Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage...

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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

Transcript of Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage...

Page 1: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 2: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 3: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 4: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 5: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 6: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 7: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 8: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 9: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 10: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 11: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 12: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 13: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 14: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 15: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 16: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 17: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 18: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 19: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 20: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 21: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 22: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 23: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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

Page 24: Joint Estimation of Parameters of Mortgage Portfolio · Joint Estimation of Parameters of Mortgage Portfolio Jaroslav Dufek, Martin Sm d, Petr Gapko Data Factors Statistics Estimation

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