Supplement to. Asymptotic Inference about Predictive Accuracy using High Frequency Data


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1 Supplemen o Asympoic Inference abou Predicive Accuracy using High Frequency Daa Jia Li Deparmen of conomics Duke Universiy Andrew J. Paon Deparmen of conomics Duke Universiy This Version: Ocober 7, 2014 Absrac This supplemen conains hree appendices. Appendix A conains proofs of resuls in he main ex. Appendix B provides deails for he sepwise procedures discussed in Secion 5 of he main ex. Appendix C conains some addiional simulaion resuls. Conac address: Deparmen of conomics, Duke Universiy, 213 Social Sciences Building, Box 90097, Durham NC , USA. mail: and 1
2 Supplemenal Appendix A: Proofs of main resuls In his appendix, we prove he resuls in he main ex. Resuls in Secions 3 and 4 of he main ex are proved in Appendix A.1. These proofs use hree echnical lemmas (Lemmas A1 A3), which we prove in Appendix A.2. Below, we use K o denoe a generic consan, which may change from line o line bu does no depend on he ime index. A.1 Proofs for Secions 3 and 4 We prove Theorems and Proposiion 4.1 in his secion. Throughou his appendix, we denoe X = X b sds + 0 σ s dw s, X = X X, (A.1) where he process b s is defined in Assumpion HF. Below, for any process Z, we denoe he ih reurn of Z in day by Z = Z τ() Z τ( 1). Proof of Theorem 3.1. Denoe β = σ τ( 1) W/d 1/2. Observe ha for m = 2/p and m = 2/(2 p), g( X/d 1/2 ) g(β ) p [( K 1 + β pq + X/d 1/2 pq) X/d 1/2 β p] ( [( K 1 + β pqm + X/d 1/2 pqm )]) 1/m ( X/d 1/2 β pm) 1/m Kd p/2, (A.2) where he firs inequaliy follows he meanvalue heorem, he Cauchy Schwarz inequaliy and condiion (ii); he second inequaliy is due o Hölder s inequaliy; he hird inequaliy holds because of condiion (iii) and X/d 1/2 β 2 Kd. Hence, g( X/d 1/2 ) g(β ) p Kd 1/2, which furher implies n Î(g) g ( ) β d Kd p 1/2. (A.3) Below, we wrie ρ ( ) in place of ρ ( ; g) for he sake of noaional simpliciy. Le ζ = g(β ) ρ(c τ( 1) ). By consrucion, ζ forms a maringale difference sequence. By condiion (iv), for all i, [(ζ ) 2 ] [ρ(c τ( 1) ; g 2 )] K. Hence, n ζ d 2 = n [(ζ ) 2 ]d 2 Kd, yielding n n ζ d ζ d Kd p 2 1/2. (A.4) 2
3 In view of (A.3) and (A.4), i remains o show Firs noe ha n ρ(c s )ds ρ ( ) c τ( 1) d Kd p 1/2. (A.5) n ρ (c s ) ds ρ ( ) n c τ( 1) d = τ() τ( 1) ( ρ (cs ) ρ ( c τ( 1) )) ds. (A.6) We hen observe ha, for all s [τ(, i 1), τ(, i)] and wih m = 2/p and m = 2/ (2 p), ρ (c s ) ρ ( ) c p τ( 1) ( K [(1 + c s pq/2 + c τ( 1) pq/2) c s c τ( 1) p]) 1/p ( [( K 1 + σ s pqm + σ τ( 1) pqm )]) 1/pm ( cs c τ( 1) pm) 1/pm Kd 1/2, (A.7) where he firs inequaliy follows from he meanvalue heorem, he Cauchy Schwarz inequaliy and condiion (ii); he second inequaliy is due o Hölder s inequaliy; he hird inequaliy follows from condiion (iii) and he sandard esimae c s c τ( 1) 2 Kd. From here (A.5) follows. This finishes he proof. Q..D. We now urn o he proof of Theorem 3.2. Recalling (A.1), we se ĉ τ() = 1 k k d 1 +j ( +jx )( +j X ). (A.8) The proof of Theorem 3.2 relies on he following echnical lemmas, which are proved in Appendix A.2. Lemma A1. Le w 2 and v 1. Suppose (i) Assumpion HF holds for some k 2wv and (ii) k d 1/2 as. Then [ ĉ w { ] 1/2 v Kd τ() c τ() Fτ() Kd w/4 in general, if σ is coninuous. (A.9) Lemma A2. Le w 1 and v 1. Suppose (i) Assumpion HF holds for some k 2wv and (ii) k d 1/2 as. Then [ ] ĉ τ() c w τ() F τ() v Kd w/2. (A.10) 3
4 Lemma A3. Le w 1. Suppose Assumpion HF hold wih k 2w. We have ĉ τ() ĉ τ() w Kd θ(k,w,ϖ,r), where θ (k, w, ϖ, r) = min {k/2 ϖ (k 2w) w, 1 ϖr + w(2ϖ 1), w(ϖ 1/2) + (1 ϖr) min{w/r, (k w)/k}}. (A.11) Proof of Theorem 3.2. Sep 1. Throughou he proof, we denoe [ F τ() ] by i [ ]. Consider he decomposiion: Î (g) I (g) = 4 R j, where R 1 = R 2 = R 3 = R 4 = n k i=0 n k i=0 n k i=0 n k ( ) g(ĉ τ() ) g(c τ()) g(c τ() ) (ĉ τ() c τ()) d g(c τ() ) (ĉ τ() c τ())d g(c τ() )d g(c s )ds (g(ĉ τ() ) g(ĉ τ() ))d ; i=0 (A.12) (A.13) (A.14) (A.15) noe ha in he firs wo lines of he above display, we have reaed ĉ τ() and c τ() as heir vecorized versions so as o simplify noaions. In his sep, we show ha R 1 p By Taylor s expansion and condiion (i), { Kd 1/(2p) Kd 1/2 in general, if σ is coninuous. (A.16) R 1 K K n k i=0 n k i=0 d (1 + c τ() q 2 + ĉ τ() q 2 ) ĉ τ() c τ() 2 d ((1 + c τ() q 2 ) ĉ τ() c τ() 2 + ĉ τ() c τ() q). (A.17) 4
5 Le v = q/2 and v = q/(q 2). Noice ha [(1 + c τ() q 2 ) p ĉ τ() c τ() 2p] K (1 + cτ() q 2 ) p v { 1/2 Kd in general, Kd p/2 when σ is coninuous, i ĉ τ() c τ() 2p v (A.18) where he firs inequaliy follows from repeaed condiioning and Hölder s inequaliy, and he second inequaliy is derived by using Lemma A1 wih w = 2p. Applying Lemma A1 again (wih w = qp and v = 1), we derive ĉ τ() c τ() qp Kd 1/2 and, when σ is coninuous, he bound can be improved as Kd qp/4 Kd p/2. The claim (A.16) hen follows from (A.17). Sep 2. In his sep, we show ha R 2 p Kd 1/2. (A.19) Denoe ζ i = g(c τ() ) (ĉ τ() c τ()), ζ i = i [ζ i ] and ζ i = ζ i ζ i. Noice ha ζ i = g(c τ() ) i [ĉ τ() c τ() ]. By condiion (i) and he Cauchy Schwarz inequaliy, ζ i K(1 + c τ() q 1 ) i [ĉ τ() c τ() ]. Observe ha, wih v = q and v = q/(q 1), ζ p i K Kd p/2, 1 + c τ() p(q 1) v i [ĉ τ() c τ()] p v (A.20) where he firs inequaliy is by Hölder s inequaliy, and he second inequaliy is derived by using Lemma A2 (wih w = p). Hence, Nex consider ζ i. Firs noice ha n k i=0 ζ id p Kd 1/2. (A.21) ζ 2 i K ζ i 2 (A.22) [ K (1 + c τ() q 1 ) 2 ĉ τ() c τ() 2] (A.23) K 1 + c τ() 2(q 1) v i ĉ τ() c τ() 2 v (A.24) Kd 1/2, (A.25) where he firs inequaliy is obvious; he second inequaliy follows from condiion (i) and he Cauchy Schwarz inequaliy; he hird inequaliy is by repeaed condiioning and Hölder s inequal 5
6 iy; he fourh inequaliy is derived by applying Lemma A1 (wih w = 2). Furher noice ha ζ i and ζ l are uncorrelaed whenever i l k. By he Cauchy Schwarz inequaliy and he above esimae, as well as condiion (ii), n k i=0 ζ i d 2 n k Kk i=0 ζ i 2 d 2 Kd. (A.26) Therefore, n k i=0 ζ i d 2 Kd 1/2. This esimae, ogeher wih (A.21), implies (A.19). Sep 3. Consider R 3 in his sep. Le v = 2/p and v = 2/(2 p). Noice ha for s [τ(, i 1), τ(, i)], [ g(c s ) g(c τ( 1) ) p K (1 + cτ() p(q 1) + cs p(q 1) ) ] cs c τ( 1) p (A.27) K 1 + c τ() p(q 1) + cs p(q 1) v c s c τ( 1) p v (A.28) Kd p/2. Hence, g(c s ) g(c τ( 1) ) p Kd 1/2. This esimae furher implies Sep 4. By a meanvalue expansion and condiion (i), (A.29) R 3 p Kd 1/2. (A.30) g(ĉ τ() ) g(ĉ τ() ) K(1 + ĉ τ() q 1 ) ĉ τ() ĉ τ() + K ĉ τ() ĉ τ() q. (A.31) By Lemma A3, ĉ τ() ĉ τ() q Kd θ(k,q,ϖ,r). (A.32) Le m = k/2(q 1) and m = k/(k 2(q 1)). By Hölder s inequaliy and Lemma A3, ] [(1 + ĉ τ() q 1 ) ĉ τ() ĉ τ() (1 + ĉ τ() q 1 ) ĉ τ() ĉ τ() Kd θ(k,m,ϖ,r)/m. m m (A.33) Therefore, we have R 4 Kd min{ θ(k,q,ϖ,r), θ(k,m,ϖ,r)/m}. (A.34) We now simplify he bound in (A.34). Noe ha he condiion k (1 ϖr)/(1/2 ϖ) implies, 6
7 for any w 1, { k/2 ϖ (k 2w) w 1 ϖr + w(2ϖ 1), w(ϖ 1/2) + (1 ϖr) (k w)/k 1 ϖr + w(2ϖ 1), (A.35) and, recalling m = k/(k 2(q 1)), (1 ϖr + m(2ϖ 1)) /m 1 ϖr + q(2ϖ 1). (A.36) Using (A.35) and q 2 r, we simplify θ (k, q, ϖ, r) = 1 ϖr + q(2ϖ 1); similarly, θ (k, m, ϖ, r) = min{1 ϖr + m(2ϖ 1), m (1/r 1/2)}. We hen use (A.36) o simplify (A.34) as R 4 Kd min{1 ϖr+q(2ϖ 1),1/r 1/2}. (A.37) Combining (A.16), (A.19), (A.30) and (A.37), we readily derive he asserion of he heorem. Q..D. Proof of Theorem 3.3. Sep 1. For x, y R d, we se { k (y, x) = g (y + x) g (x) g (y) By Taylor s heorem and condiion (ii), k (y, x) K h (y, x) K h (y, x) = g (y + x) g (x) g (y) g (y) x1 { x 1}. 2 2 ( ) y qj 1 x + x qj 1 y, ( y q j 2 x 2 + x q j 1 y + y q j 1 x 1 { x >1} ). (A.38) (A.39) We consider a process (Z s ) s [,] ha is given by Z s = X s X τ( 1) when s [τ(, i 1), τ(, i)). We define Z s similarly bu wih X replacing X; recall ha X is defined in (A.1). We hen se Z s = Z s Z s. Under Assumpion HF, we have { v [0, k] [sups [τ( 1),τ()] Z s v ] Kd v/2, v [2, k] [sup s [τ( 1),τ()] Z s v F τ( 1) ] Kd, (A.40) where he firs line follows from a classical esimae for coninuous Iô semimaringales, and he second line is derived by using Lemmas and in Jacod and Proer (2012). 7
8 By Iô s formula, we decompose Ĵ (g) J (g) = g (Z s ) b s ds + 1 d j,l 2 2 g (Z s ) c jl,s ds j,l=1 + ds h (Z s, δ (s, z)) λ (dz) + g (Z s ) σ s dw s R + k (Z s, δ (s, z)) µ (ds, dz). R (A.41) Below, we sudy each componen in he above decomposiion separaely. Sep 2. In his sep, we show g (Z s ) b s ds Kd 1/2. (A.42) p Le m = 2/p and m = 2/ (2 p). Observe ha, for all s [τ(, i 1), τ(, i)], p g (Z s ) 2 b s p K Z s qj 1 b s K K 2 2 Kd 1/2, 1/p ( Z s (q j 1)pm ) 1/pm ( b s pm ) 1/pm ( Z s 2(q j 1) ) 1/2 ( b s pm ) 1/pm (A.43) (A.44) (A.45) (A.46) where he firs inequaliy is due o condiion (ii) and he CauchySchwarz inequaliy; he second inequaliy is due o Hölder s inequaliy; he hird inequaliy follows from our choice of m; he las inequaliy follows from (A.40). The claim (A.42) hen readily follows. Sep 3. In his sep, we show 1 2 d j,l=1 j,l 2 g (Z s ) c jl,s ds Kd 1/2. (A.47) p By a componenwise argumen, we can assume ha d = 1 wihou loss of generaliy and suppress 8
9 he componen subscrips in our noaion below. Le m = 2/(2 p). We observe 2 g (Z s ) c s p K K 2 ( [ Z s p(qj 2) c s p]) 1/p (A.48) 2 ( ( Z s 2(qj 2) ) 1/2 c s pm ) 1/pm (A.49) Kd 1/2, (A.50) where he firs inequaliy follows from condiion (ii); he second inequaliy is due o Hölder s inequaliy and our choice of m ; he las inequaliy follows from (A.40). The claim (A.47) is hen obvious. Sep 4. In his sep, we show By (A.39) and δ (s, z) Γ (z), h (Z s, δ (s, z)) K Hence, by condiion (iii), 2 ds R h (Z s, δ (s, z)) λ (dz) Kd 1/2. (A.51) p ( Z s q j 2 Γ (z) 2 + Γ (z) q j 1 Z s + Z s q j 1 Γ (z) 1 {Γ(z)>1} ). (A.52) R h (Z s, δ (s, z)) λ (dz) K By (A.40), for any s [τ(, i 1), τ(, i)], 2 ( Z s q j 2 + Z s + Z s q j 1 ). h (Z s, δ (s, z)) λ (dz) R 2 2 ( ( ) K Z s 2(q 1/2 j 2) + ( Z s 2) 1/2 + ( ) ) Z s 2(q 1/2 j 1) Kd 1/2. (A.53) (A.54) The claim (A.51) hen readily follows. Sep 5. In his sep, we show g (Z s ) σ s dw s 2 Kd 1/2. (A.55) 9
10 By he Burkholder Davis Gundy inequaliy, 2 g (Z s ) σ s dw s [ ] K g (Z s ) 2 σ s 2 ds [ K g ( ] Z s) 2 σs 2 ds + K + K [ n τ() τ( 1) [ n τ() τ( 1) g (Zs ) g ( Z s ) 2 ] στ( 1) 2 ds ) 2 ] σ s σ τ( 1) 2 ds g (Z s ) g ( Z s. (A.56) We firs consider he firs erm on he majoran side of (A.56). By Hölder s inequaliy, we have, for s [τ(, i 1), τ(, i)], [ g ( Z s) 2 σs 2] K 2 ( Z s 2q j ) (qj 1)/q j ( ) σ s 2q 1/qj j Kd q 1 1, (A.57) where he second inequaliy is due o (A.40). This esimae implies [ g ( ] Z s) 2 σs 2 ds Kd. Now urn o he second erm on he majoran side of (A.56). Observe ha [ n K K +K τ() 2 g (Z s ) g ( Z s ) 2 ] σ τ( 1) 2 ds τ( 1) [ n τ() τ( 1) [ 2 n τ() Z s τ( 1) [ 2 n τ() Z s τ( 1) ( Z s 2(q j 2) + Z ) 2(q j 2) Z 2 ] σ τ( 1) 2 ds s 2(q j 2) Z 2 ] σ τ( 1) 2 ds s 2(q j 1) ] σ τ( 1) 2 ds. s (A.58) (A.59) By repeaed condiioning and (A.40), we have 2 [ n τ() Z s τ( 1) 2(q j 1) ] σ τ( 1) 2 ds Kd. (A.60) 10
11 Moreover, by Hölder s inequaliy and (A.40), for s [τ(, i 1), τ(, i)], [ Z s 2(q j 2) Z s 2 σ ] τ( 1) 2 ( Z s ) 2q (qj 2)/q j ( j Z s Kd q j 2 d 1/q j. 2q j ) 1/qj ( σ τ( 1) 2q j ) 1/qj (A.61) Therefore, 2 [ n τ() τ( 1) Z s 2(q j 2) Z 2 ] στ( 1) 2 ds Kd. (A.62) s Combining (A.59) (A.62), we have [ n τ() τ( 1) g (Zs ) g ( Z s ) 2 ] στ( 1) 2 ds Kd. (A.63) We now consider he hird erm on he majoran side of (A.56). By he meanvalue heorem and condiion (ii), [ g (Zs ) g ( Z s ) 2 ] σs σ τ( 1) 2 2 [ K Z s 2(q j 2) Z s 2 σ ] s σ τ( 1) 2 + K 2 [ Z 2(q j 1) ] σs σ τ( 1) 2. s (A.64) By Hölder s inequaliy and (A.40), [ Z s 2(q j 1) σ ] s σ τ( 1) 2 ( Z ) 2q (qj 1)/q j j ( ) σ s σ τ( 1) 2q 1/qj j s Kd (q j 1)/q j d 1/q j Kd. (A.65) Similarly, [ Z s 2(q j 2) Z s 2 σ ] s σ τ( 1) 2 ( Z s ) 2q (qj 2)/q j ( j Z ) 2q 1/qj j ( ) σ s σ τ( 1) 2q 1/qj j Kd q j 2 d 1/q j d 1/q j Kd. s (A.66) 11
12 Combining (A.64) (A.66), we have [ g (Zs ) g ( Z s) 2 σs σ τ( 1) 2 ] Kd. (A.67) Hence, [ n τ() τ( 1) g (Z s ) g ( Z s ) 2 ] σ s σ τ( 1) 2 ds Kd. (A.68) We have shown ha each erm on he majoran side of (A.56) is bounded by Kd ; see (A.58), (A.63) and (A.68). The esimae (A.55) is now obvious. Sep 6. We now show R k (Z s, δ (s, z)) µ (ds, dz) Kd 1/2. (A.69) 2 By Lemma in Jacod and Proer (2012), (A.39) and Assumpion HF, 2 k (Z s, δ (s, z)) µ (ds, dz) R 2 [ K ( Z ) ] s qj 1 δ (s, z) + Z s δ (s, z) q 2 j 1 λ (dz) K 2 Kd, ds [ ds R R ( ) ] Z s 2(qj 1) Γ (z) 2 + Z s 2 Γ (z) 2(q j 1) λ (dz) (A.70) which implies (A.69). Sep 7. Combining he esimaes in Seps 2 6 wih he decomposiion (A.41), we derive Ĵ(g) J (g) p Kd 1/2 as waned. Q..D. Proof of Theorem 3.4. Define Z s as in he proof of Theorem 3.3. By applying Iô s formula o ( X)( X) for each i, we have he following decomposiion: RV QV = Z s b sds ds Z s δ (s, z) 1 { δ(s,z) >1} λ (dz) R Z s (σ s dw s ) + 2 Z s δ (s, z) µ (ds, dz). R (A.71) Recognizing he similariy beween (A.71) and (A.41), we can use a similar (bu simpler) argumen as in he proof of Theorem 3.3 o show ha he L p norm of each componen on he righhand 12
13 side of (A.71) is bounded by Kd 1/2. The asserion of he heorem readily follows. Q..D. Proof of Theorem 3.5. Sep 1. Recall (A.1). We inroduce some noaion BV = n π n n 2 d 1/2 X d 1/ X d, ζ 1,i = d 1/2 X d 1/ X, ζ 2,i = d 1/2 X d 1/ X, R 1 = n ζ 1,id, R 2 = n ζ 2,id. (A.72) I is easy o see ha BV BV K(R 1 + R 2 ). By Lemmas and in Jacod and Proer (2012), [ d 1/ X p F τ() ] Kd (p/r) 1 p/2. Moreover, noe ha d 1/2 X p K d 1/2 X p + K d 1/2 X p K. (A.73) By repeaed condiioning, we deduce ζ i,1 p Kd (1/r) (1/p) 1/2, which furher yields R 1 p Kd (1/r) (1/p) 1/2. Now urn o R 2. Le m = p /p and m = p / (p p). Since pm k by assumpion, we use Hölder s inequaliy and an argumen similar o ha above o derive ( ζ p 2,i d 1/2 X pm) 1/pm ( d 1/ X pm ) 1/pm Kd (1/r) (1/p ) 1/2. (A.74) Hence, R 2 p Kd (1/r) (1/p ) 1/2. Combining hese esimaes, we deduce BV BV p K R 1 p + K R 2 p Kd (1/r) (1/p ) 1/2. (A.75) Sep 2. In his sep, we show BV c s ds Kd 1/2. (A.76) p For j = 0 or 1, we denoe β,j = σ τ( 1) d 1/2 +j +jw and λ,j = d 1/2 +j +jx β,j. Observe ha BV n n π β n 1 2,0 β,1 d n ( ) (A.77) K d 1/2 X λ,1 + λ,0 β,1 d. 13
14 Le m = 2/p and m = 2/ (2 p). By Hölder s inequaliy and Assumpion HF, d 1/2 X λ,1 p ( d 1/2 X pm ) 1/pm ( λ,1 pm ) 1/pm (A.78) Kd 1/2, (A.79) where he second inequaliy follows from d 1/2 X q K for each q [0, k] and λ,j 2 Kd +j. Similarly, λ,0 β,1 p Kd 1/2. Combining hese esimaes, we have BV n n π β n 1 2,0 β,1 d Kd p 1/2. (A.80) Le ξ i = (π/2) β,0 β,1, ξ i = [ ] ξ i F τ( 1) and ξ i = ξ i ξ i. Under Assumpion HF wih k 4, ξ i 2 ξ i 2 K. Moreover, noice ha ξ i is F τ(+1) measurable and [ξ i F τ( 1) ] = 0. Therefore, ξ i is uncorrelaed wih ξ l whenever i l 2. By he CauchySchwarz inequaliy, n ξ i d 2 n Kd ξ i 2 d Kd. (A.81) By direc calculaion, ξ i = c τ( 1). By a sandard esimae, for any s [τ(, i 1), τ(, i)], we have c s c τ( 1) p Kd 1/2 and, hence, n ξ id c s ds Kd 1/2. (A.82) p Combining (A.80) (A.82), we derive (A.76). Sep 3. We now prove he asserions of he heorem. We prove par (a) by combining (A.75) and (A.76). In par (b), BV coincides wih BV because X is coninuous. The asserion is simply (A.76). Q..D. Proof of Theorem 3.6. We only consider ŜV + for breviy. To simplify noaion, le g (x) = {x} 2 +, x R. We also se k(y, x) = g(y + x) g(y) g(x). I is elemenary o see ha k(y, x) K x y for x, y R. We consider he decomposiion n n g ( X) = g ( X ) n + g ( X ) n + k( X, X ). (A.83) 14
15 By Theorem 3.1 wih I (g) ρ(c s; g)ds = (1/2) c sds, we deduce n g ( X ) I (g) Kd 1/2. (A.84) p Hence, when X is coninuous (so X = X ), he asserion of par (b) readily follows. Now consider he second erm on he righhand side of (A.83). We define a process (Z s ) s [,] as follows: Z s = X s X τ( 1) when s [τ(, i 1), τ(, i)). Since r 1 by assumpion, Z is a finievariaional process. Observe ha n g( X ) g (δ (s, z)) µ (ds, dz) R p = k (Z s, δ (s, z)) µ (ds, dz) R p K Z s Γ(z)µ (ds, dz) [ R ] K Z s p Γ (z) p λ (dz) Kd, ds R p [( ) p ] + K ds Z s Γ (z) λ (dz) R (A.85) where he equaliy is by Iô s formula (Theorem II.31, Proer (2004)); he firs inequaliy is due o k(y, z) K x y ; he second and he hird inequaliies are derived by repeaedly using Lemma of Jacod and Proer (2012). I hen readily follows ha n g ( X ) g (δ (s, z)) µ (ds, dz) Kd 1/p Kd 1/2. (A.86) R p Nex, we consider he hird erm on he righhand side of (A.83). Le m = p /p and m = p /(p p). We have k( X, X ) p K ( X pm ) ( 1/pm X pm ) 1/pm Kd 1/2+1/p, (A.87) where he firs inequaliy is due o k(y, x) K x y and Hölder s inequaliy; he second inequaliy holds because Assumpion HF holds for k pp /(p p) and X p Kd. Hence, n k( X, X ) Kd 1/p 1/2. (A.88) p The asserion of par (a) readily follows from (A.83), (A.84), (A.86) and (A.88). Q..D. 15
16 Proof of Proposiion 4.1. (a) Under H 0, [ f T ] = χ. By Assumpions A1 and C, (a T ( f T χ), a T S d d T ) (ξ, S). By he coninuous mapping heorem and Assumpion A2, ϕ T ϕ (ξ, S). By Assumpion A3, φ T α. Now consider H 1a, so Assumpion B1(ii) is in force. Under H 1a, he nonrandom sequence a T ([ f j,t ] χ j) diverges o + ; by Assumpion C, a T ([ f j,t ] χ j) diverges o + as well. Hence, by Assumpion A1 and Assumpion B1(ii), ϕ T diverges o + in probabiliy. Since he law of (ξ, S) is igh, he law of ϕ (ξ, S) is also igh by Assumpion A2. Therefore, z T,1 α = O p (1). I is hen easy o see φ T 1 under H 1a. The case wih H 2a can be proved similarly. (b) Under H 0, a T ( f T χ) a T ( f T [ f T ]). Le φ T = 1{ϕ(a T ( f T [ f T ]), a T S T ) > z T,1 α }. By monooniciy (Assumpion B1(i)), φ T φ T. Following a similar argumen as in par (a), φ T α. Then lim sup T φ T α readily follows. The case under H a follows a similar argumen as in par (a). Q..D. A.2 Proofs of echnical lemmas Proof of Lemma A1. Sep 1. We ouline he proof in his sep. For noaional simpliciy, we denoe i ξ = [ξ F τ() ] for some generic random variable ξ; in paricular, i ξ w is undersood as i [ ξ w ]. Le α i = ( X )( X ) c τ( 1) d. We decompose ĉ τ() c τ() = ζ 1,i + ζ 2,i, where ζ 1,i = k 1 k (c τ(+j 1) c τ() ), ζ 2,i = k 1 k d 1 +j α i+j. (A.89) In Seps 2 and 3 below, we show i ζ 1,i w v i ζ2,i w v { Kd 1/2 Kd w/4 in general, { Kd + Kk w/2 Kd w/2 if σ is coninuous, + Kk w/2 in general, if σ is coninuous. (A.90) (A.91) The asserion of he lemma hen readily follows from condiion (ii) and w 2. Sep 2. We show (A.90) in his sep. Le ū = τ(, i + k 1) τ(, i). Since ū = O(d 1/2 ), we can assume ū 1 wihou loss. By Iô s formula, c can be represened as c = c R bs ds + σ s dw s 0 2σ s δ (s, z) µ (ds, dz) + 0 R δ (s, z) δ (s, z) µ (ds, dz), (A.92) 16
17 for some processes b s and σ s ha, under condiion (i), saisfy b s wv + σ s wv K. (A.93) By (A.92), where ζ 1,i w sup c τ()+u c τ() w K u [0,ū] 4 ξ l,i, ξ 1,i = sup u [0,ū] τ()+u τ() bs ds w, ξ 2,i = sup u [0,ū] τ()+u τ() σ s dw s w, ξ 3,i = sup u [0,ū] τ()+u τ() R 2σ s δ (s, z) µ (ds, dz) w, ξ 4,i = sup u [0,ū] τ()+u δ τ() R (s, z) δ (s, z) µ (ds, dz) w. l=1 (A.94) (A.95) By (A.93), i is easy o see ha i [ξ 1,i ] v ξ 1,i v Kū w. Moreover, i [ξ 2,i ] v ξ 2,i v Kū w/2, where he second inequaliy is due o he Burkholder Davis Gundy inequaliy. By Lemma in Jacod and Proer (2012) and condiion (i), [ τ()+ū i [ξ 3,i ] K i τ() R ( τ()+ū +K i τ() σ s w δ (s, z) w λ (dz) ds R ] σ s 2 δ (s, z) 2 λ (dz) ds ) w/2 [ ] ( τ()+ū ) w/2 τ()+ū K i σ s w ds + K i σ s 2 ds. τ() τ() (A.96) (A.97) (A.98) Hence, i [ξ 3,i ] v Kū. By Lemma in Jacod and Proer (2012) and condiion (i), [ τ()+ū i [ξ 4,i ] K i τ() R [( τ()+ū +K i τ() δ (s, z) 2w λ (dz) ds R ] δ (s, z) 2 λ (dz) ds ) w ] (A.99) (A.100) Kū. (A.101) Hence, i [ξ 4,i ] v Kū. Combining hese esimaes wih (A.94), we derive (A.90) in he general case as desired. Furhermore, when σ is coninuous, we have ξ 3,i = ξ 4,i = 0 in (A.94). The asserion of (A.90) in he coninuous case readily follows. Sep 3. In his sep, we show (A.91). Le α i = i 1[α i ] and α i = α i α i. We can hen 17
18 decompose ζ 2,i = ζ 2,i + ζ 2,i, where ζ 2,i = k 1 Iô s formula, i is easy o see ha k d 1 +j α i+j and ζ 2,i = k 1 [ τ(+j) α i+j K i+j 1 (X s X τ(+j 1) )(b s) ds] τ(+j 1) [ τ(+j) + i+j 1 (c s c τ(+j 1) )ds] τ(+j 1). By Jensen s inequaliy and repeaed condiioning, k d 1 +j α i+j. By (A.102) τ(+j) w i α i+j w K i (X s X τ(+j 1) )(b s) ds τ(+j 1) τ(+j) w +K i (c s c τ(+j 1) )ds. (A.103) τ(+j 1) Since condiional expecaions are conracion maps, we furher have i α i+j w v K +K i τ(+j) τ(+j 1) τ(+j) w (X s X τ(+j 1) )(b s) ds w (c s c τ(+j 1) )ds. v τ(+j 1) v (A.104) By sandard esimaes, he firs erm on he majoran side of (A.104) is bounded by Kd 3w/2 +j. Following a similar argumen as in Sep 2, we can bound he second erm on he majoran side of (A.104) by Kd w+1 +j in general and by Kd3w/2 +j if σ is coninuous. Hence, i α i+j w v Kd w+1 +j, and he bound can be improved o be Kd 3w/2 +j when σ is coninuous. By Hölder s inequaliy and he riangle inequaliy, i ζ 2,i w { Kd in general, v Kd w/2 when σ is coninuous. (A.105) Now consider ζ 2,i. Noice ha (α i+j ) 1 j k forms a maringale difference sequence. Using he Burkholder Davis Gundy inequaliy and hen Hölder s inequaliy, we derive i ζ 2,i w Kk w/2 1 k d w +j i α i+j w. (A.106) Moreover, noice ha i α i+j w v α i+j w v Kd w +j. Hence, i ζ 2,i w v Kk w/2. 18
19 Combining his esimae wih (A.105), we have (A.91). This finishes he proof. Q..D. Proof of Lemma A2. Sep 1. Recall he noaion in Sep 1 of he proof of Lemma A1. In his sep, we show ha By (A.92), for each j 1, [ [ ] τ(+j 1) i cτ(+j 1) c τ() = i bs ds + τ() By condiions (i,ii) and Hölder s inequaliy, we have i ζ 1,i w v Kd w/2. (A.107) τ(+j 1) τ() R δ (s, z) δ (s, z) λ (dz) ds ]. (A.108) i [ cτ(+j 1) c τ() ] w v K (k d ) w Kd w/2. (A.109) We hen use Hölder s inequaliy and Minkowski s inequaliy o derive (A.107). Sep 2. Similar o (A.102), we have [ τ(+j) i [α i+j ] K i (X s X τ(+j 1) )(b s) ds] τ(+j 1) [ τ(+j) + i (c s c τ(+j 1) )ds] τ(+j 1). (A.110) (A.111) Noice ha [ τ(+j) w i (X s X τ(+j 1) )(b s) ds] τ(+j 1) v τ(+j) w K (X s X τ(+j 1) )(b s) ds Kd 3w/2 +j, v τ(+j 1) (A.112) where he firs inequaliy is due o Jensen s inequaliy; he second inequaliy follows from sandard esimaes for coninuous Iô semimaringales (use Hölder s inequaliy and he Burkholder Davis Gundy inequaliy). Similar o (A.109), we have i [ cs c τ(+j 1) ] w v Kd w +j for s [τ(, i + j 1), τ(, i + j)]. We hen use Hölder s inequaliy o derive [ τ(+j) w i (c s c τ(+j 1) )ds] Kd 2w τ(+j 1) +j. v (A.113) Combining (A.112) and (A.113), we deduce i [α i+j ] w v Kd 3w/2 +j. Hence, by Hölder s inequaliy, i [ζ 2,i ] w v Kd w/2. This esimae, ogeher wih (A.107), implies he asserion of he lemma. Q..D. 19
20 Proof of Lemma A3. We denoe u +j = ᾱd ϖ +j. We shall use he following elemenary inequaliy: for all x, y R d and 0 < u < 1: (x + y) (x + y) 1 { x+y u} xx K( x 2 1 { x >u/2} + y 2 u 2 + x ( y u)). (A.114) Applying (A.114) wih x = +j X, y = +j X and u = u +j, we have ĉ τ() ĉ τ() K(ζ 1 + ζ 2 + ζ 3 ), where ζ 1 = k 1 ζ 2 = k 1 ζ 3 = k 1 k k k d 1 +j +jx 2 1 { +j X >u +j /2} d 1 +j ( +jx u +j ) 2 d 1 +j +jx ( +j X u +j ). Since k 2w, by Markov s inequaliy and +j X k Kd k/2 +j, we have +j X 2 1 { +j X >u +j /2} w K +jx k u k 2w Kd k/2 ϖ(k 2w) +j. +j (A.115) (A.116) (A.117) (A.118) Hence, ζ 1 w Kd k/2 ϖ(k 2w) w. By Corollary 2.1.9(a,c) in Jacod and Proer (2012), we have for any v > 0, [( ) v ] +j X d ϖ 1 +j (1 ϖr) min{v/r,1} Kd +j. (A.119) Applying (A.119) wih v = 2w, we have [( d ϖ +j +jx 1) 2w ] Kd+j 1 ϖr. ζ 2 w Kd 1 ϖr+w(2ϖ 1). We now urn o ζ 3. Le m = k/w and m = k/ (k w). Observe ha +j X ( u 1 +j +jx 1) w { K +j X wm } 1/m { [ ( u 1 +j +jx 1) wm]} 1/m w/2+(1 ϖr) min{w/r,(k w)/k} Kd+j, Therefore, (A.120) where he firs inequaliy is by Hölder s inequaliy; he second inequaliy is obained by applying 20
21 (A.119) wih v = wm. Therefore, ζ 3 w w(ϖ 1/2)+(1 ϖr) min{w/r,(k w)/k} Kd. Combining he above bounds for ζ j w, j = 1, 2 or 3, we readily derive he asserion of he lemma. Q..D. 21
22 Supplemenal Appendix B: xensions: deails on sepwise procedures B.1 The SepM procedure In his subsecion, we provide deails for implemening he SepM procedure of Romano and Wolf (2005) using proxies, so as o complee he discussion in Secion 5.2 of he main ex. Recall ha we are ineresed in esing k pairs of hypoheses Muliple SPA { H j,0 : [f j,+τ H j,a : lim inf T [ ] 0 for all 1, f j,t ] > 0, 1 j k. (B.1) We denoe he es saisic for he jh esing problem as ϕ j,t ϕ j (a T ft, a T S T ), where ϕ j (, ) is a measurable funcion. The SepM procedure involves criical values ĉ 1,T ĉ 2,T, where ĉ l,t is he criical value in sep l. Given hese noaions, we can describe Romano and Wolf s SepM algorihm as follows. 1 Algorihm 1 (SepM). Sep 1. Se l = 1 and A 0,T = {1,..., k}. Sep 2. Compue he sepl criical value ĉ l,t. Rejec he null hypohesis H j,0 if ϕ j,t > ĉ l,t. Sep 3. If no (furher) null hypoheses are rejeced or all hypoheses have been rejeced, sop; oherwise, le A l,t be he index se for hypoheses ha have ye been rejeced, ha is, A l,t = {j : 1 j k, ϕ j,t ĉ l,t }, se l = l + 1 and hen reurn o Sep 2. To specify he criical value ĉ l,t, we make he following assumpion. Below, α (0, 1) denoes he significance level and (ξ, S) is defined in Assumpion A1 in he main ex. Assumpion SepM: For any nonempy nonrandom A {1,..., k}, he disribuion funcion of max j A ϕ j (ξ, S) is coninuous a is 1 α quanile c(a, 1 α). Moreover, here exiss a sequence of P esimaors ĉ T (A, 1 α) such ha ĉ T (A, 1 α) c(a, 1 α) and ĉ T (A, 1 α) ĉ T (A, 1 α) whenever A A. The sepl criical value is hen given by ĉ l,t = ĉ T (A l 1,T, 1 α). Noice ha ĉ 1,T ĉ 2,T in finie samples by consrucion. The boosrap criical values proposed by Romano and Wolf (2005) verify Assumpion SepM. The following proposiion describes he asympoic properies of he SepM procedure. We remind he reader ha Assumpions A1, A2, B1 and C are given in he main ex. 1 The presenaion here unifies Algorihms 3.1 (nonsudenized SepM) and Algorihm 4.1 (sudenized SepM) in Romano and Wolf (2005). 22
23 Proposiion B1. Suppose ha Assumpions C and SepM hold and ha Assumpions A1, A2 and B1 hold for each ϕ j ( ), 1 j k. Then (a) he null hypohesis H j,0 is rejeced wih probabiliy ending o one under he alernaive hypohesis H j,a ; (b) Algorihm 1 asympoically conrols he familywise error rae (FW) a level α. Proof. By Assumpions A1 and C, (a T ( f T [ f T ]), a d T S T ) (ξ, S). (B.2) The proof can hen be adaped from ha in Romano and Wolf (2005). The deails are given below. Firs consider par (a), so H j,a is rue for some j. By (B.2) and Assumpion B1(ii), ϕ j,t diverges o + in probabiliy. By Assumpion SepM, i is easy o see ha ĉ l,t forms a igh sequence for fixed l. Hence, ϕ j,t > ĉ l,t wih probabiliy ending o one. From here he asserion in par (a) follows. Now urn o par (b). Le I 0 = {j : 1 j k, H 0,j is rue} and FW T = P(H j,0 is rejeced for some j I 0 ). If I 0 is empy, FW T = 0 and here is nohing o prove. We can hus suppose ha I 0 is nonempy wihou loss of generaliy. By par (a), all false hypoheses are rejeced in he firs sep wih probabiliy approaching one. Since ĉ T (I 0, 1 α) ĉ 1,T, lim sup T FW T = lim sup T P ( ϕ j (a T ft, a ) T S T ) > ĉ T (I 0, 1 α) for some j I 0 ( lim sup P ϕ j (a T ( f ) T [ f T ]), a T S T ) > ĉ T (I 0, 1 α) for some j I 0 T ( ) = lim sup P max ϕ j (a T ( f T [ f T j I T ]), a T S T ) > ĉ T (I 0, 1 α) 0 ( ) = P max ϕ j (ξ, S) > c(i 0, 1 α) j I 0 = α. This is he asserion of par (b). Q..D. B.2 Model confidence ses In his subsecion, we provide deails for consrucing he model confidence se (MCS) using proxies. In so doing, we complee he discussion in Secion 5.3 of he main ex. Below, we denoe he paper of Hansen, Lunde, and Nason (2011) by HLN. Recall ha he se of superior forecass is defined as M { j { 1,..., k } : [f j,+τ ] [f l,+τ ] for all 1 l k } and 1, 23
24 and he se of asympoically inferior forecass is given by M { j { 1,..., k } : ( ) lim inf [ f T l,t ] [ f j,t ] > 0 for some (and hence any) l M }. The formulaion above slighly generalizes HLN s seing by allowing for daa heerogeneiy. Under (mean) saionariy, M coincides wih HLN s definiion of MCS; in paricular, i is nonempy and complemenal o M. In he heerogeneous seing, M may be empy and he union of M and M may be inexhausive. We avoid hese scenarios by imposing Assumpion MCS1: M is nonempy and M M = {1,..., k}. We now describe he MCS algorihm. We firs need o specify some es saisics. Below, for any subse M {1,..., k}, we denoe is cardinaliy by M. We consider he es saisic ϕ M,T = ϕ M (a T ft, a T S T ), where ϕ M (, ) = max j M ϕ j,m (, ), and, as in HLN (see Secion here), ϕ j,m (, ) may ake eiher of he following wo forms: for u R k and 1 j k, u i u j max, where s ij = s ji (0, ) for all 1 i k, i M sij ϕ j,m (u, s) = M 1 i M u i u j, where s j (0, ). sj We also need o specify criical values, for which we need Assumpion MCS2 below. We remind he reader ha he variables (ξ, S) are defined in Assumpion A1 in he main ex. Assumpion MCS2: For any nonempy nonrandom M {1,..., k}, he disribuion of ϕ M (ξ, S) is coninuous a is 1 α quanile c(m, 1 α). Moreover, here exiss a sequence of esimaors P ĉ T (M, 1 α) such ha ĉ T (M, 1 α) c(m, 1 α). Wih ĉ T (M, 1 α) given in Assumpion MCS2, we define a es φ M,T = 1{ϕ M,T > ĉ T (M, 1 α)} and an eliminaion rule e M = arg max j M ϕ j,m,t, where ϕ j,m,t ϕ j,m (a T ft, a T S T ). The MCS algorihm, when applied wih he proxy as he evaluaion benchmark, is given as follows. Algorihm 2 (MCS). Sep 1: Se M = {1,..., k}. Sep 2: if M = 1 or φ M,T = 0, hen sop and se M T,1 α = M; oherwise coninue. Sep 3. Se M = M \ e M and reurn o Sep 2. 24
25 The following proposiion summarizes he asympoic propery of MT,1 α. In paricular, i shows ha he MCS algorihm is asympoically valid even hough i is applied o he proxy insead of he rue arge. Proposiion B2. Suppose Assumpions A1, C, MCS1 and MCS2. Then (5.5) in he main ex holds, ha is, lim inf T (M M T,1 α ) ( ) 1 α, P MT,1 α M = 1. Proof. Under Assumpions A1 and C, we have (a T ( f T [ f T ]), a T S T ) d (ξ, S). For each M {1,..., k}, we consider he null hypohesis H 0,M : M M and he alernaive hypohesis H a,m : M M. Under H 0,M, ϕ M,T = ϕ M (a T ft, a T S T ) = ϕ M (a T ( f T [ f T ]), a T S T ), and, d hus, by he coninuous mapping heorem, ϕ M,T ϕ M (ξ, S). Therefore, by Assumpion MCS2, φ M,T α under H 0,M. On he oher hand, under H a,m, ϕ M,T diverges in probabiliy o + and hus φ M,T 1. Moreover, under H a,m, P(e M M ) 0; his is because sup j M M ϕ j,m,t is eiher igh or diverges in probabiliy o, bu ϕ M,T diverges o + in probabiliy. The asserions hen follow he same argumen as in he proof of Theorem 1 in HLN. Q..D. 25
26 Supplemenal Appendix C: Addiional simulaion resuls C.1 Sensiiviy o he choice of runcaion lag in longrun variance esimaion In Tables 1 6, we presen resuls on he finiesample rejecion frequencies of he Giacomini and Whie (2006) ess (GW) using he approaches of Newey and Wes (1987) and Kiefer and Vogelsang (2005) o conduc inference; we denoe hese wo approaches by NW and KV. In he main ex, we use a runcaion lag of 3P 1/3 for NW and 0.5P for KV when compuing he longrun variance. Below we furher consider using P 1/3 and 5 (for all P ) for NW, and 0.25P and P for KV. Overall, we confirm ha feasible ess using proxies have finiesample rejecion raes similar o hose of he infeasible es using he rue arge. Tha is, he negligibiliy resul is likely in force. More specifically, we find ha he GW KV approach has conservaive o good size conrol across various seings provided ha he sample size is sufficienly large (P = 1000 or 2000). In conras, he performance of he GW NW es is less robus: wih hese choices of runcaion lags his es rejecs oo ofen in Simulaions A and B, and coninues o overrejec in Simulaion C. These resuls confirm insighs from he lieraure on inconsisen longrun variance esimaion; see Kiefer and Vogelsang (2005), Müller (2012) and references herein. C.2 Disagreemen beween feasible and infeasible es indicaors In Tables 7 9, we repor he disagreemen on es decisions (i.e., rejecion or nonrejecion) beween infeasible ess based on he rue arge variable and feasible ess based on proxies. In view of he size disorion of he GW NW es, we only consider he GW KV es, wih m = 0.5P, for breviy. The seing is he same as ha in Secion 6 of he main ex. In he columns headed Weak we repor he finiesample rejecion frequency of he feasible es minus ha for he infeasible es. Under he heory developed in Secion 4, which ensures weak negligibiliy, he differences should be zero asympoically. 2 In he columns headed Srong we repor he proporion of imes in which he feasible and infeasible rejecion indicaors disagreed. If srong negligibiliy, in he sense of commen (ii) o Proposiion 4.1, holds, hen his proporion should be zero asympoically. As noed in he main ex, he weak negligibiliy resul holds well across all hree simulaion designs, wih he differences repored in hese columns are almos all zero o wo decimal places, excep for he lowes frequency proxy. The resuls for srong negligibiliy are more mixed: in Simulaions A and B we see evidence in suppor of srong negligibiliy, while for Simulaion C we observe some disagreemen. 2 Posiive (negaive) values indicae ha he feasible es based on a proxy rejecs more (less) ofen han he corresponding infeasible es based on he rue arge variable. 26
27 GW NW (m = 5) GW NW (m = P 1/3 ) Proxy RV+1 P = 500 P = 1000 P = 2000 P = 500 P = 1000 P = 2000 R = 250 True Y = 5 sec = 1 min = 5 min = 30 min R = 500 True Y = 5 sec = 1 min = 5 min = 30 min R = 1000 True Y = 5 sec = 1 min = 5 min = 30 min Table 1: Giacomini Whie es rejecion frequencies for Simulaion A. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy, and m is he runcaion lag in he longrun variance esimaion. 27
28 GW KV (m = 0.25P ) GW KV (m = P ) Proxy RV+1 P = 500 P = 1000 P = 2000 P = 500 P = 1000 P = 2000 R = 250 True Y = 5 sec = 1 min = 5 min = 30 min R = 500 True Y = 5 sec = 1 min = 5 min = 30 min R = 1000 True Y = 5 sec = 1 min = 5 min = 30 min Table 2: Giacomini Whie es rejecion frequencies for Simulaion A. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy, and m is he runcaion lag in he longrun variance esimaion. 28
29 GW NW (m = 5) GW NW (m = P 1/3 ) Proxy BV+1 P = 500 P = 1000 P = 2000 P = 500 P = 1000 P = 2000 R = 250 True Y = 5 sec = 1 min = 5 min = 30 min R = 500 True Y = 5 sec = 1 min = 5 min = 30 min R = 1000 True Y = 5 sec = 1 min = 5 min = 30 min Table 3: Giacomini Whie es rejecion frequencies for Simulaion B. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy, and m is he runcaion lag in he longrun variance esimaion. 29
30 GW KV (m = 0.25P ) GW KV (m = P ) Proxy BV+1 P = 500 P = 1000 P = 2000 P = 500 P = 1000 P = 2000 R = 250 True Y = 5 sec = 1 min = 5 min = 30 min R = 500 True Y = 5 sec = 1 min = 5 min = 30 min R = 1000 True Y = 5 sec = 1 min = 5 min = 30 min Table 4: Giacomini Whie es rejecion frequencies for Simulaion B. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy, and m is he runcaion lag in he longrun variance esimaion. 30
31 GW NW (m = 5) GW NW (m = P 1/3 ) Proxy RC +1 P = 500 P = 1000 P = 2000 P = 500 P = 1000 P = 2000 R = 250 True Y = 5 sec = 1 min = 5 min = 30 min R = 500 True Y = 5 sec = 1 min = 5 min = 30 min R = 1000 True Y = 5 sec = 1 min = 5 min = 30 min Table 5: Giacomini Whie es rejecion frequencies for Simulaion C. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy, and m is he runcaion lag in he longrun variance esimaion. 31
32 GW KV (m = 0.25P ) GW KV (m = P ) Proxy RC +1 P = 500 P = 1000 P = 2000 P = 500 P = 1000 P = 2000 R = 250 True Y = 5 sec = 1 min = 5 min = 30 min R = 500 True Y = 5 sec = 1 min = 5 min = 30 min R = 1000 True Y = 5 sec = 1 min = 5 min = 30 min Table 6: Giacomini Whie es rejecion frequencies for Simulaion C. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy, and m is he runcaion lag in he longrun variance esimaion. 32
33 P = 500 P = 1000 P = 2000 Proxy RV+1 Weak Srong Weak Srong Weak Srong R = 250 = 5 sec = 1 min = 5 min = 30 min R = 500 = 5 sec = 1 min = 5 min = 30 min R = 1000 = 5 sec = 1 min = 5 min = 30 min Table 7: Giacomini Whie es rejecion indicaor disagreemen frequencies for Simulaion A. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy. Columns headed Weak repor he difference beween he feasible and infeasible ess rejecion frequencies. Columns headed Srong repor he proporion of simulaions in which he feasible and infeasible ess disagree. 33
34 P = 500 P = 1000 P = 2000 Proxy BV+1 Weak Srong Weak Srong Weak Srong R = 250 = 5 sec = 1 min = 5 min = 30 min R = 500 = 5 sec = 1 min = 5 min = 30 min R = 1000 = 5 sec = 1 min = 5 min = 30 min Table 8: Giacomini Whie es rejecion indicaor disagreemen frequencies for Simulaion B. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy. Columns headed Weak repor he difference beween he feasible and infeasible ess rejecion frequencies. Columns headed Srong repor he proporion of simulaions in which he feasible and infeasible ess disagree. 34
35 P = 500 P = 1000 P = 2000 Proxy RC +1 Weak Srong Weak Srong Weak Srong R = 250 = 5 sec = 1 min = 5 min = 30 min R = 500 = 5 sec = 1 min = 5 min = 30 min R = 1000 = 5 sec = 1 min = 5 min = 30 min Table 9: Giacomini Whie es rejecion indicaor disagreemen frequencies for Simulaion C. The nominal level is 0.05, R is he lengh of he esimaion sample, P is he lengh of he predicion sample, is he sampling frequency for he proxy. Columns headed Weak repor he difference beween he feasible and infeasible ess rejecion frequencies. Columns headed Srong repor he proporion of simulaions in which he feasible and infeasible ess disagree. 35
36 References Giacomini, R., and H. Whie (2006): Tess of condiional predicive abiliy, conomerica, 74(6), Hansen, P. R., A. Lunde, and J. M. Nason (2011): The Model Confidence Se, conomerica, 79(2), pp Jacod, J., and P. Proer (2012): Discreizaion of Processes. Springer. Kiefer, N. M., and T. J. Vogelsang (2005): A New Asympoic Theory for HeeroskedasiciyAuocorrelaion Robus Tess, conomeric Theory, 21, pp Müller, U. (2012): HAC Correcions for Srongly Auocorrelaed Time Series, Discussion paper, Princeon Universiy. Newey, W. K., and K. D. Wes (1987): A Simple, Posiive Semidefinie, Heeroskedasiciy and Auocorrelaion Consisen Covariance Marix, conomerica, 55, Proer, P. (2004): Sochasic Inegraion and Differenial quaions. SpringerVerlag, Berlin, 2nd edn. Romano, J. P., and M. Wolf (2005): Sepwise muliple esing as formalized daa snooping, conomerica, 73(4),