Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin ˇ Sm´ ıd, Petr Gapko Data Factors Statistics Estimation of σ Likelihood Asymptotics Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin ˇ Sm´ ıd, Petr Gapko Czech Academy of Sciences CMS 2017 Bergamo, 1.6.2017 Jaroslav Dufek, Martin ˇ Sm´ ıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
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Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Joint Estimation of Parameters of MortgagePortfolio
Jaroslav Dufek, Martin Smıd, Petr Gapko
Czech Academy of Sciences
CMS 2017 Bergamo, 1.6.2017
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
1 Data
2 Factors
3 Statistics
4 Estimation of σ
5 Likelihood
6 Asymptotics
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Data
Detailed loan data are usually (more than) confidential
∆Ic x x x x x x x∆Yc x x x x x∆Ir x x x x x∆Yr x x x x x
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Estimation of σ
So far, σ (standard deviation of individual “collateral price”factor) was set speculatively and θ (VECM parameters) wereestimated
We want to estimate θ and σ jointly
Obvious candidate: MLE
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Estimation of σ
So far, σ (standard deviation of individual “collateral price”factor) was set speculatively and θ (VECM parameters) wereestimated
We want to estimate θ and σ jointly
Obvious candidate: MLE
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Estimation of σ
So far, σ (standard deviation of individual “collateral price”factor) was set speculatively and θ (VECM parameters) wereestimated
We want to estimate θ and σ jointly
Obvious candidate: MLE
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Reminder from Previous Presentation
We observe Gt , . . . ,Gn, Q1, . . . ,Qn and where Gt ’s and Qt ’s aretransformations of factors It ,Yt , respectively.
We want to predict G ’s and Q’s
Solution
We need to transform G ’s and Q’s into I ’s and Y ’sThen predict I ’s and Y ’s (by a VECM model)And finally transform I ’s and Y ’s back into G ’s and Q’s
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Reminder from Previous Presentation
We observe Gt , . . . ,Gn, Q1, . . . ,Qn and where Gt ’s and Qt ’s aretransformations of factors It ,Yt , respectively.
We want to predict G ’s and Q’s
Solution
We need to transform G ’s and Q’s into I ’s and Y ’sThen predict I ’s and Y ’s (by a VECM model)And finally transform I ’s and Y ’s back into G ’s and Q’s
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Reminder from Previous Presentation
We observe Gt , . . . ,Gn, Q1, . . . ,Qn and where Gt ’s and Qt ’s aretransformations of factors It ,Yt , respectively.
We want to predict G ’s and Q’s
Solution
We need to transform G ’s and Q’s into I ’s and Y ’sThen predict I ’s and Y ’s (by a VECM model)And finally transform I ’s and Y ’s back into G ’s and Q’s
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Reminder from Previous Presentation
We observe Gt , . . . ,Gn, Q1, . . . ,Qn and where Gt ’s and Qt ’s aretransformations of factors It ,Yt , respectively.
We want to predict G ’s and Q’s
Solution
We need to transform G ’s and Q’s into I ’s and Y ’s
Then predict I ’s and Y ’s (by a VECM model)And finally transform I ’s and Y ’s back into G ’s and Q’s
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Reminder from Previous Presentation
We observe Gt , . . . ,Gn, Q1, . . . ,Qn and where Gt ’s and Qt ’s aretransformations of factors It ,Yt , respectively.
We want to predict G ’s and Q’s
Solution
We need to transform G ’s and Q’s into I ’s and Y ’sThen predict I ’s and Y ’s (by a VECM model)
And finally transform I ’s and Y ’s back into G ’s and Q’s
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Reminder from Previous Presentation
We observe Gt , . . . ,Gn, Q1, . . . ,Qn and where Gt ’s and Qt ’s aretransformations of factors It ,Yt , respectively.
We want to predict G ’s and Q’s
Solution
We need to transform G ’s and Q’s into I ’s and Y ’sThen predict I ’s and Y ’s (by a VECM model)And finally transform I ’s and Y ’s back into G ’s and Q’s
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Likelihood Computation
Ith Gt
The log-likelihood function of transformation
l(g, θ, σ) =n∑
i=1
(log fi (h
−1(g;σ); θ) + log∣∣∣ 1h′(h−1(g;σ);σ)
∣∣∣)f is density corresponding to VECMθ parameters of the VECMσ variance of collateral individual factor
h(σ, ι) = ϕ(− ισ )− exp{ι+ 1
2σ2}ϕ(− ι
σ − σ), ϕ− normal c.d.f.
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Conditions for consistence
R0 Condition of identification
θ 6= θ0 ⇒ f (•|θ) 6= f (•|θ0)
R1 ExpectationThe expectation
E log f (yt , θ) =
∫ ∞−∞
log f (yt , θ)f (yt , θ0)dy
exists.
Additional conditions for consistence
Θ is a compact subset of Rn
E[supθ∈Θ ln |f (z |θ)|] <∞
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Condition for asymptotic normality
R2 DifferentiabilityThe log-likelihood function log LT (θ) is at least twicecontinuously differentiable in an open interval around θ0.
Additional conditions for asymptotic normality:
θ ∈ interior(Θ)f (z |θ) > 0 on a neighborhood N of θ0∫
supθ,σ∈N ‖∇θf (z |θ)‖dz <∞,∫supθ,σ∈N ‖∇θθf (z |θ)‖dz <∞
J : = E(∇θ ln f (z |θ0)(∇θ ln f (z |θ0)T ) exists and is non-singularE supθ∈N ‖∇θθf (z |θ)× ln f (z |θ)‖ <∞
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Joint estimation of VECM and σ
σc : Original 0.13, estimated 0.0, likelihood ratio test ∗ ∗ ∗significant
σr : Original 0.05, estimated 0.03, LR insignificant
I residential I commercial
0.2 0.4 0.6 0.8 1.0
400
450
0.2 0.4 0.6 0.8 1.0
400
450
Likelihood function in σ with optimal θσ 0.01 0.02 0.03 0.04 0.05 0.06
I residental 481.78 481.87 481.96 481.81 481.26 480.74
I commercial 469.27 468.81 468.02 466.93 465.53 463.83
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio
Joint Estimationof Parameters of
MortgagePortfolio
Jaroslav Dufek,Martin Smıd,Petr Gapko
Data
Factors
Statistics
Estimation of σ
Likelihood
Asymptotics
Discussion
Interpretation
Cointegration of I indicates connection between LGD and macro
Lack of cointegration of Y indicates “exogenity” of default rate
Jaroslav Dufek, Martin Smıd, Petr Gapko Joint Estimation of Parameters of Mortgage Portfolio