A Macro-Finance Model of the Term Structure: the Case for a Quadratic Yield Model

Transcription

1 Title page Outline A Macro-Finance Model of the Term Structure: the Case for a 21, June Czech National Bank

2 Structure of the presentation Title page Outline Structure of the presentation: Model Formulation Filtering & Estimation Conclusions & Results

3 Macroeconomic models of the yield curve Add macro to the finance arbitrage-free models affine models, (Ang-Piazzesi, 23, 26; Rudebusch-Wu, 28); Arbitrage-free Nelson-Siegel model (Christensen-Diebold-Rudebusch, 28); Bond pricing in DSGE models (Rudebusch-Swanson-Wu, 26); Applications: forecasting of inflation and real activity; central bank liquidity facilities; great moderation and great conundrum.

4 The spread on the money market 6 Policy Rate and Pribor 5 % Policy Rate Pribor 1M Pribor 3M Pribor 6M Pribor 1Y 1 2M1 21M1 22M1 23M1 24M1 25M1 26M1 27M1 28M1 29M1 2 Spread between Pribor and Policy Rate % M1 21M1 22M1 23M1 24M1 25M1 26M1 27M1 28M1 29M1

5 The spread between government bonds and the 3M Pribor 8 Yields % Pribor 3M Pribor 1Y Yield 3Y Yield 5Y Yield 1Y 1 2M1 21M1 22M1 23M1 24M1 25M1 26M1 27M1 28M1 29M1 3.5 Spreads vis-a-vis 3M pribor % M1 21M1 22M1 23M1 24M1 25M1 26M1 27M1 28M1 29M1

6 The decomposition of the spread 1.8 Spread 3M - 2W Spread 3M-2W Spread 1M-2W p.b M11 26M11 27M11 28M11 29M Spread 1Y - 2W Spread 1Y-2W Spread 1M-2W p.b..5 25M11 26M11 27M11 28M11 29M Spread 1Y - 2W Spread 1Y-2W Spread 3M-2W p.b..5 25M11 26M11 27M11 28M11 29M11

7 The Fama-Bliss regression The Fama-Bliss regression (AER, 1987): Excess Return t+1 = α + βspread t + ε t. β = is implied by the expectation hypothesis of the term structure. The recently introduced macro-finance term structure models (affine latent models) have problems if β. Econometric studies usually find that on the low end of the yield curve β =, i.e., the expectation hypothesis approximatively holds.

8 Results of the Fama-Bliss regression for the Czech money market Rolling regression (18 months window) of the Fama-Bliss regression on the Czech data suggest that: up to the second half of 28, β = β since then. See next two figures: 1 Recursive estimation of the coefficient β for various maturities, 2 The dynamic prediction of the arbitrage-free version of the dynamic Nelson - Siegel model

9 Recursive estimation of Fama-Bliss βs β t : 2M-1M β t : 3M-1M β t : 6M-1M β t : 6M-3M β t : 9M-3M β t : 1Y-3M

10 Theoretical formulation Numerical filtering State equation As in affine models, the underlying set of macroeconomic factors follows a VAR process: X t+1 = ΦX t + κ + Ψε t+1. The log of the pricing kernel is defined as follows: m t+1 = δ + ΓX t + Xt T X t + ς t ε t+1 + κη t+1, hence the short-term interest rate i t is given by: i t = E t m t V tm t+1 = δ + ΓX t + X T t X t 1 2 (ςt ς + κ 2 ).

11 Theoretical formulation Numerical filtering Recursion The price of a k-period bond p k t is guessed in the form of: p k t = A k + B T k + X T t C k X t, with A 1 = δ 1 2 (ςt ς + κ 2 ), B 1 = Γ, C 1 =. Under log-normality: ( pt k = E t m t+1 + p k+1 t+1 ) + 1 ( 2 V t i k t = k 1 log p k t. m t+1 + p k+1 t+1 ),

12 Theoretical formulation Numerical filtering Recursion (cont.) The undetermined coefficient technique yields the following recursion: ( [ ]) A k = δ + A k 1 + 2tr C k 1 ΨΨ T [ [ ]] κ 2 + ς T ς + Bk 1 T ΨΨT B k 1 + 2tr Ψ T C k 1 ΨΨ T C k 1 Ψ, ( ) ) Bk T = Γ + Bk 1 T Φ + 2 (ς + Bk 1 T Ψ Ψ T Ck 1 T Φ, ( ) C k = + Φ T C k 1 Φ + 2Φ T C k 1 ΨΨCk 1 T Φ. Note that if C 1 = is symmetric, so are C k.

13 Theoretical formulation Numerical filtering A Formulation of the Empirical Model X t = ΦX t 1 + Ψε t, i k t = A k k BT k k X t + X T t C k k X t + ν it. We observe i k t and perhaps some elements of X t. A non-linear state space system has been obtained, we need to evaluate the likelihood (estimation), to filter the state (forecasting), and so on...

14 Non-linear state space models Theoretical formulation Numerical filtering There are various choices of filtering non-linear state space models: Extended Kalman filter: based on local linearization of the state and observation equations; Gaussian sum filter: global filter; the probability distributions being approximated by a convex combination of gaussian pdfs; Partcile filter: global filter; the distribution of states approximated using MC techniques; Unscented filter: local filter based on unscented transformation.

15 The first-order approximation Theoretical formulation Numerical filtering Consider X N(µ, σ 2 ) random variable and Y = X 2. The first-order approximation (EKF) of Y works as follows: Ỹ 1 = µ 2 + Y X X =µ(x µ) = µ 2 + 2µ(X µ), hence EỸ 1 = µ 2 < EY = µ 2 + σ 2 VỸ 1 = 4µ 2 VY = 2σ 4 + µ 2 σ 2. The first order approximation is biased and the EKF may yield a biased estimation of the states.

16 The unscented filter Theoretical formulation Numerical filtering The unscented filter tries to approximate the mean and the variance of the non-linear transform more precisely than the first -order approximation. Three possible ways: 1 unscented transform a kind of quadrature; 2 Monte Carlo integration; 3 sometimes exact integration possible. Otherwise, the standard Kalman filter formulae apply. = thus this is an approximate filter, but hopefully more precise than the EKF.

17 The unscented transformation Theoretical formulation Numerical filtering How does the unscented transformation work? 1 Define a set of points: {x (±i) x (±i) = x ± x i }; 2 Compute y (±i) = f (x (±i) ); 3 Set ȳ = i w iy (±i) ; P y = i w i(y (±i) ȳ)(y (±i) ȳ) T. The unscented transformation x i = n 1 w (P x ) 1/2 i and w i = 1 w 2n. For large-dimensional problems, x i could be randomly drawn from N(, P x ), and w i = n 1.

18 Algorithms competition Theoretical formulation Numerical filtering I did a Monte Carlo study, which suggests that: the unscented filter can compete with the particle filter in precision, but is faster; the EKF and the Gaussian sum filter are slow and inaccurate. Conclusion It is feasible to estimate and filter the quadratic yield model in a reasonable time using the unscented filtering.