Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage...
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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
Thus, we took the US nationwide data
mortgage delinquency (= PD)charge-off (= PD × LGD)
for
residential mortgagescommercial mortgages
Thus, we in fact examined a hypothetical nationwide USportfolio
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
The Data - Losses
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.1 0.11 0.12
1992 1998 2004 2010 2016
Qr
0 0.02 0.04 0.06 0.08
0.1 0.12 0.14
1992 1998 2004 2010 2016
Qc
0 0.05
0.1 0.15
0.2 0.25
0.3 0.35
1992 1998 2004 2010 2016
LGDr
0 0.05
0.1 0.15
0.2 0.25
0.3 0.35
0.4 0.45
1992 1998 2004 2010 2016
LGDc
Qr, LGDr - residential, Qc, LGDc - commercial
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
The Data - Factors (Naive Dynamics)
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2 2.1 2.2 2.3
1992 1998 2004 2010 2016
Yr
-0.4-0.35
-0.3-0.25
-0.2-0.15
-0.1-0.05
0 0.05
1992 1998 2004 2010 2016
Ir
1 1.2 1.4 1.6 1.8
2 2.2 2.4
1992 1998 2004 2010 2016
Yc
-0.6-0.5-0.4-0.3-0.2-0.1
0 0.1 0.2
1992 1998 2004 2010 2016
Ic
Yr, Ir - residential, Yc, Ic - commercial
σ speculatively determined from volatility of HPIr , HPIc respectively,as σr = 0.05, σc = 0.13.
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
Y Explanation Candidates
Y roughly identified with disposable wealth ofhousehold/companymacro candidate variables
GDP (both residential and commercial)PI - personal income (residential)IP - industrial production (commercial)
vars suggested by literatureHPI - house price index (Case et. al. 1995)
Y residential Y commercial
9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8
1992 1998 2004 2010 2016
GDP
9.1
9.2
9.3
9.4
9.5
9.6
9.7
1992 1998 2004 2010 2016
PI
4.8 4.9
5 5.1 5.2 5.3 5.4 5.5
1992 1998 2004 2010 2016
HPIr
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
2 2.1 2.2 2.3
1992 1998 2004 2010 2016
Yr
9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8
1992 1998 2004 2010 2016
GDP
4.55 4.6
4.65 4.7
4.75 4.8
4.85 4.9
1992 1998 2004 2010 2016
IP
4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9
1992 1998 2004 2010 2016
HPIc
1 1.2 1.4 1.6 1.8
2 2.2 2.4
1992 1998 2004 2010 2016
Yc
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
I Explanation Candidates
I roughly identified with house price index (HPI)
macro candidate variables
HPIr / HPIc - residential / commercial price indices
vars suggested by literature
U - unemployment, GDP growth (Qi at. al. 2009)
I residential I commercial
4.8 4.9
5 5.1 5.2 5.3 5.4 5.5
1992 1998 2004 2010 2016
HPIr
1.3 1.4 1.5 1.6 1.7 1.8 1.9
2 2.1 2.2 2.3
1992 1998 2004 2010 2016
U
-0.025-0.02
-0.015-0.01
-0.005 0
0.005 0.01
0.015 0.02
1992 1998 2004 2010 2016
d_GDP
-0.4-0.35
-0.3-0.25
-0.2-0.15
-0.1-0.05
0 0.05
1992 1998 2004 2010 2016
Ir
4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9
1992 1998 2004 2010 2016
HPIc
4.55 4.6
4.65 4.7
4.75 4.8
4.85 4.9
1992 1998 2004 2010 2016
IP
-0.025-0.02
-0.015-0.01
-0.005 0
0.005 0.01
0.015 0.02
1992 1998 2004 2010 2016
d_GDP
-0.6-0.5-0.4-0.3-0.2-0.1
0 0.1 0.2
1992 1998 2004 2010 2016
Ic
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
Statistical Analysis
According to ADF test, none of
Yr , Ir ,Yc , Ic ,GDP,PI , IP,U,HPIr
is either stationary or trend stationary (HPIc was not treatedbecause its series is shorter and explanation power poor)
Dynamics of the first differences could be studied
However, we also tested for cointegration (long term dynamics).
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
Statistical Analysis - Cointegration
Engle-Granger tests
F. Significant regressors Unit root of residuals LagsIc U PI IP GDP HPIr rejected 0Ir PI IP GDP HPIr rejected 0
Yc U PI IP HPIr not rejected 4Yc U PI IP HPIr not rejected 4
⇒ only Ix found cointegrated with macro vars
Tests remain positive after removal of PI and U.
Johansen testVAR involving
VAR involving Ic Ir Yc Yr IP GDP HPIr
Lag selection - HQC: 2, BIC: 1, AIC: max, LR: max - 2 chosen
Johansen test (with exogenous FEDRate, ∆U and ∆PI - Trace:3, LMax: 2 - 3 chosen
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
VECM model
Partial VECM model estimated (Johansen 1992) with lag 2:
∆Ft = αβ′(Ft−1,Mt−1)+Φ∆Ft−1+ΨEt−1+Et , rank(β) = 3
Ft = (Ict , Irt ,Yct ,Yrt),Mt = (IPt ,GDPt ,HPIrt),Et = (FEDRatet ,∆Ut ,∆PIt ,∆IPt ,∆GDPt ,∆HPIrt)
Insignificant regressors removed
Resulting model∆Ict−1 ∆Yct−1 ∆Irt−1 ∆Yrt−1 α1 α2 α3 FEDRt−1 ∆GDPt−1 ∆HPIrt−1
∆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