Asimpleproceduretoestimatekstructuralparametersonconditionallyendogenousvariableswithoneconditionallymeanindependentinstrumentinlinearmodels MunichPersonalRePEcArchive Süß,Philipp

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Munich Personal RePEc Archive

A simple procedure to estimate k

structural parameters on conditionally endogenous variables with one

conditionally mean independent instrument in linear models

Süß, Philipp

Goethe University Frankfurt

10 February 2015

Online at https://mpra.ub.uni-muenchen.de/62030/

MPRA Paper No. 62030, posted 11 Feb 2015 14:29 UTC

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❆ s✐♠♣❧❡ ♣r♦❝❡❞✉r❡ t♦ ❡st✐♠❛t❡ k str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ♦♥

❝♦♥❞✐t✐♦♥❛❧❧② ❡♥❞♦❣❡♥♦✉s ✈❛r✐❛❜❧❡s ✇✐t❤ ♦♥❡ ❝♦♥❞✐t✐♦♥❛❧❧②

♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥t ✐♥str✉♠❡♥t ✐♥ ❧✐♥❡❛r ♠♦❞❡❧s

P❤✐❧✐♣♣ ❙üÿ✯

❋❡❜r✉❛r② ✶✵✱ ✷✵✶✺

❆❜str❛❝t

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❡ ♣r♦♣♦s❡s ❛ s✐♠♣❧❡ ♣r♦❝❡❞✉r❡ t♦ ❡st✐♠❛t❡k♣❛r❛♠❡t❡rs ♦❢ ✐♥t❡r❡st ✐♥ ❛ ❧✐♥❡❛r

♠♦❞❡❧ ✇✐t❤ ♣♦t❡♥t✐❛❧❧②k❝♦♥❞✐t✐♦♥❛❧❧② ❡♥❞♦❣❡♥♦✉s ✈❛r✐❛❜❧❡s ♦❢ ✐♥t❡r❡st ❛♥❞m❡♥❞♦❣❡♥♦✉s ❝♦♥tr♦❧

✈❛r✐❛❜❧❡s ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❛t ❧❡❛st ♦♥❡ ✐♥str✉♠❡♥t❛❧ ✈❛r✐❛❜❧❡ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❝♦♥❞✐t✐♦♥❛❧

♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥❝❡✳

❑❡②✇♦r❞s✿ ■♥str✉♠❡♥t❛❧ ✈❛r✐❛❜❧❡s❀ ❈♦♥❞✐t✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ ❛ss✉♠♣t✐♦♥❀ ❯♥❞❡r✐❞❡♥t✐✜❡❞ ♠♦❞❡❧

❏❊▲ ❈❧❛ss✐✜❝❛t✐♦♥✿ ❈✷✻

✯ ●♦❡t❤❡ ❯♥✐✈❡rs✐t② ❋r❛♥❦❢✉rt✱ ❉❡♣❛rt♠❡♥t ♦❢ ❆♣♣❧✐❡❞ ❊❝♦♥♦♠❡tr✐❝s ❛♥❞ ■♥t❡r♥❛t✐♦♥❛❧ ❊❝♦♥♦♠✐❝

P♦❧✐❝②✱ ●rü♥❡❜✉r❣♣❧❛t③ ✶ ✻✵✸✷✸ ❋r❛♥❦❢✉rt ❛♠ ▼❛✐♥✱ ●❡r♠❛♥② ❚❡❧✳✿ ✰✹✾✲✻✾✲✼✾✽✲✸✹✽✹✶❀ ❊✲♠❛✐❧✿

P❤✐❧✐♣♣✳❙✉❡ss❅✇✐✇✐✳✉♥✐✲❢r❛♥❦❢✉rt✳❞❡

✶

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✶ ■♥tr♦❞✉❝t✐♦♥

■♥ ❛♥ ❛tt❡♠♣t t♦ ✉♥❝♦✈❡r ❝❛✉s❛❧ ❡✛❡❝ts ❛♣♣❧✐❡❞ ❡♠♣✐r✐❝❛❧ r❡s❡❛r❝❤❡rs ❢r❡q✉❡♥t❧② r❡s♦rt t♦ ✐♥str✉♠❡♥t❛❧

✈❛r✐❛❜❧❡s ❡st✐♠❛t✐♦♥ ✉s✐♥❣ t❤❡ ✷❙▲❙ ❡st✐♠❛t♦r t♦ s♦❧✈❡ ❡♥❞♦❣❡♥❡✐t② ✐ss✉❡s✳ ❚❤❡ ✇❡❧❧ ❦♥♦✇♥ ♦r❞❡r

❝♦♥❞✐t✐♦♥ ❢♦r ✐❞❡♥t✐✜❝❛t✐♦♥ st❛t❡s t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ✐♥str✉♠❡♥ts ❤❛s t♦ ❜❡ ❣r❡❛t❡r t❤❛♥ ♦r ❡q✉❛❧

t♦ t❤❡ ♥✉♠❜❡r ♦❢ ❡♥❞♦❣❡♥♦✉s ✈❛r✐❛❜❧❡s✳ ❚❤✐s ♥♦t❡ ♣r♦♣♦s❡s ❛ ♣r♦❝❡❞✉r❡ t♦ ❡st✐♠❛t❡ k ♣❛r❛♠❡t❡rs

♦❢ ✐♥t❡r❡st ✐♥ ❛ ❧✐♥❡❛r ♠♦❞❡❧ ✇✐t❤ ♣♦t❡♥t✐❛❧❧② k ❝♦♥❞✐t✐♦♥❛❧❧② ❡♥❞♦❣❡♥♦✉s ✈❛r✐❛❜❧❡s ♦❢ ✐♥t❡r❡st ❛♥❞

m ❡♥❞♦❣❡♥♦✉s ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡s ✇✐t❤ ❛t ❧❡❛st ♦♥❡ ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t ✐♥str✉♠❡♥t❛❧ ✈❛r✐❛❜❧❡ z

❛♥❞ ❞❡s❝r✐❜❡s ❛ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❝♦♥s✐st❡♥❝②✳ ■♥ ❞✐s❝✉ss✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥❝❡

❛ss✉♠♣t✐♦♥ ❙t♦❝❦ ✭✷✵✶✵✮ st❛t❡s✱ t❤❛t ✏❢♦❝✉s s❤✐❢t❡❞ ❢r♦♠ ❞❡✈❡❧♦♣✐♥❣ ❛ ❢✉❧❧ ♠♦❞❡❧ ❢♦ry t♦ ❡st✐♠❛t✐♥❣ ❛ s✐♥❣❧❡ ❡✛❡❝t ✇❡❧❧✑✳ ❋♦❧❧♦✇✐♥❣ t❤✐s r❡❛s♦♥✐♥❣✱ s✉♣♣♦s❡ ❛ r❡s❡❛r❝❤❡r ✐s ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ♣❛r❛♠❡t❡rsβ♦♥k

✈❛r✐❛❜❧❡s ❞❡♥♦t❡❞ ❜②x❛♥❞ ♦❜t❛✐♥❡❞m❝♦♥tr♦❧ ✈❛r✐❛❜❧❡sw❛s ✇❡❧❧ ❛s ❛t ❧❡❛st ♦♥❡ ✐♥str✉♠❡♥t❛❧ ✈❛r✐❛❜❧❡

z✳ ❆ss✉♠❡ t❤❛t t❤❡ r❡s❡❛r❝❤❡r ❤❛s ❛ r❛♥❞♦♠ s❛♠♣❧❡ ♦❢ {(yi, xi, wi, zi)}^N_i₌₁✱ t❤❡ ♥❡❝❡ss❛r② ♠♦♠❡♥ts

❡①✐st ❛♥❞ t❤❡ tr✉❡ ♠♦❞❡❧ ✐s ❧✐♥❡❛r ✐♥ ♣❛r❛♠❡t❡rs✳

y=x^′β+w^′γ+u ✭✶✮

❆s ✐s ✇❡❧❧ ❦♥♦✇♥✱ ✐❢u✐s ♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢x❣✐✈❡♥w❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♠❡❛♥ ♦❢u✐s ❧✐♥❡❛r ✐♥w

✭E(u|x, w) =w^′δ✮✱ t❤❡♥ t❤❡ ❖▲❙ ❡st✐♠❛t♦r ✭βˆ✮ ❢♦r t❤❡ ❝♦❡✣❝✐❡♥t ✈❡❝t♦rβ ✐s ❝♦♥s✐st❡♥t✱ ❛ss✉♠✐♥❣ t❤❡

r❛♥❦ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✳^✶ ■❢ t❤❡ ❡rr♦r t❡r♠ ✭u✮ ✐s ♥♦t ♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ✈❛r✐❛❜❧❡s ♦❢ ✐♥t❡r❡st ✭x✮

❝♦♥❞✐t✐♦♥❛❧ ♦♥ t❤❡ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡s ✭w✮✱ t❤❡♥ ❖▲❙ ✐s ❣❡♥❡r❛❧❧② ✐♥❝♦♥s✐st❡♥t✳ ❚❤❡ ❢❛✐❧✉r❡ ♦❢ ❝♦♥❞✐t✐♦♥❛❧

♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥❝❡ ❧❡❛❞s ♠❛♥② r❡s❡❛r❝❤❡rs t♦ ❝♦♥s✐❞❡r ✐♥str✉♠❡♥t❛❧ ✈❛r✐❛❜❧❡s str❛t❡❣✐❡s✳ ■t ✐s ❡❛s② t♦

s❤♦✇ t❤❛t t❤❡ t❤❡ ❡rr♦r ✭u✮ ♥❡❡❞ ♥♦t ❜❡ ♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ✐♥str✉♠❡♥t❛❧ ✈❛r✐❛❜❧❡s ✭z✮✳ ■♥ t❤❡ ❥✉st✲

❛♥❞ ♦✈❡r✐❞❡♥t✐✜❡❞ ❝❛s❡✱ ✐t s✉✣❝❡s ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ✷❙▲❙ ❡st✐♠❛t♦r t❤❛tu✐s ♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥t

♦❢z❣✐✈❡♥w❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♠❡❛♥ ♦❢u✐s ❧✐♥❡❛r ✐♥w✭E(u|z, w) =w^′δ✮✳ ❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤❡ ♥♦t❡

✐s str✉❝t✉r❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ t❤❡ ♥❡①t s❡❝t✐♦♥ ■ ♣r♦✈✐❞❡ ❛ ♣r♦♦❢ ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ✷❙▲❙ ❡st✐♠❛t♦r

✐♥ t❤❡ ❥✉st✲ ❛♥❞ ♦✈❡r✐❞❡♥t✐✜❡❞ ❝❛s❡✳ ■♥ t❤❡ t❤✐r❞ s❡❝t✐♦♥ ■ ❛❞❛♣t t❤❡ ♣r♦♦❢ ❢♦r t❤❡ ✏✉♥❞❡r✐❞❡♥t✐✜❡❞✑ ❝❛s❡✱

♣r♦♣♦s❡ t❤❡ ♠❛✐♥ ♣r♦❝❡❞✉r❡ ❛♥❞ ❞❡s❝r✐❜❡ ❛ ✭str♦♥❣❡r✮ ❝♦♥❞✐t✐♦♥❛❧ ✐♥❞❡♣❡♥❞❡♥❝❡ ❛ss✉♠♣t✐♦♥✱ ✇❤✐❝❤ ✐s s✉✣❝✐❡♥t ❢♦r ❝♦♥s✐st❡♥❝②✳ ❚❤❡ ♠❛✐♥ ♠❡t❤♦❞ ✐s ❜r✐❡✢② ♦✉t❧✐♥❡❞ ❛s ❢♦❧❧♦✇s✳ ❆ss✉♠❡ t❤❡r❡ ❛r❡k✈❛r✐❛❜❧❡s

♦❢ ✐♥t❡r❡st ❛♥❞ ♦♥❡ ✈❛❧✐❞ ✐♥str✉♠❡♥t❛❧ ✈❛r✐❛❜❧❡ ❢♦r ❛❧❧k✈❛r✐❛❜❧❡s✳ ❚♦ ♦❜t❛✐♥ ❝♦♥s✐st❡♥t ❡st✐♠❛t❡s ❢♦r ❛❧❧

k♣❛r❛♠❡t❡rs ♣❡r❢♦r♠ ks❡♣❛r❛t❡ ✷❙▲❙ r❡❣r❡ss✐♦♥s ❢♦r ❡❛❝❤xj j∈ {1, ..., k}✉s✐♥❣ z ❛s ❛♥ ✐♥str✉♠❡♥t

❢♦rxj ❛♥❞x₋j, w s❡r✈✐♥❣ ❛s t❤❡✐r ♦✇♥ ✐♥str✉♠❡♥ts✳ ❚❤❡ ❧❛st s❡❝t✐♦♥ ❝♦♥❝❧✉❞❡s✳

✷ Pr♦♦❢ ❥✉st✲ ❛♥❞ ♦✈❡r✐❞❡♥t✐✜❡❞ ❝❛s❡

Pr♦♦❢ ♦❢plim( ˆβ₂SLS) =β ✐❢E(u|w, z) =E(u|w) =w^′δ✱ ✐♥ t❤❡ ❝❛s❡dim(z)≥dim(x)✿

❙t❛❝❦✐♥❣ ❛❧❧ ◆ ♦❜s❡r✈❛t✐♦♥s✱ t❤❡ ♠♦❞❡❧ ✐s ❣✐✈❡♥ ❜②✿

Y = Xβ+W γ+U ✭✷✮

✶❋rö❧✐❝❤ ✭✷✵✵✽✮ ❡♠♣❤❛s✐③❡s t❤❛t t❤❡ ❧✐♥❡❛r✐t② ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ♠❡❛♥ ♦❢ u ✐♥ w ✭E(u|x, w) = w^′δ✮ ✐s ✈✐t❛❧ ❢♦r

❝♦♥s✐st❡♥❝②✳ ❲♦♦❧❞r✐❞❣❡ ✭✷✵✵✺✮ s❤♦✇s t❤❛t t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❝❡rt❛✐♥ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡s ❝❛♥ ❧❡❛❞ t♦ ❛ ❢❛✐❧✉r❡ ♦❢ t❤❡ ❝♦♥❞✐t✐♦♥❛❧

♠❡❛♥ ✐♥❞❡♣❡♥❞❡♥❝❡ ❛ss✉♠♣t✐♦♥✳

✷

(4)

❚❤❡ ✷❙▲❙ ❡st✐♠❛t♦r ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛♥ ❖▲❙ ❡st✐♠❛t♦r ♦❢ ❛ r❡❣r❡ss✐♦♥ ♦❢ y ♦♥ xˆ ❛♥❞ w✱ ✇❤❡r❡ xˆ

❞❡♥♦t❡s t❤❡ ♣r❡❞✐❝t❡❞ ✈❛❧✉❡ ❢r♦♠ t❤❡ ❧✐♥❡❛r ♣r♦❥❡❝t✐♦♥ ❢r♦♠x♦♥z❛♥❞w✭ˆx=z^′θˆ₁+w^′θˆ₂✮✳ ❉❡♥♦t✐♥❣

ˆ

e❛s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ r❡s✐❞✉❛❧✱ ♦♥❡ ❝❛♥ ✇r✐t❡x= ˆx+ ˆe✳ ❆❞❞✐♥❣ ❛♥❞ s✉❜tr❛❝t✐♥❣Xβˆ t♦ t❤❡ ❡q✉❛t✐♦♥

❛❜♦✈❡ ②✐❡❧❞s✿

Y = ˆXβ+W γ+ (U+ (X−X)β)ˆ ✭✸✮

❯s✐♥❣ t❤❡ ❋r✐s❝❤ ❲❛✉❣❤ ▲♦✈❡❧❧ t❤❡♦r❡♠ t❤❡ ❝♦❡✣❝✐❡♥ts βˆ₂SLS ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ♣❛rt✐❛❧❧✐♥❣ ♦✉t W

❢r♦♠Xˆ ❛♥❞Y ✳ ▲❡tMW ❞❡♥♦t❡ t❤❡ r❡s✐❞✉❛❧ ♠❛❦❡r ♠❛tr✐① ♦❢W ❣✐✈❡♥ ❜②MW =I−W(W^′W)⁻¹W^′✳

❚❤❡ ✷❙▲❙ ❡st✐♠❛t♦r ✐s t❤❡♥✿

βˆ₂SLS = ((MWX)ˆ ^′(MWXˆ))⁻¹(MWX)ˆ ^′(MWY)

= ( ˆX^′MWX)ˆ ⁻¹Xˆ^′MW(Xβ+W γ+U)

= ( ˆX^′MWX)ˆ ⁻¹Xˆ^′MWXβ+ 0 + ( ˆXMWXˆ)⁻¹Xˆ^′MWU

❚❤❡ t❤✐r❞ ❧✐♥❡ ✐s ❞✉❡ t♦MWW = 0✳ ◆❡①t ■ ♦❜t❛✐♥ t❤❡ ♣r♦❜❛❜✐❧✐t② ❧✐♠✐t ♦❢ t❤❡ ❧❛st t❡r♠ ✐♥ t❤❡ ❡q✉❛t✐♦♥

❛❜♦✈❡✿

plim(( ˆX^′MWXˆ)⁻¹Xˆ^′MWU) = plim(N⁻¹Xˆ^′MWX)ˆ ⁻¹plim(N⁻¹Xˆ^′MWU)

= A·E( ˆXMWE(U|W, Z))

= A·E( ˆXMWW δ)

= 0

❚❤❡ t❤✐r❞ ❧✐♥❡ ✐s ❞✉❡ t♦ t❤❡ ❦❡② ❛ss✉♠♣t✐♦♥ E(u|w, z) =E(u|w) =w^′δ ❛♥❞ r❛♥❞♦♠ s❛♠♣❧✐♥❣✳ ❚❤❡

❢♦✉rt❤ ❧✐♥❡ ✐s ❞✉❡ t♦MWW = 0✳

■t r❡♠❛✐♥s t♦ s❤♦✇ t❤❛t

plim(( ˆX^′MWXˆ)⁻¹Xˆ^′MWXβ) = β plim(N⁻¹Xˆ^′MWXˆ)⁻¹plim(N⁻¹Xˆ^′MWX)·β = β

■❢ t❤❡ ✜rst t✇♦ t❡r♠s ♦♥ t❤❡ ▲❍❙ ❛r❡ ❡q✉❛❧✱ t❤❡♥plim( ˆβ) =β✳ ❯s❡Xˆ =X−E✳ˆ

plim(N⁻¹Xˆ^′MWX) =ˆ plim(N⁻¹Xˆ^′MWX)−plim(N⁻¹Xˆ^′MWE)ˆ

Eˆ ✐s t❤❡ ✈❡❝t♦r ♦❢ r❡s✐❞✉❛❧s ❢r♦♠ ❛ r❡❣r❡ss✐♦♥ ♦❢x♦♥z ❛♥❞w.❍❡♥❝❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♦❢ t❤❡ ♣r❡❞✐❝t❡❞

✈❛❧✉❡s ❛♥❞ t❤❡ r❡s✐❞✉❛❧s ✐s ③❡r♦ ❛♥❞ ❛♥② r❡❣r❡ss♦r ✐s ✉♥❝♦rr❡❧❛t❡❞ ✇✐t❤ t❤❡ r❡s✐❞✉❛❧ ✭s❡❡ ❋❖❈➫s ♦❢ ❖▲❙✮✳

N⁻¹Xˆ^′MWEˆ = N⁻¹Xˆ^′(I−W(W^′W)⁻¹W^′) ˆE

= N⁻¹Xˆ^′Eˆ+ ˆX^′W(W^′W)⁻¹N⁻¹W^′Eˆ

= 0 + ˆX^′W(W^′W)⁻¹0

= 0

❚❤✐s ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢ ✐♥ t❤❡ ❥✉st✲ ❛♥❞ ♦✈❡r✐❞❡♥t✐✜❡❞ ❝❛s❡✳

✸

(5)

✸ Pr♦♦❢ ✏✉♥❞❡r✐❞❡♥t✐✜❡❞✑ ❝❛s❡✱ t❤❡ ♠❛✐♥ ♣r♦❝❡❞✉r❡ ❛♥❞ ❡①❛♠♣❧❡

❚❤❡ ♠❛✐♥ ♣r♦❝❡❞✉r❡ ✉s❡❞ t♦ ♦❜t❛✐♥ ❝♦♥s✐st❡♥t ❡st✐♠❛t❡s ❢♦r βˆj j ∈ {1, ..., k} ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❘✉♥

k s❡♣❛r❛t❡ ✷❙▲❙ r❡❣r❡ss✐♦♥s ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ✐♥str✉♠❡♥ts✳ ❋♦r t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ j✬t❤

❝♦❡✣❝✐❡♥t ✭βˆj✮ ✉s❡z❛s ❛♥ ✐♥str✉♠❡♥t ❢♦rxj ❛♥❞x₋j✱ws❡r✈✐♥❣ ❛s t❤❡✐r ♦✇♥ ✐♥str✉♠❡♥ts✳ x₋j❞❡♥♦t❡s

❛❧❧ x^′s ✇✐t❤♦✉t xj✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ✐s s✉✣❝✐❡♥t ❢♦r t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ ❡❛❝❤ βˆj ✭plim( ˆβj) = βj ∀j✮✿

E(u|z, x₋j, w) =x^′₋_jδ₋j,1+w^′δ₋j,2∀j. ✭✹✮

❚❤❡ ♣r♦♦❢ ♦❢ ❝♦♥s✐st❡♥❝②plim( ˆβj) =βj ✐s ❥✉st ❛ ♠❛tt❡r ♦❢ r❡❞❡✜♥✐t✐♦♥✳ ▲❡t W₋^new_j = [W, X₋j]❜❡ t❤❡

♥❡✇ ✏W✑ ❛♥❞ t❤❡ ♣r♦♦❢ ❛❜♦✈❡ ❛♣♣❧✐❡s✳ ❚❤❡ q✉❡st✐♦♥ ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥E(u|z, x₋j, w) =x^′₋_jδ₋j,1+w^′δ₋j,2

∀j ❤♦❧❞s✱ ❞❡♣❡♥❞s ♦♥ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ❛t ❤❛♥❞✳ ❆♥ ❡①❛♠♣❧❡ ✇❤❡r❡ t❤❡ ❛ss✉♠♣t✐♦♥ s❡❡♠s ♣❧❛✉s✐❜❧❡ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❈♦♥s✐❞❡r t❤❡ ❣♦❛❧ ♦❢ ❡st✐♠❛t✐♥❣ t❤❡ ✏❝❛✉s❛❧✑ ❡✛❡❝t ♦❢ ✈❛r✐♦✉s t②♣❡s ♦❢ ❡♠♣❧♦②♠❡♥t ✭❢♦r

❡①❛♠♣❧❡ ✐♥❞✉str✐❛❧ ❛♥❞ s❡r✈✐❝❡ s❡❝t♦rs ❡♠♣❧♦②♠❡♥t✮ ♦♥ ✈✐♦❧❡♥t ❝r✐♠❡s ✐♥ ❛ ❧✐♥❡❛r s❡tt✐♥❣✳ ❆ss✉♠❡ t❤❡

❡rr♦r ❝♦♥t❛✐♥s ❝r✐♠✐♥♦❣❡♥✐❝ tr❛✐ts✱ ✇❤✐❝❤ ❧♦✇❡r ❡♠♣❧♦②♠❡♥t ♣r♦s♣❡❝ts ❛♥❞ ❤❛✈❡ ❛ st✐♠✉❧❛t✐♥❣ ❡✛❡❝t ♦♥

✈✐♦❧❡♥t ❝r✐♠❡s✳ ❈♦♥s✐❞❡r ✜r♠ ❜❛♥❦r✉♣t❝✐❡s ❛s ❛♥ ✐♥str✉♠❡♥t✳ ■♥ t❤✐s ❡①❛♠♣❧❡✱ t❤❡ ❝♦♥❞✐t✐♦♥ ✇♦✉❧❞

❤♦❧❞ ✐❢ t❤❡ ♥✉♠❜❡r ♦❢ ❜❛♥❦r✉♣t❝✐❡s ✐s ✉♥r❡❧❛t❡❞ t♦ ❝r✐♠✐♥♦❣❡♥✐❝ tr❛✐ts ❝♦♥❞✐t✐♦♥❛❧ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢

❡♠♣❧♦②❡❡s ✐♥ t❤❡ s❡r✈✐❝❡ s❡❝t♦r ❛♥❞ t❤❛t t❤❡ ♥✉♠❜❡r ♦❢ ❜❛♥❦r✉♣t❝✐❡s ❛r❡ ✉♥r❡❧❛t❡❞ t♦ ❝r✐♠✐♥♦❣❡♥✐❝

tr❛✐ts ❝♦♥❞✐t✐♦♥❛❧ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❡♠♣❧♦②❡❡s ✐♥ t❤❡ ✐♥❞✉str✐❛❧ s❡❝t♦r ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ s❡❡♥ ❛s ♣❧❛✉s✐❜❧❡✳

✹ ❈♦♥❝❧✉s✐♦♥

❚❤❡ ♥♦t❡ s❤♦✇❡❞ t❤❛t t❤❡r❡ ❛r❡ ❝❛s❡s ✇❤❡r❡ ♦♥❡ ✐♥str✉♠❡♥t❛❧ ✈❛r✐❛❜❧❡ ✐s s✉✣❝✐❡♥t ❢♦r t❤❡ ❡st✐♠❛t✐♦♥

♦❢ k≥ 1 str✉❝t✉r❛❧ ♣❛r❛♠❡t❡rs ✐♥ ❛ ❧✐♥❡❛r ♠♦❞❡❧✳ ❆ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❝♦♥s✐st❡♥❝② ✇❛s s♣❡❧❧❡❞

♦✉t ✇❤✐❝❤ ♠❛② ❤❡❧♣ ❛♣♣❧✐❡❞ r❡s❡❛r❝❤❡rs ❛ss❡s✐♥❣ t❤❡ ♣❧❛✉s✐❜✐❧✐t② ♦❢ t❤❡r❡ r❡s✉❧ts ✇❤❡♥ ❡st✐♠❛t✐♥❣ t❤❡

k❡✛❡❝ts s❡♣❛r❛t❡❧②✳

❘❡❢❡r❡♥❝❡s

❬✶❪ ❋rö❧✐❝❤✱ ▼✳✱ ✭✷✵✵✽✮✿ P❛r❛♠❡tr✐❝ ❛♥❞ ◆♦♥♣❛r❛♠❡tr✐❝ ❘❡❣r❡ss✐♦♥ ✐♥ t❤❡ Pr❡s❡♥❝❡ ♦❢ ❊♥❞♦❣❡♥♦✉s

❈♦♥tr♦❧ ❱❛r✐❛❜❧❡s✱ ■♥t❡r♥❛t✐♦♥❛❧ ❙t❛t✐st✐❝❛❧ ❘❡✈✐❡✇ ✷✵✵✽ ✭✷✮✱ ✷✶✹✲✷✷✼

❬✷❪ ❙t♦❝❦✱ ❏✳ ❍✳✱ ✭✷✵✶✵✮✿ ❚❤❡ ❖t❤❡r ❚r❛♥s❢♦r♠❛t✐♦♥ ✐♥ ❊❝♦♥♦♠❡tr✐❝ Pr❛❝t✐❝❡✿ ❘♦❜✉st ❚♦♦❧s ❢♦r ■♥❢❡r✲

❡♥❝❡✱ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝ P❡rs♣❡❝t✐✈❡s✱ ✷✵✶✵ ✭✷✮✱ ✽✸✲✾✹

❬✸❪ ❲♦♦❧❞r✐❞❣❡✱ ❏✳ ▼✳✱ ✭✷✵✵✺✮✿ ❱✐♦❧❛t✐♥❣ ■❣♥♦r❛❜✐❧✐t② ♦❢ tr❡❛t♠❡♥t ❜② ❝♦♥tr♦❧❧✐♥❣ ❢♦r t♦♦ ♠❛♥② ❢❛❝t♦rs✱

✷✵✵✺ ❊❝♦♥♦♠❡tr✐❝ ❚❤❡♦r② ✷✵✵✺ ✭✷✶✮✱ ✶✵✷✻✲✶✵✷✽

✹