Modeling and Forecasting the Term Structure of Government Bond Yields

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Ἀκελδαμά Αργυριάδης
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1 Modeling and Forecasting the Term Structure of Government Bond Yields Francis X. Diebold University of Pennsylvania, NBER and Morgan Stanley Investment Management First Annual Risk Management Institute Research Conference, Capital Flows and Asset Prices: International Dimensions of Risk Ritz-Carlton, Singapore, July 2007

3 Yield Curve Data Monthly CRSP Fixed maturities (months): 3, 6, 9, 12, 15, 18, 21, 24, 30, 36, 48, 60, 72, 84, 96, 108, and

4 Advances in Arbitrage-Free Modeling Cross sectional flavor (e.g., HJM, 1992 Econometrica) Time series flavor (e.g., Dai and Singleton, 2000 JF) What about forecasting?

5 We Take a Classic but Ad Hoc Yield Curve Model and: C show that it has a modern interpretation C show that it s flexible C show that it fits well C show that it forecasts well

6 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

7 The Nelson-Siegel Yield Curve has a Modern Interpretation f t (τ) ' β 1t % β 2t e &λ t τ % β 3t λ t τe &λ t τ Original: y t (τ) ' b 1t % b 2t 1&e &λ t τ λ t τ & b 3t e &λ t τ Ours: y t (τ) ' β 1t % β 2t 1&e &λ t τ λ t τ 1&e &λ t τ % β 3t &e &λ t τ λ t τ level, slope, curvature

8 Factor Loadings in the Nelson-Siegel Model 1 β Loadings β β τ (Maturity, in Months)

9 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

10 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

persistent (4) Spread dynamics are much less persistent (5) Short

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

12 6.6 Fitted Yield Curve I Yield Curve on 2/28/ Yield (Percent) M a turity (M o nths )

13 The Model Forecasts Well: Modeling and Forecasting Level, Slope and Curvature Estimated level, slope, and curvature factors: ˆβ 1t, ˆβ 2t, ˆβ 3t

14 Nelson-Siegel with AR(1) Factors ŷ t%h/t (τ) ' ˆβ 1,t%h/t % ˆβ 2,t%h/t 1&e &λτ λτ % ˆβ 3,t%h/t 1&e &λτ λτ &e &λτ ˆβ i,t%h/t ' ĉ i % ˆγ i ˆβ it, i ' 1, 2, 3

15 Random Walk ŷ t%h/t (τ) ' y t (τ)

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

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

18 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)

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

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

21 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)] )

22 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)] )

23 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)] )

24 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)] )

25 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

26 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

27 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

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 Slope Regression Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months NA NA NA NA NA 1 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 Fama-Bliss Forward Regression Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 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 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

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 VAR(1) on Yield Levels Maturity (τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months 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 Changes 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 ECM(1) with one Common Trend 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 Two Common Trends 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(3) with Three Common Trends Maturity ( τ) Mean Std. Dev. RMSE ˆρ(12) ˆρ(24) 3 months year years years years

36 State-Space Representation (Diebold, Rudebusch, and Aruoba, 2006) C One-step exact maximum-likelihood estimation C Optimal extraction of latent factors C Optimal point and interval forecasts C Incorporation of conditional heteroskedasticity

37 Important Case: Inclusion of Macro and Liquidity Variables (Diebold, Rudebusch, and Aruoba, 2006) where

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

39 Term Structures of Credit Spreads (Diebold, Li, and Yue, in progress)

40 Yield Curves Across Countries and Time

41 Single-Country Models

42 Estimated Country Level Factors Estimated Country Slope Factors

43 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

44 Multi-Country Model, I

45 Multi-Country Model, II

46 State Space Representation

49 Estimates of Global Model Global Level Factor Global Slope Factor

50 Extracted Global Level Factor Extracted Global Slope Factor

51 Country Level Factors

52 Country Slope Factors

53 Variance Decompositions

54 Variance Decompositions, 1985: :08 Variance Decompositions, 1995: :08

55 Background Diebold, F.X., Piazzesi, M. and Rudebusch, G.D. (2005), Modeling Bond Yields in Finance and Macroeconomics, American Economic Review, 95, Diebold, F.X. and Li, C. (2006), Forecasting the Term Structure of Government Bond Yields, Journal of Econometrics, 130, 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, Diebold, F.X., Li, C. And Yue, V.Z. (2007), Global Yield Curve Dynamics and Interactions: A Generalized Nelson-Siegel Approach, Manuscript, University of Pennsylvania.

56 Extensions Heteroskedasticity, confidence tunnels, and density forecasts Freedom from arbitrage

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