(Arbitrage-Free, Practical) Modeling of Term Structures of Government Bond Yields
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1 (Arbitrage-Free, Practical) Modeling of Term Structures of Government Bond Yields
2 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.
3 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
4
5 Incompletely-Satisfying Advances in Arbitrage-Free Modeling Cross sectional flavor (e.g., HJM, 1992 Econometrica) Time series flavor (e.g., Vasicek, 1977 JFE)
6 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)
7
8 The Model Fits Well
9
10 Factor Loadings in the Nelson-Siegel Model 1 β Loadings β β τ (Maturity, in Months)
11 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
12 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
13 The Model is Easily Fit, and Fits Well OLS with fixed
14 6.6 Fitted Yield Curve I Yield Curve on 2/28/ Yield (Percent) M a turity (M o nths )
15 The Model Forecasts Well (Dynamic Nelson-Siegel with AR(1) Factors)
16 Random Walk ŷ t%h/t (τ) ' y t (τ)
17 Slope Regression ŷ t%h/t (τ)&y t (τ) ' ĉ(τ) % ˆγ(τ)(y t (τ)&y t (3))
18 Fama-Bliss Forward Regression ŷ t%h/t (τ)&y t (τ) ' ĉ(τ) % ˆγ(τ)(f h t (τ)&y t (τ))
19 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)
20 AR(1) on Yield Levels ŷ t%h/t (τ) ' ĉ % ˆγy t (τ)
21 VAR(1) on Yield Levels ŷ t%h/t (τ) ' ĉ % ˆΓy t (τ)
22 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)] )
23 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)] )
24 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)] )
25 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)] )
26 1-Month-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(1) ˆρ(12) 3 months year years years years Random Walk Maturity (τ) Mean Std. Dev. RMSE ˆρ(1) ˆρ(12) 3 months year years years years
27 6-Month-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(6) ˆρ(18) 3 months year years years years Random Walk Maturity (τ) Mean Std. Dev. RMSE ˆρ(6) ˆρ(18) 3 months year years years years
28 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years Random Walk Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years
29 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years Slope Regression Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months NA NA NA NA NA 1 year years years years
30 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years Fama-Bliss Forward Regression Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years
31 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years Cochrane-Piazzesi Forward Regression Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months NA NA NA NA NA 1 year years years years
32 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years VAR(1) on Yield Levels Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years
33 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years VAR(1) on Yield Changes Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years
34 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years ECM(1) with one Common Trend Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years
35 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years ECM(1) with Two Common Trends Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years
36 1-Year-Ahead Forecast Error Analysis, Nelson-Siegel with AR(1) Factor Dynamics Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years ECM(3) with Three Common Trends Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years
37 Out-of-Sample Forecasting, DNS vs. Random Walk Month-Ahead Month-Ahead Month-Ahead RMSE(DNS) / RMSE(Ra ndom Walk) RMSE(DNS) / RMSE(Ra ndom Walk) RMSE(DNS) / RMSE(Ra ndom Walk) Maturity Matur ity Maturity
38 Term Structures of Credit Spreads
39 Generalized Duration (Diebold, Ji and Li, 2006) Discount bond: Coupon bond:
40 Dynamic Nelson-Siegel has a Natural State-Space Structure
41 Compactly (Diebold, Rudebusch and Aruoba, 2006) where
42 State-Space Representation Powerful framework One-step exact maximum-likelihood estimation Optimal extraction of latent factors Optimal point and interval forecasts
43 More Heteroskedasticity, confidence tunnels, density forecasts Regime switching Bayesian estimation and analysis
44 Inclusion of Macro and Policy Variables (Diebold, Rudebusch and Aruoba, 2006) where
45 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
46 Yield Curves Across Countries and Time
47 Single-Country Models
48 Estimated Country Level Factors Estimated Country Slope Factors
49 Principal Components Analysis Level Factors PC1 PC2 PC3 PC4 Eigenvalue Variance Prop Cumulative Prop Slope Factors PC1 PC2 PC3 PC4 Eigenvalue Variance Prop Cumulative Prop
50 Multi-Country Model, I
51 Multi-Country Model, II
52 State Space Representation
53
54
55 Estimates of Global Model Global Level Factor Global Slope Factor
56 Extracted Global Level Factor Extracted Global Slope Factor
57 Country Level Factors
58 Country Slope Factors
59 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
60 Affine Equilibrium (Duffie-Kan, 1996,...) Risk-neutral dynamics:, where Freedom from arbitrage requires: where and solve Duffie-Kan ODEs,
61 Making DNS Arbitrage-Free and find and s.t. Duffie-Kan ODE is satisfied
62 Solution: Duffie-Kan ODE:
63 Propositon (Arbitrage-Free Nelson Siegel) Suppose that the instantaneous risk-free rate is: with risk-neural state dynamics:
64 Then yield dynamics are arbitrage-free:, where has Nelson-Siegel form:...
65 ...and where the yield adjustment term is:
66 Factor Dynamics Under the Physical Measure Essentially affine risk premium (Duffee, 2002): Same dynamic structure under the P measure:
67 Yield Data and Model Estimation January 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
68 Yield adjustment term in basis points AFNS indep. factor AFNS corr. factor Maturity in years Figure 1: Yield-Adjustment Terms for AFNS Models.
69 Yield in percent Empirical mean yields Indep. factor DNS mean yields Corr. factor DNS mean yields Indep. factor AFNS mean yields Corr. factor AFNS mean yields 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.
70 Models (e.g., Independent-Factor) 1. DNS Independent (Diebold and Li, 2006)
71 2. AFNS Independent
72 Out-of-Sample Forecasting, Independent Case AFNS v.s DNS Month-Ahead Month-Ahead Month-Ahead RMSE(AFDNS) / RMSE(DNS) RMSE( AFDNS) / RMSE(DNS) RMSE( AFDNS) / RMSE(DNS) Maturity Maturity Maturity
73 Figure 3: Out-of-Sample Root Mean Squared Forecast Error Ratios.
74 Forecast Horizon in Months Model h=6 h=12 3-Month Yield DNS indep DNS corr AFNS indep AFNS corr Year Yield DNS indep DNS corr AFNS indep AFNS corr Year Yield DNS indep DNS corr AFNS indep AFNS corr Year Yield DNS indep DNS corr AFNS indep AFNS corr Year Yield DNS indep DNS corr AFNS indep AFNS corr Year Yield DNS indep DNS corr AFNS indep AFNS corr
75 Forecast Horizon in Months Maturity/Model h=6 h=12 6-Month Yield Random Walk Preferred A 0 (3) AFNS indep Year Yield Random Walk Preferred A 0 (3) AFNS indep Year Yield Random Walk Preferred A 0 (3) AFNS indep
76 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
77
78 Conclusions AFNS delivers tractable rigorous modeling AFNS delivers rigorous tractable modeling AF restrictions may help forecasts
79 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?
80 Unrestricted DNS Table: Unrestricted DNS Parameter Estimates L t 1 S t 1 C t 1 U L t (0.020) (0.022) (0.020) (1.423) S t (0.027) (0.023) (0.025) (2.031) C t (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.
81 DNS with Risk-Neutral Dynamics a Univariate AR, A = 0 a a 33 a Curvature added, A = 0 a 22 a a 33 a λ restriction, A = 0 1 a 23 a a 33 a Cross λ restriction, A = 0 1 a 23 a a All restrictions, A = 0 1 a 23 a a 23
82 Table: Tests of DNS Risk-Neutral Restrictions Likelihood ratio of restricted models Test statistics P-value Univariate AR Curvature added λ restriction Cross λ restriction All restrictions Granger causality test C t 1 does not Granger cause S t S t 1 does not Granger cause C t
83 Unrestricted AFNS Table: The General AFNS Model Parameter Estimates L t 1 S t 1 C t 1 U L t (0.017) (0.018) (0.016) (1.450) S t (0.023) (0.023) (0.020) (1.561) C t (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.
84 AFNS with Risk-Neutral Dynamics κ Univariate AR, K P = 0 κ κ 33 κ Curvature added, K P = 0 κ 22 κ κ 33 κ λ restriction, K P = 0 1 κ 23 κ κ 33 κ Cross λ restriction, K P = 0 1 κ 23 κ κ All restrictions, K P = 0 1 κ 23 κ κ 23
85 Table: Tests of AFNS Risk-Neutral Restrictions Likelihood ratio of restricted models Test statistics P-value Univariate AR Curvature added λ restriction Cross λ restriction All restrictions Granger causality test C t 1 does not Granger cause S t S t 1 does not Granger cause C t
86 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 univariate AR curvature added λ restriction cross λ restrictions all restrictions 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.
87 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 univariate AR curvature added λ restriction cross λ restrictions all restrictions
88 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 univariate AR curvature added λ restriction cross λ restrictions all restrictions
89 DNS 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 unrestricted VAR univariate AR curvature added λ restriction cross λ restrictions all restrictions
90 DNS 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 unrestricted VAR univariate AR curvature added λ restriction cross λ restrictions all restrictions
91 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 univariate AR curvature added λ restriction cross λ restrictions all restrictions
92 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 univariate AR curvature added λ restriction cross λ restrictions all restrictions
93 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 univariate AR curvature added λ restriction cross λ restrictions all restrictions
94 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 unrestricted VAR univariate AR curvature added λ restriction cross λ restrictions all restrictions
95 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 unrestricted VAR univariate AR curvature added λ restriction cross λ restrictions all restrictions
96 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 univariate AR unrestricted VAR curvature added λ restriction cross λ restrictions all restrictions
97 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 univariate AR unrestricted VAR curvature added λ restriction cross λ restrictions all restrictions
98 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).
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