(Arbitrage-Free, Practical) Modeling of Term Structures of Government Bond Yields

Background Diebold and Li (2006), Forecasting the Term Structure of Government Bond Yields, J. Econometrics. Diebold, Rudebusch and Aruoba (2006), The Macro-economy and the Yield Curve: A Dynamic Latent Factor Approach, J. Econometrics. Diebold, Li and Yue (2008), Global Yield Curve Dynamics and Interactions: A Generalized Nelson-Siegel Approach, J. Econometrics. Diebold, Piazzesi and Rudebusch (2005), Modeling Bond Yields in Finance and Macroeconomics, American Economic Review. Christensen, Diebold and Rudebusch, G.D. (2009), The Affine Arbitrage- Free Class of Nelson-Siegel Term Structure Models, Econometrics Journal. Christensen, Diebold and Rudebusch, G.D. (2010, revised), An Arbitrage- Free Generalized Nelson-Siegel Term Structure Model, Mimeo.

Definitions and Notation P t (τ) ' e &τ y t (τ) f t (τ) '&P ) t (τ) / P t (τ) y t (τ) ' 1 f τ m t (u)du τ 0

Incompletely-Satisfying Advances in Arbitrage-Free Modeling Cross sectional flavor (e.g., HJM, 1992 Econometrica) Time series flavor (e.g., Vasicek, 1977 JFE)

We Take a Classic Yield Curve Model (Nelson-Siegel) and: Show that it has a modern interpretation Show that its flexible, fits well, and forecasts well Explore a variety of implications and extensions Make it arbitrage-free (yet still tractable)

The Model Fits Well

Factor Loadings in the Nelson-Siegel Model 1 β 1 0.8 Loadings 0.6 0.4 β 2 0.2 β 3 0 0 20 40 60 80 100 120 τ (Maturity, in Months)

Level y t (4) ' β 1t Empirical measure: y t (120) Slope y t (4) & y t (0) '&β 2t Empirical measure: y t (120) & y t (3) Curvature 2y t (mid) & y t (0) & y t (4) 2y t (24) & y t (0) & y t (4) ' 2y t (24) & 2β 1 & β 2..3β 3t 2y t (24) & y t (3) & y t (120) '.00053β 2t %.37β 3t

The Model is Flexible Yield curve facts: (1) Average curve is increasing and concave (2) Many shapes (3) Yield dynamics are persistent (4) Spread dynamics are much less persistent (5) Short rates are more volatile than long rates (6) Long rates are more persistent than short rates

The Model is Easily Fit, and Fits Well OLS with fixed

6.6 Fitted Yield Curve I Yield Curve on 2/28/97 6.4 Yield (Percent) 6.2 6 5.8 5.6 5.4 5.2 5 0 20 40 60 80 100 120 M a turity (M o nths )

The Model Forecasts Well (Dynamic Nelson-Siegel with AR(1) Factors)

Random Walk ŷ t%h/t (τ) ' y t (τ)

Slope Regression ŷ t%h/t (τ)&y t (τ) ' ĉ(τ) % ˆγ(τ)(y t (τ)&y t (3))

Fama-Bliss Forward Regression ŷ t%h/t (τ)&y t (τ) ' ĉ(τ) % ˆγ(τ)(f h t (τ)&y t (τ))

Cochrane-Piazzesi Forward Regression ŷ t%h/t (τ)&y t (τ) ' ĉ(τ) % ˆγ 0 (τ) y t (τ) % ˆγ k (τ) j 9 k'1 f 12k t (12)

AR(1) on Yield Levels ŷ t%h/t (τ) ' ĉ % ˆγy t (τ)

VAR(1) on Yield Levels ŷ t%h/t (τ) ' ĉ % ˆΓy t (τ)

VAR(1) on Yield Changes ẑ t%h/t ' ĉ % ˆΓz t z t / [y t (3)&y t&1 (3), y t (12)&y t&1 (12), y t (36)&y t&1 (36), y t (60)&y t&1 (60), y t (120)&y t&1 (120)] )

ECM(1) with One Common Trend ẑ t%h ' ĉ% ˆΓz t z t / [y t (3)&y t&1 (3) y t (12)&y t (3) y t (36)&y t (3) y t (60)&y t (3) y t (120)&y t (3)] )

ECM(1) with Two Common Trends ẑ t%h/t ' ĉ % ˆΓz t z t / [y t (3)&y t&1 (3) y t (12)&y t&1 (12) y t (36)&y t (3) y t (60)&y t (3) y t (120)&y t (3)] )

ECM(1) with Three Common Trends ẑ t%h ' ĉ% ˆΓz t z t / [y t (3)&y t&1 (3) y t (12)&y t&1 (12) y t (36)&y t&1 (36) y t (60)&y t (3) y t (120)&y t (3)] )

1-Month-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(1) ˆρ(12) 3 months -0.045 0.170 0.176 0.247 0.017 1 year 0.023 0.235 0.236 0.425-0.213 3 years -0.056 0.273 0.279 0.332-0.117 5 years -0.091 0.277 0.292 0.333-0.116 10 years -0.062 0.252 0.260 0.259-0.115 Random Walk Maturity (τ) Mean Std. Dev. RMSE ˆρ(1) ˆρ(12) 3 months 0.033 0.176 0.179 0.220 0.053 1 year 0.021 0.240 0.241 0.340-0.153 3 years 0.007 0.279 0.279 0.341-0.133 5 years -0.003 0.276 0.276 0.275-0.131 10 years -0.011 0.254 0.254 0.215-0.145

6-Month-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(6) ˆρ(18) 3 months 0.083 0.510 0.517 0.301-0.190 1 year 0.131 0.656 0.669 0.168-0.174 3 years -0.052 0.748 0.750 0.049-0.189 5 years -0.173 0.758 0.777 0.069-0.273 10 years -0.251 0.676 0.721 0.058-0.288 Random Walk Maturity (τ) Mean Std. Dev. RMSE ˆρ(6) ˆρ(18) 3 months 0.220 0.564 0.605 0.381-0.214 1 year 0.181 0.758 0.779 0.139-0.150 3 years 0.099 0.873 0.879 0.018-0.211 5 years 0.048 0.860 0.861 0.008-0.249 10 years -0.020 0.758 0.758 0.019-0.271

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 Random Walk Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.416 0.930 1.019-0.118-0.109 1 year 0.388 1.132 1.197-0.268-0.019 3 years 0.236 1.214 1.237-0.419 0.060 5 years 0.130 1.184 1.191-0.481 0.072 10 years -0.033 1.051 1.052-0.508 0.069

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 Slope Regression Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months NA NA NA NA NA 1 year 0.896 1.235 1.526-0.187-0.024 3 years 0.641 1.316 1.464-0.212 0.024 5 years 0.515 1.305 1.403-0.255 0.035 10 years 0.362 1.208 1.261-0.268 0.042

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 Fama-Bliss Forward Regression Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.942 1.010 1.381-0.046-0.096 1 year 0.875 1.276 1.547-0.142-0.039 3 years 0.746 1.378 1.567-0.291 0.035 5 years 0.587 1.363 1.484-0.352 0.040 10 years 0.547 1.198 1.317-0.403 0.062

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 Cochrane-Piazzesi Forward Regression Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months NA NA NA NA NA 1 year -0.162 1.275 1.285-0.179-0.079 3 years -0.377 1.275 1.330-0.274-0.028 5 years -0.529 1.225 1.334-0.301-0.021 10 years -0.760 1.088 1.327-0.307-0.020

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 VAR(1) on Yield Levels Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months -0.276 1.006 1.043-0.219-0.099 1 year -0.390 1.204 1.266-0.322-0.058 3 years -0.467 1.240 1.325-0.345-0.015 5 years -0.540 1.201 1.317-0.348-0.012 10 years -0.744 1.060 1.295-0.328-0.010

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 VAR(1) on Yield Changes Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.717 1.072 1.290-0.068-0.127 1 year 0.704 1.240 1.426-0.223-0.041 3 years 0.627 1.341 1.480-0.399 0.051 5 years 0.559 1.281 1.398-0.459 0.070 10 years 0.408 1.136 1.207-0.491 0.072

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 ECM(1) with one Common Trend Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.738 0.982 1.228-0.163-0.123 1 year 0.767 1.143 1.376-0.239-0.072 3 years 0.546 1.203 1.321-0.278-0.013 5 years 0.379 1.191 1.250-0.278-0.003 10 years 0.169 1.095 1.108-0.224 0.009

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 ECM(1) with Two Common Trends Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.778 1.037 1.296-0.175-0.129 1 year 0.868 1.247 1.519-0.286-0.033 3 years 0.586 1.186 1.323-0.288-0.034 5 years 0.425 1.155 1.231-0.304-0.014 10 years 0.220 1.035 1.058-0.274 0.015

1-Year-Ahead Forecast Error Analysis, 1990.01-2000.12 Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.150 0.724 0.739-0.288 0.001 1 year 0.173 0.823 0.841-0.332-0.004 3 years -0.123 0.910 0.918-0.408 0.015 5 years -0.337 0.918 0.978-0.412 0.003 10 years -0.531 0.825 0.981-0.433-0.003 ECM(3) with Three Common Trends Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 0.810 0.951 1.249-0.245-0.082 1 year 0.786 1.261 1.486-0.248-0.064 3 years 0.613 1.453 1.577-0.289 0.028 5 years 0.306 1.236 1.273-0.246-0.069 10 years 0.063 1.141 1.143-0.191-0.086

Out-of-Sample Forecasting, DNS vs. Random Walk 1.1 1-Month-Ahead 1.1 6-Month-Ahead 1.1 12-Month-Ahead RMSE(DNS) / RMSE(Ra ndom Walk) 1.0 0.9 0.8 0.7 RMSE(DNS) / RMSE(Ra ndom Walk) 1.0 0.9 0.8 0.7 RMSE(DNS) / RMSE(Ra ndom Walk) 1.0 0.9 0.8 0.7 0 2 4 6 8 10 12 Maturity 0 2 4 6 8 10 12 Matur ity 0 2 4 6 8 10 12 Maturity

Term Structures of Credit Spreads

Generalized Duration (Diebold, Ji and Li, 2006) Discount bond: Coupon bond:

Dynamic Nelson-Siegel has a Natural State-Space Structure

Compactly (Diebold, Rudebusch and Aruoba, 2006) where

State-Space Representation Powerful framework One-step exact maximum-likelihood estimation Optimal extraction of latent factors Optimal point and interval forecasts

More Heteroskedasticity, confidence tunnels, density forecasts Regime switching Bayesian estimation and analysis

Inclusion of Macro and Policy Variables (Diebold, Rudebusch and Aruoba, 2006) where

One-Step vs. Two-Step One-Step: calibrated fixed but estimated Time-varying (structured) Two-Step: calibrated Time-varying (unstructured) Two-step proves appealing for tractability Fixed linked to absence of arbitrage

Yield Curves Across Countries and Time

Single-Country Models

Estimated Country Level Factors 14 12 10 8 6 4 2 0 86 88 90 92 94 96 98 00 02 04 Estimated Country Slope Factors 6 4 2 0-2 -4-6 86 88 90 92 94 96 98 00 02 04

Principal Components Analysis Level Factors PC1 PC2 PC3 PC4 Eigenvalue 3.64 0.27 0.10 0.05 Variance Prop. 0.91 0.05 0.03 0.01 Cumulative Prop. 0.91 0.96 0.99 1.00 Slope Factors PC1 PC2 PC3 PC4 Eigenvalue 1.99 1.01 0.69 0.30 Variance Prop. 0.50 0.25 0.17 0.08 Cumulative Prop. 0.50 0.75 0.92 1.00

Multi-Country Model, I

Multi-Country Model, II

State Space Representation

Estimates of Global Model Global Level Factor Global Slope Factor

Extracted Global Level Factor Extracted Global Slope Factor

Country Level Factors

Country Slope Factors

Two Approaches to Yield Curves I. Dynamic Nelson-Siegel (Diebold-Li,...) Popular in practice Level, slope, curvature Easy to estimate, with good fits and forecasts But, does not enforce absence of arbitrage II. Affine Equilibrium (Duffie-Kan,...) Popular in theory Enforces absence of arbitrage But, difficult to estimate and evaluate

Affine Equilibrium (Duffie-Kan, 1996,...) Risk-neutral dynamics:, where Freedom from arbitrage requires: where and solve Duffie-Kan ODEs,

Making DNS Arbitrage-Free and find and s.t. Duffie-Kan ODE is satisfied

Solution: Duffie-Kan ODE:

Propositon (Arbitrage-Free Nelson Siegel) Suppose that the instantaneous risk-free rate is: with risk-neural state dynamics:

Then yield dynamics are arbitrage-free:, where has Nelson-Siegel form:...

...and where the yield adjustment term is:

Factor Dynamics Under the Physical Measure Essentially affine risk premium (Duffee, 2002): Same dynamic structure under the P measure:

Yield Data and Model Estimation January 1987 - December 2002 Sixteen maturities (in years):.25,.5,.75, 1, 1.5, 2, 3, 4, 5, 7, 8, 9, 10, 15, 20, 30 Linear, Gaussian state space structure Y Estimate using Kalman filter

Yield adjustment term in basis points 100 80 60 40 20 0 AFNS indep. factor AFNS corr. factor 0 5 10 15 20 25 30 Maturity in years Figure 1: Yield-Adjustment Terms for AFNS Models.

Yield in percent 5.0 5.5 6.0 6.5 7.0 7.5 Empirical mean yields Indep. factor DNS mean yields Corr. factor DNS mean yields Indep. factor AFNS mean yields Corr. factor AFNS mean yields 0 5 10 15 20 25 30 Time to maturity in years Figure 2: Mean Yield Curves. We show the empirical mean yield curve, and the independent- and correlated-factor DNS and AFNS model mean yield curves.

Models (e.g., Independent-Factor) 1. DNS Independent (Diebold and Li, 2006)

2. AFNS Independent

Out-of-Sample Forecasting, Independent Case AFNS v.s DNS 1.1 1-Month-Ahead 1.1 6-Month-Ahead 1.1 12-Month-Ahead RMSE(AFDNS) / RMSE(DNS) 1.0 0.9 0.8 0.7 RMSE( AFDNS) / RMSE(DNS) 1.0 0.9 0.8 0.7 RMSE( AFDNS) / RMSE(DNS) 1.0 0.9 0.8 0.7 0.6 0 4 8 12 16 20 24 28 32 0.6 0 4 8 12 16 20 24 28 32 0.6 0 4 8 12 16 20 24 28 32 Maturity Maturity Maturity

Figure 3: Out-of-Sample Root Mean Squared Forecast Error Ratios.

Forecast Horizon in Months Model h=6 h=12 3-Month Yield DNS indep 96.87 173.39 DNS corr 87.43 166.91 AFNS indep 91.63 164.70 AFNS corr 88.49 161.94 1-Year Yield DNS indep 103.25 170.85 DNS corr 102.71 173.14 AFNS indep 98.49 163.46 AFNS corr 98.63 165.50 3-Year Yield DNS indep 92.22 135.24 DNS corr 99.55 145.82 AFNS indep 86.99 126.95 AFNS corr 90.64 135.79 5-Year Yield DNS indep 87.87 122.09 DNS corr 94.95 132.40 AFNS indep 82.41 112.85 AFNS corr 88.15 124.87 10-Year Yield DNS indep 74.71 105.02 DNS corr 79.48 112.37 AFNS indep 67.48 92.39 AFNS corr 90.21 123.89 30-Year Yield DNS indep 71.35 96.90 DNS corr 72.71 99.68 AFNS indep 48.06 61.97 AFNS corr 71.38 96.75

Forecast Horizon in Months Maturity/Model h=6 h=12 6-Month Yield Random Walk 40.0 48.4 Preferred A 0 (3) 36.5 42.1 AFNS indep 34.0 41.3 2-Year Yield Random Walk 65.2 76.2 Preferred A 0 (3) 56.6 60.0 AFNS indep 54.3 59.0 10-Year Yield Random Walk 66.9 81.5 Preferred A 0 (3) 63.6 73.8 AFNS indep 60.7 71.8

Incorporating Additional Factors Svensson (1995): ( 1 e λ 1 τ ( 1 e λ 1 τ y(τ) = β 1+β 2 )+β ) ( 3 e λ 1τ 1 e λ 2 τ ) +β 4 e λ 2τ +ε(τ) λ 1τ λ 1τ λ 2τ Dynamic Svensson: y t(τ) = L t+s t ( 1 e λ 1 τ λ 1 τ ) +C 1 t ( 1 e λ 1 τ ) ( e λ 1τ 1 e +Ct 2 λ 2 τ ) e λ 2τ +ε t(τ) λ 1 τ λ 2 τ Generalized Dynamic Svensson: y t (τ) = L t +S 1 t ( 1 e λ 1 τ λ 1 τ ) +S 2 t ( 1 e λ 2 τ λ 2 τ ) +C 1 ( 1 e λ 1 τ t λ 1 τ e λ 1 τ ) +C 2 ( 1 e λ 2 τ t e λ 2 τ ) +ε t (τ) λ 2 τ

Conclusions AFNS delivers tractable rigorous modeling AFNS delivers rigorous tractable modeling AF restrictions may help forecasts

Onward II: Forecasting Christensen-Diebold-Rudebusch (First draft 2007): Mixed evidence that AFNS forecasts better than DNS Duffee (2009): AFNS should not forecast better than DNS - Interpretation from our vantage point: AFNS and DNS have identical P-dynamics - What if we impose the Q-restrictions? (Diebold-Hua, 2010) - What if we shrink toward the Q-restrictions?

Unrestricted DNS Table: Unrestricted DNS Parameter Estimates L t 1 S t 1 C t 1 U L t 0.984 0.000-0.004 7.093 (0.020) (0.022) (0.020) (1.423) S t 0.004 0.934 0.078-3.023 (0.027) (0.023) (0.025) (2.031) C t 0.027 0.054 0.883-1.345 (0.056) (0.047) (0.040) (1.379) Note: Each row presents coefficients from the transition equation for the respective state variable. Standard errors appear in parenthesis. Asterisks denote significance at the 5% level.

DNS with Risk-Neutral Dynamics a 11 0 0 Univariate AR, A = 0 a 22 0 0 0 a 33 a 11 0 0 Curvature added, A = 0 a 22 a 23 0 0 a 33 a 11 0 0 λ restriction, A = 0 1 a 23 a 23 0 0 a 33 a 11 0 0 Cross λ restriction, A = 0 1 a 23 a 23 0 0 1 a 23 1 0 0 All restrictions, A = 0 1 a 23 a 23 0 0 1 a 23

Table: Tests of DNS Risk-Neutral Restrictions Likelihood ratio of restricted models Test statistics P-value Univariate AR 25.696 0.000 Curvature added 1.876 0.866 λ restriction 2.664 0.850 Cross λ restriction 2.754 0.907 All restrictions 6.148 0.631 Granger causality test C t 1 does not Granger cause S t 23.710 0.000 S t 1 does not Granger cause C t 1.580 0.210

Unrestricted AFNS Table: The General AFNS Model Parameter Estimates L t 1 S t 1 C t 1 U L t 0.987 0.005-0.010 7.309 (0.017) (0.018) (0.016) (1.450) S t 0.007 0.933 0.082-2.560 (0.023) (0.023) (0.020) (1.561) C t 0.015 0.040 0.901-1.669 (0.026) (0.047) (0.040) (1.260) Note: Each row presents coefficients from the transition equation for the respective state variable. Standard errors appear in parenthesis. Asterisks denote significance at the 5% level.

AFNS with Risk-Neutral Dynamics κ 11 0 0 Univariate AR, K P = 0 κ 22 0 0 0 κ 33 κ 11 0 0 Curvature added, K P = 0 κ 22 κ 23 0 0 κ 33 κ 11 0 0 λ restriction, K P = 0 1 κ 23 κ 23 0 0 κ 33 κ 11 0 0 Cross λ restriction, K P = 0 1 κ 23 κ 23 0 0 1 κ 23 10 7 0 0 All restrictions, K P = 0 1 κ 23 κ 23 0 0 1 κ 23

Table: Tests of AFNS Risk-Neutral Restrictions Likelihood ratio of restricted models Test statistics P-value Univariate AR 13.420 0.037 Curvature added 8.340 0.138 λ restriction 2.380 0.882 Cross λ restriction 10.100 0.183 All restrictions 9.000 0.342 Granger causality test C t 1 does not Granger cause S t 15.005 0.000 S t 1 does not Granger cause C t 1.956 0.164

DNS 1-Month-Ahead Forecasts Output-of-sample 1-month-ahead forecasting rankings 3-month 1-year 2-year 3-year 5-year 10-year 30-year unrestricted VAR 5 2 5 6 5 6 6 univariate AR 6 6 6 5 6 5 5 curvature added 4 1 3 3 3 3 2 λ restriction 3 4 4 2 4 4 3 cross λ restrictions 2 3 2 4 2 2 4 all restrictions 1 5 1 1 1 1 1 We estimate all models recursively from 1987:1 to the time forecast is made, beginning in 1997:1 and extending through 2002:12. We define forecast errors at t + h as y t+h (τ) ŷ t+h/t (τ) and report the rankings in terms of root mean square errors. 1 is best, and 6 is worst.

DNS 6-Month-Ahead Forecasts Output-of-sample 6-month-ahead forecast rankings 3-month 1-year 2-year 3-year 5-year 10-year 30-year unrestricted VAR 5 6 6 6 6 6 6 univariate AR 6 5 5 4 2 2 2 curvature added 4 4 2 2 3 3 3 λ restriction 3 3 3 3 4 4 4 cross λ restrictions 2 2 4 5 5 5 5 all restrictions 1 1 1 1 1 1 1

DNS 1-Year-Ahead Forecasts Output-of-sample 1-year-ahead forecast rankings 3-month 1-year 2-year 3-year 5-year 10-year 30-year unrestricted VAR 6 6 6 6 6 6 6 univariate AR 5 5 4 3 3 2 3 curvature added 2 2 2 2 2 3 2 λ restriction 4 4 3 4 4 4 4 cross λ restrictions 3 3 5 5 5 5 5 all restrictions 1 1 1 1 1 1 1

AFNS 1-Month-Ahead Forecasts Output-of-sample 1-month-ahead forecast rankings 3-month 1-year 2-year 3-year 5-year 10-year 30-year unrestricted VAR 6 6 6 6 6 6 6 univariate AR 1 2 4 4 5 5 4 curvature added 2 1 1 1 3 2 1 λ restriction 3 4 5 5 4 4 3 cross λ restrictions 5 5 3 3 2 3 5 all restrictions 4 3 2 2 1 1 2

AFNS 6-Month-Ahead Forecasts Output-of-sample 6-month-ahead forecast rankings 3-month 1-year 2-year 3-year 5-year 10-year 30-year unrestricted VAR 6 6 6 2 2 2 2 univariate AR 2 2 2 3 3 3 3 curvature added 3 3 3 4 4 4 4 λ restriction 4 4 4 5 5 5 5 cross λ restrictions 5 5 5 6 6 6 6 all restrictions 1 1 1 1 1 1 1

AFNS 1-Year-Ahead Forecasts Output-of-sample 1-year-ahead forecast rankings 3-month 1-year 2-year 3-year 5-year 10-year 30-year unrestricted VAR 6 6 6 6 6 6 6 univariate AR 5 5 4 3 3 2 2 curvature added 2 2 2 2 2 3 3 λ restriction 4 4 5 4 4 4 4 cross λ restrictions 3 3 3 5 5 5 5 all restrictions 1 1 1 1 1 1 1

AFNS 2-Year-Ahead Forecasts Output-of-sample 2-year-ahead forecast rankings 3-month 1-year 2-year 3-year 5-year 10-year 30-year random walk 4 4 2 2 2 2 2 univariate AR 1 2 3 3 3 4 4 unrestricted VAR 7 7 7 7 7 7 7 curvature added 2 3 4 4 4 3 3 λ restriction 6 6 6 6 5 6 5 cross λ restrictions 5 5 5 5 6 5 6 all restrictions 3 1 1 1 1 1 1

AFNS 3-Year-Ahead Forecasts Output-of-sample 3-year-ahead forecast rankings 3-month 1-year 2-year 3-year 5-year 10-year 30-year random walk 6 2 2 2 2 2 2 univariate AR 1 4 4 4 3 4 4 unrestricted VAR 7 7 7 7 7 7 7 curvature added 2 3 3 3 4 3 3 λ restriction 4 5 5 5 5 5 5 cross λ restrictions 3 6 6 6 6 6 6 all restrictions 5 1 1 1 1 1 1

Onward I: Tractable AF Modeling... Joslin-Singleton-Zhu (2009) obtain a larger class of tractable AF models. AFNS is one case. Krippner (2009) shows that AFNS is an optimal approximation to the generic Gaussian affine model of Dai and Singleton (2002).