University of Groningen Life cycle behavior under uncertainty van Ooijen, Raun IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2016 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): van Ooijen, R. (2016). Life cycle behavior under uncertainty: Essays on savings, mortgages and health [Groningen]: University of Groningen, SOM research school Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 03-05-2018
Life cycle behavior under uncertainty Essays on savings, mortgages and health PhD thesis to obtain the degree of PhD at the University of Groningen on the authority of the Rector Magnificus Prof. E. Sterken and in accordance with the decision by the College of Deans. This thesis will be defended in public on Thursday 21 January 2016 at 12:45 hours by Raun van Ooijen born on 7 November 1983 in Oss
Supervisor Prof. R.J.M. Alessie Co-supervisor Dr. A.S. Kalwij Assessment Committee Prof. M. De Nardi Prof. R.H. Koning Prof. A.H.O. van Soest
y k y k+1 [y k, y k+1 ] : k = 1,..., 5 (, 15], ( 15, 5], ( 5, 5], (5, 15], [15, ) p p(1 p)
y k y k+1
Y Y R R = 0 R = 1 (R = 1) (R = 1) (R = 1 X) = Φ(X β), R X Φ P r(r = 1) R
Fi r(y) = P i(y y R = r) p r ik, k = 1,, 5 pr ik = p(y k Y i y k+1 R = r) r = {0, 1} p r ik F r ik(y k ) = P r i (Y y k R = r) k l=1 p r il. i µ r i σr i µ r i, σ r i 6 k=1 [ ( Fik(y r ln(yik ) µ r )] 2 i k ) Φ σi r, µ r i = E[ln(Y i ) R = r] σi r2 = [ln(y i ) R = r] E[Y i R] = exp(µ r i + 1 2 σr i 2 ) [Y i R] = exp(2µ r i + 2σr i 2 ) exp(2µ r i + σr i 2 ) Y Y X = x [Y X = x] = E[Y 2 X = x] (E[Y X = x]) 2 E(Y 2 X = x) = E[E(Y 2 R, X = x)] = E(Y 2 R = 0, X = x)p r(r = 0 X = x) + E(Y 2 R = 1, X = x)p r(r = 1 X = x) E(Y 2 R = r, X = x) = [Y R = r, X = x] + (E[Y R = r, X = x]) 2 Y E(Y X = x) =
E[E(Y R, X = x)] = P r(r = 0 X = x)e(y X = x, R = 0) + P r(r = 1 X = x)e(y X = x, R = 1)
s 5 s i = β 0 + δ a ai + β 1 i + β 2 µ i + x iθ + ϵ i, a=2 a µ i x i
s i = β 0 + 5 δ a ai + β 1 i + β 2 µ i + β 3 0 i + β 4 µ 0 i + x iθ + ϵ i, a=2 i 1 i 0 i 0 i
ˆβ1 / s =.005.382/.121 =.0158
Expects a reform (fraction) Uncertain about a refrom (fraction) 0.2.4.6.8 1 Election Election Follow up survey Spring Agreement Questionnaire Election Election 2003w1 2005w1 2007w1 2009w1 2011w1 2013w1 2015w1 < < < < < <
P r(r = 1) Density 0 5 10 15.2.4.6.8 1 Pr(reform) < < < < < >
H 0 : µ 1 = µ 2 = µ 3 H 0 : µ = µ H 0 : 1 = 2 = 3 H 0 : =
a b SD( t t 1 ) t 2012Q1 2008Q1
µ µ 0 µ 0 < < < < < R 2
%
< >
< >
Density 0.2.4.6.8 -.5 0.5 1 1.5 2 2.5 3 3.5 Linear prediction (mortgage risk)
> 100%
χ 2 p p p χ 2 p p p χ 2 p p p
χ 2 p p p χ 2 p p p χ 2 p p p
χ 2 F p
χ 2 F p χ 2 F p χ 2 F p χ 2 F p
χ 2 F p χ 2 F p χ 2 F p χ 2 F p
p p χ 2 p
µ 1 µ 2
t t 1 t 2
t t + 2
% %
300 (a) Married 300 (b) Widowed 250 250 1000 euro 200 150 100 1000 euro 200 150 100 50 50 0 0 65 70 75 80 85 90 65 70 75 80 85 90 Age Age 300 Married 300 Widowed 250 250 1000 euro 200 150 100 1000 euro 200 150 100 50 50 0 0 65 70 75 80 85 90 65 70 75 80 85 90 Age Age 300 Married 300 Widowed 250 250 1000 euro 200 150 100 1000 euro 200 150 100 50 50 0 65 70 75 80 85 90 Age 0 65 70 75 80 85 90 Age
200 Married 200 Widowed 150 150 1000 euro 100 1000 euro 100 50 50 0 0 65 70 75 80 85 90 65 70 75 80 85 90 Age Age 200 Married 200 Widowed 150 150 1000 euro 100 1000 euro 100 50 50 0 0 65 70 75 80 85 90 65 70 75 80 85 90 Age Age 70 Married 70 Widowed 60 60 Ownership rate % 50 40 30 20 Ownership rate % 50 40 30 20 10 10 0 65 70 75 80 85 90 Age 0 65 70 75 80 85 90 Age
70 Married 70 Widowed 60 60 Ownership rate % 50 40 30 20 Ownership rate % 50 40 30 20 10 10 0 0 65 70 75 80 85 90 65 70 75 80 85 90 Age Age 70 Married 70 Widowed 60 60 Ownership rate % 50 40 30 20 Ownership rate % 50 40 30 20 10 10 0 0 65 70 75 80 85 90 65 70 75 80 85 90 Age Age 70 Married 70 Widowed 60 60 Ownership rate % 50 40 30 20 Ownership rate % 50 40 30 20 10 10 0 65 70 75 80 85 90 Age 0 65 70 75 80 85 90 Age
300 250 200 Widowed Upper income tertile Middle income tertile Bottom income tertile 1000 euro 150 100 50 0 65 70 75 80 85 90 Age 120 100 Homeowner Renter Widowed 80 1000 euro 60 40 20 0 65 70 75 80 85 90 Age
t t + 2 % t t + 2 % t t + 2 t t + 2
t t + 2 % t t + 2 % t t + 2 t t + 2
1000 euro 300 250 200 150 100 Net Worth (mean) 1000 euro 300 250 200 150 100 Net Worth (median) No No No No Minor Minor No Minor Severe No Minor/Severe Dead No Severe Severe 50 50 0 2005 2006 2007 2008 2009 0 2005 2006 2007 2008 2009 120 Net Financial Wealth (mean) 120 Net Financial Wealth (median) 100 100 80 80 1000 euro 60 40 1000 euro 60 40 20 20 0 2005 2006 2007 2008 2009 0 2005 2006 2007 2008 2009 60 60 Ownership % 50 40 30 20 10 Fraction of total assets % 50 40 30 20 10 0 2005 2006 2007 2008 2009 0 2005 2006 2007 2008 2009
e e < < < >
SRH SRH SRH SRH SRH
SRH SRH SRH SRH SRH SRH
SRH SRH SRH SRH x SRH SRH SRH
SRH SRH SRH SRH SRH SRH SRH SRH x
SRH x y x β ŷ = x ˆβ SRH β β SRH ˆβ
i t x it yit x it y it = x itβ + ε it, t = 1,..., T, ε it SRH x it ε it x it N(0, 1) β ε it ε it c i u it ε it = c i + u it c i NID(0, σc 2 ) u it N(0, σu) 2 cov(c i, u it ) = 0, t = 1,..., T. σ 2 u = 1 σ 2 c var(ε it ) = 1 u it u it u it = γu it 1 + ζ it ζ it NID(0, σ 2 ζ). var(ε it ) = 1 σ 2 ζ = σ2 u (1 γ 2 ) = (1 σ 2 c ) (1 γ 2 )
SRH SRH it y it SRH it = L λ g L 1 < y it λ g L, L = 1,, 5; g = 1,..., G, λ g = (λ g 1, λg 2, λg 3, λg 4 ) g λ g 0 = λg 5 = g β β SRH θ = (β, σc 2, γ) g SRH λ g t, t = 1,..., 4 y i1 = x i1β g 1 + ε i1 y i2 = x i2β g 2 + ε i2 y i3 = x i3β g 3 + ε i3 y i4 = x i4β g 4 + ε i4, G = 5
ε i = (ε i1, ε i2, ε i3, ε i4 ) x i = (x i1, x i2, x i3, x i4 ) 0 1 ρ g 21 ρ g 31 ρ g 41 ε i x i NID 0 0, ρ g 21 1 ρ g 32 ρ g 42 ρ g 31 ρ g 32 1 ρ g 43. 0 ρ g 31 ρ g 32 ρ g 43 1 g ξ g = (η g 1,..., ηg 4, ) ρg η g t = (β g t, λ g t ), t = 1,..., 4 ρ g = (ρ g 21, ρg 31, ρg 41, ρg 32, ρg 42, ρg 43 ) η g 1 =... = ηg 4 = ηg = (β g, λ g ) θ g = (β g, λ g, ρ g ) β g = β ρ g = ρ ε it c i u it ρ σ 2 c γ ρ 21 = ρ 32 = ρ 43 = (1 γ)σ 2 c + γ ρ 31 = ρ 42 = (1 γ 2 )σ 2 c + γ 2 ρ 41 = (1 γ 3 )σ 2 c + γ 3. γ σ 2 c β ŷ it = x it ˆβ + c i + ũ it. x ˆβ it SRH
c i + ũ it x ˆβ it c i ˆσ c 2 ũ i1 1 ˆσ c 2 ζ it, t = 2,, T (1 ˆγ 2 )(1 ˆσ c 2 ) ũ it u it SRH
SRH SRH SRH SRH SRH SRH SRH SRH SRH
SRH SRH SRH
SRH β γ σc 2
SRH SRH yit β σ 2 c γ y it y it 1 (1 γ) σ 2 c + γ = 0.70 σ 2 c γ ˆσ 2 c = 0.161 ˆγ = 0.856 yit SRH SRH β SRH
Pr (SRH it = l SRH it 1 = k, x it, x it 1 ) = Pr (ˆλg l 1 x ˆβ it < ε it < ˆλ g l x ˆβ ˆλ g it k 1 x ˆβ it 1 < ε it 1 < ˆλ g k x ˆβ ) it 1 = P (ˆλ g l 1 x ˆβ it < ε it < ˆλ g l x ˆβ, ˆλ g it k 1 x ˆβ it 1 < ε it 1 < ˆλ g k x ˆβ, it 1 ˆρ 12 ) P (ˆλ g k 1 x ˆβ it 1 < ε it 1 < ˆλ g k x ˆβ), it 1 k, l = 1,..., 5 ˆρ 12 = (1 ˆγ) ˆσ 2 c + ˆγ
SRH
SRH SRH SRH SRH SRH N = 24, 486 N = 16, 720
N = 16, 720 t 1 \ t t 1 \ t N = 163, 695
N = 16, 720 λ g1,1 λ g1,2 λ g1,3 λ g1,4 λ g2,1 λ g2,2 λ g2,3 λ g2,4 λ g3,1 λ g3,2 λ g3,3 λ g3,4 λ g4,1 λ g4,2 λ g4,3 λ g4,4 λ g5,1 λ g5,2 λ g5,3 λ g5,4 γ σc 2 G = 5
N = 16, 720 λ g1,1 λ g1,2 λ g1,3 λ g1,4 λ g2,1 λ g2,2 λ g2,3 λ g2,4 λ g3,1 λ g3,2 λ g3,3 λ g3,4 λ g4,1 λ g4,2 λ g4,3 λ g4,4 λ g5,1 λ g5,2 λ g5,3 λ g5,4 γ σc 2 G = 5
SRH N = 16, 720 t \ t + 1 t \ t + 1
t \ t + 1
t \ t + 1
-2-1.5-1 -.5 0 Males Females 20 40 60 80 100 Age
Males -2-1.5-1 -.5 0 Higher Lower Intermediate 20 30 40 50 60 70 80 90 100 Age Females -2-1.5-1 -.5 0 Higher Lower Intermediate 20 30 40 50 60 70 80 90 100 Age
[28, 41] [22, 511; 40, 000]
R 2
t t λ D (t X, v) = λ D 0 (t) exp(x β D + v), X K D D β D λ D 0 ( ) v t t λ R (t Z, w) = λ R 0 (t) exp(z β R + w), Z K R R X β R λ R 0 ( ) w
m λ C m(t X, u) = λ m 0 (t) exp(x β m C + u), βc m K D λ m 0 ( ) u (v = v i ) = p i N i=1 p i = 1 N N
β D β R λ D 0 ( ) λ R 0 ( ) f(v) g(w) βc m λm 0 ( ) h m (u) m k(v, w)
exp(β D ) n + 1 n
p
p p p
survivor function 0.0 0.2 0.4 0.6 0.8 1.0 duration until 1st claim disability duration 95% conf.interval 500 1000 1500 2000 2500 3000 duration in days
exp(βd) exp(β C m ) m = 1, 2, 3 p βd β C m (u = u 1) = 1 (u = u2) = p γ p p p p u1 u2 p γ exp (βd) p exp (β C 1 ) p exp (β2 C ) p exp (β3 C ) p 2
exp (β D ) p exp (β D ) p 2 p p v 1 v 2 p γ exp(β D ) p β D (v = v 1 ) = 1 (v = v 2 ) = p γ
exp(βd) p βd (v = v1) = 1 (v = v2) = p γ p p p p v1 v2 p γ exp (βd) p exp (βd) p exp (βd) p exp (βd) p 2
exp (β D ) p exp (β D ) p 2 p p v 1 v 2 p γ exp(β D ) p β D (v = v 1 ) = 1 (v = v 2 ) = p γ
β C m p β C m m = 1, 2, 3 (u = u1) = 1 (u = u2) = p γ p p p u1 u2 p γ exp (β C 1 ) p exp (β2 C ) p exp (β3 C ) p 2
exp (β R ) p exp (β R ) p exp (β R ) p exp (β R ) p exp (β R ) p 2 p p p p p w 1 w 2 w 3 p 1 p 2 γ exp(β R ) p β R (w = w i ) = p i γ
exp (β R ) p exp (β R ) p 2 p p w 1 w 2 w 3 p 1 p 2 γ exp(β R ) p β R (w = w i ) = p i γ
λ D (t X, v) = λ D 0 (t) exp(x β D + v), S D (t X, v) = ( t 0 ) λ D (s X, v)ds. D k k = 1,..., n D X k v v N D v {v 1,..., v N D} δk D = 0 D k L = n D k=1 (N D p l S D (D k X k, v l )λ D (D k X k, v l ) δd k l=1 p l = (v = v l ) l = 1,..., N D ), λ R (t Z, w) = λ R 0 (t) exp(z β R + w), S R (t Z, w) = ( t 0 ) λ R (s Z, w)ds. R jk j k = 1,..., n R Z jk w w N R w {w 1,..., w N R} k j k δjk R = 0 R jk
L = n R k=1 (N R j k ) p l S R (R jk Z jk, w l )λ R (R jk Z jk, w l ) δr jk, l=1 j=1 p l = (w = w l ) l = 1,..., N R λ C m(t X, u) = λ m 0 (t) exp(x β m C +u), S C m(t X, u) = ( t 0 ) λ C m(s X, u)ds. C km k = 1,..., n D m X k u m u m N C m u m {u m 1,..., u m N C } m = 1 m = 2 m = 3 δ C km = 1 k m δkm C = 0 m L = n D k=1 ( N m C ) p lm Sm(C C km X k, u m l )λ C m(c km X k, u m l ) δc km, l=1 p lm = (u m = u m l ) l = 1,..., N C m
z p
p p