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102-34 5 67 8.902 +:; < ! += >public.econ.duke.edu/~psarcidi/tesis_07_03_05.pdf9 Mz z M z M uR 9{Nx z ` sx

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Page 1: 102-34 5 67 8.902 +:; < ! += >public.econ.duke.edu/~psarcidi/tesis_07_03_05.pdf9 Mz z M z M uR 9{Nx z ` sx

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1

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s.t. ζi,s,t = Yi,s,t + (1 + r)Ai,s,t−1 −Ai,s,táLeMè

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lnYi,s,t+1 = µs,t+1 (Xi,s,t+1) + θi,1αs,t+1,1 + θi,2αs,t+1,2 + εi,s,t+1.

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Ö�Ë4É Ç¨ÍÆ�Ó²É�ʦÃ}l'É�È�Ã7ÍÇ ÉÌ`×,Í�ÅsÌDÈ�ϨÅ�Ì�æÒ!Ç8Í�Î�È�ϨÅ�ÎwÉÑ/È2ÍÆ`Ê7ÍMÉ×RæÇ,Ò¨Ü�i�ÍÆ�Ó²É�ʦÃ}lsæÇ,Ò�ÍǨÅ�Ê7ÍMÉ×RæÇ,Ò�ÈqÉ�ÚMÅsÌ«Ñ�É�Æ2ŲÍ�Î�ÖfÍÈ�Ï4Ø,Æ2ÍÖ,Ê7Å/Ó�Ì�Üs Ð,Ø,ØfÍ!Ì2Å`È�Ï�É�È�é�ÅmǨÍÆ�Ó²É�ʦÃ}l�Å`È�ϨÅ`Ê7ÍMÉ×RæÇ,Ò�æÇSÈ�ϨÅDߨÆ2Ì�È�ØfÅ/Æ�Ã7Í[×�Ì2Í�È�Ï�É�È

αh,1 = 1Ü

k�Ì2Ì�Ð,Ó�æÇ,Ò�È�Ï�É�ÈXÃ7ÌDæǨ×,Å/ØfÅ/Ǩ×,Å/Ç�ÈaÍ�Î�È�ϨÅ�Å/Æ�Æ2ÍÆ`È2Å/Æ�Ó�Ì

{Ui,h,t}Tt=1

Î ÍÆ�Ó È�ϨÅ�ÑsÍ'ÛÉ�Æ�à É�ǨÑsÅ�Ó²É�È�Æ�Ã|æÕÍ�Î�Ï,æÒ!Ï8Ì2Ñ2ϨÍ�ÍÊ�Ê7ÍÒÅ�É�Æ�Ç,æÇ,ÒMÌ�Î Æ2ÍÓ)È�ϨÅ`רÉ�ÈqÉRÜ s ÍʦÛ�æÇ,Ò�È�ϨÅDÌ�ËRÌ�È2Å/ÓJÍ�Î�ÅYy�Ð�É�È�Ã7ÍǨÌ�È�Ï�É�È�ÑsÍÓ�ÅsÌ�Î Æ2ÍÓ ÅYy�Ð�É�È�æÇ,Ò�È�ϨÅDרÉ�ÈqÉ*á Ê7Å9Î È�Ï�É�Ǩ×�Ì�Ã7×,Å'è1È2ÍaÈ�ϨÅÈ�ϨÅsÍÆ2Å/È�Ã7Ñ�É�Ê+ÑsÍ'ÛÉ�Æ�à É�ǨÑsÅmØ,Æ2Ås×RÃ7Ñ/È2Ås×-Ö[ËEÈ�ϨÅDÎNÉÑ/È2ÍÆ~Ì�È�Æ�ШÑ/È�Ð,Æ2Å«é�Å«ÍÖ,ÈqÉ�æÇ-È�ϨÅ`ÎwÉÑ/È2ÍÆ�Ê7ÍMÉ×RæÇ,ÒM̪ÍÇ�È�ϨÅDߨÆ2Ì�È�ÎNÉÑ/È2ÍÆ�Î Æ2ÍÓ

cov(lnYi,h,t, lnYi,h,t′ | X

)

cov(lnYi,h,1, lnYi,h,t′ | X

) =αh,tαh,t′σ

αh,t′σ2θ

= αh,t, t 6= t′.

¥ ½ ê1�Qø�ÿsïNð�;ñ�ëqïNð ú9ÿªúbù,ï���ø�íëqî�ëqì�ø�ïNø�îNü�úbù�ï���ø=KZí�ü õ'ñ���ð ñML+ñ�ú9ü ï�ù�ó�ÿ�ñ�ïNðòúbÿDð ü�ø�ü ï�ë��Qôòð üL�Qø��Dð ÿDëbíQí!ø�ÿ��'ð�*$P��¥Ý¾ ��ø�ø7�5úsø�QîNð ÷6¦ZP§�¯�¯�¨4�

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·�æÛMÅ/Ç-È�ϨÅDÊ7ÍMÉ×RæÇ,ÒMÌ~ÎwÍÆ~É�ʦÊ<È�æÓ�Å`ØUÅ/Æ�Ã7Í�×,Ì�ÞRé�Å«Ñ�É�Ç-È�ϨÅ/Ç-Æ2ÅsÑsÍ'ÛMÅ/Æ~È�ϨÅDÛ!É�Æ�à É�ǨÑsÅmÍ�ΣÈ�ϨÅ�ÎwÉÑ/È2ÍÆ�Î Æ2ÍÓ È�ϨūÅYy[Ð�É�È�Ã7ÍǨÌ

σ2θ =

cov (lnYi,h,1, lnYi,h,t | X)

αh,t, t = 1, ...T.

Ý�ϨÅ`Û!É�Æ�à É�ǨÑsÅsÌ~Í�ΣÈ�ϨÅDÐ,Ç,Ã^y�ШÅ/ǨÅsÌqÌ2ÅsÌ�ÎwÍÆ~É�ʦÊtÉ�Æ2Å`Æ2ÅsÑsÍ'ÛMÅ/Æ2Ås×�Î Æ2ÍÓ

var (lnYi,h,t | X) − α2h,tσ

2θ = σ2

εh,t.

é�ϨÅ/Æ2Å�é�Å�Ú�ǨÍ�é È�ϨÅ�Ê7Å9Î È�Ï�É�Ǩ×�Ì�Ã7×,Å�Î Æ2ÍÓFÈ�ϨÅ�רÉ�ÈqÉ�É�Ǩ×�È�ϨÅ�Ø,Æ2ÅsÑsÅsÅs×RæÇ,ÒmÉ�Æ�Ò!Ð,Ó�Å/Ç�È�Ü7b�ΨÖfÍÈ�ÏθiÉ�Ǩ×

{εi,h,t}Tt=1

É�Æ2Å�ǨÍÆ�Ó²É�ʦʦË×RÃ7Ì�È�Æ�æÖ,Ð,È2Ås× È�ϨÅ/æÆ�×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍǨÌ~É�Æ2ŪÃ7×,Å/Ç�È�Ã|ß�Ås×*Ö[˲Æ2ÅsÑsÍ�ÛMÅ/Æ�æÇ,Ò�È�ϨÅ/æÆ�Û!É�Æ�à É�ǨÑsÅsÌ�É�Ç¨× È�ϨÅ�ÉÌ2Ì2Ð,Ó�Ø,È�Ã7ÍÇ È�Ï�É�È�È�ϨÅ/æÆ�Ó�Å�É�ǨÌ�É�Æ2Ål�Å/Æ2Í,Ü�i�ÍÆ�Ó²É�ʦæÈ�ËÕÃ7Ì`ǨÍÈ`Æ2ÅYy[Ð,æÆ2Ås×4È2Í*Ã7×,Å/Ç�È�Ã|Î ËÕÈ�ϨÅ�×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍǨÌ�Ü���Ì�æÇ,Ò-Ý�ϨÅsÍÆ2Å/Ó�h æÇdkªØ,ØUÅ/Ǩ×RÃ|æCà!Þ<È�ϨÅ�×RÃ7Ì2È�Æ�æÖ,Ð,È�Ã7ÍǨÌÍ�Î+ÖfÍÈ�Ï

θiÉ�Ǩ×

{εi,h,t}Tt=1

Ñ�É�ÇSÖUÅ�ǨÍÇ,Ø�É�ÆqÉ�Ó�Å/È�Æ�Ã7Ñ�É�ʦʦË�Ã7×,Å/Ç�È�Ã|ß�Ås×<Ü:i�ÍÈ�Ã7ÑsÅDÈ�Ï�É�È�ÞRé�Ï,æÊ7Å`é�Å`Ñ�É�Ç�ÎwÍÆ�Ó ÑsÍ�Û!É�Æ�à É�ǨÑsÅsÌ�ÎwÍÆ�Ï,æÒ!ÏÌ2ÑqϨÍ[ÍÊ5Å�É�Æ�Ç,æÇ,ÒM̪Í�ÛMÅ/Æ�È�æÓ�Å!Þ¨é�Å«Ñ�É�Ç-ǨÅ/ÛMÅ/Æ�Î ÍÆ�Ó ÑsÍ'ÛÉ�Æ�à É�ǨÑsÅsÌ�Í�ΣÅ�É�Æ�Ç,æÇ,ÒMÌ�ÉÑ/Æ2Í!Ì2Ì�Ì2ÑqϨÍ[ÍʦæÇ,ÒEÊ7Å/ÛMÅ/Ê7Ì�Ì�æǨÑsÅ«Å�É�Æ�Ç,æÇ,ÒMÌ�É�Æ2ÅDǨÍÈÍÖ¨Ì2Å/Æ�ÛMÅs×-ÍÇSÖUÍÈ�Ï-Ì2ÑqϨÍ[ÍʦæÇ,Ò²Ê7Å/ÛMÅ/Ê7Ì�ÎwÍÆ~É�Ç�ËMÍǨÅ!Ü

·�æÛMÅ/Ç�È�ϨÅ`ǨÍÆ�Ó²É�ʦÃ}l'É�È�Ã7ÍǨÌ�é�Å�n�ШÌ�È~Ó²É×,Å«È2Í�È�ϨÅ`Ï,æÒ!ÏIÌ2Ñ2ϨÍ�ÍÊ+Ì2Ë�Ì�È2Å/Ó Í�Î1Å�É�Æ�Ç,æÇ,ÒMÌ�Þ�Ó²É�Ú[æÇ,Ò*É�Ì�æÓ�Ã¦Ê É�Æ�Ì2Å/È~Í�Î�ǨÍÆ�Ó²É�Ê|çÃ}l'É�È�Ã7ÍǨÌ�È2ͲÈ�ϨÅ�ÑsÍʦÊ7Å/ÒMÅ�Ì�Ë�Ì2È2Å/ÓÙé�ÍÐ,Ê7×ÕÉ�Ó�ÍÐ,Ç�ÈDÈ2Í Ì2Å/È�È�æÇ,Ò È�ϨÅmÌ�æÒ!ÇCáwÉ�Ǩ×SÓ²É�Ò!Ç,æÈ�Ш×,Å'èªÍ�Î1È�ϨÅ�Ð,ǨÍÖ¨Ì2Å/Æ�ÛMÅs×IÑsÍ�Û!É�Æ�à É�ǨÑsÅaÍ�ÎÅ�É�Æ�Ç,æÇ,ÒM̪ÖUÅ/È�é�ÅsÅ/Ç4ÑsÍʦÊ7Å/ÒMÅ�É�Ǩ×SÏ,æÒ!ÏIÌ2ÑqϨÍ[ÍÊ�Å�É�Æ�Ç,æÇ,ÒMÌ�Ü�Ý5ÍEÌ2ÅsÅ«È�Ï,Ã7Ì'ިǨÍÈ�Ã7ÑsÅ«È�Ï�É�È�È�ϨūÐ,ǨÍÖ¨ÌqÅ/Æ�ÛMÅs×�ÑsÍ'ÛÉ�Æ�à É�ǨÑsÅaÍ�Î1Å�É�Æ�Ç,æÇ,ÒMÌæÇ�Ï,æÒ!ÏSÌ2ÑqϨÍ[ÍÊ�É�Ǩ×-ÑsÍʦÊ7Å/ÒMÅ`æÇ-ØfÅ/Æ�Ã7Í[×4àDÃ7Ì

cov (lnYi,h,1, lnYi,c,1 | X) = αc,1σ2θ

á�à��Mè

Ì2ÍÕÌ2Å/È�È�æÇ,Òαc,1 = 1

é�ÍÐ,Ê7×CæÓ�ØfÍ!Ì2Å�ÉIÌ�È�Æ2ÍÇ,Ò4Æ2ÅsÌ�È�Æ�Ã7Ñ/È�Ã7ÍÇCÈ�Ï�É�È�È�ϨÅ�ÑsÍ�Û!É�Æ�à É�ǨÑsÅ�Ã7ÌaØfÍ!Ì�æÈ�æÛMÅ-É�Ç¨× ß,æRÅs× Ö[Ë$È�ϨÅ*Û!É�Æ�à É�ǨÑsÅ×,Å/È2Å/Æ�Ó�æǨÅs× Ã¦Ç È�ϨÅSÏ,æÒ!Ï Ì2ÑqϨÍ[ÍÊ`Ì�ËRÌ�È2Å/Ó-Ü�Ý�ϨÅsÍÆ2Å/Óõj$Ã¦Ç È�ϨÅ�kªØ,ØfÅ/Ǩ×RÃ|æ Ì�ϨÍ�é�Ì È�Ï�É�È*Æ2ÅsÌ�È�Æ�Ã7Ñ/È�Ã7ÍǨÌ�Í�ÎmÈ�Ï,Ã7Ì Ç�É�È�Ð,Æ2ÅÕ×,ÍǨÍÈ�ǨÅsÅsסÈ2ÍÕÖfÅEæÓ�ØfÍ!Ì2Ås×<Ü¡Ý�ϨÅ�Ì�ËRÌ�È2Å/Ó Î ÍÆ�ÑsÍʦÊ7Å/ÒMÅ�Å�É�Æ�Ç,æÇ,ÒM̲Ã7Ì�Ã7×,Å/Ç�È�Ã|ß�Ås× é�æÈ�ϨÍÐ,ȲÉ×,×RæÈ�Ã7ÍÇ�É�Ê~ǨÍÆ�Ó²É�ʦÃ}l'É�È�Ã7ÍǨÌ�Ã|Î�È�ϨÅ×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍÇIÍ�Î

θiÃ7Ì�ǨÍǨÌ�Ë[Ó�Ó�Å/È�Æ�Ã7Ñ!Ü

b�ÎθiÏ�ÉÌ�ÉDÌ�Ë[Ó�Ó�Å/È�Æ�Ã7Ñ~×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍDzÌ�Ð,Ø,ØfÍ!Ì2Å�È�Ï�É�È�ÞMÖfÅsÌ�Ã7×,Å�È�ϨÅ�רÉ�ÈqÉDÍÇEÅ�É�Æ�Ç,æÇ,ÒMÌ�Þ�ÑsÍǨÌ�Ð,Ó�Ø,È�Ã7ÍÇ É�Ǩ×�Ì2Ñ2ϨÍ�ÍʦæÇ,ÒmÑqϨÍÃ7ÑsÅsÌ

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Mi,j = µMj(XMi,j

)+ θiα

Mj + εMi,j , j = 1, ...J,

áLegfMè

é�ϨÅ/Æ2ÅεMi =

(εMi,1, .., ε

Mi,J

) Ã7ÌDɲÛMÅsÑ/È2ÍÆ�Í�Î1Ó�Å�É�ÇKl�Å/Æ2ÍEÓmÐ,È�Ð�É�ʦʦËSæǨ×,Å/ØfÅ/Ǩ×,Å/Ç�È`ÆqÉ�Ǩ×,ÍÓ Û!É�Æ�à É�Ö,Ê7ÅsÌ�ÜεMiÃ7ÌDÉÌ2Ì�Ð,Ó�Ås×�È2ͲÖfÅ

æǨ×,Å/ØfÅ/Ǩ×,Å/Ç�ȪÍ�ÎθiÉ�Ǩ×-Í�Î

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ωiÎ ÍÆ~É�ʦÊ

tÉ�Ǩ×

s.

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É�Æ2Å`Ã7×,Å/Ç[È�Ã|ß�Ås×�Î Æ2ÍÓ

αMj =cov

(lnYi,h,1,Mi,j | X,X

M)

σ2θ

, j = 1, ..., J.

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cov(lnYi,c,t,Mi,j | X,X

M)

= αc,tαMj σ

2θ ,

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εMi,j

}Jj=1

É�Ǩ×{εi,c,t}

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E

1 + r

1 + ρ

(ζi,t+1ηt (Ki,t, ξi,t)

ζi,tηt (Ki,t+1, ξi,t+1)

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b<ÉÌ2Ì�Ð,Ó�Å�È�Ï�É�ÈKi,t

É�Ǩ×Ki,t+1

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1 + r

1 + ρ

(ζi,t+1ηt (Ki,t, ξi,t)

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é�Å«ÍÖ,ÈqÉ�æÇ

ln

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ζi,t

)=

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é�ϨÅ/Æ2Åπi,t = ξi,t+1 − ξi,t −

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b«ÉÌ2Ì2Ð,Ó�Å�È�Ï�É�ÈaÐ,ǨÅ/Ó�Ø,Ê7Í�ËMÅsסØfÅsÍØ,Ê7Å�Ï�ÉsÛMÅ*Å�É�Æ�Ç,æÇ,ÒMÌ�Í�Î�l�Å/Æ2Í,Üt~�É�Æ�Ç,æÇ,ÒMÌmÎwÍÆmæǨ×RæÛ�Ã7×RÐ�É�Ê7Ìaé�ϨÍIÉ�Æ2ÅEÓ�Ã7Ì2Ì�æÇ,ÒSÎ ÍÆaÍÈ�ϨÅ/ÆÆ2Å�ÉÌ2ÍǨÌ`ÖfÅsÌ�Ã7×,ÅsÌDǨÍǨÅ/Ó�Ø,Ê7Í'Ë[Ó�Å/Ç[È�É�Æ2Å�æÓ�Ø,Ð,È2Ås×4ШÌ2æÇ,ÒSÉ é�Å/æÒ!Ï�È2Ås× ÉsÛMÅ/ÆqÉ�ÒMÅ�Í�Î�Å�É�Æ�Ç,æÇ,ÒMÌ`æÇ4È�ϨÅ�Ì�Ð,Æ�Æ2ÍÐ,Ǩ×RæÇ,ÒSËMÅ�É�Æ2Ì`é�æÈ�Ï×,ÅsÑ/ʦæÇ,æÇ,Ò`é�Å/æÒ!Ï[È2Ì�Ò!æÛMÅ/Ç�Ö�Ë�É�Ç�~�Ø�É�ǨÅsÑ2Ï,Ç,æÚMÍ'ÛaÚMÅ/Æ�ǨÅ/ÊNÜ�����b�Ǩ×RæÛ[Ã7×RÐ�É�ʨÅ�É�Æ�Ç,æÇ,ÒṂʦÃ|ÎwÅ�Ñ/Ë�Ñ/Ê7ÅsÌ�É�Æ2Å�È�ϨÅ/Ç�Ì�æÓ�Ø,ʦÃ|ß�Ås×aÈ2ÍVjªØUÅ/Æ�Ã7Í�×,Ì�Üa¨ÍÆ�Å�ÉÑ2Ï�ØfÅ/Æ�Ã7Í�×<Þ[Ê7ÍÒ�Å�É�Æ�Ç,æÇ,ÒMÌ~É�Æ2Å�Ñ�É�Ê7Ñ/Ð,Ê É�È2Ås×SÉÌ�È�ϨÅ�Ê7ÍÒaÍ�Î5È�ϨÅ�Ø,Æ2ÅsÌ2Å/Ç[È�ÛÉ�ʦШÅDÍ�Î+Å�É�Æ�Ç,æÇ,ÒMÌ�ÎwÍÆ�È�ϨŪØfÅ/Æ�Ã7Í�×E×RÃ7Ì2ÑsÍÐ,Ç[È2Ås×SÉ�Èh)� Ü

a¨ÍƪÅ�ÉÑ2ÏSÌ2ÑqϨÍ[ÍʦæÇ,Ò²Ê7Å/ÛMÅ/Ês = {h, c}

É�Ǩ×*ÎwÍÆ�Å�ÉÑ2ÏSØUÅ/Æ�Ã7Í�×�Í�ΣÅ�É�Æ�Ç,æÇ,ÒMÌt = {1, ..., 5}

b�ߨǨ×-È�Ï�É�È�Ó�Í[×,Å/ʦæÇ,ÒlnYi,s,t

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lnYi,s,t = Xiβs,t + θi,1αs,t,1 + θi,2αs,t,2 + εi,s,t.

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(αh,1,2)Ã7Ì�ǨÍÆ�Ó²É�ʦÃ}l�Ås×-È2Í-à!Ü

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Costi = Ziγ + θi,1αcost,1 + θi,2αcost,2 + ωi.

Ý�ϨÅZiæǨÑ/ʦШ×,Å�È�ϨÅ

XiæÇ�Å�É�Æ�Ç,æÇ,ÒMÌ�Ø,ʦШÌ+ÛÉ�Æ�à É�Ö,Ê7ÅsÌ�È�Ï�É�È£ÉQÄfÅsÑ/È+È�ϨÅ�Ì2ÑqϨÍ[ÍʦæÇ,ÒD×,ÅsÑ/Ã7Ì�Ã7ÍÇaÅ9æRÑ/ʦШÌ�æÛMÅ/ʦËmʦæÚMÅ�ÎNÉ�Ó�æʦË`Ö�ÉÑ2Ú�Ò!Æ2ÍÐ,Ǩ×<Ü

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θi,l ∼

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j=1

πl,jf(θi,l;µl,j, σ

2l,j

)

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) Ü� q äX�,åm���Xu����)�g���pug�

F(M | XM

) u��Uä2�(���SâL��ä��D� â ���µMj

(XM

) u�r�ä��V� q ä�s���yByDu��X�Hu� XM ������� q ä���uâ1�&�7�â-s��Z�9âJt��Q� âJu��u�UM � FUM (uM )

ã

� 5W8�8��w� b�×,Å/Ç�È�Ã|ß�Ñ�É�È�Ã7ÍÇ¡Í�Î�È�ϨŲÓ�Å�É�Ç8Î Ð,ǨÑ/È�Ã7ÍǨÌ�Í'ÛMÅ/Æ�È�ϨÅ/æÆaÌ�Ð,Ø,ØfÍÆ�ÈmÃ7Ì«È�Æ�æÛ�à É�Ê�Ì2æǨÑsÅ�é�Å ÍÖ¨Ì2Å/Æ�ÛMÅMj

Î ÍÆmÅ�ÉÑqÏXM É�Ǩ×

Ñ�É�ÇIÆ2ÅsÑsÍ�ÛMÅ/ÆDÈ�ϨÅmÓ²É�Æ�Ò!æÇ�É�Ê�×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍÇ4Í�ÎUMj

ܪÝ�ϨÅmæÇ[È2Å/Æ2ÑsÅ/Ø,È2ÌmÉ�Æ2Å«Æ2ÅsÑsÍ'ÛMÅ/Æ2Ås×IÎ Æ2ÍÓ ÉÌ2Ì�Ð,Ó�Ås×�l�Å/Æ2ÍEÓ�Å�É�ÇIÍ�ÎUMj

ܪÝ�ϨÅn�ÍæÇ�È�ÎwÍʦÊ7Í�é�ÌEæÓ�Ó�Ås×Rà É�È2Å/Ê¦Ë Ì�æǨÑsÅ

Pr(M < m | XM

)= FUM

(m− µM

(XM

)) Ö�Ë ÉÌ2Ì2Ð,Ó�Ø,È�Ã7ÍÇ�áZk~ç4eMè9Ü Ý�ϨÅ/Ç�Þ�Î Æ2ÍÓáZk~ç4hMèDé�ÅEÑ�É�Ç8ߨǨ×$É�Ç

XM = xMé�ϨÅ/Æ2Å

µM(xM)

= kÉ�Ǩ×

kÃ7Ì�É

J×RæÓ�Å/ǨÌ�Ã7ÍÇ�É�Ê�ÛMÅsÑ/È2ÍÆ�Üw´�Å/È

m = k − µM(xM)

Ì2ÍPr(M < m | XM = xM

)= F (k) .

s æǨÑsÅ�È�ϨÅ�ØfÍæÇ�ÈωÃ7Ì�É�Æ�Ö,æÈ�ÆqÉ�Æ�ËMÞªé�ÅÕÑ�É�Ç ÛÉ�Æ�Ë Ã¦È È2ÍHÃ7×,Å/Ç�È�Ã|Î Ë È�ϨÅ�Î Ð,ʦÊGn�ÍæÇ�È

×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍÇ�Ü�

F9.='B8>5<'�"��� sws��E�²ä$� q �g�:� q ä2��ä��7är-�g�&��ä��¦äX��äX�&��s[u�p� ������� �����S� ������� ��âwã ä�ã¦å"� q ä��vuâ1�&�"�wu��D�âZ� â2ug�6s9u�� X,Z,U,W � q u��}����D�T� q �g�:� q ä9�vug�Ý��uCx�â1�4�pr����bâL�g� âJu������2ä2ä��}u����â1� âJu�� q u��}�Ws �

�������� Support(φ (Z) , µs (X)) = Support (φ (Z)) × Support (µs (X))

ã[�W�.���bâL�g� âJu������2ä2ä �®L¾ ���Qð ü�ëqüwüwó�ì�íQïNð úbÿ�ñ�ëqÿ�ø�ëqüwðòôöõp�!ø�îNø�ôòë«*'ø��Dëqÿ���ð ü�ú9ÿQôöõªì�ë��Qø�ù�úbî5ñ�úbÿ#�9ø�ÿ�ð ø�ÿQñ�ø��®Z¿ êZÿ�ëqôòô%ñ�ëbüwø�ü�Q��ðöï���ë��-�QðöïNð ú9ÿëqôfëqüwüwó�ì�íQïNð úbÿ�ü���fø�ñ�ëqÿaîNø�ôòë«*aë��-�QðöïNðN�/ø�üwø�íë2î�ë��Qðòô ðöï õaëbÿ���ð8�Qø�ÿ�ïNðöù�õ�ù�ó�ÿ�ñ�ïNðòúbÿ�ü1úbùUï���ø�ù�úbîNì y = µ (X, U) ��õó�üwð ÿQ÷�ï���ø�ëqÿëqô õ'üwð ü5ðòÿ6/Dëqï�'�þ�ð ÿ�¦ª©«U�U�[�¨4�

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z+äX�µs (X) = (µs,1 (X) , ..., µs,T (X))

ã � sws��E�²ä$� q �g�

Support(φ (Z)) ⊇ Support (W ) ,

Support (µs (X)) ⊇ Support (Us)

� q ä��¨åA� q äV�²ä����2���E���Y� â2ug�6s µs,t (X)���2äDâL��ä��&� â ~�ä��Au��¡� q ä�s���yByDu��X�1u� X ã � �¡s�u�å7� q ä���uâ1�&�7�â�sX�Z�bâ_t��Q� â2ug�¢u� Us â�sp�>u��CyQ�g�� �

�²äX�Z�9âZ�(�g�Ý���²âL��ä��&� â ~�ä��¤��u��t = 1, ..., T

�vu���ä��)� q s = h, cã

� 5W8�8��w� �ªÇ¨×,Å/Æ�È�ϨÅ`ÑsÍǨ×RæÈ�Ã7ÍǨÌ�Í�Î�È�ϨūÈ�ϨÅsÍÆ2Å/Ó-Þ,é�Å«Ñ�É�Ç-ߨǨ×�ʦæÓ�æÈ~Ì2Å/È2ÌZ−É�Ǩ×SÉ

Z+ Ì2ШÑ2Ï-È�Ï�É�È

Pr(S = 1 | Z ∈ Z−

)= 0

É�Ǩ×

Pr(S = 1 | Z ∈ Z+

)= 1

é�ϨÅ/Æ2Å�é�ÅDÑ�É�Ç�Ì�È�æʦÊ%Î Æ2ÅsÅ/ʦ˲ÑqÏ�É�Ç,ÒMÅDÈ�Ϩŵs (X)

ÜAb�ÇEÈ�ϨÅ�ʦæÓ�æÈ�Ì2Å/È2Ì�Þ[È�ϨÅDÑsÍǨ×RæÈ�Ã7ÍǨÌ�Í�Î+Ý�ϨÅsÍÆ2Å/Ó � É�Æ2Å�ÌbÉ�È�Ã7Ì�ß�Ås×�É�Ǩײé�Å�Ñ�É�ÇÉ�Ø,Ø,ʦË*æÈ�Î ÍÆ�Å�ÉÑqÏ�Ì�ËRÌ�È2Å/Ó Í�Î�Ì2Ñ2ϨÍ�ÍʦæÇ,ÒEÅYy[Ð�É�È�Ã7ÍǨÌ

s�

a,Ð,Æ�È�ϨÅ/ÆDÉÌ2Ì�Ð,Ó�ÅDÈ�Ï�É�È

�X�p�£��Support

(µs (X) , µM

(XM

))⊇ Support

(Us, U

M)

b�×,Å/Ç�È�Ã|ß�Ñ�É�È�Ã7ÍÇSÍ�Î5È�ϨÅGn�ÍæÇ�È�×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍÇ�Í�Î (Us, U

M) Î ÍÆ�Å�ÉÑ2Ï

sÃ7Ì�Ì�È�ÆqÉ�æÒ!Ï[È�Î ÍÆ�é�É�Æ2×-Ì�æǨÑsÅ!Þ�æÇ*È�ϨŪʦæÓ�æÈ�Ì2Å/È�Þ�é�Å�Ñ�É�Ç ÎwÍÆ�Ó

È�ϨÅ`Ê7Å9Î È�Ï�É�Ǩ×-Ì�Ã7×,Å`Í�Î

o�Æ (M ≤ m,Ys ≤ ys | X

M , X)

= FUM ,Us

(m− µM

(XM

), ys − µs (X)

)

é�Ï,Ã7Ñ2Ï�é�Å«Ñ�É�Ç�È�ÆqÉÑsÅ«Ö�Ë�Ñ2Ï�É�Ç,Ò!æÇ,ÒysÉ�Ǩ×

mÜ1Ý�ϨÅDæÇ�È2Å/Æ2ÑsÅ/Ø,È2Ì�Í�Î

µs (X)É�Ǩ×

µM(XM

) É�Æ2Å�ß,æRÅs×�Î Æ2ÍÓ È�ϨÅmÉÌqÌ�Ð,Ó�Ø,È�Ã7ÍÇÈ�Ï�É�È�È�ϨÅ`Ó�Å�É�ǨÌ�á ÍÆ�Ó�Ås×Rà É�ǨÌbè�Í�Î

UM , UsÉ�Æ2ÅVl�Å/Æ2Í,Ü

v ϨÅ/ÇSÈ�ϨÅ`Ð,ǨÍÖ¨ÌqÅ/Æ�ÛÉ�Ö,Ê7ÅsÌDÉ�Æ2Å«Æ2Å/Ø,Æ2ÅsÌ2Å/Ç[È2Ås×SæÇSÈ2Å/Æ�Ó�̪Í�Î1ÅYy[Ð�É�È�Ã7ÍǨÌaá�à!à'è�É�Ç¨× áLegfMè9Þ¨È�ϨÅmǨÅ9æ�È�È�ϨÅsÍÆ2Å/Ó�Ì�Ì�ϨÍ'é;È�Ï�É�È~é�ÅÑ�É�Ç�ǨÍÇ,Ø�É�ÆqÉ�Ó�Å/È�Æ�Ã7Ñ�É�ʦʦË-Ã7×,Å/Ç[È�Ã|Î Ë È�ϨÅ`×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍǨÌ~Í�ΣÈ�ϨÅ�ÎNÉÑ/È2ÍÆ2Ì~É�Ǩ×*È�ϨÅ`Ð,Ç,Ã^y[ШÅ/ǨÅsÌ2Ì2ÅsÌ�ÉÌ�é�Å/ʦÊ+ÉÌ�È�ϨÅDÎNÉÑ/È2ÍÆ�Ê7ÍMÉ×RæÇ,ÒMÌ�Üb�ߨÆ2Ì�È�Ì�ÈqÉ�È2ÅmÉaÈ�ϨÅsÍÆ2Å/Ó È�Ï�É�È�é�æʦÊ�ÖUÅDШÌqÅ9Î Ð,ÊfÎwÍÆ�È�Ï,Ã7Ì�Ø,Ð,Æ�ØfÍ!Ì2Å!Ü

F9.='B8>5<'�"¤� z5äX�Q1

�g�D�Q2

t2ä��2x1u|�����D�{u��¥r-���9âZ�BtX�¦ä�s�� q �g�¦sY�g� â-sÝ���

Q1 = θ +R1

Q2 = θ +R2

x q ä��2ä θ,R1���D�

R2�g��ä|���m�Z�.���1�5��â1�D��ä§y,ä��D��äX�&��x�âZ� q E (θ) < ∞, E (R1) = 0, E (R2) = 0,

� q äT�}u����â1� âJu��6spu�[¨>�6t9â1�¨â�© s

e�x

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� q äXug��ä�� ����äªsX�g� â-s-~�ä��6��u���ä���� q �����D�Bu��«r-���9âZ�BtX�¦ä|���D�w� q äX� q �Cr�ä��>u���r-�g�¨â�s q â1�4�a�4��ã ä�ã � � q �g��)����ä��bâ-s�� âL�¤���m�D�Y� âJu��6s�ãS� q ä��¨å� q ä$�����J�âÝ�����B��ä��6s9â1� âNä�s[u� θ,R1

���D�R2

���2ä«âL��ä��D� â ~�ä(��ã

� 5W8�8��w� s ÅsÅ2uªÍÈ�Ê É�Æ2Ì�Ú�Ã�á�à��gÜ)æ!è9Ü�

` ÍǨÌ�Ã7×,Å/Æ+ШÌ�æÇ,ÒªÍÇ,ʦË`È�ϨÅ1æÇRÎ ÍÆ�Ó²É�È�Ã7ÍÇ�ÍÇmÓ�Å�ÉÌ�Ð,Æ2Å/Ó�Å/Ç[È2Ì�É�Ǩ×mÅ�É�Æ�Ç,æÇ,ÒMÌ+æÇ�Å�ÉÑ2ÏaÌ2Ñ2ϨÍ�ÍʦæÇ,ÒDÌ�ÈqÉ�È2Å!ÜBa¨ÍÆ£É�Ò!æÛMÅ/ÇaÌ2Ñ2ϨÍ�ÍʦæÇ,ÒÊ7Å/ÛMÅ/Ê

sé�Å`Ï�É�ÛMÅmÉ�Ì�ËRÌ�È2Å/Ó Í�ΣÅYy[Ð�É�È�Ã7ÍǨÌ

M1 = µM1(XM

1

)+ θαM1 + εM1ÜÜÜ

MJ = µMJ(XMJ

)+ θαMJ + εMJ

lnYs,1 = µs,1 (Xs,1) + θαs,1 + εs,1ÜÜÜ

lnYs,T = µs,T (Xs,T ) + θαs,T + εs,T

áLeghMè

Ý�ϨÅ�È2ÍÈqÉ�Ê1Ç[Ð,ÓmÖfÅ/ÆmÍ�Î�ÅYy[Ð�É�È�Ã7ÍǨ̫Ã7ÌDÒ!æÛMÅ/Ç8Ö[ËJ · T

Ü�´�Å/ÈLÖUÅaÈ�ϨÅ�È2ÍÈqÉ�Ê1Ç[Ð,ÓmÖfÅ/ÆmÍ�Î�ÎNÉÑ/È2ÍÆ2Ì�Ü s ÐQ�²Ñ/Ã7Å/Ç�ÈaÑsÍǨ×RæÈ�Ã7ÍǨÌ`ÎwÍÆ

Ã7×,Å/Ç�È�Ã|ß�Ñ�É�È�Ã7ÍÇIé�æʦÊ�ÖfÅ

L ≤ 2 · J · T + 1,

È�Ï�É�È�È�ϨÅDÊ7ÍMÉ×RæÇ,ÒMÌ~É�Æ2Å`Ì�ШÑqÏ�È�Ï�É�ȪÉ�È�Ê7Å�ÉÌ�È�È�é�Í�ÅYy[Ð�É�È�Ã7ÍǨÌ�×,Å/ØUÅ/Ǩ×�ÍÇ,ʦË*ÍÇ�È�ϨÅ�ߨÆ2Ì�È�ÎNÉÑ/È2ÍÆ�Þ¨É�È�Ê7Å�ÉÌ�È�È�é�Í�×,Å/ØUÅ/Ǩ×�ÍÇ,ʦË*ÍÇÈ�ϨÅaߨÆ2Ì�ÈmÉ�Ǩ×8ÌqÅsÑsÍǨ×IÎwÉÑ/È2ÍÆ�Þ5É�Ǩ×$Ì2Í*ÍÇ4Ð,Ø8È2Í*È�ϨÅaߨÆ2Ì�È

L − 1ÎNÉÑ/È2ÍÆ2ÌmÉ�Ǩ×8É�È`Ê7Å�ÉÌ�È`È�Ï,Æ2ÅsŲÅYy�Ð�É�È�Ã7ÍǨ̫×,Å/ØfÅ/Ǩ×8ÍÇ$É�ʦÊ

L

ÎNÉÑ/È2ÍÆ2Ì�Ü s ШÑ2Ï¡É�Ç$É�Æ�ÆqÉ�Ç,ÒMÅ/Ó�Å/Ç�Èaé�ÍÐ,Ê7×4ÖfÅ�Ó�ÍÈ�æÛÉ�È2Ås×$Ö�ËIÈ�ϨŲÉÌ2Ì�Ð,Ó�Ø,È�Ã7ÍǨÌ�É�ÖUÍÐ,È«È�ϨÅEÉ�Æ�Æ�æÛÉ�Ê1Í�Î�æÇRÎ ÍÆ�Ó²É�È�Ã7ÍÇHÓ²É×,Å�æÇÈ�ϨÅ`È2Å9æRÈ�Ü

b�Î�é�Å`Æ2Å�É�Æ�ÆqÉ�Ç,ÒMÅ`È�ϨūÅYy[Ð�É�È�Ã7ÍǨÌ�æÇ$áLeghMè�É�Ǩ×�Ø,Ð,È�È�ϨÅ�ÎNÉÑ/È2ÍÆ�Ê7ÍMÉ×RæÇ,ÒMÌ�æÇ�ÉaÓ²É�È�Æ�Ã|æ<Þ,æÈ�é�ÍÐ,Ê7×-Ï�ÉsÛMÅ«È�ϨÅDÎ ÍÆ�Ó

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´�ÍMÉ×RæÇ,ÒMÌ�Î ÍÆ�ÎNÉÑ/È2ÍÆ

θ1 θ2 θ3 ... θL−1 θL

6= 0 0 0 ... 0 0

6= 0 0 0 ... 0 0

6= 0 6= 0 0 ... 0 0

6= 0 6= 0 0 ... 0 0

... ... ... ... ... ...

6= 0 6= 0 6= 0 ... 6= 0 0

6= 0 6= 0 6= 0 ... 6= 0 0

6= 0 6= 0 6= 0 ... 6= 0 6= 0

6= 0 6= 0 6= 0 ... 6= 0 6= 0

6= 0 6= 0 6= 0 ... 6= 0 6= 0

áLe�x[è

i�Í�é«Þ,ÉÌ2Ì�Ð,Ó�ŪÈ�Ï�É�ÈLM

ÎNÉÑ/È2ÍÆ2Ì�Å/Ç�È2Å/Æ�È�ϨÅ~Ó�Å�ÉÌ2Ð,Æ2Å/Ó�Å/Ç�È�ÅYy[Ð�É�È�Ã7ÍǨÌ�É�Ǩ×<Þ[é�æÈ�ϨÍÐ,È�Ê7Í!Ì2Ì�Í�Î�ÒMÅ/ǨÅ/ÆqÉ�ʦæÈ�ËMÞ�Ì�Ð,Ø,ØUÍ!ÌqÅ~é�Å~Ê7Í�ÍÚEÉ�È

s = h.�(×

F9.='B8>5<'�"­¬ ¨��Xu�� � q äT���D�����Wsbâ-saâ1�®� q ä}u���äX�[s)� ��� �g�D�¯� ��� x�ä q �CräT���g���Su�� F (UM , Uh | X,XM) ���D���Wsws��E�²ä|� q �g�

UM , Uh q �CsI�¤�Y�)�Y�_ug��s��Z���Q���Z�E�2ä��2ä�yD�2ä�ssä��D���g� â2ug�ø�Ws«â1�°� �{��� ã+� q ä��,åG� q ä2��u��)�!âÝ�4�Cs{αMj

}Jj=1

, {αh,t}Tt=1

���2ä�âL��ä��&� â ~�ä����/y�_u�u��Uä��>u������g��â?±��g� âJu�����u���ä���� q �Y�)�Y�_u���ã)� q äI�����J�â1�D���7�â-s��Z�9âJt��Q� âJu��6s[u� {θl}Ll=1 ,

{εMj

}Jj=1

���D�{εh,t}

Tt=1

����ä��>u��CyQ�g�� ��²äX�Z�9âZ�(�g�Ý���²âL��ä��&� â ~�ä����Ws|x�ä��1�7ã

� 5W8�8��w� Ý+Í�Å�ÉÌ2Å`ÍÇ-ǨÍÈqÉ�È�Ã7ÍÇ�b1ÒMÅ/ǨÅ/Æ�Ã7Ñ�É�ʦʦË�é�Æ�æÈ2Å`È�ϨūÅYy[Ð�É�È�Ã7ÍǨÌ�æÇHáLeghMè�ÉÌ

yk =

L∑

l=1

θlδk,l + ek.

s Í,ÞfÃ|Î�ÎwÍÆDÅ9æ¨É�Ó�Ø,Ê7ÅlnYh,t

Ã7̪È�ϨÅ�ß,Î È�ÏIÅYy[Ð�É�È�Ã7ÍÇ�Þy5 = lnYh,t − µh,t (X)

Þ%È�ϨÅaÊ7ÍMÉ×RæÇ,Ò-ÍÇIÈ�ϨÅaÈ�Ï,æÆ2×SÎNÉÑ/È2ÍÆαh,t,3 = δ5,3

É�Ǩ×εh,t = e5

Üi�ÍÈ�Ã7ÑsÅ`È�Ï�É�È�Þ�Ì2æǨÑsÅ`È�ϨÅ`ÎNÉÑ/È2ÍÆ�Ï�ÉÌ�ǨÍ�Ç�É�È�Ð,ÆqÉ�Ê+Ì2Ñ�É�Ê7Å!Þ,é�ūǨÅsÅs×SÈ2Í�Ì2Å/È�æÈ�á È�Ï�É�ȪÃ7Ì

δθ = κδ θκ

ÎwÍÆ~É�Ç�Ë-ÑsÍǨÌ�ÈqÉ�Ç[Èκè9ÜGv8Å

É�Ê7Ì2ÍaǨÅsÅs× È2ÍaǨÍÆ�Ó²É�ʦÃ}l�Å`È�ϨÅDÌ�æÒ!Ç�Í�Î+È�ϨÅDÅ9ÄfÅsÑ/È�Í�Î5È�ϨÅ�ÎNÉÑ/È2ÍÆ�Ì�æǨÑsÅ!Þ[Î ÍÆ�Å9æ¨É�Ó�Ø,Ê7Å!ÞRÏ�ÉsÛ�æÇ,Ò�Ó�ÍÆ2Å`Í�Î�ÎNÉÑ/È2ÍÆlÉ�Ǩ×

δl,k > 0Ã7Ì

ÅYy[Ð,æÛÉ�Ê7Å/Ç[ȪÈ2ÍEÏ�ÉsÛ�æÇ,ÒEÊ7ÅsÌq̪Í�Î�È�ϨÅmÎwÉÑ/È2ÍÆ`É�Ǩ×δl,k < 0

ܪÝ5ÍEØ,æÇ�×,Í'é�ÇÕÌ2æÒ!ÇIÉ�Ǩ×SÌqÑ�É�Ê7Å!Þ�é�ÅmǨÍÆ�Ó²É�ʦÃ}l�Å�ÍǨÅ�Ê7ÍMÉ×RæÇ,ÒEÈ2ÍEÍǨÅÎwÍÆ�Å�ÉÑ2Ï-ÎNÉÑ/È2ÍÆ�Ü® Ú �5üm�!ø�ù�úbîNø��bðöù-�UøUë2îNøD��ð ô ô ðòÿQ÷5ïNú�ì�ëqþ/ø�ëbÿ�ë��-�QðöïNð ú9ÿëqôsÿ�úqîNì�ëbô ðN'�ëqïNð ú9ÿ����Uø%ñ�ëbÿ�ð5�'ø�ÿsïNðöù�õ�ï��Qø�ù|ëbñ�ïNúqî,ü ïwîNó�ñ�ïNóQîNø,��ð ï��Qú9ó'ï�ï���ø%ë��-�QðöïNðN�/ø¨üwø�íë2î�ë��Qðòô ðöï õëbüwüwóQì�íQïNð ú9ÿªó�üwð ÿ�÷�ø�²sóëqïNð úbÿ�ü�úbùRï��Qø�ù|úqîNì y = µ (X, θ′α + ε) ù|ú9ô ô ú���ð ÿ�÷�ï���ø�ëqÿëqô õ'üwð ü5ðòÿªý�ø�ñ�þ�ì�ëbÿ���/Dë2ï�'�þ�ð ÿE�-?£ë��9ë2îwîNú-��ëbÿ��¼M+î�'�ó뼦ª©«U�U«¬Y¨4�

egÜ

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s ÈqÉ�Æ�ÈDÖ�Ë�ÈqÉ�Ú[æÇ,Ò*È�ϨÅmߨÆ2Ì�ÈDÈ�é�Í ÅYy[Ð�É�È�Ã7ÍǨ̫É�Ǩ×4É�Ç�ËIÍÈ�ϨÅ/ÆDÅYy[Ð�É�È�Ã7ÍÇk > 2

É�Ǩ×IǨÍÆ�Ó²É�ʦÃ}l�Åδ1,1 = 1

Ü s æǨÑsÅ�é�ÅaÚ[ǨÍ'éÈ�ϨÅ/æÆ7n�ÍæÇ�È~×RÃ7Ì2È�Æ�æÖ,Ð,È�Ã7ÍÇSé�ÅmÑ�É�Ç�Ã7×,Å/Ç�È�Ã|Î Ë È�ϨūÊ7ÍMÉ×RæÇ,ÒMÌ~ÍÇ�ÎNÉÑ/È2ÍÆ�ÍǨÅDÎwÍÆ�È�ϨūÌ2ÅsÑsÍǨ×-ÅYy[Ð�É�È�Ã7ÍÇ-Ö�ËEÎwÍÆ�Ó�æÇ,Ò

cov (yk, y2)

cov (yk, y1)= δ2,1

É�Ǩ×�È�ϨÅDÊ7ÍMÉ×RæÇ,ҲΠÍÆ�È�ϨÅkthÅYy[Ð�É�È�Ã7ÍÇ�Î Æ2ÍÓ

cov (yk, y2)

cov (y1, y2)= δk,1.

s æǨÑsÅ�È�ϨÅ~Ñ2ϨÍÃ7ÑsÅ�Í�ÎUÈ�ϨÅkthÅYy�Ð�É�È�Ã7ÍÇEÃ7Ì�É�Æ�Ö,æÈ�ÆqÉ�Æ�Ë�é�Å�Ñ�É�DzÃ7×,Å/Ç[È�Ã|Π˲É�ʦÊ�Í�Î<È�ϨÅ�Ê7ÍMÉ×RæÇ,ÒMÌ�ÎwÍÆ�ÎwÉÑ/È2ÍÆ�ÍǨÅ!ÜØv æÈ�ϲÈ�ϨÅ�Ê7ÍMÉ×RæÇ,ÒMÌ

ÍÇ-Ï�É�Ǩ×<Þ,é�Å«Ñ�É�Ç�È�ϨÅ`ÈqÉ�ÚMÅ«ÅYy[Ð�É�È�Ã7ÍÇ$àmÉ�Ǩ×-É�Ç[Ë�ÅYy�Ð�É�È�Ã7ÍÇkÉ�Ǩ×*ÎwÍÆ�Ó

y1 = θ1 + e1

yk

δk,1= θ1 +

L∑

l=2

θlδk,l

δk,1+

ek

δk,1.

��Ì�æÇ,ÒCÝ�ϨÅsÍÆ2Å/Ó h é�Å4Ñ�É�ÇAǨÍÇ,Ø�É�ÆqÉ�Ó�Å/È�Æ�Ã7Ñ�É�Ê¦Ê¦Ë Ã7×,Å/Ç[È�Ã|Î Ë È�ϨÅ4×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍǨÌ�Í�Îθ1, e1

É�Ç¨× ∑Ll=2 θl

δl,kδ1,k

+ ek

δ1,k

áwÉ�Ç¨× Í�Î∑L

l=2 θlδl,k + ekè9Ü

v8ÅaǨÍ�éOÈqÉ�ÚMÅaÅYy�Ð�É�È�Ã7ÍǨÌ2h É�Ǩ×�x*É�Ǩ×ÕÌ2ÍÓ�Å�É�Æ�Ö,æÈ�ÆqÉ�Æ�Ë�ÅYy[Ð�É�È�Ã7ÍÇk > 4.

i�ÍÆ�Ó²É�ʦÃ}lsæÇ,Òδ3,2 = 1

é�Å�Ñ�É�ÇIÃ7×,Å/Ç[È�Ã|Î ËSÈ�ϨÅÊ7ÍMÉ×RæÇ,ÒMÌ~ÍÇ-ÎwÉÑ/È2ÍƤe�ÍÇ�È�ϨūÆ2Å/Ó²É�æÇ,æÇ,ÒEÅYy[Ð�É�È�Ã7ÍǨÌ�Ö�ËEÎwÍÆ�Ó�æÇ,Ò

cov (yk − θ1δk,1, y4 − θ1δ4,1)

cov (y3 − θ1δ3,1, y4 − θ1δ4,1)= δk,2

É�Ǩ×�ÈqÉ�ÚMÅ«ÅYy[Ð�É�È�Ã7ÍǨÌ3É�Ǩ×

k

y3 − θ1δ3,1 = θ2 + e3

yk − θ1δk,1

δk,2= θ2 +

L∑

l=3

θlδk,l

δk,2+

ek

δk,2.

kªØ,Ø,ʦËSÈ�ϨÅsÍÆ2Å/Ó h*É�Ǩ×ÕÃ7×,Å/Ç�È�Ã|Î ËSÈ�ϨÅ�×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍǨÌmÍ�Îθ2,e3

É�Ç¨× ∑Ll=3 θl

δk,l

δk,2+ ek

δk,2

ǨÍÇ,Ø�É�ÆqÉ�Ó�Å/È�Æ�Ã7Ñ�É�ʦʦËMÜ���ËSØ,Æ2Í�ÑsÅsÅs×RæÇ,ÒÌ2ÅYy[ШÅ/Ç�È�à É�ʦʦË�é�Å~Ñ�É�DzÃ7×,Å/Ç[È�Ã|Π˲É�ʦÊUÍ�Î<È�ϨÅ�Ê7ÍMÉ×RæÇ,ÒMÌ�É�Ǩ×�ǨÍÇ,Ø�É�ÆqÉ�Ó�Å/È�Æ�Ã7Ñ�×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍǨÌ�Í�ÎfÈ�ϨÅ�Ó�Í[×,Å/Ê�Î ÍÆ1Ó�Å�ÉÌ2Ð,Æ2Å/Ó�Å/Ç�È2Ì�É�Ǩ×Ì2ÑqϨÍ[ÍʦæÇ,Ò

s = h�

a,Æ2ÍÓ)È�ϨÅ~Ý�ϨÅsÍÆ2Å/Ó x¨ÞMé�Å~ǨÍ�é Ï�ÉsÛMÅ�Ú�ǨÍ�é�Ê7Ås×RÒMÅ�Í�Î<È�ϨÅ�ǨÍÇ,Ø�É�ÆqÉ�Ó�Å/È�Æ�Ã7Ñ�×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍÇ Í�Î<È�ϨÅ�ÎNÉÑ/È2ÍÆ2Ì�É�ǨײÐ,Ç,Ã^y�ШÅ/ǨÅsÌqÌ2ÅsÌÉÌ�é�Å/ʦÊ�ÉÌ~È�ϨūÊ7ÍMÉ×RæÇ,ÒMÌ~Î ÍÆ�Ó�Å�ÉÌ2Ð,Æ2Å/Ó�Å/Ç�È2Ì`É�Ǩ×SÌ2ÑqϨÍ[ÍʦæÇ,Ò

s = h.v8ūǨÅ9æRÈ~È�Ð,Æ�ÇSÍÐ,ÆDÉ�È�È2Å/Ç�È�Ã7ÍÇIÈ2Í�Ã7×,Å/Ç�È�Ã|ß�Ñ�É�È�Ã7ÍÇÕÈ2Í�È�ϨÅ

É�Ç�É�Ê7ÍÒMÍШÌ�Ì�ËRÌ�È2Å/Ó)Í�Î�ÅYy�Ð�É�È�Ã7ÍǨÌ�æÇ4áLeghMè�ÎwÍÆ�È�ϨŪÑsÍʦÊ7Å/ÒMÅ�ás = c

è9Ü7i�ÍÈ�Ã7ÑsÅ~È�Ï�É�È�ÞRÌ�æǨÑsÅ�È�ϨŪÓ�Å�ÉÌ�Ð,Æ2Å/Ó�Å/Ç�È�ÅYy[Ð�É�È�Ã7ÍǨÌ�×,ÍmǨÍÈ×,Å/ØfÅ/Ǩ×SÍÇ-È�ϨÅ�Ì2Ñ2ϨÍ�ÍʦæÇ,Ò²Ê7Å/ÛMÅ/Ê�Ñ2ϨÍ!Ì2Å/Ç�Þ�È�ϨÅ/Ë�É�Æ2ÅmÃ7×,Å/Ç�È�Ã|ß�Ås×SÎ Æ2ÍÓ ÍÐ,Æ~Ø,Æ2Å/Û�Ã7ÍШ̪É�Æ�Ò!Ð,Ó�Å/Ç�È`É�Ǩ×SÌqͲÉ�Æ2Å`È�ϨÅm×RÃ7Ì2È�Æ�æÖ,Ð,È�Ã7ÍǨÌÍ�ΣÈ�ϨūÅ/Ê7Å/Ó�Å/Ç�È2̪Í�Î

θÜ

e)æ

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Ý�É�ÚMÅDÉ�Ç[Ë�È�é�ÍaÅYy[Ð�É�È�Ã7ÍǨÌ�Î Æ2ÍÓ)È�ϨŪÑsÍʦÊ7Å/ÒMÅ�Ì�ËRÌ�È2Å/ÓJÉ�Ç¨× ÍǨÅ�Í�Î�È�ϨŪÅYy[Ð�É�È�Ã7ÍǨÌ�È�Ï�É�È�×,Å/ØfÅ/Ǩ×EÍÇ,ʦ˲ÍÇθ1Ü7a�æÆ2Ì�È�ÉÌ2Ì�Ð,Ó�Å

È�Ï�É�ȪÉ�È�Ê7Å�ÉÌ�ȪÍǨūÍ�Î�È�ϨÅDÈ�Ï,Æ2ÅsÅmÅYy�Ð�É�È�Ã7ÍǨ̪ÑsÍÆ�Æ2ÅsÌ�ØUÍǨ×,Ì~È2Í�É�Ó�Å�ÉÌ�Ð,Æ2Å/Ó�Å/Ç[È�Þ�ÌqÉ�ËMjÜ�Ý�ϨÅ/Ç�ÞRÎ Æ2ÍÓ

cov(Mj ,

ʦÇYc,t | X

M , X)

= αMj αc,tσ2θ1

é�ŲÑ�É�Ç4Ã7×,Å/Ç�È�Ã|Î Ëαc,t

Î ÍÆmÉ�Ç[Ët = 1, ..., T.

bNÎ�Þ<ÍÇ4È�ϨÅ�ÍÈ�ϨÅ/Æ`Ï�É�Ǩ×<Þ5ǨÍÇ,ŲÍ�Î�È�ϨÅ�ÅYy[Ð�É�È�Ã7ÍǨ̫Ã7Ì«É Ó�Å�ÉÌ�Ð,Æ2Å/Ó�Å/Ç[È�ÅYy[Ð�É�È�Ã7ÍÇé�ÅDÑ�É�ÇSÉ�Ø,Ø,ʦË�È�ϨÅDǨÅ9æRÈ�È�ϨÅsÍÆ2Å/Ó È�Ï�É�È�Ì2ϨÍ�é�Ì�È�Ï�É�È�Ã|Î5È�ϨÅD×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍÇ-Í�Î

θÃ7Ì�ǨÍǨÌ�Ë�Ó�Ó�Å/È�Æ�Ã7Ñ!Þ,é�ÅDÑ�É�Ç*Ã7×,Å/Ç�È�Ã|Π˲È�ϨŪÎNÉÑ/È2ÍÆ

Ê7ÍMÉ×RæÇ,ÒMÌ~Å/ÛMÅ/ÇSÃ|Î�È�ϨÅ/Æ2ÅmÉ�Æ2ūǨÍ�Ó�Å�ÉÌ�Ð,Æ2Å/Ó�Å/Ç�È2Ì�Ü

F9.='B8>5<'�"�£ �&�W²!äI�³��ä��fä��bâL�[s��WsX��ä��´u�2� q �2ä2ä�äXµY�Q�g� âJu��6s[x�â1� q ���Y�)�Y�_ug��s��Z���Q���Z�E�2ä

y1 = θ1δ1,1 + e1

y2 = θ1δ2,1 + e2

y3 = θ1δ1,3 +

L∑

l=2

θlδ3,l + e3.

x�âZ� q θ �Ar�ä��Y�_u��¶u�2�²ä��g�¢±'ä��Xu����m�Z�.���1�5�EâÝ�D�[ä�y,ä��D�[ä��&���X���Y�_u���s|x�â1� q ²#�>u�x7�¶�!â�sX�Z�bâ_t��Q� â2u��¯u�����D� e1, e2 �g�D� e3 �g�¡s�u���ä(���±'äX��uEâ1�D��ä�y,äX�D��ä��&�·u��ä���� q ug� q ä������D��u�2� q ä"�X���Y�_u���s θ ã[� q äX�,å1â �

�����/¸W E(θk1)6= 0

�vu��k����u��)�EâÝ�D��äJ��ä��

� q ä��¹u#�)�â1�4�Ws δ1,1, δ2,1 ����� δ3,1 �g��ä²âL��ä��&� â ~�ä������D��svuSâ�s�� q ä|�>u��Cy.����g��äY�Z�bâL�T�!â�sX�Z�bâ_t��Q� â2u��ºu��� q ä��E��â2µX�¨ä��Uäswssäs {ej}2j=1

���D��u� ∑Ll=2 θlδ3,l + e3.

� 5W8�8��w� b�æʦʦШÌ�È�ÆqÉ�È2Å`È�ϨÅmÑ�ÉÌ2ÅDæÇ-é�Ï,Ã7Ñ2Ïk = 3

Î ÍÆ�ÑsÍǨ×RæÈ�Ã7ÍÇ¡áZk�ç4ÜMè9Ü1Ý�ϨÅmÅ9æ�È2Å/ǨÌ�Ã7ÍÇ�Ã7Ì~ÍÖ�Û�Ã7ÍШÌ�Ü"a¨ÍÆ�Ó

cov (y1, y2) = δ1,1δ2,1σ2θ1

cov (y1, y3) = δ1,1δ3,1σ2θ1

cov (y2, y3) = δ2,1δ3,1σ2θ1

É�Ǩ×-Ì2ÍʦÛMÅ�ÎwÍÆ

δ21,1 =cov (y1, y2) cov (y1, y3)

cov (y2, y3) σ2θ1

.

Ý�ϨÅ/Ç�Î Æ2ÍÓ

E(y21yj)

= δ21,1δ1,jE(θ31

)

eg»

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é�Å~Ã7×,Å/Ç�È�Ã|Î Ëδj,1Î ÍÆ�É�ʦÊ

j > 1Ü9·DÍæÇ,Ò�Ö�ÉÑ2Ú�È2Í

cov (y1, y2)é�Å~Æ2ÅsÑsÍ�ÛMÅ/Æ

δ1,1.Ý�ϨÅ~×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍǨÌ�Í�Î<È�ϨÅ�Ð,Ç,Ã^y[ШÅ/ǨÅsÌ2Ì2ÅsÌ1Î ÍʦÊ7Í'é

Ö�Ë�×,ÅsÑsÍÇ[ÛMÍʦÐ,È�Ã7ÍÇ�Üb�×,Å/Ç�È�Ã|ß�Ñ�É�È�Ã7ÍÇÕÍ�Î�È�ϨÅDé�ϨÍÊ7ÅmÌ�Ë�Ì2È2Å/Ó Î ÍÆ

s = cÎ ÍʦÊ7Í'é�Ì�Ö�Ë-É�Ø,Ø,ʦË�æÇ,Ò�È�ϨūÌqÉ�Ó�ÅDÊ7ÍÒ!Ã7Ñ«Í�ΣÈ�ϨÅsÍÆ2Å/Ó�x�ÌqÅYy�ШÅ/Ç[È�à É�ʦʦËMÜ

þ-ù"Ì&ÓØà)ÐJI:ÏQá�à)Ð�ÒØÓ¹Ò�ý2àgË"Ì�]TÒBÑ-à�Þ9ßGÓ"Ïmà)Ð�ÒBÓ

v4Æ�æÈ2Å`È�ϨūÑsÍʦÊ7Å/ÒMÅ�É�È�È2Å/ǨרÉ�ǨÑsÅmÑsÍǨ×RæÈ�Ã7ÍÇ áLÜMè�ÉÌ

E (Gc,1 (I1) −Gh,1 (I1) − φ (Z) − θλ− ω | I0) > 0.

´�Å/Èθ0 ×,Å/ǨÍÈ2ÅmÈ�ϨūÅ/Ê7Å/Ó�Å/Ç�È2Ì�Í�Î θ æǨÑ/ʦШ×,Ås×-æÇ-È�ϨÅmÉ�ÒMÅ/Ç[È� öÌ�æÇRÎwÍÆ�Ó²É�È�Ã7ÍÇ�Ì2Å/È�É�È�È�ϨÅ`È�æÓ�Å`È�ϨūÌqÑ2ϨÍ�ÍʦæÇ,Ò²×,ÅsÑ/Ã7Ì�Ã7ÍÇSÃ7Ì�Ó²É×,Å

É�Ǩ×�Ê7Å/Èλ0 ×,Å/ǨÍÈ2Å`È�ϨūÌ�Ð,Ö[ÛMÅsÑ/È2ÍƪÍ�Î λ ÉÌ2ÌqÍ[Ñ/à É�È2Ås×*é�æÈ�Ï-È�ϨÅ/Ó-Ü�Â�Å9ߨǨÅ

E(G∗

c,1 (I1) −G∗

h,1 (I1) | I0

)= µV (X) + τ

(X, θ0

)

é�Ï,Ã7Ñ2Ï Ã7Ì«Ú�ǨÍ�é�Ç¡Ì�æǨÑsÅ É�ʦÊ�Í�Î�È�ϨŲÅ/Ê7Å/Ó�Å/Ç[È2Ì�Í�ÎE(G∗

c,1 (I1) −G∗

h,1 (I1) | I0

) É�Æ2ŲÚ[ǨÍ'é�Ç$Î Æ2ÍÓ ÍÐ,Æ�Ø,Æ2Å/Û[Ã7ÍШÌ�É�Ç�É�ʦËRÌ�Ã7Ì�ÜÝ�ϨūÅsÑsÍǨÍÓ�Å/È�Æ�Ã7Ñ/à É�ÇIÏ�ÉÌ�רÉ�ÈqÉ�ÍÇ�È�ϨÅ`Ê7Å9Î È�Ï�É�Ǩ×-Ì�Ã7×,Å`Í�Î

o�Æ(S = c|X,Z) =

o�Æ (τ(X, θ0

)− θ0λ0 − ω > φ (Z) − µV (X)

).

F9.='B8>5<'�"¤»k� sws��E�²ä�� q �g�¤� q ä��2ä��7är-���&�Dä��7ä��²ä��&��s+u��� ������� q u��}��ã�¼ q ���4��ä)� ���§��� svut� q �g��� q ä�â1�D��ä�y,äX�D��ä��D�bäAu�� Z ������ q ä�ä�����ug�$��ä����[s q u�� �Cs2�wu��D�âZ� â2ug�D���3u�� X ã

z+äX�Ze

t2ä2� q ämäX�¦ä��²ä��D��s�u� Z � q �g�"���2ä6��ug�9��yQ�g����u� X ��äX½)�Y�5�.��ä�� � �g�D�����E�X� q ä��2�Ws�s��m�²ä6� q �g�¦x�ä$�(���K��ä�~9�Uä φ (ze, x)

��u��2���Ý�ByQ�â1��s(ze, x)

â1�t� q ä�s��/y{yDug���1u�� Z ã � s�x�â1� q ���1�>�!â�sY�X��äY��ä$� q uâZ�bä·y���uBtX�¦ä��³s2� q ä�sX�(���7ä$�Uä2ä��CsV�_u�t2ä�ssäX��ã � sws��m�²äV� q �)�¾�¿)À<Á!Â

var(τ(x, θ0

)− θ0λ0 − ω

)= 1

�vug�X = x

ã� q äX�,å φ (Z)

åλ0 �g�D�|� q ä2��u��CyQ�������²äX�Z�9âZ���â-s��Z�9â_t��m� âJu���u� ω �g��ämâL��ä��&� â ~�ä���ã

� 5W8�8��w� ÂDÅ9ߨǨÅ

Υ(X, θ0

)= τ

(X, θ0

)− θ0λ0 − ω

É�Ǩ×*ß,æX = x.

Ý�ϨÅmÍÖ¨Ì2Å/Æ�ÛMÅs×�Ø,Æ2ÍÖ�É�Ö,æʦæÈ�Ë�È�Ï�É�È~È�ϨūÉ�ÒMÅ/Ç�ȪÑqϨÍ[Í!ÌqÅsÌ�ÑsÍʦÊ7Å/ÒMÅEá ÑsÍǨ×RæÈ�Ã7ÍÇ�É�Ê�ÍÇX = x

É�Ǩ×-ШÌ�æÇ,Ò|k~ç3æ!è�Ã7Ì

Pr(Υ(x, θ0

)> φ (Ze, x) − µV (x)

)

é�ϨÅ/Æ2ŵV (x)

Ã7Ì�ÉmÚ�ǨÍ�é�Ç�ÑsÍǨÌ�ÈqÉ�Ç�Èmá ÑsÍǨ×RæÈ�Ã7ÍÇ�É�Ê�ÍÇX = x

è9Ü7�ªÌ2æÇ,Ò�È�ϨÅ�á ÑsÍǨ×RæÈ�Ã7ÍÇ�É�Ê�ÍÇXè�æǨ×,Å/ØfÅ/Ǩ×,Å/ǨÑsÅ`Í�Î

θ0λ0 +ω

É�Ǩ×Zeé�Å�Ñ�É�ÇÕÈ�ϨÅ/ÇIШÌ2Å«È�ϨÅ�É�Ç�É�ʦË�Ì2Ã7Ì�Í�Î"qÕÉ�È�lsÚ�æǡá�à��g�geMè�È2ͲÃ7×,Å/Ç[È�Ã|Î Ë

φ (Ze, x)É�Ǩ×�È�ϨÅa×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍÇ4Í�Î

Υ(x, θ0

),É�ʦÊ

eg�

Page 30: 102-34 5 67 8.902 +:; < ! += >public.econ.duke.edu/~psarcidi/tesis_07_03_05.pdf9 Mz z M z M uR 9{Nx z ` sx

Í�ΣÈ�Ï,Ã7Ì�ÑsÍǨ×RæÈ�Ã7ÍÇ�É�Ê+ÍÇX = x.

i�Å9æ�È�Þ£Ì�È�æʦÊ�ÑsÍǨ×RæÈ�Ã7ÍÇ�É�Ê�ÍÇX = x

Þ�é�Å Ñ�É�Ç$ÎwÍÆ�Ó È�ϨÅ$n�ÍæÇ�È�×RÃ7Ì�È�Æ�æÖ,Ð,È�Ã7ÍÇ Í�ΪÓ�Å�ÉÌ�Ð,Æ2Å/Ó�Å/Ç�È2Ì�Þ1Ê7ÍÒ�Å�É�Æ�Ç,æÇ,ÒMÌ�É�Ǩ×8È�ϨÅÑ2ϨÍÃ7ÑsÅmæǨ×,Å9æ*Ì2æǨÑsÅ`Ú[ǨÍ'éAÈ�ϨÅDÊ7Å9Î È�Ï�É�Ǩ×-Ì�Ã7×,Å

o�Æ (UM ≤ m− µM

(XM

), Us ≤ ys − µs (x) | X = x, S = s, Ze = ze

) o�Æ(S = s | Ze = ze)

=

∫ m−µM(XM)

−∞

∫ ys−µs(ex)

−∞

∫∞

φ(ze,ex)−µV (ex)f(UM , Us,Υ

(x, θ0

))dΥ(x, θ0

)dUsdU

M

É�Ǩסé�ÅSÑ�É�Ç È�ÆqÉÑsÅ*æÈ�Ö�Ë¡Û!É�Æ�Ë[æÇ,Òm, y, zE .

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b�Ç�È�Ï,Ã7ÌDÌ2ÅsÑ/È�Ã7ÍǶb�×RÃ7Ì2Ñ/ШÌq̪Ì2ÍÓ�Å�Í�Î�È�ϨÅaÑsÍÓ�Ø,Ð,ÈqÉ�È�Ã7ÍÇ�É�Ê1Ã7Ì2Ì2ШÅsÌDÉÌqÌ2Í[Ñ/à É�È2Ås×�é�æÈ�Ï4ÅsÌ�È�æӲÉ�È�æÇ,Ò*È�ϨÅaÓ�Í[×,Å/ÊNÜ6b�Ì�ÈqÉ�Æ�ÈDé�æÈ�ÏIÈ�ϨÅÌ2ÍʦÐ,È�Ã7ÍÇSÍ�ΣÈ�Ϩū×RË�Ç�É�Ó�Ã7Ñ`Ø,Æ2ÍÒ!ÆqÉ�Ó Ã¦ÇSÅYy[Ð�É�È�Ã7ÍÇHáLjMè�Ð,Ǩ×,Å/ƪ×RÃ|ÄfÅ/Æ2Å/Ç[È�æÇRÎ ÍÆ�Ó²É�È�Ã7ÍÇIÌ2Å/È2Ì�ÜÝ�É�ÚMÅ«É�Ç Ã¦Ç¨×RæÛ�Ã7×RÐ�É�Ê<é�ϨÍaÏ�ÉÌ�É�ʦÆ2Å�É×RËEÑ2ϨÍ!ÌqÅ/Ç�Ï,Ã7Ì�Ì2Ñ2ϨÍ�ÍʦæÇ,Ò�Ê7Å/ÛMÅ/Ê

sÜ7a¨ÍÆ�Ì�æÓ�Ø,ʦÃ7Ñ/æÈ�ËMÞ�Ì�Ð,Ø,Ø,Æ2ÅsÌ2Ì�È�ϨÅ�æǨ×RæÛ[Ã7×RÐ�É�Ê<æǨ×,Å9æ

i

É�Ǩ×�È�ϨÅXæÇ-Ó�Å�É�ÇSÅ�É�Æ�Ç,æÇ,ÒMÌ~ÌqÍ�È�Ï�É�È

µs,t = µs,t (Xs,t) .

´�Å/È

Rs,t (µs,t, θ, εs,t, As,t−1) = Ys,t + (1 + r)As,t−1

×,Å/ǨÍÈ2Å`È�ϨÅ`Æ2ÅsÌqÍÐ,Æ2ÑsÅs̪ÉsÛ!É�Ã¦Ê É�Ö,Ê7Å`È2Í�È�ϨÅ�É�ÒMÅ/Ç�ȪÉ�È�È�æÓ�ÅtÜ

s ÈqÉ�Æ�È�é�æÈ�ÏSÈ�ϨÅ�Ê ÉÌ�È�ØfÅ/Æ�Ã7Í[×TÉ�Ǩ×IÉÌ2Ì�Ð,Ó�Å«È�Ï�É�È

ItÑsÍÇ[ÈqÉ�æǨÌDÉ�ʦÊ�È�ϨÅmÅ/Ê7Å/Ó�Å/Ç[È2ÌDÍ�Î1È�ϨÅ

Lç�×RæÓ�Å/ǨÌ�Ã7ÍÇ�É�Ê�ÛMÅsÑ/È2ÍÆ

θÜ�Ý�ϨÅ

Û!É�ʦШÅDÎ Ð,ǨÑ/È�Ã7ÍÇIÉ�ÈTÃ7Ì�Ò!æÛMÅ/Ç-Ö�Ë*È�ϨÅDÐ,È�æʦæÈ�Ë È�ϨÅ�É�ÒMÅ/Ç�ȪÍÖ,ÈqÉ�æǨÌ�Î Æ2ÍÓ ÑsÍǨÌ2Ð,Ó�æÇ,Ò²Ï,Ã7Ì�Æ2Å/Ó²É�æÇ,æÇ,ÒEÆ2ÅsÌ2ÍÐ,Æ2ÑsÅsÌ

V ∗

s,T = Gs,T (Rs,T (µs,T , θ, εs,T , As,T−1)) = u (Rs,T ) .

k�ÈT − 1

á Ì�È�æʦÊ5ÉÌqÌ�Ð,Ó�æÇ,Ò�È�ϨÅ`é�ϨÍÊ7Å`ÛMÅsÑ/È2ÍÆθÃ7Ì�æÇ

Itè�È�ϨÅmÉ�ÒMÅ/Ç[È� öÌ~ÍØ,È�æÓ�Ã}l'É�È�Ã7ÍÇIØ,Æ2ÍÖ,Ê7Å/Ó Ã7Ì

maxAs,T−1u (Rs,T−1 (µs,T−1, θ, εs,T−1, As,T−2) −As,T−1) +

1

1 + ρ

∫Gs,T (Rs,T (µs,T , θ, εs,T , As,T−1)) dF (εs,T ) ,

Ì2Í�È�ϨÅmÉ�ÒMÅ/Ç[È~Å9æRØUÅsÑ/È2Ì�È�ϨūÐ,Ç,Ú[ǨÍ'é�Çεs,T

ÍÐ,È�Ü1Ý�ϨūÌ2ÍʦÐ,È�Ã7ÍÇSÈ2Í�È�Ï,Ã7Ì�Ø,Æ2ÍÖ,Ê7Å/Ó Ã7Ì~É�Î Ð,ǨÑ/È�Ã7ÍÇ

A∗

s,T−1 = Γs,T−1 (Rs,T−1 (µs,T−1, θ, εs,T−1, As,T−2) , µs,T + θαs,T )

È�Ï�É�È�ÅsÌ�ÈqÉ�Ö,ʦÃ7Ì�ϨÅsÌ�Þ�Î ÍÆ�É`Ò!æÛMÅ/Ç�É�Ó�ÍÐ,Ç�È�Í�Î�Æ2ÅsÌ2ÍÐ,Æ2ÑsÅsÌ�É�Ç¨× É«Ò!æÛMÅ/ÇEÓ�Å�É�Ç Ê7ÍÒ�Å�É�Æ�Ç,æÇ,ÒMÌ�É�ÈTÞMϨÍ'é Ó�ШÑ2Ï*È�ϨŪÉ�ÒMÅ/Ç[È�é�æʦÊ%ÌqÉ�ÛMÅ

ÎwÍÆ�È�ϨÅ`ǨÅ9æ�È�ØfÅ/Æ�Ã7Í�×*Û[à ÉAs,T−1

Ü�Ý�ϨūÉÌ2Ì2Í�Ñ/à É�È2Ås×�ÛÉ�ʦШÅ�Î Ð,ǨÑ/È�Ã7ÍÇ-Ã7Ì

V ∗

s,T−1 = Gs,T−1 (Rs,T−1 (µs,T−1, θ, εs,T−1, As,T−2) , µs,T + θαs,T ) =

u(Rs,T−1 −A∗

s,T−1

)+

1

1 + ρ

∫Gs,T

(Rs,T

(µs,T , θ, εs,T , A

s,T−1

))dF (εs,T )

o�Æ2Í�ÑsÅsÅs×RæÇ,Ò�Ì2ÅYy[ШÅ/Ç�È�à É�Ê¦Ê¦Ë é�Å�ߨǨ×*È�Ï�É�È�ÞRÃ|ÎθÃ7Ì�Ú[ǨÍ'é�Ç�Ö�ËEÈ�ϨÅDÉ�ÒMÅ/Ç[ȪÉ�È

t+ 1Þ�È�ϨÅDØfÍʦÃ7Ñ/Ë É�Ç¨× Û!É�ʦШÅ~Î Ð,ǨÑ/È�Ã7ÍǨÌ�×,ÅsÌ2Ñ/Æ�æÖ,æÇ,Ò

È�ϨūÌ2ÍʦÐ,È�Ã7ÍÇIÉ�Èt+ 1

É�Æ2Å

A∗

s,t+1 = Γs,t+1

(Rs,t+1, {µs,τ + θαs,τ}

Tτ=t+2

)

V ∗

s,t+1 = Gs,t+1

(Rs,t+1, {µs,τ + θαs,τ}

Tτ=t+2

).

h)æ

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i�ÍÈ�Ã7ÑsÅ�È�Ï�É�È�È�ϨţÇ�Ð,Ó�ÖUÅ/Æ�Í�Î,É�Æ�Ò!Ð,Ó�Å/Ç[È2Ì5æÇ`È�ϨÅ�Î Ð,ǨÑ/È�Ã7ÍÇ«Ò!Æ2Í'é�Ì5Ì�æǨÑsÅ1é�Å�ǨÅsÅs×`È2Í�ÚMÅsÅ/ØmÈ�ÆqÉÑ2Ú`Í�Î,É�ʦÊÎ Ð,È�Ð,Æ2Å{µs,τ + θαs,τ}

Tτ=t+2

É�Ǩ×�ǨÍÈ�ÍÇ,ʦË*ǨÅ9æRÈ�ØUÅ/Æ�Ã7Í�×> öÌ�Üs Ð,Ø,ØfÍ!Ì2Å«È�Ï�É�È�È�ϨūæÇRÎwÍÆ�Ó²É�È�Ã7ÍÇ4Ì2Å/È�Ø,Æ2ÍØfÍ!Ì2Ås×�Ã7Ì~Ì�ШÑqÏ�È�Ï�É�È`É�È~È�æÓ�Å

tÈ�ϨÅaÉ�ÒMÅ/Ç�È`×,Í�ÅsÌ�ǨÍÈ�Ú[ǨÍ'é;È�ϨÅ�Ê ÉÌ�ȪÅ/Ê7Å/Ó�Å/Ç[È�Í�Î

θÜ7b�Ç È�Ï,Ã7̪Ñ�ÉÌ2Å!ÞRÈ�ϨÅmÉ�ÒMÅ/Ç[È� öÌ~ÑsÍǨÌ�Ð,Ó�Ø,È�Ã7ÍÇÕÉ�ʦÊ7Í[Ñ�É�È�Ã7ÍÇ�Ø,Æ2ÍÖ,Ê7Å/Ó Ã7Ì

maxAs,tu(Rs,t

(µs,t, {θl}

L−1l=1 , εs,t, As,t−1

)−As,t

)+

1

1 + ρ

∫ ∫Gs,t+1 (Rs,t+1 (µs,t+1, θ, εs,t+1, As,t)) dF (εs,t) dF (θL)

É�Ǩ×�È�ϨÅ�Î Ð,ǨÑ/È�Ã7ÍǨÌ�È�Ï�É�È~×,ÅsÌqÑ/Æ�æÖUÅ`È�ϨÅ`ÌqÍʦÐ,È�Ã7ÍÇSÍ�Î�È�ϨūØ,Æ2ÍÖ,Ê7Å/Ó É�Æ2Å

A∗

s,t = Γs,t

Rs,t

(µs,t, {θl}

L−1l=1 , εs,t, As,t−1

),

{µs,τ +

L−1∑

l=1

θlαs,τ,l

}T

τ=t+1

V ∗

s,t = Gs,t

Rs,t

(µs,t, {θl}

L−1l=1 , εs,t, As,t−1

),

{µs,τ +

L−1∑

l=1

θlαs,τ,l

}T

τ=t+1

.

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θLÉQÄ<ÅsÑ/È�æÇ,Ò

Å�É�Æ�Ç,æÇ,ÒM̪ΠÍÆτ ≤ t

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Ã7Ì�Ì2ÍʦÛMÅs×aÎwÍÆ1É~Ò!Æ�Ã7×�Í�ÎUØUÍæÇ[È2Ì�ÍÇRs,t

É�Ç¨× {µs,τ +

∑L−1l=1 θlαs,τ,l

}Tτ=t+1

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É�Ç¨× {µs,τ +

∑L−1l=1 θlαs,τ,l

}Tτ=t+1

Ü�¢�èzªÇ¨ÑsÅ~È�ϨÅ�Î Ð,ǨÑ/È�Ã7ÍǨÌ1È�Ï�É�È�ÌqÍʦÛMÅ�È�ϨŪÉ�ʦÊ7Í�Ñ�É�È�Ã7ÍDzØ,Æ2ÍÖ,Ê7Å/Ó É�Æ2Å�ÎwÍÐ,Ǩ×2��u��¼���â�r�ä���r-�g�5�¨ä�u�¤� q ä�yQ�������²äX��ä���s�u�¼� q ä¼�pu���ä��|Þ

È�ϨŠʦæÚMÅ/ʦæϨÍ[Í�סÑ�É�ÇHÖfÅ*Å/ÛÉ�ʦÐ�É�È2Ås×<Ü4Ý�ϨÅ*ÑsÍÇ[È�Æ�æÖ,Ð,È�Ã7ÍÇ È2ÍSÈ�ϨŠʦæÚMÅ/ʦæϨÍ�Í[סÍ�Î�É�ǡæǨ×RæÛ�Ã7×RÐ�É�Ê�é�ϨÍIÑqϨÍ[Í!ÌqÅsÌ�Þ£ÎwÍÆaÅ9æ¨É�Ó�Ø,Ê7Å!Þ

S = cÃ7Ì(³ �

Θ

∏5t=1 fεc,t (lnYc,t − µc,t (X) − θαc,t|θ,Xt)Pr (S = c|Z, θ)

∏4t=1 fξt

(lnζt − lngc,t (It) −Ktυt|θ,K

) dF (θ) .

~�Û!É�ʦÐ�É�È�Ã7ÍÇCÍ�Î�È�ϨÅ�ʦæÚMÅ/ʦæϨÍ�Í[×HÆ2ÅYy�Ð,æÆ2ÅsÌ�È�ϨŠÅsÑsÍǨÍÓ�Å/È�Æ�Ã7Ñ/à É�ÇCÈ2ÍSÌ2ÍʦÛMÅEÈ�ϨŲ×RË�Ç�É�Ó�Ã7ѲØ,Æ2ÍÒ!ÆqÉ�Ó Ã¦Ç ÍÆ2×,Å/Æ�È2Í�Å/ÛÉ�ʦÐ�É�È2ÅEÈ�ϨÅÌ2ÑqϨÍ[ÍʦæÇ,Ò8Ì2Å/Ê7ÅsÑ/È�Ã7ÍÇCØ,Æ2ÍÖ�É�Ö,æʦæÈ�Ë¡É�ǨסÈ�ϨÅ*ÑsÍǨÌ�Ð,Ó�Ø,È�Ã7ÍÇCØfÍʦÃ7Ñ/Ë8Î Ð,ǨÑ/È�Ã7ÍÇ

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°Lÿ ���Qø4÷bîNð8� ð üSëbñ�ïNóëbô ôöõ ÷9ø�ÿQø�î�ëqïNø�� ðòÿ ïNø�îNì�ü*úqù�ï���øÕóQÿ��'ø�îNôöõQð ÿQ÷û�bëqîNðòë��Qôòø�ü X, θ, ε, A ëbÿ�� ï���ø�üwøÕí!ú9ð ÿsïNü-ó�üwø��CïNú ù|úqîNì Rs,tëbÿ��

n

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Table 3.1Tests for Information Set Misspecification

Additional Consumption ConsumptionSchooling Choice

Parameters Age 19 – 24 Age 25 – 30 Age 19 – 24

- - - - - 0.325489Std. Error - - - - - 0.000341

- - - - - -0.199844Std. Error - - - - - 0.000419

0.809959 0.784294 0.653787 0.866148 0.909762 -1.099710Std. Error 0.018729 0.026117 0.025914 0.018270 0.034298 0.002360

1.837480 0.835822 1.829688 1.175653 0.953176 0.036775Std. Error 0.104920 0.052729 0.053899 0.107930 0.050305 0.000062

- - - - - -0.639739Std. Error - - - - - 0.001273

θ2

not known at periods 1 and 2 θ2

not known at period 1θ

1and θ

2not known at

schooling decision dateSchooling

Choice SchoolingChoice

θ1

θ1θ1

θ2

θ2θ2

θ1θ2

Let g(I) be the function describing the predicted choice in the model as a function of the assumed information set I. We then add the left outfactors to the choice function (after the agent integrates them out) and test whether their associated parameters are different from zero.

Table 3.2Tests for Misspecification: Model Selection Criteria

-18395.18 -18013.03 -16864.99 -16820.04AIC: 18636.18 18254.03 17107.99 17060.04BIC: 19388.18 19006.03 17866.22 17808.92

In both cases a smaller number means we favor the selection of that model

θ2

not known atperiods 1 and 2

θ2

not known atperiod 1

θ1 and θ2 not known atschooling decision

date

θ1

and θ2

alwaysknown

Loglikelihood:

AIC (Akaike Information Criteria) = -Loglikelihood + Number of Parameters

BIC (Schwarz Criteria) = -Loglikelihood + 0.5*Number of Parameters*log(n)

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Table 4.1Estimated Parameters for Test Equations

Word Knowledge Coding Speed

Variable Estimate Std Err. Estimate Std Err. Estimate Std Err. Estimate Std Err. Estimate Std Err.Constant -1.5481 0.1560 -1.2621 0.1160 -0.7885 0.1481 -1.6816 0.1725 -1.6597 0.1939

Mother's Education 0.0443 0.0412 0.0313 0.0149 0.0307 0.0837 0.0245 0.0040 0.0241 0.0079Father's Education 0.0350 0.0332 0.0564 0.0116 0.0322 0.0877 0.0079 0.0103 0.0729 0.0223

Family Income in 1979 0.0008 0.0014 0.0009 0.0013 -0.0001 0.0015 0.0041 0.0014 0.0019 0.0016Broken Home -0.0395 0.0029 -0.0317 0.0025 -0.0697 0.0055 0.0687 0.0054 0.0001 0.0522

Number of Siblings -0.0193 0.1160 -0.0339 0.1481 -0.0065 0.1725 -0.0220 0.1939 -0.0271 0.0040

Lived in South at age 14 -0.1074 0.0149 -0.0715 0.0837 0.0127 0.0040 -0.1302 0.0079 -0.1034 0.0075Urban at age 14 0.0539 0.0116 -0.0414 0.0877 -0.0039 0.0103 0.0111 0.0223 0.0312 0.0021

Enrolled at test date 0.2789 0.0013 0.1555 0.0015 0.1396 0.0014 0.1281 0.0016 0.5244 0.0284Age at test date -0.0092 0.0025 0.0123 0.0055 -0.0278 0.0054 -0.0052 0.0522 -0.0695 0.0104

0.1650 0.1481 0.1001 0.1725 0.1204 0.1939 0.1472 0.0040 0.2444 0.0129

1.0000 0.0000 0.8089 0.0198 0.9482 0.0240 0.6470 0.0188 1.0067 0.0283

Variance 0.2692 0.0094 0.1491 0.0072 0.1903 0.0081 0.4320 0.0099 0.2983 0.0103

ArithmeticReasoning

ParagraphCompletion

MathematicsKnowledge

Highest gradecompleted at test dateθ1

Table 4.2

19-24 25-30 31-36 37-51 52-65Variable Estimate Std Err. Estimate Std Err. Estimate Std Err. Estimate Std Err. Estimate Std Err.Constant 0.2444 0.0121 0.4136 0.0195 0.4647 0.0126 1.3905 0.0099 1.3591 0.0125

Born 1916-1925 --- --- --- --- --- --- --- --- -0.0482 0.0014

Born 1926-1935 --- --- --- --- 0.0312 0.0009 0.1486 0.0004 -0.1571 0.0066Born 1936-1945 --- --- 0.1811 0.0068 0.2702 0.0090 0.1807 0.0099 -0.0070 0.0375

Born 1946-1955 --- --- 0.1472 0.0049 0.1286 0.0040 0.1182 0.0069 --- ---Born 1956-1965 -0.2540 0.0085 -0.0010 0.0164 0.0799 0.0024 0.2035 0.0010 --- ---

Born 1966-1975 -0.4220 0.0125 --- --- --- --- --- --- --- ---

0.0497 0.0012 0.1558 0.0033 0.1941 0.0042 0.5294 0.0015 0.3277 0.0072

1.0000 0.0000 1.2020 0.0198 1.1347 0.0171 1.0684 0.0154 1.5766 0.0184Parameters of the MixtureMean 1 0.0693 0.0019 0.0194 0.0009 -0.3157 0.0135 0.5403 0.0474 -0.5322 0.0546

Mean 2 0.2314 0.0085 2.8930 0.1207 -0.1633 0.0075 0.0332 0.0013 2.7518 0.2757Mean 3 -0.7134 0.0291 -0.9019 0.0832 0.1175 0.0042 -1.2263 0.0411 0.0616 0.0119

Variance 1 0.1628 0.0063 0.0410 0.0028 0.9943 0.0009 0.0132 0.0043 0.3399 0.0043Variance 2 0.0509 0.0067 0.4528 0.0028 0.0336 0.0037 0.0185 0.0031 0.0577 0.0051

Variance 3 0.9007 0.0059 0.3883 0.0044 0.0306 0.0030 0.0051 0.0011 0.0514 0.0089

Mixture Weight 1 0.4658 0.0130 0.9635 0.0040 0.1316 0.0085 0.0196 0.0011 0.1147 0.0218Mixture Weight 2 0.3692 0.0103 0.0038 0.0004 0.2154 0.0068 0.9462 0.0031 0.0024 0.0015

Mixture Weight 3 0.1650 0.0046 0.0327 0.0037 0.6530 0.0093 0.0342 0.0021 0.8829 0.0216

Estimated Parameters for High School Earnings Equation1

θ1

θ2

1The earnings equation is Yt,s=exp(Xβt,s+θαt,s+εt,s), where X is the vector of cohort dummies listed above, θ� � � � � � �� �� � � � �� � �� β � � � �

α � � � � � � � � � � � � � � �

� � � � � � � �� � �ε � �

� � � � � � � �� � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Table 4.3

19-24 25-30 31-36 37-51 52-65Variable Estimate Std Err. Estimate Std Err. Estimate Std Err. Estimate Std Err. Estimate Std Err.Constant -0.2139 0.0152 0.6618 0.0165 1.3663 0.0183 1.7136 0.0189 1.6125 0.0257

Born 1916-1925 --- --- --- --- --- --- --- --- -0.0184 0.0004

Born 1926-1935 --- --- --- --- 0.2595 0.0080 0.3285 0.0089 0.1486 0.0046Born 1936-1945 --- --- -0.0221 0.0008 -0.3322 0.0111 0.2687 0.0091 0.2888 0.0093

Born 1946-1955 --- --- -0.0293 0.0009 -0.3837 0.0115 0.2316 0.0090 --- ---Born 1956-1965 -0.6844 0.0323 -0.1157 0.0037 -0.4214 0.0125 0.1313 0.0038 --- ---

Born 1966-1975 -0.4480 0.0189 --- --- --- --- --- --- --- ---

0.2366 0.0071 0.1517 0.0034 0.2798 0.0073 0.6109 0.0182 1.2268 0.0260

0.4239 0.0113 1.2005 0.0181 1.3503 0.0191 1.1936 0.0250 0.9068 0.0245Parameters of the MixtureMean 1 0.0103 0.0004 -0.4446 0.0222 -0.0877 0.0029 0.2096 0.0096 -0.1456 0.0132

Mean 2 -0.5618 0.0177 -0.3110 0.0227 0.0180 0.0006 -0.1002 0.0034 0.2514 0.0191Mean 3 0.4487 0.0148 0.1355 0.0048 0.0045 0.0012 -0.0225 0.0029 -0.3865 0.0456

Variance 1 0.2448 0.0102 0.9631 0.0037 0.9625 0.0037 0.2558 0.0081 0.0212 0.0094Variance 2 0.9471 0.0061 0.1133 0.0067 0.0358 0.0062 0.0040 0.0007 0.0055 0.0019

Variance 3 0.2594 0.0069 0.0383 0.0033 0.0526 0.0078 0.0189 0.0031 0.0201 0.0082

Mixture Weight 1 0.3986 0.0016 0.0846 0.0114 0.0955 0.0056 0.1252 0.0101 0.3102 0.0115Mixture Weight 2 0.2711 0.0014 0.1936 0.0104 0.3180 0.0059 0.0846 0.0081 0.4888 0.0164

Mixture Weight 3 0.3303 0.0012 0.7218 0.0123 0.5865 0.0074 0.7902 0.0130 0.2010 0.0073

Estimated Parameters for College Earnings Equation1

θ1

θ2

1The earnings equation is Yt,s=exp(Xβt,s+θαt,s+εt,s), where X is the vector of cohort dummies listed above, θ� � � � � � �� �� � � � �� � �� β � � � �

α � � � � � � � � � � � � � � �

� � � � � � � �� � �ε � �

� � � � � � � �� � � � � � � �� � � � � � � � � � � � � � � � � � � � � � � � � � � ��

Table 4.4

Variable Estimate Std Err.Mother's education -0.0832 0.2133Father's education -0.0773 0.1961Broken home 0.2236 0.0072Number of siblings 0.0641 0.0042Urban at age 14 -0.0105 0.0004Tuition 0.0648 0.0238Born 1916-1925 0.4398 0.0338Born 1926-1935 0.5511 0.0423Born 1936-1945 0.1887 0.0135Born 1946-1955 0.1749 0.0115Born 1956-1965 0.0750 0.0052Born 1966-1975 0.6468 0.0468

-0.2662 0.00910.1715 0.0068

Estimated Parameters for CostEquation

θ1

θ2

Table 4.5Estimated Parameters for Consumption Measurement Error

(Coefficient of Relative Risk Aversion = 2.15)

19-24 25-30 31-36 37-51Variable Estimate Std Err. Estimate Std Err. Estimate Std Err. Estimate Std Err.Constant 0.4495 0.0163 0.3827 0.0138 0.4002 0.0245 0.0441 0.0090

Married -0.2693 0.0209 -0.0608 0.0031 0.0671 0.0046 0.0017 0.1521

Number of Children -0.0252 0.0032 -0.0774 0.0052 -0.0267 0.0010 0.0303 0.0068Variance 0.2650 0.0054 0.1858 0.0032 0.3158 0.0085 0.4437 0.0213

Table 4.6Estimated Parameters for Factor Distributions

Factor 1 (Ability) Factor 2Variable Estimate Std Err. Estimate Std Err.Mean 1 0.3479 0.0009 0.0678 0.0004

Mean 2 -0.3342 0.0072 -0.1377 0.0019

Mean 3 -0.7909 0.0355 -1.3221 0.0633Variance 1 0.0518 0.0000 0.0502 0.0000

Variance 2 0.3145 0.0113 0.1659 0.0078Variance 3 1.7457 0.0069 1.1787 0.0314

Mixture Weight 1 0.5217 0.0052 0.7972 0.0067

Mixture Weight 2 0.4310 0.0051 0.1808 0.0071Mixture Weight 3 0.0474 0.0009 0.0220 0.0011

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Present value of earnings from age 19 to 65 discounted using an interest rate of 3%. L et (Y0,Y

1) denote

potential outcomes in the high school and college sectors, respectively. L et S=0 denote choice of the highschool sector and S=1 denote choice of the college sector. Define the ex-post return as R =(Y

1-Y

0)/Y

0 .

f(r|S=0) denotes the density function of returns for people who choose high school (solid line) and f(r|s=1)denotes the density of ex-post returns for college graduates (dashed line).

-1 -0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 1Density of realized monetary returns conditional on choice

return

High schoolCollege

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Log of present value of earnings from age 19 to 24 discounted using an interest rate of 3%. Let (Y0,Y

1) denote

potential outcomes in high school and college sectors, respectively. Let S=0 denote high school sector,and S=1 denote college sector. Define observed earnings as Y=SY

1+(1−S)Y

0. Finally, let f(log(y)) denote

the density function of observed earnings. Here we plot the density functions f generated from the data(the solid curve), against that predicted by the model (the dashed line). We use kernel density estimationto smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 2.1Densities of fitted and actual log present value of earnings

for period 1 for overall sample

log earnings

FittedActual

Log of present value of earnings from age 25 to 30 discounted using an interest rate of 3%. Let (Y0,Y

1) denote

potential outcomes in high school and college sectors, respectively. Let S=0 denote high school sector,and S=1 denote college sector. Define observed earnings as Y=SY

1+(1−S)Y

0. Finally, let f(log(y)) denote

the density function of observed earnings. Here we plot the density functions f generated from the data(the solid curve), against that predicted by the model (the dashed line). We use kernel density estimationto smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 2.2Densities of fitted and actual log present value of earnings

for period 2 for overall sample

log earnings

FittedActual

Log of present value of earnings from age 31 to 36 discounted using an interest rate of 3%. Let (Y0,Y

1) denote

potential outcomes in high school and college sectors, respectively. Let S=0 denote high school sector,and S=1 denote college sector. Define observed earnings as Y=SY

1+(1−S)Y

0. Finally, let f(log(y)) denote

the density function of observed earnings. Here we plot the density functions f generated from the data(the solid curve), against that predicted by the model (the dashed line). We use kernel density estimationto smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 2.3Densities of fitted and actual log present value of earnings

for period 3 for overall sample

log earnings

FittedActual

Log of present value of earnings from age 37 to 52 discounted using an interest rate of 3%. Let (Y0,Y

1) denote

potential outcomes in high school and college sectors, respectively. Let S=0 denote high school sector,and S=1 denote college sector. Define observed earnings as Y=SY

1+(1−S)Y

0. Finally, let f(log(y)) denote

the density function of observed earnings. Here we plot the density functions f generated from the data(the solid curve), against that predicted by the model (the dashed line). We use kernel density estimationto smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 2.4Densities of fitted and actual log present value of earnings

for period 4 for overall sample

log earnings

FittedActual

Log of present value of earnings from age 52 to 65 discounted using an interest rate of 3%. Let (Y0,Y

1) denote

potential outcomes in high school and college sectors, respectively. Let S=0 denote high school sector,and S=1 denote college sector. Define observed earnings as Y=SY

1+(1−S)Y

0. Finally, let f(log(y)) denote

the density function of observed earnings. Here we plot the density functions f generated from the data(the solid curve), against that predicted by the model (the dashed line). We use kernel density estimationto smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 2.5Densities of fitted and actual log present value of earnings

for period 5 for overall sample

log earnings

FittedActual

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Log of present value of earnings from age 19 to 24 discounted using an interest rate of 3%. Let Y0 denote.

the present value of high school earnings for this period. Here we plot the density functions f(log(y0)|S=0)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.1Densities of fitted and actual log present value of earningsfor period 1 for people who choose to graduate high school

log earnings

FittedActual

Log of present value of earnings from age 25 to 30 discounted using an interest rate of 3%. Let Y0 denote.

the present value of high school earnings for this period. Here we plot the density functions f(log(y0)|S=0)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.2Densities of fitted and actual log present value of earningsfor period 2 for people who choose to graduate high school

log earnings

FittedActual

Log of present value of earnings from age 31 to 36 discounted using an interest rate of 3%. Let Y0 denote.

the present value of high school earnings for this period. Here we plot the density functions f(log(y0)|S=0)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3.3Densities of fitted and actual log present value of earningsfor period 3 for people who choose to graduate high school

log earnings

FittedActual

Log of present value of earnings from age 37 to 51 discounted using an interest rate of 3%. Let Y0 denote.

the present value of high school earnings for this period. Here we plot the density functions f(log(y0)|S=0)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 3.4Densities of fitted and actual log present value of earningsfor period 4 for people who choose to graduate high school

log earnings

FittedActual

Log of present value of earnings from age 52 to 65 discounted using an interest rate of 3%. Let Y0 denote.

the present value of high school earnings for this period. Here we plot the density functions f(log(y0)|S=0)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 3.5Densities of fitted and actual log present value of earningsfor period 5 for people who choose to graduate high school

log earnings

FittedActual

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Log of present value of earnings from age 19 to 24 discounted using an interest rate of 3%. Let Y1 denote.

the present value of college earnings for this period. Here we plot the density functions f(log(y1)|S=1)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 4.1Densities of fitted and actual log present value of earnings

for period 1 for people who choose to graduate college

log earnings

FittedActual

Log of present value of earnings from age 25 to 30 discounted using an interest rate of 3%. Let Y1 denote.

the present value of college earnings for this period. Here we plot the density functions f(log(y1)|S=1)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4.2Densities of fitted and actual log present value of earnings

for period 2 for people who choose to graduate college

log earnings

FittedActual

Log of present value of earnings from age 31 to 36 discounted using an interest rate of 3%. Let Y1 denote.

the present value of college earnings for this period. Here we plot the density functions f(log(y1)|S=1)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4.3Densities of fitted and actual log present value of earnings

for period 3 for people who choose to graduate college

log earnings

FittedActual

Log of present value of earnings from age 37 to 51 discounted using an interest rate of 3%. Let Y1 denote.

the present value of college earnings for this period. Here we plot the density functions f(log(y1)|S=1)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 4.4Densities of fitted and actual log present value of earnings

for period 4 for people who choose to graduate college

log earnings

FittedActual

Log of present value of earnings from age 52 to 65 discounted using an interest rate of 3%. Let Y1 denote.

the present value of college earnings for this period. Here we plot the density functions f(log(y1)|S=1)

generated from the data (the solid curve), against that predicted by the model (the dashed line). We use kerneldensity estimation to smooth these functions.

−3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4.5Densities of fitted and actual log present value of earnings

for period 5 for people who choose to graduate college

log earnings

FittedActual

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Table 5

High School College Overall

Age 19 – 24Chi2 Statistic 134.3487 77.1154 181.9245

Critical Value* 137.7015 85.9649 204.6902

Age 25 – 30Chi2 Statistic 175.7894 133.9425 273.6058

Critical Value* 186.1458 139.9208 308.2548

Age 31 – 36Chi2 Statistic 151.8737 139.1731 288.8907

Critical Value* 179.5806 157.6099 316.8185

Age 37 – 52Chi2 Statistic 92.2219 55.8473 109.0211

Critical Value* 67.5048 68.6693 110.8980

Age 53 – 65Chi2 Statistic 25.4714 40.2154 89.6596

Critical Value* 41.3371 27.5871 64.0011

Goodness of Fit Test of Equality Between Fitted and Actual Distributions of Log Present Value of Earnings

* 95% Confidence, equiprobable bins with aprox. 10 people per bin

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Log of the present value of consumption from age 19 to 24 discounted using an interest rate of 3%.Density of fitted and actual log consumptionfor period 1

−4 −3 −2 −1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 5.1Density of fitted and actual log consumption

for period 1

log consumption

ActualFitted

Log of the present value of consumption from age 25 to 30 discounted using an interest rate of 3%.Density of fitted and actual log consumptionfor period 2

−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 5.2Density of fitted and actual log consumption

for period 2

log consumption

ActualFitted

Log of the present value of consumption from age 31 to 36 discounted using an interest rate of 3%.Density of fitted and actual log consumptionfor period 3

−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 5.3Density of fitted and actual log consumption

for period 3

log consumption

ActualFitted

Log of the present value of consumption from age 37 to 51 discounted using an interest rate of 3%.Density of fitted and actual log consumptionfor period 4

−1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 5.4Density of fitted and actual log consumption

for period 4

log consumption

ActualFitted

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Let f(θ1) denote the probability density function of factor θ

1. We allow f(θ

1) to be a mixture of normals.

Assume µ1=E(θ

1), σ

1=Var(θ

1). Let f(µ

1,σ

1) denote the density of a normal random variable with mean µ

1and variance σ

1. The solid curve is the actual density of factor θ

1, f(θ

1), while the dashed curve is

the density of a normal random variable with mean µ1 and variance σ

1. We proceed similarly for factor 2.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Figure 6.1Density of factors and their normal equivalents

factor

Factor 1Normal version of factor 1*Factor 2Normal version of factor 2*

Let f(θ1) denote the probability density function of factor θ

1. We allow f(θ

1) to be a mixture

of normals. The solid line plots the density of factor 1 conditional on choosing the high school sector,that is, f(θ

1|choice=high school). The dashed line plots the density of factor 1 conditional on choosing

the college sector, that is, f(θ1|choice=college).

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 6.2Density of Ability (Factor 1)

conditional on choice

factor 1

High SchoolCollege

Let f(θ2) denote the probability density function of factor θ

2. We allow that f(θ

2) to be a mixture

of normals. The solid line plots the density of factor 2 conditional on choosing the high school sector,that is, f(θ

2|choice=high school). The dashed line plots the density of factor 2 conditional on choosing

the college sector, that is, f(θ2|choice=college).

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 6.3Density of the Factor 2conditional on choice

factor 2

High SchoolCollege

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Table 6.1Returns to college by decile of the ability distribution

Ability Decile Choose high school Choose college

1 -0.006 0.153 0.072 0.0332 0.112 0.126 0.134 0.0673 0.192 0.111 0.207 0.0864 0.271 0.101 0.279 0.0985 0.329 0.095 0.335 0.1066 0.372 0.091 0.375 0.1127 0.413 0.087 0.414 0.1168 0.451 0.084 0.455 0.1209 0.507 0.080 0.503 0.12610 0.650 0.072 0.656 0.135

Mean Return1 Proportion2 Mean Return1 Proportion2

1Let Y0 denote lifetime earnings in high school and Y1 denote lifetime earnings incollege. Then the return to college is (Y1-Y0)/Y0.2Proportion of people who choose the schooling level indicated above and come fromthe ability decile to the left out of those who make the indicated choice. For example,15.3% of those individuals who choose high school come from the first decile of theability distribution.

Table 6.2Distribution of college earnings conditional on high school earnings

CollegeHigh School 1 2 3 4 5 6 7 8 9 10

1 0.68 0.20 0.07 0.03 0.01 0.01 0.00 0.00 0.00 0.002 0.22 0.33 0.20 0.12 0.06 0.04 0.02 0.01 0.00 0.003 0.06 0.23 0.24 0.19 0.13 0.07 0.04 0.02 0.01 0.004 0.02 0.13 0.20 0.21 0.16 0.12 0.08 0.04 0.02 0.015 0.01 0.06 0.14 0.18 0.19 0.17 0.12 0.07 0.04 0.026 0.00 0.03 0.09 0.14 0.18 0.19 0.16 0.13 0.07 0.037 0.00 0.01 0.04 0.08 0.14 0.18 0.19 0.17 0.13 0.068 0.00 0.00 0.02 0.04 0.07 0.13 0.19 0.22 0.20 0.129 0.00 0.00 0.00 0.02 0.04 0.08 0.14 0.21 0.29 0.22

10 0.00 0.01 0.01 0.01 0.01 0.03 0.05 0.11 0.24 0.54

Pr(di<Yc<di+1 |dj<Yh<dj+1) where di is the ith decile of the college lifetime earnings distribution and dj is the jth decile ofthe high school lifetime earnings distribution

Corrrelation(Yc,Yh) = 0.659171

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Let Y0 denote the present value of earnings from age 19 to 65 in the high school sector (discounted at a

3% interest rate). Let f(y0) denote its density function. The solid curve plots the fitted Y

0 density

for those agents who choose high school, that is, f(y0|choice=high school), while the dashed line shows the

counterfactual density function of Y0 for those agents who are actually college graduates, that is, f(y

0|choice=college).

0 500 1000 1500 2000 2500 3000 35000

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−3

Figure 7.1Density of present value of earnings from age 19 to 65

in the high school sector

thousands of dollars

HS (fitted)Col (counterfactual)

Let Y1 denote the present value of earnings from age 19 to 65 in the college sector (discounted at a 3%

interest rate). Let f(y1) denote its density function. The solid curve plots the counterfactual Y

1 density

for those agents who choose high school, that is, f(y1|choice=high school), while the dashed line shows the

fitted density function of Y1 for those agents who are actually college graduates, that is, f(y

1|choice=college).

0 500 1000 1500 2000 2500 3000 35000

0.2

0.4

0.6

0.8

1x 10

−3

Figure 7.2Density of present value of earnings from age 19 to 65

in the college sector

thousands of dollars

HS (counterfactual)Col (fitted)

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Table 7.1

Choice under certaintyHigh School: College

High School:81.02% 18.98%

College:22.84% 77.16%

Proportion of people who, after observing their realizedoutcomes, regret their choice

Choice underuncertainty

Average Annual Return:5.65%

Average Annual Return:11.46%

Average Annual Return7.22%

Average Annual Return9.61%

* For example, out of those that select high school under uncertainty 81% would stillchoose high school if they were to choose based on their realized earnings. Theaverage return for people who choose high school under uncertainty is 7.22%. It wouldbe 5.65% if people were to choose after their outcomes are realized.

Table 7.2

Agent’s Forecast1 of the Variance of the Present Value of Earnings2

Under Different Information Sets at Schooling Choice Date

V ar(Yh) V ar(Yc) V ar(Yc − Yh)Variance with I = Ø 213657.03 360423.15 186705.84

I1 = θ1

Variance 186568.88 215538.98 131538.31

Fraction of the variance3 with

I = Ø explained by I1

12.68% 40.20% 29.55%

I2 = θ2

Variance 90262.93 185409.75 165322.26Fraction of the variance with

I = Ø explained by I2

57.75% 48.56% 11.45%

I3 = {θ1, θ2}Variance 64535.63 49739.06 114352.54

Fraction of the variance with

I = Ø explained by I3

69.79% 88.20% 41.75%

1Variance of the unpredictable component of earnings between age 19 and 65 as predicted at age 192We use an interest rate of 3% to calculate the present value of earnings.3So the variance of the unpredictable component of high school earnings with I1 = θ1 is

(1 − 0.1268) ∗ 213657.03 = 186568.88

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L et Y0

denote the present value of earnings in the high school sector discounted at a 3% interest rate

L et I denote the agent's information set and f(Y0|I) denote the density of the present value of

earnings in high school conditioned on the information set I . We plot f(Y0|I) under no information,

with each factor in the information set, and with both factors in the information set. We use kerneldensity estimation to smooth these functions.

0 500 1000 1500 2000 25000

0.5

1

1.5

2

2.5

3

3.5

4x 10

-3

F igure 8.1Density of present value of high school earnings

under different information sets for the agent

thousands of dollars

I= ∅

I={ θ1

}

I={ θ2

}

I={ θ1

,θ2

}

L et Y1

denote the present value of earnings in the college sector discounted at a 3% interest rate

L et I denote the agent's information set and f(Y1|I) denote the density of the present value

of earnings in college conditioned on the information set I . We plot f(Y1|I) under no information,

with each factor in the information set, and with both factors in the information set. We use kerneldensity estimation to smooth these functions.

0 500 1000 1500 2000 25000

0.5

1

1.5

2

2.5

3x 10

-3

F igure 8.2Density of present value of college earningsunder different information sets for the agent

thousands of dollars

I =∅

I ={θ1

}

I ={θ2

}

I ={θ1

,θ2

}

L et Y0,Y

1denote the present value of earnings in the high school and college sectors, respectively,

discounted at a 3% interest rate. Define D=Y1-Y

0. L et I denote the agent's information set and f(d|I)

denote the density of the difference in present value of earnings in the college and high school sectorsconditioned on the information set I. We plot f(Y

0|I) under no information, with each factor in the

information set, and with both factors in the information set. We use kernel density estimation tosmooth these functions.

-500 0 500 1000 1500 20000

0.5

1

1.5

2

2.5x 10

-3

F igure 8.3Density of the difference between the present value of college and high school earnings

under different information sets for the agent

thousands of dollars

I = ∅

I ={ θ1

}

I ={ θ2

}

I ={ θ1

,θ2

}

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L et (Vh,1

,Vc,1

) denote the value functions for the high school and colleg e at period 1.

Define the ex-ante gross utility difference, D=E (Vc,1

-Vh,1

| I0) where the expectation

is taken with respect to the information available at perio d 0. T he solid line shows the density of D foragents who choose high school (i.e., f(d|choice=high school )), and the dashed line shows the density of D

for agents who choose college (i.e., f(d|choice=college))

-2 -1.5 -1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Figure 9.1Density of expected gross utility differences conditional o n choice

Utils

High schoolCollege

Let C denote psychic costs. Let f(c) denote its density function. The solid line shows the densityof psychic costs for high school graduates, that is f(c|choice=high school). The dashed line showsthe density of psychic costs for college graduates, that is, f(c|choice=college).

−4 −3 −2 −1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure 9.2Density of psychic costs

conditional on choice

Utils

High schoolCollege

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Table 8

Subsidy OverallNone 44.15%

Zero Tuition Economy 49.50%

Percentage of people who choose college when tuition is set to zero

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Variable Number of observations

ASVAB test scores 2293

Enrolled in school at test date 2438

Age at test date 2439

Highest grade completed at test date 2386

Mother's highest grade completed2

2439

Father's highest grade completed2

2439

Broken home dummy 2434

Number of siblings 2439

Lived in an urban environment at age 14 2433

In state tuition for 4 year college at age 17 2438

Education 2397Year of birth 2439

2 Includes imputed values.

Table A-1

Number of observations1 by variable

White Males from NLSY79

1We begin with a sample of 2439 white males. We drop any individuals who are missing any of the

variables listed above. This results in a sample with 2196 people. We also drop anyone for whom we

cannot impute earnings when they are missing from the survey. This produces a sample of 2167

individuals.

Table A-2Evolution of attrition over time

White males from PSIDAge 19 25 31 37 52

19 842 722 527 202 025 0 2254 1721 839 4131 0 0 2267 1143 17337 0 0 0 1661 53852 0 0 0 0 877

*For example, out of 842 individuals whoseearnings are observed at 19, 722 are alsoobserved at age 25, 527 at age 31, and so on.

Table A-3Evolution of attrition over time

Age 19 25 31 37 5219 483 420 325 122 025 0 1500 1185 570 2431 0 0 1542 761 9637 0 0 0 1076 30952 0 0 0 0 508

White male high school and collegegraduates from PSID

*For example, out of 483 individuals whoseearnings are observed at 19, 420 are alsoobserved at age 25, 325 at age 31, and so on.

Table A-4Earnings with and without imputed values1

For NLSY from age 19 to 43Observations Mean

High School and CollegeOriginal 20431 32061.32 29858.5Imputed 26646 30927.7 30536.12

High SchoolOriginal 11103 28298.47 21258.04Imputed 14601 26911.42 22043.71

CollegeOriginal 9328 36540.19 37121.4Imputed 12045 35796.25 37822.81

For PSID from age 19 to 65Observations Mean

High School and CollegeOriginal 40488 48024.54 36116.65Imputed 50929 47041.02 40626.4

High SchoolOriginal 21434 37621.36 21932.5Imputed 27307 36640.38 26673.98

CollegeOriginal 19054 59727.15 44407.22Imputed 23622 59064.15 49663.37

StdDev

StdDev

1To impute earnings, we take a weighted mean of the observed earnings foreach individual with weights given by the Epanechnikov kernel.

Table A-5

Overall High School CollegeObs Mean Obs Mean Obs Mean

1Earnings 1848 0.93 0.64 1153 1.15 0.64 695 0.58 0.45Consumption 1139 0.75 0.51 659 0.92 0.54 480 0.51 0.36

2Earnings 2865 1.84 0.92 1598 1.67 0.78 1267 2.06 1.02Consumption 898 1.63 0.96 482 1.50 0.80 416 1.78 1.10

3Earnings 2908 2.50 1.57 1551 1.96 1.07 1357 3.11 1.81Consumption 931 2.27 1.49 507 1.88 1.21 424 2.73 1.65

4Earnings 1076 7.04 4.47 523 5.27 2.60 553 8.71 5.17Consumption 107 7.23 4.40 28 4.66 1.92 79 8.14 4.67

5Earnings 509 5.58 4.25 308 4.29 2.28 201 7.56 5.60Consumption 123 7.22 4.37 62 5.65 3.28 61 8.82 4.77

Earnings and consumption for each period1

Period2 Variable3 StdDev StdDev StdDev

1 Hundreds of thousands of dollars.2 Periods are defined as ages: 19-24, 25-30, 31-36, 37-51 and 52-65.3 Earnings are defined as the present value of earnings for the ages included in the period discounted using an interest rate of 3%.Consumption is defined as the difference between available resources at the beginning of the period (earnings plus assets) anddiscounted assets the next period.

Table A-6Comparison of variables from PSID and NLSY

People born between 1957 and 1964PSID NLSY

Variable Mean Std. Dev. Mean Std. Dev.Tuition for 4 year college 2.00 0.53 2.16 0.82Number of Siblings 3.11 1.98 2.82 1.82PV Earnings from 19 to 24 1.40 0.70 0.77 0.55PV Earnings from 25 to 30 1.89 0.83 1.70 0.97PV Earnings from 31 to 36 2.63 1.65 2.37 1.67Consumption from 19 to 24 1.23 0.54 0.69 0.49Consumption from 25 to 30 1.80 0.83 1.57 0.91Consumption from 31 to 36 2.51 1.46 2.15 1.47Assets at age 25 0.15 0.54 0.13 0.43Assets at age 31 0.43 0.92 0.53 1.08Assets at age 37 0.71 0.95 0.98 1.52

Categorical Variable Proportion ProportionMother Dropped out of High School 0.175 0.195Mother High School Graduate 0.538 0.525Mother Attended Some College 0.139 0.144Mother College Graduate 0.149 0.136Father Dropped out of High School 0.256 0.251Father High School Graduate 0.371 0.353Father Attended Some College 0.124 0.147Father College Graduate 0.249 0.249College Graduate 0.401 0.453Urban at age 14 0.873 0.742Divorced Parents 0.150 0.155