SUPPLEMENTARY MATERIALS

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1 SUPPLEMENARY MAERIALS me-varyg rsk premum large cross-secoal equy daases Parck Gaglard, Elsa Ossola ad Olver Scalle hese supplemeary maerals provde he proofs of he echcal lemmas used he paper Appedx 5 ad he resuls of Moe-Carlo expermes ha vesgae he fe-sample properes of he esmaors ad es sascs Appedx 6. We also derve ferece for he cos of equy ad clude some emprcal resuls for Ford Moor, Dsey Wal, Moorola ad Soy Appedx 7. Fally, we provde some robusess checks for he emprcal aalyss Appedx 8. Appedx 5: Proofs of he echcal lemmas A.5. Proof of Lemma We have ŵ w = χ ˆv v + χ v ad ˆv v = ˆv v ˆv v. Sce v s uformly lower bouded from par, we have ŵ w C χ ˆv v C ˆv v + C prove ha he frs erm s o p s suffce o show: χ. he secod erm he RHS s o p from Lemma 4. o sup χ ˆv v = o p. 33 We use Equao 24. Sce ˆν ν = O p c, for some c > by repeag he proof of Proposo 2 wh kow weghs equal o, χ C.5, he uform boud 33 follows f we prove: ˆQ x, Cχ,, χ τ, χ 2,, S M, ad by usg Assumpo sup χ Ŝ S = O p c, 34 sup χ ˆQ x, Q x = O p c, 35 sup χ τ, τ = O p c, 36

2 for some c >. o prove he uform boud 34, we use Equao 26. As he proof of Lemma, we have sup O p,log η/2 ad sup /2 Y, = O p,log η/2 from Assumpo C. c, ad smlarly sup /2 W,, + W 2,, = /2 W 3,, = O p η/2, from Assumpos C. e ad f, respecvely. More- 4 over, ˆQ we use x, M ad χ τ, χ 2,. hus, from Assumpo C.5, boud 34 follows. o prove 35 ˆQ x, Q x = τ ˆQ, x, where W, s defed as Equao 27 ad s such ha x, W, Q sup W, = O p,log η/2 from Assumpo C. b. Fally, 36 follows from τ, τ τ, τ I, E[I, γ, χ τ, χ 2,, τ M ad by usg sup I, E[I, γ = O p,log η/2 from Assumpo C. d. A.5.2 Proof of Lemma 3 A.5.2. Par Le us wre I 2 as: I 2 = = ŵ τ 2, ŵ τ 2, + ŵ τ 2, + ŵ τ 2, ˆQ x, Y, Y, S ˆQ, x, ˆQ x Y, Y, S ˆQ, x + ˆQ x Y, Y, S ˆQ, x, ˆQ x, =: ˆQ x I ˆQ 2 x + I ˆQ 22 x + ŵ τ 2, ˆQ x Y, ˆQ x Y, S ˆQ, x, ˆQ x I 22 + I 23. ˆQ x, ˆQ x Y, ˆQ x Y, S ˆQ, x We corol he erms separaely. Proof ha I 2 = w τ 2 Y, Y, S, + Op,log / = O p + O p,log /. We use 2

3 a decomposo smlar o erm I he proof of Lemma 2: I 2 = w τ 2 + χ w τ 2, τ 2 + χ Y, Y, S, + ˆv Y, Y, S, v τ 2, Y, Y, S, χ w τ 2 Y, Y, S, =: I 2 + I 22 + I 23 + I 24. o prove I 2 = O p, ake k, l =,..., K, ad cosder ζ := w τ 2 Y,k, Y,l, S,kl,. he: = E[ζ 2 x, I, {γ } = w w j τ 2 τj 2 cov Y,k, Y,l,, Y j,k, Y j,l, x, I, γ, γ j 2,j,j, 2, 3, 4 w w j τ 2 τ 2 j cov ε, ε,2, ε j,3 ε j,4 x, γ, γ j I, I,2 I j,3 I j,4 x,kx 2,lx 3,kx 4,l. From Assumpos A. c, C.3 b ad C.4, follows E[ζ 2 = O. Hece, ζ = O p ad I 2 = O p. We ca prove ha I 22 = o p ad I 23 = o p by usg argumes smlar o erms I 2 ad I 3 he proof of Lemma 2. Fally, le us prove ha I 24 = O p,log /. Smlarly o I 4 he proof of Lemma 2, we use ˆv v = v 2 ˆv v + ˆv v 2 ˆv v 2, 37 ad Equao 24. We focus o erm: I 24 = χ v 2 τ, 3 c ˆν ˆQ x, Ŝ S ˆQ x, cˆν Y, Y, S,, he oher corbuos o I 24 ca be corolled smlarly. Now, we use Equao 26 ad rea x as a 3

4 scalar o ease oao. We have: I 24 = χ v 2 τ, 4 c ˆν ˆQ x, W,, ˆQ x, cˆν Y, Y, S, χ v 2 τ, 4 c ˆν ˆQ x, W 2,, ˆQ x, cˆν Y, Y, S, +2 χ v 2 τ, 5 c ˆν ˆQ x, W 3,, ˆQ x, Y, ˆQ x, cˆν Y, Y, S, χ v 2 τ, 6 c ˆν ˆQ 4 x, ˆQ x, ˆQ x, Y, Y, 2 ˆQ x, cˆν Y, Y, S, =: c ˆν I 24 + I I I 244 cˆν. Le us focus o erm I 24 ad prove ha s O p,log /. We have: I 24 = χ v 2 τ 4 2, ˆQ x, W,, Y, 2 erm I 24 s such ha: ad E[I 24 x, I, {γ } Cχ2, χ4 2, 2 V [I 24 x, I, {γ } Cχ4, χ8 2, 4,j χ v 2 τ, 4 2 ˆQ x, W,, S, =: I 24 + I 242., 2, 3 E[η, ε,2 ε,3 x, γ,,..., 6 covη, ε,2 ε,3, η j,4 ε j,5 ε j,6 x, γ, γ j. From Assumpos C.2, C.3 f ad C.5, we ge E[I 24 = O log / ad V [I 24 = o, whch mples I 24 = O p,log /. he oher erms makg I 24 ca be corolled smlarly, ad we ge I 24 = O p,log /. Proof ha I 22 = o p. We have: I 22 = χ v τ, 2 + χ ˆv ˆQ x, Y, ˆQ x Y, S, v τ, 2 ˆQ x, Y, ˆQ x Y, S, =: I22 + I 222. We focus o erm I 22, use Equao 27 ad rea x as a scalar o ease oao. We have: I 22 = + χ v τ, 3 ˆQ x, W, χ v τ, 2 ˆQ x, W ˆQ x Y, Y, S, ˆQ x Y, Y, S, =: I22 + I 222 ˆQ x. 4

5 Le us focus o I 22. We have: E[ I 22 2 x, I, {γ } Cχ2, χ6 2, 2,j,..., 4 W, W j, covε, ε,2, ε j,3 ε j,4 x, γ, γ j. By he Cauchy-Schwarz equaly, we ge: E[ I 22 2 {γ } Cχ 2, χ 6 2, sup E[ W, 4 γ /2 2,j, 2, 3, 4 E [ covε, ε,2, ε j,3 ε j,4 x, γ, γ j 2 γ, γ j /2. From Assumpos C. b, C.3 b, C.4 a, ad C.5, we deduce E[ I 22 2 = o, whch mples I 22 = o p. Smlar argume ca be used o prove ha he oher erms makg I 22 are o p. Proof ha I 23 = o p. hs sep uses argumes smlar as for I 22. A Par We have I 22 = ŵ τ, 2 I 22 = χ v τ 2, ˆQ x, W,, ˆQ x, + Le us frs cosder I 22. We have: ˆQ x, W,, ˆQ x,, where W,, s as Equao 26. Wre: E[ I 22 2 x, I, {γ } Cχ 4, χ 4 2, χ ˆv 2,j v τ, 2 ˆQ x, W,, ˆQ x, =: I 22 + I 222., 2 covη,, η j,2 x, γ, γ j. From Assumpos C.3 a ad C.5, follows E[ I = O log /, ad hus I 222 = O p,log /. Le us ow cosder erm I 222. We use Equao 37, ad plug he decomposos 24 ad 26. We focus o erm c 2ˆν I 222 of he resulg expaso, where: he oher erms ca be reaed smlarly. We have: I 222 = χ v 2 τ, 4 4 ˆQ x, W,, 2. E[I 222 x, I, {γ } Cχ 4, χ 4 2, 2, 2 covε 2,, ε 2, 2 x, γ, 5

6 ad V [I 222 x, I, {γ } Cχ 8, χ 8 2, 4,j, 2, 3, 4 covη, η,2, η j,3 η j,4 x, γ, γ j. From Assumpos C.3 a ad C.5, follows E[I 222 = O log /. By Assumpos C.3 d ad C.5 we ca prove ha V [I 222 = o, ad follows I 222 = O p /. A Par We have I 23 = 2 ŵ τ 3, ˆQ 3 x, W 3,, Y, + ŵ τ 4, 4 4 ˆQ x, ˆQ x, Y, 2, where W 3,, ad are as Equao 26 ad we rea x as a scalar o ease oao. By smlar argumes as par we ca prove ha I 23 = O p,log /. ˆQ 4 x, A Par v he saeme follows from Lemma -, χ τ, χ 2,, χ ad Assumpo C.5. ˆQ x, Cχ,, boud 34, S M A Par v he saeme follows from Equao 2, Lemma v, I = O p ad ŵ τ, 2 E 2 O p,log. ˆQ x, Y, Y, ˆQ x, = A.5.3 Proof of Lemma 4 [ We have P [ χ = P [τ, χ 2, + P P,. We have P, P [ I, χ 2, CN ˆQx, χ, =: P, + P 2,. Le us frs corol [ P I, τ χ 2, M, where we use τ M for all Assumpo C.4 c. he, for < δ < M /2 ad large such ha M χ 2, > δ, we [ ge he upper boud P, P I, τ δ. By usg ha [ τ = E[I, γ ad P I, τ [ [ δ = E P I, E[I, γ δ γ 6

7 sup γ [, b >. [ P I γ E[I γ δ, from Assumpo C. d follows P, = O b, for ay Le us ow cosder P 2,. By usg ˆQ x, M Assumpo C.4 a, we ge eg max ˆQ x, M, ad [ /2. hus CN ˆQx, M /2 eg m ˆQ [ x, Hece P2, P eg m ˆQ x, M/χ 2,. By usg ha eg m ˆQ x, eg m Q x ˆQ [ x, Q x, we ge P 2, P ˆQ x, Q x eg m Q x M/χ 2,. Now, le < δ eg m Q x /2 ad large such ha eg m Q x M/χ 2, > δ. he, by usg [ [ P ˆQ x, Q x δ P I, x x Q x [ δ +P τ, δ we ge P 2, [ P I, x x Q x δ + O b. he frs erm he RHS s O b by usg [ P I, x x Q x [ δ sup P I γx x Q x γ [, δ ad Assumpo C. b. he, P 2, = O b, for ay b >. A.5.4 Proof of Lemma 5 Le W γ := I γ E[I γ ad r := a for < a < η/2. Sce W γ for all γ [,, we have: ˆ sup E[ W γ 4 sup E[ W γ = sup P[ W γ δdδ r + sup P[ W γ δdδ γ [, γ [, γ [, γ [, r ˆ r + C exp { C 2 δ 2 η} dδ + C 3 exp { C 4 η} ˆ r r δ dδ r + C exp { C 2 r 2 η} + C 3 exp { C 4 η} log/r = o, ˆ from Assumpo C. d. 7

8 A.5.5 Proof of Lemma 6 By defo of S j, we have S j S j = Ŝj { Ŝ j κ} S j,j,j S j { Sj κ} S j + Ŝj { Ŝ j κ} S j { Sj κ},j =: I 3 + I 32.,j By Assumpo A.4, I 3 =,j where c := max Le us ow cosder I 32 : S j { Sj <κ} max S j q = O p δ. j S j q κ q κ q c = O p κ q δ, 38 j I 32 = Ŝj { Ŝ j κ, S j <κ} + S j { Ŝ j <κ, S j κ},j,j + Ŝj S j { Ŝ j κ, S j κ},j max Ŝj { Ŝ j κ, S j <κ} + max S j { Ŝ j <κ, S j κ} j j + max Ŝj S j { Ŝ j κ, S j κ} =: I 33 + I 34 + I 35. From Assumpo A.4, we have: Le us sudy I 33 : I 33 max By Assumpo A.4, j I 35 max Ŝj S j max S j q κ q = O p ψ c κ q. 39,j j Ŝj S j { Ŝ j κ, S j <κ} + max j S j { Sj <κ} =: I 36 + I 37. j I 37 κ q c. 4 8

9 Now ake v,. Le N ɛ := j { Ŝ j S j >ɛ}, for ɛ >, he I 36 = max max,j j Ŝj S j { Ŝ j κ, S j vκ} + max Ŝj S j max N v κ + max,j Ŝj S j { Ŝ j κ,vκ< S j <κ} j Ŝj S j c vκ q. Moreover, by he Chebyschev equaly, for ay posve sequece R we have: [ P max N ɛ R P [N ɛ R whch mples max R E[N ɛ [ N ɛ = O p 2 max P Ŝ j S j ɛ. hus,,j 2 max R,j [ P Ŝ j S j ɛ, I 36 = O p ψ 2 Ψ v κ + ψ c vκ q. 4 Fally, we cosder I 34. We have I 34 max max,j j Ŝ j S j + Ŝ j { Ŝ j <κ, S j κ} Ŝj S j max j { Sj κ} + κ max j { Sj κ} = O p ψ c κ q + c κ q. 42 Combg he resul follows. A.5.6 Proof of Lemma 7 By usg ˆε, = ε, x ˆβ β ad Ŝ j = I j, ε, ε j, x x, we have: j Ŝ j = Ŝ j I j, ε, x j + I j, ˆβ β x x j =: Ŝ j A j B j + C j, ˆβj β j x x ˆβj β j x x I j, ε j, x j ˆβ β x x 9

10 where A j = B j. he, for ay, j, we have Ŝj S j Ŝj S j + A j + B j + C j. We ge for ay ξ : where [ Ψ ξ max P Ŝ,j + max P,j Ψ ξ/4 := max,j [ ξ j S j + max P A j ξ [ + max P B j ξ 4,j 4,j 4 [ C j ξ = Ψ ξ/4 + 2P, ξ/4 + P 2, ξ/4, 43 4 [ [ P Ŝj ξ S j, P, ξ/4 := max P A j ξ, ad 4,j 4 [ P 2, ξ/4 := max P C j ξ,j 4 a Boud of Ψ ξ/4. We use ha Ŝ j S j = = τ j, We deduce:. Le us boud he hree erms he RHS of Iequaly 43. I j, ε, ε j, x x S j j I j, ε, ε j, x x E [ ε, ε j, x x γ γ j ad τj M. he: Ŝ j S j M I j, ε, ε j, x x E [ ε, ε j, x x γ γ j + τ j, τ j I j, ε, ε j, x x E [ ε, ε j, x x γ γ j. Ψ ξ/4 [ max P I j, ε, ε j, x x E [ ε, ε j, x x [,j γ γ j ξ + max P τ j, τ j 8M,j [ + max P I j, ε, ε j, x x E [ ε, ε j, x x ξ,j γ γ j 8 [ 2 max P I j, ε, ε j, x x E [ ε, ε j, x x [,j γ γ j ξ + max P τ j, τ j 8M,j =: 2P 3, + P 4,, for small ξ. We use P 3, sup γ,γ [, [ P ξ 8 ξ 8 I γi γ ε γε γ x x E [ ε γε γ x x ξ 8M ad Assumpo C. e o ge P 3, C exp { C 2ξ 2 η} + C 3ξ exp { C 4 η}, for some cosas C, C2, C 3, C 4 >. o boud P 4,, we use τ j M ad τ j, τ j τ j τ j, τ j, τ j

11 τ j τ j 2max P,j τ j, τ j τ j, τ j 2M 2 τ j, τ j, f τj, [ τ j, τ j ξ 2M 2 from Assumpo C. d we ge: [ max P τ,j j, τ j 2M 2 We deduce: ξ 8 8, for small ξ. By usg τ j, = sup γ,γ [, τ j M /2. hus, we have P 4, I j, ad τ j = E[I j, γ, γ j, [ P I γi γ E[I γi γ 2M 2 C exp { C 2ξ η } + C 3ξ /2 exp { C 4 η}. ξ 8 Ψ ξ/4 C exp { C 2ξ 2 η} + C 3ξ exp { C 4 η}. 44 b Boud of P, ξ/4. For some cosa C, we have A j Cτ j, max I j, ε, x k,l,m,k x,l x,m ˆβj β j. Le χ 3, = log a, for a >. From a smlar argume as he proof of Lemma 4, ad Assumpo C. d, we have max P [τ j, χ 3, = O b, for ay b >. hus,,j P, ξ/4 [ ξ max P τ j, max I j, ε, x,j k,l,m,k x,l x,m ˆβj β j 4C [ ξ max P [τ j, χ 3, + max P max I j, ε, x,j,j k,l,m,k x,l x,m 4χ 3, C ad τ j, χ 3, [ ξ + max P ˆβj β j,j 4χ 3, C ad τ j, χ 3, [ K + 3 max max P I j, ε, x,j k,l,m,k x,l x,m [ ξ +P ˆβj β j 4χ 3, C ad τ j, χ 3, By Assumpo C. f, [ max max P ξ I,j k,l,m j, ε, x,k x,l x,m 4χ 3, C ξ 4χ 3, C + O b. 45 { C exp C 2 ξ } η χ 3, +C 3 χ3, ξ exp { C 4 η}. 46

12 [ ξ Le us ow focus o P ˆβj β j 4χ 3, C ad τ j, χ 3,. By usg ˆβ j β j χ3, Q x I j, x ε j, + χ 3, ˆQ x,j Q x I j, x ε j, whe τ j, χ 3,, we ge [ ξ P ˆβj β j 4χ 3, C ad τ j, χ 3, [ P I j, x ε j, ξ 2 4χ 3, C χ 3, Q x [ +P ˆQ x,j Q x I j, x ε j, ξ 2 4χ 3, C χ 3, [ ξ P I j, x ε j, Q 6χ 3 3, C x +P /4 ˆQ x,j ξ Q x 6χ 3 3, C + P /4 ξ I j, x ε j, 6χ 3 3, C [ ξ 2P I j, x ε j, 6χ 3 3, C Q x + P /4 ˆQ x,j ξ Q x 6χ 3 3, C, 47 for small ξ. From Assumpo C.c, he frs probably he RHS of Iequaly 47 s such ha: [ ξ P I j, x ε j, Q 6χ 3 3, C x C exp { } C 2 ξ χ 3 η + C3 χ 3 3, 3, ξ exp { C 4 η}. 48 o boud he secod probably he RHS of Iequaly 47 we use he ex Lemma. Lemma 2 For ay wo o-sgular marces A ad B such ha A B < 2 A we have: B A 2 A 2 A B. 2

13 From Lemma 2, we ge: P /4 ˆQ x,j ξ Q x 6χ 3 3, C P ˆQx,j Q x 2 [ +P ˆQx,j Q x 2P for small ξ >. From Assumpo C.b, P /4 ξ ˆQx,j Q x 2 6χ 3 3, C Q x ˆQx,j Q x 2 2 C exp ξ 6χ 3 3, C 2 Q x { ξ 6χ 3 3, C C 2 /4 Q x /4 Q x ξ χ 3 3, η } 2 2, he, from we ge: P, ξ/4 C exp { C 2ξ η /χ 3 3, +2C 3 χ 3 3, ξ /4 exp { C 4 η}. 49 } + C 3 χ3/2 3, ξ exp { C 4 η} + O b, 5 for small ξ > ad some cosas C, C 2, C 3, C 4 >. c Boud of P 2, ξ/4. We have from Assumpo C.4 C j ˆβ β ˆβj β j sup I j, x k,l,m,p,k x,l x,m x,p j C ˆβ β ˆβj β j. hus, we have: [ P 2, ξ/4 max P C ˆβ [ ξ ξ /2 β ˆβj β j 2P ˆβ β.,j 4 4C By he same argumes as above, we ge: P 2, ξ/4 C exp { C 2ξ η /χ 3 3, } + C 3 χ3/2 3, ξ exp { C 4 η}, 5 for small ξ > ad some cosas C, C 2, C 3, C 4 >. 3

14 d Cocluso. From equales 43, 44, 5 ad 5 we deduce: Ψ ξ C exp { C2ξ 2 η} + C 3 exp { C 4 η} + O b, ξ where ξ := m{ξ, ξ/χ 3 3, }, for small ξ > ad cosas C, C 2, C 3, C 4 >. For ξ = v κ ad log κ = M, we ge ξ = ν κ for large ad η { } 2 Ψ v κ C 2 exp C2M 2 v 2 log + 2 C3 v M +O 2 b = O, for b ad M suffcely large, whe, such ha = O γ for γ >. log Fally, le us prove ha ψ = O p η. Le ɛ >. he, [ log P ψ ɛ η for large ɛ. he cocluso follows. 2 max,j log = 2 Ψ η ɛ [ log P Ŝ j S j ɛ η η log exp { C 4 η} 2 Ψ vκ = O, A.5.7 Proof of Lemma 8 Uder he ull hypohess H, ad by defo of he fed resdual ê, we have By defo of ˆQ e, follows ê = a b ˆν + ĉ ν ˆβ β = a b ν + ĉ ν ˆβ β b ˆν ν 52 = ĉ ν ˆβ β b ˆν ν. ˆQ e = 2 ŵ [ĉ ν ˆβ β 2 ˆν ν ŵ b ˆβ β ĉν + ˆν ν ŵ b b ˆν ν 2 =: ŵ [ĉ ν ˆβ β 2I7 + I 72. 4

15 Le us sudy he secod erm he RHS: I 7 = ˆν ν ŵ τ, b Y, ˆQ x,ĉν =: ˆν ν I 7 ĉ ν, where I 7 = O p by he same argumes used o corol erm I he proof of Proposo 3. We have ˆν ν = O p,log + ad ĉ ν = O p by Lemma 3 v. hus, I 7 = O p,log +. Le us ow cosder I 72. From Lemma - ad Lemma 3 v, we have I 72 = O p,log + 2. he cocluso follows. A.5.8 Proof of Lemma 9 Uder H, ad usg Equao 52, we have ê = e + ĉ ν ˆβ β b ˆν ν. By defo of ˆQ e, follows: ˆQ e = ŵ e ŵ ĉ ν + ŵ [ĉ ν ˆβ β e 2 ˆν ν ˆβ β 2 2 ˆν ν ŵ b e ŵ b ˆβ β ĉν + ˆν ν ŵ b b ˆν ν =: I 8 + I 82 + I 83 + I 84 + I 85 + I From Equaos 24 ad 26 ad smlar argumes as Seco A.2.3 c, we have I 8 = w e 2 + O p,log. By smlar argumes as for erm I he proof of Proposo 3, we have I 82 = ŵ τ, e Y, ˆQ x, ĉ ν = O p. By usg ŵ b e = w b e + O p,log = O p + O p,log ad ˆν ν = O p,log +, we ge I 83 = O p,log + +. Smlar as for I 82 we have I 85 = O p,log 3 +. From 3 ˆν ν = O p,log +, we have I 86 = O p,log + 2. he cocluso follows. 5

16 A.5.9 Proof of Lemma By applyg MN heorem 2 p.35, heorem p. 55 ad usg W, = I, we have Ab = vec Ab = b A vec I = vec [ b A vec I = vec I I m vec b A = vec I I m I W, I m vec b vec A = vec I I m I I m vec vec A b = vec I I m vec vec A b. A.5. Proof of Lemma A.5.. Assumpo APR.4 We use ha eg max A max [, we have: =,..., j= eg max Σ ε,, max a,j for ay marx A = [a j,j=,...,. he, for ay sequece γ =,..., j= where C := sup E[ε γ 2. Defe: γ [, J = { Cov[ε γ, ε γ j C γ : max m=,...,j max m=,...,j {γ j I m } 54 j= } {γ I m } = o. = he Assumpo APR.4 holds f µ Γ J =. From heorem 2.. Sou 974, s eough o show ha µ Γ max {γ I m } > ε <, for ay ε >. Now, sce max B m = o, m=,...,j m=,...,j = = we have µ Γ max {γ I m } > ε µ Γ max {γ I m } B m > ε/2, for m=,...,j m=,...,j = = large. hus, we ge: µ Γ max {γ I m } > ε J max µ Γ {γ I m } B m > ε/2, m=,...,j m=,...,j = 6 =

17 for large. o boud he probably he RHS, we use {γ I m } B m ad he Hoeffdg s equaly see Bosq 998, heorem.2 o ge: µ Γ {γ I m } B m > ε/2 2 exp ε 2 /8. = = he, sce J, we ge: µ Γ max {γ I m } > ε 2 exp ε 2 /8 <, m=,...,j ad he cocluso follows. = = A.5..2 Assumpo A. Codos a ad b are clearly sasfed uder BD., BD.3 ad BD.4. Le us ow cosder codo c. We have σ j, = E[ε γ ε γ j γ, γ j =: σ j depede of. hus, E[σ 2 j, γ, γ j /2 = σ j. By BD., BD.4 ad he Cauchy-Schwarz equaly σ j = J m= where C = sup E[ε γ 2. Hece, we ge: γ [, E E[σ 2 j, γ, γ j /2 C,j = C J m= {γ, γ j I m }E[ε γ ε γ j γ, γ j C J m= E[{γ I m } + C B m + C J m= B 2 m = O From BD.2, he RHS s O, ad codo c Assumpo A. follows. J j m= + J m= {γ, γ j I m }, E[{γ, γ j I m } J Bm 2 m=. A.5..3 Assumpo A.2 Le us cosder codo a. Uder BD. ad BD.3, we have S j = σ j Q x ad S b = lm E τ τ j w w j σ j Q x b b τ j. hs lm s fe f exss, sce from BD.4 we have,j j τ τ j w w j σ j Q x b b τ j j C σ,j, ad E σ,j = O from Assumpo A..,j,j,j 7

18 Moreover: where ξ, = = w τ Y, b = = = w τ I, x b ε, = w τ I, x b ε,. he ragular array ξ, s a margale dfferece sequece = w.r.. he sgma-feld F, = {f, ε,, γ, =,..., }. From a mulvarae verso of Corollary 5.26 Whe, he CL codo a follows f we show: E[ξ, ξ, S b, = = ξ, ξ, E[ξ, ξ, = o p, sup E[ ξ, 2+δ = O, for some δ >. =,..., Moreover, we prove he alerave characerzao of he asympoc varace-covarace marx: v S b = a.s.- lm,j w w j τ τ j τ j σ j Q x b b j. Le us check hese codos. Le G = {γ, =,..., }. We have: E[ξ, ξ, G = = =,j,j,j = ξ,, w w j τ τ j E [I, I j, x x b b j ε, ε j, γ, γ j w w j τ τ j E[I, I j, γ, γ j E[x x b b j E[ε, ε j, γ, γ j τ τ j w w j σ j Q x b b j. τ,j By akg expecao o boh sdes, codo follows. Le us ow cosder codo. Defe ζ, = ξ,,k ξ,,l E[ξ,,k ξ,,l, where ξ,,k s he k-h eleme of ξ,. Sce E[ζ, =, s eough o show V [ζ, = o, for ay k, l. We show hs for k = l, he proof for k l s smlar. For exposory purpose we om he dex k, ad we wre x 2,k x2. We have: V [ζ, = 2 V [ξ, Cov ξ,, 2 ξ,s 2, 55 8 s

19 where: ξ, 2 = w w j τ τ j I, I j, x 2 b b j ε, ε j,.,j Cosder frs he erms Covξ 2,, ξ 2,s for s. By he varace decomposo formula: Covξ 2,, ξ 2,s = E [ Covξ 2,, ξ 2,s G + Cov [ Eξ 2, G, Eξ 2,s G. We have Covξ 2,, ξ 2,s G = from he..d. assumpo over me. Moreover: E[ξ 2, G =,j w w j τ τ j τ j Q x σ j b b j = where α j = w w j τ τ j τ j b b j Q x ad Q x = E[x 2. hus: Cov [ Eξ 2, G, Eξ 2,s G = 2 J m,p=,j,k,l J m= α j σ j {γ, γ j I m },,j Cov α j σ j {γ, γ j I m }, α kl σ kl {γ k, γ l I p }. I he above sum, he erms such ha ses {, j} ad {k, l} do o have a commo eleme, vash. Cosder ow he sum of he erms such ha = k erms such ha = l, or j = k, or j = l are symmerc. herefore, le us focus o he sum S := 2 J m,p=,j,l Cov α j σ j {γ, γ j I m }, α l σ l {γ, γ l I p } = J 2 Cov α j σ j {γ, γ j I m }, α l σ l {γ, γ l I m } 2 m=,j,l J m,p=,m p,j,l E [α j σ j {γ, γ j I m } E [α l σ l {γ, γ l I p }. From BD.4, we have α j C ad σ j C. hus, we ge S = O J 2 E[{γ, γ j, γ l I m } + m=,j,l O J 2 E [{γ, γ j I m } E [{γ, γ l I p }. By usg ha E[{γ, γ j, γ l I m } = m,p=,m p,j,l,j,l O B m + 2 Bm Bm 3 ad E [{γ, γ j I m } E [{γ, γ l I p } = O B m B p +,j,l 9

20 2 BmB 2 p + B m Bp BmB 2 p 2, we ge S = O / + he RHS s o from BD.2. hus, we have show ha: uformly s. Cosder ow V [ξ 2,. By he varace decomposo formula: J m= B 2 m + J m= B 3 m + J m= B 2 m Covξ 2,, ξ 2,s = o, 56 V [ξ 2, = E [ V ξ 2, G + V [ Eξ 2, G. By smlar argumes as above, we have V [ Eξ 2, G = o uformly. Cosder ow erm E [ V ξ 2, G. We have: Moreover: V ξ, G 2 = 2 w w j w k w l τ τ j τ k τ l b b j b k b l,j,k,l Cov I, I j, x 2 ε, ε j,, I k, I l, x 2 ε k, ε l, γ, γ j, γ k, γ l Cov I, I j, x 2 ε, ε j,, I k, I l, x 2 ε k, ε l, γ, γ j, γ k, γ l. = E [I, I j, I k, I l, γ, γ j, γ k, γ l E [ε, ε j, ε k, ε l, γ, γ j, γ k, γ l E[x 4 σ j σ kl τj τ kl E[x 2 2. From he block depedece srucure BD., he expecao E [ε, ε j, ε k, ε l, γ, γ j, γ k, γ l s dffere from zero oly f a par of dces are a same block I m, ad he oher par s also a same block I p, say, possbly wh m = p. Smlarly, σ j σ kl s dffere from zero oly f γ ad γ j are he same block ad γ k ad γ l are he same block. From BD.4, we deduce ha V ξ, G 2 C J 2 {γ, γ j I m }{γ k, γ l I p }, where he double sum he elemes,j,k,l m,p= wh m p are o zero oly f he pars γ, γ j ad γ k, γ l have o eleme commo. hus: E [ V ξ, G 2 = O J 2 E[{γ, γ j, γ k, γ l I m } +O 2,j,k,l m= J,j,k,l: k,l;j k,l m,p=:m p E[{γ, γ j I m }E[{γ k, γ l I p }. 2.

21 By usg J,j,k,l m,p= ge: J,j,k,l m= J E[{γ, γ j, γ k, γ l I m } = O B m + 2 Bm Bm Bm 4 ad E[{γ, γ j I m }E[{γ k, γ l I p } = O E [ V ξ, G 2 = O + m= J m,p= 2 B m B p + 3 BmB 2 p + 4 BmB 2 p 2, we J J J Bm 2 + Bm Bm 4 m= m= m= By BD.2, max m=,..., B2 m = O, ad we ge E [ V ξ, G 2 = O. hus, we have show: V ξ, 2 = O, 57 uformly.. From 55, 56 ad 57, we ge V [ζ = o, ad codo follows. From 57 ad by usg E[ξ, 2 = O, codo follows for δ = 2. Fally, codo v follows from τ τ j w w j σ j b b j = + λ V [f λ 2 σ,j b b j ad he ex Lemma 3. τ j σ σ jj,j Lemma 3 Uder Assumpos BD.-BD.4: L = lm E,j τ j σ,j b b j = σ σ jj E[ε γε γ,j,j ˆ τ j τ j σ,j σ σ jj b b j L, P -a.s., where: J ˆ ˆ ωγdγ + lm ωγ, γ dγdγ, m= I m I m wh ωγ, γ := E[I γi γ E[ε γ 2 E[ε γ 2 bγbγ ad ωγ := ωγ, γ. he, we have proved par a. Par b follows by a sadard CL. A.5..4 Assumpo A.3 Assumpo A.3 s sasfed sce he errors are..d. ad have zero hrd mome Assumpo BD.. 2

22 A.5..5 Assumpo A.4 We have o show ha max j S j q = O p δ, for ay q, ad δ > /2. From S j = σ j Q x, ad a argume smlar o 54: max S j q C j max m=,...,j j= {γ j I m } C max m=,...,j B m +C max m=,...,j for ay q >. Le us derve probably bouds for he wo erms he RHS. From BD.2: max B m m m Le ε := δ, wh δ > /2. he: P max m=,...,j [{γ j I m } B m ε J max j= B m 2 /2 = O. m=,...,j P [{γ j I m } B m, j= [{γ j I m } B m ε j= 2J exp ε 2 /2 = o, from he Hoeffdg s equaly see Bosq 998, heorem.2, ad J. hus, we have show ha max m=,...,j [{γ j I m } B m = o p δ, ad he cocluso follows. j= A.5..6 Assumpo A.5 We have S, = σ ˆQx, ad S j = σ j Q x. Le us deoe by H = σ f, I γ, γ [,, γ, =, 2,... he formao he facor pah, he dcaors pahs ad he dvdual radom effecs. he proof s wo seps. SEP : We frs show ha codoal o H we have Υ := w τ 2 [Y, Y, S, N, Ω,,, 58 P -a.s., where S, = σ vec ˆQ x, ad Ω = lm E τ 2 w w τ j 2 j τ 2 σ 2 j [Q x Q x + Q x Q x W K+.,j j For hs purpose, we apply he Lyapuov CL for heerogeous depede arrays see Davdso 994, 22

23 heorem 23.. Wre Υ = J m= {γ I m }w τ 2 [Y, Y, S, = J J m= W m,, where W m, := J {γ I m }w τ 2 [Y, Y, S,. Codoal o H, he varables W m,, for m =,..., J are depede, wh zero mea. he cocluso follows f we prove: lm V [W m, H = Ω, P -a.s, ad, J m [ lm E W m, 3 H =, P -a.s.., J 3/2 m o show, we use: V [W m, H = J = J w w j τ 2 τj 2 Cov [Y, Y,, Y j, Y j, H,j I m { [ w w j τ 2 τj 2 E Y, Y, Y j, Y j, H,j I m S, S jj, where deoes double sum over all, j =,..., such ha γ, γ j I m. Now, we have by he,j I m depedece propery over me: [ E Y, Y, Y j, Y j, H = 2 E [ε, ε,p ε j,s ε j,q f, γ, γ j I, I,p I j,s I j,q x x s x p x q s p q = E [ ε 2 ε 2 j γ, γ j 2 I, I j, x x x x + σ 2 j 2 I j, I j,p x x x p x p p +σσ 2 jj 2 2 I, I j,s x x s x x s + σj 2 2 I j, I j,s x x s x s x s =: E [ ε 2 ε 2 j γ, γ j A, + σ 2 ja 2, + σ 2 σ 2 jja 3, + σ 2 ja 4,. Moreover, A, = j 2 s I j, x x x x = O j / 2 = O/, uformly H. Le us de- j }, 23

24 fe ˆQ x,j = I j, x x, he j A 2, = 2 I j, I j,p x x x p x p A, = A 3, = 2 p s I, I j,s x x s x x s τ 2 j, A, = vec ˆQx, vec ˆQx,j ˆQ x,j + O /, ˆQx,j + O /, ad A 4, = 2 I j, I j,s x x s x s x A, s = 2 I j, I j,s x x s x s x A, s = 2 I j, I j,s x x s x x s W K+ A, s = ˆQx,j ˆQ x,j W K+ + O /. τ 2 j, he, follows ha: V [W m, H = J τ 2 w w j j τ 2 σj 2 ˆQx,j ˆQ x,j + ˆQ x,j ˆQ x,j W K+,j I m j, +O J w w j τ 2 τ 2 j,,j I m where he O erm s uform w.r.. H. hus, we ge: V [W m, H = τ 2 w w τ j 2 j J τ 2 σ 2 j Q x Q x + Q x Q x W K+ m,j j + w w j τ 2 τj 2 σ jα 2 j + O w w j τ 2 τ 2 j, m,j I m m,j I m where he α j = τ 2 j, ˆQx,j ˆQ x,j + ˆQ x,j ˆQ x,j W K+ τ 2 j Q x Q x + Q x Q x W K+ are o uformly, j, ad w w j τ 2 τ 2 j τ 2 j σj 2 = + λ Σ τ τ j σ f λ 2 j 2 τj 2. σ σ jj he, po follows from 24

25 , whch s proved by smlar ar- σ σ jj τ τ j σj 2 τ 2 L, P -a.s., where L = lm,j j σ σ jj gumes as Lemma 3. J 3/2 Now, Le us ow prove po. We have: m [ E W m, 3 H [ E Y, Y, 3 H E [ 3/2 3/2 = 3,j τ τ j τ 2 j w τ 2 m I m 3 w τ 2 m I m σ 2 j [ /3 3 E Y, Y, 3 H + S, sup E [ [ 3 E Y, 6 H = E Y, Y, H [ Y, Y, 3 H /3 + sup,..., 6 I,...I,6 E [ε,...ε,6 γ x x 2 x 3 x 4 x 5 x 6. S 3,. By he depedece propery, he o-zero erms E [ε,...ε,6 γ volve a mos 3 dffere me dces, [ whch mples ogeher wh BD.4 ha sup E Y, Y, 3 H = O, P -a.s. Smlarly sup S, = O, P -a.s. hus, we ge: J 3/2 J m= [ E W m, 3 H C 3/2 he, po follows from he ex Lemma 4. Lemma 4 Uder Assumpos BD.-BD.4: 3/2 J m= J m= 3 {γ I m }. 3 {γ I m }, P -a.s. SEP 2: We show ha 58 mples he asympoc ormaly codo Assumpo A.4. Ideed, from 58 we have: lm P [ α Υ z H z = Φ,, α Ωα for ay α R 2K+ ad for ay z R, ad P -a.s. We ow apply he Lebesgue domaed covergece heorem, by usg ha he sequece of radom varables P [α Υ z H are such ha P [α Υ z H, uformly ad. We coclude ha, for ay α R 2K+, z R: sce Φ z α Ωα lm P [ α Υ z = lm E P [ α Υ z H = Φ,, s depede of he formao se H. he cocluso follows. z, α Ωα 25

26 A.5. Proof Lemma 2 Wre: B A = [ A I A A B A = { [I A A B I } A, ad use ha, for a square marx C such ha C <, we have I C = I + C + C 2 + C ad I C I C + C C C. hus, we ge: B A A A B A A B A 2 A B A A B 2 A 2 A B, A f A B < 2 A. A.5.2 Proof of Lemma 3 Le us deoe ξ,j = σ j b b j = w γ, γ j. We have ξ,j = ξ + ξ,j. By he LLN τ j σ σ jj,j j we ge ξ = ˆ ωγ ωγdγ, P -a.s.. Le us ow cosder he double sum ξ,j. he proof proceeds hree seps. SEP : We frs prove ha For hs purpose, wre ξ,j = j X m, where X m := ωγ, γ j {γ, γ j I m }, by usg block- depedece. he, we have: j J m= J ξ,j = L + o p, where L := lm m= ˆ I m ˆ j I m ωγ, γ dγdγ. j 26

27 E[X m = whch mples: j ˆ ˆ E[ωγ, γ j {γ, γ j I m } = ωγ, γ dγdγ =: ω m, I m I m Moreover: E j ξ,j = J m= ω m L. V [X m = 2 E [ωγ, γ j ωγ k, γ l {γ, γ j, γ k, γ l I m } E[X m 2 j k l = [ 2 3 ω 2 2 m + O 3 Bm 3 + O 2 Bm 2 2 ω m 2 = OBm 4 + OBm 3 + OBm, 2 ad: CovX m, X p = 2 E [ωγ, γ j ωγ k, γ l {γ, γ j I m }{γ k, γ l I p } E[X m E[X p j k l = 2 [ 2 3 ω m ω p 2 ω m ω p = OB 2 mb 2 p, for m p, whch mples: V j ξ,j = J V [X m + J m= m,p=,m p CovX m, X p = o, from BD.2. he, Sep follows. SEP 2: here exss a radom varable L such ha ξ,j L, P -a.s.. o show hs saeme, we j use ha he eve whch seres ξ,j coverges s a al eve for he..d. sequece γ. Ideed, j we have ha ξ,j coverges f, ad oly f, ξ,j coverges, for ay eger N. he, by he j,j N, j Kolmogorov zero-oe law, he eve whch seres ξ,j coverges has probably eher or. he 27 j

28 laer case however s excluded by Sep. herefore, he sequece ξ,j coverges wh probably, ad Sep 2 follows. SEP 3: We have L = L, wh probably. Ideed, by Seps ad 2 follows ξ,j L = o p j ad ξ,j L = o p. hese equaos mply ha L L = o p, whch holds f ad oly f L = L j wh probably sce L ad L are depede of. j A.5.3 Proof of Lemma 4 he proof s smlar o he oe of Lemma 3 ad we gve oly he ma seps. J 3 3/2 {γ I m } = o p. Ideed, we have: E m= 3/2 J m= 3 {γ I m } = 3/2 from Assumpo BD.2, ad we ca show V J m=,j,k 3/2 E [γ, γ j, γ k I m = O J m= Frs, we prove ha J 3/2 Bm 3 m= = o, 3 {γ I m } = o. Secod, by usg he moooe covergece heorem ad he Kolmogorov zero-oe law, we ca show ha sequece J 3 {γ I m } coverges wh probably. hrd, we coclude ha he lm s wh 3/2 m= probably. Appedx 6: Moe-Carlo expermes I hs seco, we perform smulao exercses o balaced ad ubalaced paels order o sudy he properes of our esmao ad esg approaches. We pay parcular aeo o he eraco bewee pael dmesos ad fe samples sce we face codos lke = o 3 for ferece wh ˆν, ad = o 2 for ferece wh ˆQ e ad ˆQ a, he heorecal resuls. he smulao desg mmcs he emprcal feaures of our daa. he balaced case serves as bechmark o udersad whe s o suffcely large w.r.. o apply he heory. he ubalaced case shows ha we ca explo he gudeles foud for he balaced case whe we subsue he average of he sample szes of he dvdual asses,.e., 28

29 a kd of operave sample sze, for. o summarze our Moe Carlo fdgs, we do o face ay fe sample dsoros for he ferece wh ˆν whe =, ad = 5, ad wh ˆQ e ad ˆQ a whe =, ad = 35. I lgh of hese resuls, we do o expec o face sgfca ferece bas our emprcal applcao. A.6. Balaced pael We smulae S daases of excess reurs from a ucodoal oe-facor model CAPM, we esmae he parameer ν, ad compue he es sascs. A smulaed daase cludes: a vecor of erceps a s R, a vecor of facor loadgs b s R, ad a varace-covarace marx Ω s R. A each smulao s =,..., S, we radomly draw 9, 94 asses from he emprcal sample ha comprses 9, 94 dvdual socks. he asses are lsed by dusral secors. We use he classfcao proposed by Ferso ad Harvey 999. he vecor b s s composed by he esmaed facor loadgs for he radomly chose asses. A each smulao, we buld a block dagoal marx Ω s wh blocks machg dusral secors. he elemes of he ma dagoal of Ω s correspod o he varaces of he esmaed resduals of he dvdual asses. he off-dagoal elemes of Ω s are covaraces compued by fxg correlaos wh a block equal o he average correlao of he dusral secor compued from he 9, 94 9, 94 hresholded varace-covarace marx of esmaed resduals. Hece we ge a seg le wh he block depedece case developed Appedx 4. I order o sudy he sze ad power properes of our procedure, we se he values of he erceps a s accordg o four daa geerag processes: DGP: he rue parameer s ν =.% ad he a s are geeraed uder he ull hypohess H : a s = ; DGP2: he rue parameer s he emprcal esmae of ν, ν = 2.57%, ad he a s ull hypohess H : a s = bs ν ; are geeraed uder he DGP3: he a s are geeraed uder he alerave hypohess H a : a s =.5b s +.5 ν, where ν = 2.57%; DGP4: he a s are geeraed uder he hree-facor alerave hypohess: H a : a s = b s,3 ν,3 where b s,3 R3 ad ν,3 = [2.92%,.63%, 9.96% are esmaes for he Fama-Frech model o he 29

30 CRSP daase. DGP ad DGP2 mach wo dffere ull hypoheses. he ull hypohess for DGP assumes ha he facor comes from a radable asse, ad for DGP2 ha does o. DGP3 ad DGP4 mach wo dffere alerave hypoheses as suggesed by MacKlay 995. DGP3 s a o rsk-based alerave. I represes a devao from CAPM, whch s urelaed o rsk: we ake he oe-facor model calbraed o he daa wh erceps devag from he o arbrage resrco. DGP4 s a rsk-based alerave. I represes a devao from CAPM, whch comes from mssg rsk facors: we ake erceps from a hree-facor model calbraed o he daa, ad he we esmae a oe-facor model. Le us defe he smulaed excess reurs R, s of asse a me as follows R s, = a s + b s f + ε s,, for =,...,, ad =,...,, 59 where f s he marke excess reur ad ε s, s he error erm. he error vecors εs are depede across me ad Gaussa wh mea zero ad varace-covarace marx Ω s. We apply our esmao approach o every smulaed daase of excess reurs. We esmae he parameer ν ad we compue he sascs descrbed Seco 2.5 of he paper. Sce he pael s balaced, we do o eed o fx χ 2,. We oly use χ, = 5. However, hs rmmg level does o affec he umber of asses he smulaos. I order o compue he hresholded esmaor of he varace-covarace marx of ˆν, amely Σ ν see Proposo 4 he paper, ad he hresholded varace esmaor Σ ξ for he es sascs, we fx he parameer M equal o.78, ha s used he emprcal applcao. We defe he parameer M usg a cross-valdao mehod as proposed Bckel ad Leva 8. We buld radom subsamples from he CRSP sample. For each subsample, we mmze a rsk fuco ha explos he dfferece bewee a hresholded varace-covarace marx ad a arge varace-covarace marx see Bckel ad Leva 8 for deals. I order o udersad how our esmao approach works for dffere fe samples, we perform exercses combg dffere values of he cross-secoal dmeso ad he me dmeso. able 5 repors esmao resuls for esmaor ˆν, ad for he bas-adjused esmaor ˆν B, uder DGP ad 2. he resuls clude he bas of boh esmaors, he varace ad he Roo Mea Square Error RMSE of esmaor ˆν B, ad he coverage of he 95% cofdece erval for parameer ν based o Proposo 4. he 3

31 bas of esmaor ˆν s decreasg absolue value wh me seres sze ad s raher sable w.r.. crosssecoal sze. he aalycal bas correco s raher effecve, ad he bas of esmaor ˆν B s small. For sace, for sample szes = 5 ad =, uder DGP 2 he bas of esmaor ˆν B s equal o.3, whch absolue value s abou % of he rue value of he parameer ν = he varace of esmaor ˆν B s decreasg w.r.. boh me-seres ad cross-secoal sample szes ad. hese feaures reflec he large sample dsrbuo of he esmaors derved Proposo 3. he coverage of he cofdece ervals s close o he omal level 95% across he cosdered desgs ad sample szes. I able 6, we dsplay he rejeco raes for he es of he ull hypohess ν = radable facor. hs ull hypohess s sasfed DGP, ad he rejeco raes are raher close o he omal sze 5% of he es, wh a slgh overrejeco. I DGP 2, parameer ν s dffere from zero, ad he es feaures a power equal o %. ables 7 ad 8 repor he resuls for he ess of he ull hypoheses H : a γ = ad H : a γ = b γ ν, respecvely. he es sascs are based o ˆQ a ad ˆQ e as defed Proposo 5. DGP sasfes he ull hypohess for boh ess. For = 5, we observe a oversze, ha s creasg w.r.. crosssecoal sze. he me seres dmeso = 5 s lkely oo small compared o cross-secoal sze = ad hs combao does o reflec he codo = o 2 for he valdy of he asympoc Gaussa approxmao of he sascs. For = 5 sead, he rejeco raes of he ess are que close o he omal sze. DGP 2 sasfes he ull hypohess of he es based o ˆQ e, bu correspods o a alerave hypohess for he es based o ˆQ a. he former sasc feaures a smlar behavour as uder DGP, whle he power of he laer sasc s creasg w.r... Fally, he power of boh sascs uder he "o rsk-based" ad "rsk-based" aleraves DGP 3 ad DGP 4 s very large, wh rejeco raes close o % for all cosdered combaos of sample szes ad. 3

32 able 5: Esmao of ν, balaced case = 5 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Basˆν Basˆν B Varˆν B RMSEˆν B Coverage = 5 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Bas ˆν Basˆν B Varˆν B RMSEˆν B Coverage able 6: es of ν =, balaced case = 5 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Rejeco rae = 5 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Rejeco rae

33 able 7: es of he ull hypohess H : a γ =, balaced case = 5 DGP DGP 2 DGP 3 DGP 4 5,,5 5,,5 5,,5 5,,5 Sze/Power = 5 DGP DGP 2 DGP 3 DGP 4 5,,5 5,,5 5,,5 5,,5 Sze/Power able 8: es of he ull hypohess H : a γ = b γ ν, balaced case = 5 DGP DGP 2 DGP 3 DGP 4 5,,5 5,,5 5,,5 5,,5 Sze/Power = 5 DGP DGP 2 DGP 3 DGP 4 5,,5 5,,5 5,,5 5,,5 Sze/Power A.6.2 Ubalaced pael Le us repea smlar exercses as he prevous seco, bu wh ubalaced characerscs for he smulaed daases. We roduce hese characerscs hrough a marx of observably dcaors I s R. he marx gahers he dcaor vecors for he radomly chose asses. We fx he maxmal sample sze = 546 as he emprcal applcao. I he ubalaced seg, he excess reurs R, s of asse a me s: R s, = a s + b s f + ε s,, f I s, =, for =,...,, ad =,...,, 6 where I, s s he observably dcaor of asse a me. 33

34 I ables 9 ad, we provde he operave cross-secoal ad me-seres sample szes he Moe- Carlo repeos for rmmg χ, = 5 ad four dffere levels of rmmg χ 2,. More precsely, able 9 we repor he average umber χ of reaed asses across smulaos, as well as he mmum m χ ad he maxmum max χ across smulaos. For he lowes level of rmmg χ 2, = /2, all asses are kep all smulaos, whle for he level of rmmg χ 2, = /6 o average we keep abou wo hrds of he asses. I able, we repor he average across asses of he, ha are he average me-seres sze for asse across smulaos, as well as he m ad he max of he. Sce he dsrbuo of for a asse s rgh-skewed, we also repor he average across asses of he meda. For rmmg level χ 2, = /6, he average mea me-seres sze s abou 8 mohs, whle he average meda me-seres sze s 4 mohs. I able, we dsplay he resuls for esmaors ˆν ad ˆν B. he bas adjusme reduces subsaally he bas for esmao of parameer ν. For rmmg level χ 2, = /6, he coverage of he cofdece erval s close o he omal sze 95% for all cosdered cross-secoal szes, whle for χ 2, = /2 he coverage deeroraes wh creasg cross-secoal sze. I comparso wh able 5, he bas ad varace of esmaor ˆν B are larger ha he oes obaed he balaced case wh me-seres sze = 5. However, for rmmg level χ 2, = /6, he resuls are smlar o he oes wh = 5 able 5. I fac, hs me-seres sze of he balaced pael reflecs he operave sample szes for ha rmmg level observed able. Smlar commes apply for able 2, where we repor he resuls for he es of he hypohess ν =. For rmmg level χ 2, = /6, he sze of he es s close o he omal level 5% uder DGP, ad he he power s % uder DGP 2. I ables 3 ad 4, we dsplay he resuls for he ess based o ˆQ a ad ˆQ e, respecvely. For rmmg level χ 2, = /, we observe a oversze, ha creases wh he cross-secoal dmeso. We ge a smlar behavour wh more severe oversze wh lower rmmg levels o repored. We expec hese fdgs from he resuls he prevous seco. Ideed, for rmmg level χ 2, = /, he operave me-seres sample sze able s aroud mohs, ad ables 7 ad 8, for a balaced pael wh = 5, he sascs are overszed. For rmmg level χ 2, = /24 wh operave sze of abou 35 mohs, he oversze of he sascs s moderae. Fally, he power of he sascs s very large also he ubalaced case, ad close o %. 34

35 able 9: Operave cross-secoal sample sze rmmg level χ 2, = 2 χ 2, = 6, 3, 6, 9,, 3, 6, 9, χ, 3, 6, 9, 66 2, 4, 6, m χ, 3, 6, 9, 6,9 3,9 5,9 max χ, 3, 6, 9, 7 2, 4, 6, rmmg level χ 2, = χ 2, = 24, 3, 6, 9,, 3, 6, 9, χ 4,25 2,4 3, ,25 m χ 35, 2,3 3,5 37 8, max χ 44,3 2,5 3, ,3 able : Operave me-seres sample sze rmmg level χ 2, = 2 χ 2, = 6 χ 2, = χ 2, = 24 mea m max meameda

36 able : Esmao of ν, ubalaced case rmmg level: χ 2, = 2 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Basˆν Basˆν B Varˆν B RMSEˆν B Coverage rmmg level: χ 2, = 6 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Basˆν Basˆν B Varˆν B RMSEˆν B Coverage

37 able 2: es of ν =, ubalaced case rmmg level: χ 2, = 2 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Rejeco rae rmmg level: χ 2, = 6 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Rejeco rae able 3: es of he ull hypohess H : β γ =, ubalaced case rmmg level: χ 2, = DGP DGP 2, 3, 6, 9,, 3, 6, 9, Sze/Power DGP 3 DGP 4, 3, 6, 9,, 3, 6, 9, Sze/Power rmmg level: χ 2, = 24 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Sze/Power DGP 3 DGP 4, 3, 6, 9,, 3, 6, 9, Sze/Power

38 able 4: es of he ull hypohess H : β γ = β 3 γ ν, ubalaced case rmmg level: χ 2, = DGP DGP 2, 3, 6, 9,, 3, 6, 9, Sze/Power DGP 3 DGP 4, 3, 6, 9,, 3, 6, 9, Sze/Power rmmg level: χ 2, = 24 DGP DGP 2, 3, 6, 9,, 3, 6, 9, Sze/Power DGP 3 DGP 4, 3, 6, 9,, 3, 6, 9, Sze/Power Appedx 7: Cos of equy We ca use he resuls Seco 3 for esmao ad ferece o he cos of equy codoal facor models. We ca esmae he me varyg cos of equy CE, = r f, + b,λ of frm wh ĈE, = r f, + ˆb,ˆλ, where r f, s he rsk-free rae. We have see Appedx 7. ĈE, CE, = ψ,e 2 ˆβ β + Z b, Wp,K vec [ˆΛ Λ + o p, 6. where ψ, = λ Z, λ Z, Sadard resuls o OLS mply ha esmaor ˆβ s asympocally ormal, ˆβ β N, τ Q x, S Q x,, ad depede of esmaor ˆΛ. he, from Proposo 38

39 9, we deduce ha ĈE, CE, N, Σ CE,, codoally o Z, where Σ CE, = τ ψ,e 2Q x, S Q x, E 2ψ, + Z b, Wp,K Σ Λ W K,p Z b,. Fgure 4 plos he pah of he esmaed aualzed coss of equy for Ford Moor, Dsey, Moorola ad Soy. he cos of equy has rse remedously durg he rece subprme crss. A.7. Proof of Equao 6 We have: [ ˆb,ˆλ = r Z Z ˆB ˆΛ [ +r Z Z, Ĉ ˆΛ = [ Z Z vec ˆB ˆΛ + Z Z, vec [Ĉ ˆΛ. hus, we ge: ĈE, CE, = Z Z [ vec ˆB ˆΛ = Z Z [ˆΛ I p vec [ ˆB B + Z Z, [ˆΛ I q vec [Ĉ C vec [Ĉ ˆΛ vec [ C Λ + I p B vec [ˆΛ Λ + I p C vec [ˆΛ Λ. vec [ B Λ + Z Z, By usg ha ˆΛ = Λ + o p ad vec [ˆΛ [ˆΛ Λ = W p,k vec Λ, Equao 6 follows. 39

40 Fgure 4: Pah of esmaed aualzed coss of equy 6 CE of Ford Moor CE of Dsey Wal CE of Moorola CE of Soy he fgure plos he pah of esmaed aualzed coss of equy for Ford Moor, Dsey Wal, Moorola ad Soy ad her powse cofdece ervals a 95% probably level. We also repor he average codoal esmae sold horzoal le. he vercal shaded areas deoe recessos deermaed by he Naoal Bureau of Ecoomc Research NBER. 4

41 Appedx 8: Robusess checks I hs seco, we perform several checks o evaluae he robusess of he emprcal resuls repored he paper. I parcular, we esmae he pahs of he me-varyg rsk prema ad we compue he es sascs by: a. Assumg several asse prcg models as basele specfcao; b. Usg several ses of asse-specfc srumes Z, ; c. Usg several ses of commo srumes Z ; d. Assumg ha he me-varyg beas b, deped oly o he asse-specfc srumes. I able 5, we provde he deals of he codoal specfcaos for he four exercses. We use he followg abbrevaos. For commo srumes, we deoe by s he erm spread, ds he defaul spread, ad dvy he dvded yeld. he dvded yeld s provded by CRSP. For asse-specfc srumes, we deoe by mc, he marke capalzao, bm, he book-o-marke, ad d, he reur of he correspodg dusry porfolo. For each exercse, whe o explcly dcaed able 5, he specfcao s he four-facor model, he vecor of commo srumes s Z = [, s, ds ad he asse-specfc srume s he scalar Z, = bm,. able 5 repors he operave rmmed populao of dvdual socks ad he umber of regressors he frs-pass me seres regresso for each exercse ha we mpleme. Ideed, he populao of dvdual socks chages depedg o he asse prcg model Exercse a as a effec of he rmmg codos: he umber of asses decreases as he umber K of facors creases. Moreover, by usg he four-facor model as basele ad modfyg he ses of srumes, he umber of asses decreases as he umber of regressors he frs pass creases see Exercse c. We frs prese codoal esmaes of rsk prema by usg several asse prcg models as basele Exercse a. Pael A of Fgure 5 compares he esmaed me-varyg pahs of marke rsk prema whe we assume he four-facor model show Seco 4 ad he CAPM. Pael B compares he esmaes ˆλ m, for he four-facor model ad he Fama-Frech model. he pahs look very smlar. he dscrepacy bewee he esmaes of he CAPM ad he four-facor model s explaed by he hree facors sze, value ad momeum facor ha we roduce he four-facor model. Fgure 6 plos he esmaed me-varyg 4

42 pahs of rsk prema for he sze ad value facors compued o he four-facor model ad o he Fama- Frech model. he rsk premum for he sze facor s very smlar for he wo models. he value rsk premum for he Fama-Frech model akes slghly smaller values ha ha for he four-facor model ad exhbs a couer-cyclcal pah. Overall, he codoal esmaes of he rsk prema are sable wh respec o he asse prcg model ha s assumed for he excess reurs. Fgures 7 ad 8 plo he esmaes of he rsk prema by adopg several ses of asse-specfc srumes Z, Exercse b. We do o modfy he se of commo srumes Z compared o Seco 4 of he paper. I Fgure 7, we ge he esmaes by seg he scalar Z, equal o he marke capalzao of frm. I Fgure 8, we se Z, equal o he mohly reurs of he dusry porfolo for he dusry asse belogs o. We use he 48 Fama-Frech dusry porfolos. he rsk prema pahs look very smlar o he resuls Seco 4. he resuls for he ess of he asse prcg resrcos for he codoal specfcaos Exercse b are repored able 6, upper pael. he es sascs rejec he ull hypoheses a 5% level. he me-varyg pahs of he rsk prema showed Fgures 9 ad are compued by modfyg he se of commo srumes Z = [, Z Exercse c. I Fgure 9, Z s a bvarae vecor ha cludes he defaul spread ad he dvded yeld. he pahs of he rsk prema for marke, value ad momeum facors look smlar o he resuls Seco 4. However, he rsk premum for he sze facor feaures a very sable paer ha does o correspod o he ucodoal esmae. I Fgure, vecor Z cludes he erm spread, he defaul spread, ad he dvded yeld. he pahs of he rsk prema look smlar o he resuls Seco 4. Iroducg he dvded yeld creases he dscrepacy bewee he ucodoal esmaes ad he average over me of codoal esmaes for he sze ad momeum facors w.r.. he resuls show Fgure. O he corary, hs dscrepacy s smaller for he value premum. Moreover, he rsk premum of he momeum facor akes larger values ha ha Fgure. We also oce ha cludg he dvded yeld amog he commo srumes decreases he umber of socks afer rmmg. he es sascs rejec he ull hypohess a 5% level see able 6, mddle pael. Fally, we cosder codoal specfcaos whch he me-varyg beas are lear fucos of asse specfc srumes Z, oly Exercse d. he rsk prema are modelled va commo srumes Z = [, s, ds as usual. I Fgure, Z, s a bvarae vecor ha cludes he cosa ad he book-o-marke equy of frm. I Fgure 2, vecor Z, cludes he cosa ad he reur of he 42

43 dusry porfolo as asse-specfc srume. he pahs of he rsk prema for he four facors Fgure look more volale w.r.. he pahs Fgure. he rsk prema for marke, sze ad value facors Fgure 2 look smlar o he resuls Seco 4. he rsk premum for he momeum facor feaures a less sable paer, albe s cofdece ervals look smlar o ha Fgure. I able 6, lower pael, he es sasc does o rejec he asse prcg resrco H : β γ = β 3 γ ν for he codoal specfcao wh me-varyg beas depedg o book-o-marke equy. able 5: Operave cross-secoal sample sze χ, umber of facors K ad srumes q ad p ad frs-pass regressors d he four exercses of robusess checks χ K p q d χ K p q d Exercse a. Exercse c. CAPM 5, Z = [, ds, dvy, Fama-Frech model 4, Z = [, ds, s, dvy Exercse b. Exercse d. Z, = mc, 3, Z, = [, bm, 6, Z, = d, 4, Z, = [, d, 6,

44 able 6: es resuls for asse prcg resrcos es of he ull hypohess H : β γ = β3 γ ν es of he ull hypohess H : β γ = Exercse b. Z, = mc, Z, = d, Z, = mc, Z, = d, χ = 3, 835 χ = 4, 748 χ = 3, 835 χ = 4, 748 es sasc p-value.... Exercse c. Z = [, ds, dvy Z = [, ds, s, dvy Z = [, ds, dvy Z = [, ds, s, dvy χ =, 7 χ = 667 χ =, 7 χ = 667 es sasc p-value Exercse d. Z, = [, bm, Z, = [, d, Z, = [, bm, Z, = [, d, χ = 6, 8 χ = 6, 43 χ = 6, 8 χ = 6, 43 es sasc p-value We compue he sascs /2 Σ ξ ˆξ based o ˆQe ad ˆQa defed Proposo 5 for χ dvdual socks o es he ull hypoheses H : β γ = β3 γ ν ad H : β γ =. he able repors he sascs ad her p-values whe we use several ses of asse-specfc srumes Z, Exercse b ad commo srumes Z Exercse c, ad whe me-varyg beas are fucos of he asse-specfc srumes oly Exercse d. 44

45 Fgure 5: Pah of esmaed aualzed rsk prema for he marke facor 6 Pael A Pael B Pael A plos he pahs of esmaed aualzed marke rsk prema ˆλ m, compued by usg he four-facor model h red le ad he CAPM hck blue le. Pael B plo he pahs of marke rsk prema ˆλ m, esmaed by assumg he four-facor model h red le ad he Fama-Frech model hck blue le. he powse cofdece ervals a 95% level are also dsplayed. he vercal shaded areas deoe recessos deermed by he Naoal Bureau of Ecoomc Reasearch NBER. 45

46 Fgure 6: Pah of esmaed aualzed rsk prema for he sze ad value facors 6 Pael A Pael B he fgure plos he pahs of esmaed aualzed rsk prema ˆλ smb, Pael A ad ˆλ hml, Pael B compued by usg he four-facor model h red le ad he Fama-Frech model hck blue le. he powse cofdece ervals a 95% level are also dsplayed. he vercal shaded areas deoe recessos deermed by he Naoal Bureau of Ecoomc Reasearch NBER. 46

47 Fgure 7: Pah of esmaed aualzed rsk prema compued usg Z, = mc, ˆλm, ˆλsmb, ˆλhml, ˆλmom, he fgure plos he pah of esmaed aualzed rsk prema ˆλm,, ˆλsmb,, ˆλhml, ad ˆλmom, ad her powse cofdece ervals a 95% level whe marke capalzao s used as asse-specfc srume. he vecor of commo srumes s Z = [, s, ds. We also dsplay he ucodoal esmae dashed horzoal le ad he average codoal esmae sold horzoal le. We cosder all socks as base asses = 9, 936 ad χ = 3, 835. he vercal shaded areas deoe recessos deermed by he Naoal Bureau of Ecoomc Reasearch NBER. 47

48 Fgure 8: Pah of esmaed aualzed rsk prema compued usg Z, = d, ˆλm, ˆλsmb, ˆλhml, ˆλmom, he fgure plos he pah of esmaed aualzed rsk prema ˆλm,, ˆλsmb,, ˆλhml, ad ˆλmom, ad her powse cofdece ervals a 95% level whe he reurs of dusry porfolos are used as asse-specfc srume. he vecor of commo srumes s Z = [, s, ds. We also dsplay he ucodoal esmae dashed horzoal le ad he average codoal esmae sold horzoal le. We cosder all socks as base asses = 9, 936 ad χ = 4, 748. he vercal shaded areas deoe recessos deermed by he Naoal Bureau of Ecoomc Reasearch NBER. 48

49 Fgure 9: Pah of esmaed aualzed rsk prema compued usg Z = [, ds, dvy ˆλm, ˆλsmb, ˆλhml, ˆλmom, he fgure plos he pah of esmaed aualzed rsk prema ˆλm,, ˆλsmb,, ˆλhml, ad ˆλmom, ad her powse cofdece ervals a 95% level whe defaul spread ad dvded yeld are used as commo srumes. he sock specfc srume s booko-marke equy. We also dsplay he ucodoal esmae dashed horzoal le ad he average codoal esmae sold horzoal le. We cosder all socks as base asses = 9, 936 ad χ =, 7. he vercal shaded areas deoe recessos deermed by he Naoal Bureau of Ecoomc Reasearch NBER. 49

50 Fgure : Pah of esmaed aualzed rsk prema compued usg Z = [, ds, s, dvy ˆλm, ˆλsmb, ˆλhml, ˆλmom, he fgure plos he pah of esmaed aualzed rsk prema ˆλm,, ˆλsmb,, ˆλhml, ad ˆλmom, ad her powse cofdece ervals a 95% level whe defaul spread, erm spread ad dvded yeld are used as commo srumes. he sock specfc srume s book-o-marke equy. We also dsplay he ucodoal esmae dashed horzoal le ad he average codoal esmae sold horzoal le. We cosder all socks as base asses = 9, 936 ad χ = 667. he vercal shaded areas deoe recessos deermed by he Naoal Bureau of Ecoomc Reasearch NBER. 5

51 Fgure : Pah of esmaed aualzed rsk prema wh me-varyg beas modelled va Z, = [, bm, ˆλm, ˆλsmb, ˆλhml, ˆλmom, he fgure plos he pah of esmaed aualzed rsk prema ˆλm,, ˆλsmb,, ˆλhml, ad ˆλmom, ad her powse cofdece ervals a 95% level whe me-varyg beas are lear fucos of he book-o-marke srume oly. he rsk prema vecor volves he commo srumes Z = [, s, ds. We also repor he ucodoal esmae dashed horzoal le ad he average codoal esmae sold horzoal le. We cosder all socks = 9, 936 ad χ = 6, 8 as base asses. he vercal shaded areas deoe recessos deermed by he Naoal Bureau of Ecoomc Reasearch NBER. 5

52 Fgure 2: Pah of esmaed aualzed rsk prema wh me-varyg beas modelled va Z, = [, d, ˆλm, ˆλsmb, ˆλhml, ˆλmom, he fgure plos he pah of esmaed aualzed rsk prema ˆλm,, ˆλsmb,, ˆλhml, ad ˆλmom, ad her powse cofdece ervals a 95% level whe me-varyg beas are lear fucos of dusry porfolo reurs. he rsk prema vecor volves he commo srumes Z = [, s, ds. We also repor he ucodoal esmae dashed horzoal le ad he average codoal esmae sold horzoal le. We cosder all socks = 9, 936 ad χ = 6, 43 as base asses. he vercal shaded areas deoe recessos deermed by he Naoal Bureau of Ecoomc Reasearch NBER. 52