Estimation of non-Gaussian Affine Term Structure Models - NYU Stern

51 Pages·2012·0.78 MB·English

Model Estimation Identifyingrestrictions Results Conclusion
Estimation of non-Gaussian Affine
Term Structure Models
Drew D. Creal Jing Cynthia Wu
UniversityofChicago,BoothSchoolofBusiness
NewYorkUniversity,SternSchoolofBusiness
PRELIMINARY
December 2012
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Motivation
Affine term structure models are useful for
(cid:73) understanding the joint dynamics of the yield curve;
(cid:73) describing the discount factor for financial markets;
(cid:73) bridging macroeconomics and finance together;
(cid:73) informing monetary policy.
Why do we need to go beyond Gaussian models?
(cid:73) zero lower bound
(cid:73) time-varying volatility
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Estimation of ATSMs
(cid:73) Kim and Orphanides (2005):
... the likelihood function seems to have multiple
inequivalent local maxima which have similar likelihood
values but substantially different implications...
(cid:73) Duffee (2002):
The QML functions for these models have a large
number of local maxima.
(cid:73) Ang and Piazzesi (2003):
... difficulties associated with estimating a model with
many factors using maximum likelihood when yields are
highly persistent.
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Literature
Earlier work on estimating non-Gaussian models
(cid:73) Dai and Singleton (2000), Duffee (2002)
(cid:73) Collin-Dufresne, Goldstein and Jones (2008/2009),
Ait-Sahalia and Kimmel (2010)
(cid:73) Le, Singleton, and Dai (2010)
Recent work on estimating Gaussian models
(cid:73) Joslin, Singleton, and Zhu (2011)
(cid:73) Christensen, Diebold, and Rudebusch (2011)
(cid:73) Hamilton and Wu (2012)
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Contribution
Propose a new estimation approach for non-Gaussian
ATSMs.
(cid:73) Reduce parameter space by concentrating out P parameters.
(cid:73) Provide analytical gradient to improve numerical behavior.
(cid:73) Works for ANY rotation/identification scheme.
(cid:73) For Gaussian models, this approach generalizes Joslin,
Singleton, and Zhu (2011), and Hamilton and Wu (2012).
Extensions:
(cid:73) Imposing constraints on parameters is straightforward.
(cid:73) Adding macroeconomic variables is simple.
(cid:73) Accommodating more general dynamics, e.g. AR(p).
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Outline
Model
Estimation
Identifying restrictions
Results
Conclusion
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Model Estimation Identifyingrestrictions Results Conclusion
Outline
Model
Estimation
Identifying restrictions
Results
Conclusion
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Model Estimation Identifyingrestrictions Results Conclusion
Dynamics of the state vector under P
State vector x = (g(cid:48),h(cid:48))(cid:48) has dynamics:
t t t
g = µ +Φ g +Φ h +Σ ε +ε ,
t+1 g g t gh t gh h,t+1 g,t+1
H
(cid:88)
Σ Σ(cid:48) = Σ Σ(cid:48) + Σ Σ(cid:48) h ,
g,t g,t g g g,i g,i it
i=1
h ∼ N.C.-Gamma(ν ,Φ(cid:48) h ,σ ), i = 1,...,H.
i,t+1 i h,i t h,i
(cid:73) ε ∼ N(cid:0)0,Σ Σ(cid:48) (cid:1)
g,t+1 g,t g,t
(cid:73) g and h are G ×1 and H ×1 vectors.
t t
(cid:73) Σ and Σ are lower triangular.
g g,i
(cid:73) h is a discrete-time Cox Ingersoll Ross (1985) process.
it
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Autoregressive gamma process
The non-Gaussian state variables h follow
t
h |h ∼ N.C.-Gamma(ν ,Φ(cid:48) h ,σ ), i = 1,...,H.
i,t+1 t i h,i t h,i
Details
Conditional mean and variance are linear in h :
t
E[h |h ] = ν σ +Φ(cid:48) h i = 1,...,H
i,t+1 t i h,i h,i t
V[h |h ] = ν σ2 +2σ Φ(cid:48) h
i,t+1 t i h,i h,i h,i t
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Autoregressive gamma process
Conditional mean and variance are linear in h
t−1
E[h |h ] = ν σ +Φ(cid:48) h i = 1,...,H
i,t+1 t i h,i h,i t
V[h |h ] = ν σ2 +2σ Φ(cid:48) h
i,t+1 t i h,i h,i h,i t
Stacked together
E[h |h ] = µ +Φ h
t+1 t h h t
 ν σ2 +2σ Φ(cid:48) h ··· 0 
1 h,1 h,i h,1 t
V[ht+1|ht] =  ... ... ... 
0 ··· ν σ2 +2σ Φ(cid:48) h
H h,H h,H h,H t
Note: µ = ν σ
h,i i h,i
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