survey evidence:
  - Survey of Professional Forecasters and Fed/SMP/PDS surveys indicate slow revision of long-run interest rate expectations after the crisis; markets slow to revise equilibrium fed funds rate.
- Model projections (conditional on end-2017 information):
  - Unemployment rate projected to drop to around 3.5 percent by 2020, then start rising again.
  - Risk-free rate projected to approach 3 percent by 2021.
  - 10-year rate projected to reach 3.8 percent by 2020-2021 (noted as in line with IMF WEO projections for the United States).
- Interpretation:
  - Including macroeconomic trends and cyclical gaps supports more realistic medium- to long-run forecasts where structural expectations adjust slowly.
  - Medium-term forecasts are driven importantly by interactions of unemployment and inflation gaps with trend dynamics.

### Conclusions and policy-relevant takeaways
- A macroeconomically informed state-space term structure model:
  - Exploits both cross-sectional and temporal information in the term structure.
  - Adds cyclical dynamics and separable trend components to support realistic medium- and long-run forecasting.
  - Produces smoother and more stable estimates of expected long-term risk-free rates, consistent with slow-moving structural beliefs (inflation expectations, potential growth).
- Practical implications:
  - Model suggests that term premia estimates broadly align with other studies while delivering more stable long-run interest rate forecasts—a key input for macroeconomic forecasting and policy-making.
  - Forecasts of short- and long-term interest rates and cyclical gaps are close to those from large-scale macroeconomic models used by CBO, IMF, and private forecasters.

*Source: IMF Working Paper "2. Correlation between Yield Curve Factors and Macroeconomic Variables" (wp18140), content from the supplied PDF chapter.*

### REFERENCES

### REFERENCES

### References list

- Adrian, Tobias, Richard K. Crump, and Emanuel Moench, “Pricing the Term Structure with Linear Regressions,” Journal of Financial Economics, Vol. 110, No. 1 (October 2013): 110–38.
- Bauer, Michael D., and Thomas Mertens. 2018. “Economic Forecasts with the Yield Curve” FRBSF Economic Letter 2018-07 (March 5).
- Bauer, Michael D., and Glenn D. Rudebusch. 2016. “Resolving the Spanning Puzzle in Macro-Finance Term Structure Models.” Federal Reserve Bank of San Francisco Working Paper 201–01.
- Bauer, Michael D., and Glenn D. Rudebusch. 2017. “Interest Rates Under Falling Stars” Federal Reserve Bank of San Francisco Working Paper 2017–16.
- Christensen, Jens H.E., and Glenn D. Rudebusch. 2012. “The Response of Interest Rates to UNITED STATES and U.K. Quantitative Easing.” Economic Journal, Vol. 122, pp. F385–F414.
- Cieslak, A., and P. Povala. 2011. “Understanding Bond Risk Premia,” Working paper, Kellogg School of Management.
- Cochrane, John. 2011. “Presidential Address: Discount Rates,” The Journal of Finance, Vol. 66: 1047–1108.
- Crump, Richard., Eusepi, Stefano, and Moench, Emanuel. 2018. “The Term Structure of Expectations and Bond Yields,” Federal Reserve Bank of New York Staff Reports, No. 775. May 2016; revised April 2018.
- Diebold, F.X. and Li, C. 2006. “Forecasting the Term Structure of Government Bond Yields,” Journal of Econometrics, 130, 337–64.
- Diebold, F.X., Rudebusch, G.D. and Aruoba, B. 2006. “The Macroeconomy and the Yield Curve: A Dynamic Latent Factor Approach,” Journal of Econometrics, 131, 309–38.
- Duffee, Gregory. 2002. Term Premia and Interest Rate Forecasts in Affine Models. The Journal of Finance, Vol. 57: pp. 405–43.
- Duffee, Gregory. 2013. “Forecasting Interest Rates.” Chapter 7 in Handbook of Economic Forecasting, Vol. 2A, edited by Graham Elliott and Allan Timmermann. Amsterdam: Elsevier, pp. 385–426.
- Faust, Jon and Jonathan Wright. 2013. “Forecasting Inflation” in Handbook of Economic Forecasting (G. Elliott and A. Timmermann (eds.)), Volume 2A, Elsevier.
- GFSR. 2018. “IMF Global Financial Stability Report,” Chapter 1. April 2018. International Monetary Fund, Washington, D.C.
- Gurkaynak, Refet. 2005. “Using Federal Funds Futures Contracts for Monetary Policy Analysis,” FEDS Working Paper, 2005–29.
- Gürkaynak, Refet S., and Jonathan H. Wright. 2012. “Macroeconomics and the Term Structure.” Journal of Economic Literature, 50 (2): 331-67.
- Joslin, Scott, Anh Le, and Kenneth J. Singleton. 2013. “Why Gaussian Macro-Finance Term Structure Models Are (Nearly) Unconstrained Factor-VARs,” Journal of Financial Economics, Vol. 109, pp. 604–622.
- Joslin, Scott, Marcel Priebsch, and Kenneth J. Singleton. 2014. “Risk Premiums in Dynamic Term Structure Models with Unspanned Macro Risks,” Journal of Finance, Vol. 69, pp. 1197–1233.
- Kozicki, Sharon & Tinsley, P. A., 2001. “Term Structure Views of Monetary Policy under Alternative Models of Agent Expectations,” Journal of Economic Dynamics and Control, Vol. 25: pp. 149–184.
- Meyer, Lawrence. 2002. Rules and Discretion, At the Owen Graduate School of Management, Vanderbilt University, Nashville, Tennessee January 16. www.federalreserve.gov/boarddocs/speeches/2002/200201162/default.htm
- Rudebusch, G. R., and T. Wu, 2008, “A Macro-Finance Model of the Term Structure, Monetary Policy and the Economy,” The Economic Journal, Vol. 118, pp. 906-926.
- Swanson, Eric T. 2006. “Have Increases in Federal Reserve Transparency Improved Private Sector Interest Rate Forecasts?” Journal of Money, Credit and Banking, Vol. 38, No. 3, pp. 791–819.
- Wright, Jonathan H. 2011. “Term Premia and Inflation Uncertainty: Empirical Evidence from an International Panel Dataset.” American Economic Review, Vol. 101 No. 4, pp. 1514-34.
- Yellen, Janet. 2016. Macroeconomic Research After the Crisis. at “The Elusive ‘Great’ Recovery: Causes and Implications for Future Business Cycle Dynamics” 60th annual economic conference sponsored by the Federal Reserve Bank of Boston, Boston, Massachusetts. www.federalreserve.gov/newsevents/speech/yellen20161014a.htm

*Source: wp18140 - REFERENCES*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18140.pdf_