SUPPLEMENT TO BOOTSTRAPPING REALIZED VOLATILITY (Econometrica, Vol. 77, No. 1, January, 2009, )

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1 Econometrca Supplementary Materal SUPPLEMENT TO BOOTSTRAPPING REALIZED VOLATILITY Econometrca, Vol. 77, No. 1, January, 009, BY SÍLVIA GONÇALVES AND NOUR MEDDAHI THIS SUPPLEMENT IS ORGANIZED asfollows.frst,wentroducesomenotaton. Second, we provde auxlary lemmas and ter proofs used to derve te cumulant expansons n Appendx A of te paper. Last, we prove Proposton. and part c of Proposton.3, wc were not ncluded n Appendx B. NOTATION Recall tat σ σ du < and for any q>0, σ q 1 u 1 q/ 1/ σ q/ 1 q/ 1/ σ q,wereσ q σ q/. Note tat n general σ q σ q 1 σ q du.weletσ 0 u qp σ q /σ p q/p for any q p > 0. Wen σ q s replaced wt σ q we wrte σ qp. Smlarly, R qp R q /R p q/p.weletμ q = E Z q,were Z N01 and q>0 and note tat μ = 1, μ = 3, μ 6 = 15, and μ 8 = 105. Snce μ = 1, we can wrte σ = μ σ, wc wll be convenent for provng te results for te wld bootstrap WB. Wrte were ˆ 1/ V T = S = S 1 + U 1/ V 1 R μ σ S 1 Vˆ V and U V and V = Var 1 R = μ μ σ. Te proof of Lemma S. below reles eavly on te fact tat 1/ R μ σ = r μ σ and Vˆ V = μ μ μ r μ σ 1/ 1 were for any q>0, r q μ q σ q are condtonally on σ ndependent wt zero mean snce r = σ u,wereu..d. N01 Smlarly, let S 1 R E R / V U 1 Vˆ V /V, were V = Var 1/ R and ˆ V s a consstent estmator of V.TenT = 009 Te Econometrc Socety DOI: /ECTA5971 V

2 S. GONÇALVES AND N. MEDDAHI S 1 + U 1/. For te ndependent and dentcally dstrbuted..d. bootstrap, V = R R and ˆ V = R R For te WB, V = μ μ R and Vˆ = μ μ /μ R. Fnally, note tat trougout we wll use k to denote a sum were all ndces dffer, for example, k kj k AUXILIARY LEMMAS LEMMA S.1: Let q, p, and s be postve even ntegers. It follows tat S1 1/ σ q σ p j = +q+p/ σ q σ p σ q+p S 1/ l σ q σ p j σ s l = 3+q+p+s/ σ q p σ σ s +q+p+s/ σ q+p + 1+q+p+s/ σ q+p+s σ s + σ q+s σ p + σ q σ p+s LEMMA S.: Under Assumpton H, condtonally on σ, a1 a a3 a a5 a6 a7 E r q = μ q σ q V Var 1/ R = μ μ σ E [ R μ σ 3] = μ 6 3μ μ + μ 3 σ 6 E [ R μ σ ] = 3 μ μ σ + 3 μ 8 μ μ 6 + 1μ μ 6μ 3μσ 8 E [ R μ σ Vˆ V ] = μ μ μ 6 μ μ σ 6 μ E [ R μ σ Vˆ V ] = μ μ μ 8 μ μ μ μ 6 + μ μ σ 8 E [ R μ σ 3 Vˆ V ] = 3 μ μ μ 6 μ μ μ σ σ 6 + O3 as 0

3 a8 a9 a10 a11 a1 BOOTSTRAPPING REALIZED VOLATILITY 3 E [ R μ σ Vˆ V ] [ = μ 3 μ μ 6 3μ μ + μ 3μ 6 μ μ σ 6 ] μ + 6μ 8 μ μ μ 6 + μ μ μ μ σ σ 8 + O as 0 E [ R μ σ Vˆ V ] = μ μ μ μ 10 μ μ 6 μ μ 8 + μ μ σ 10 = O as 0 E [ R μ σ Vˆ V ] = μ μ μ μ μ μ 8 μ σ σ 8 + μ 6 μ μ σ 6 + O 3 as 0 E [ R μ σ 3 Vˆ V ] = O 3 + O as 0 E [ R μ σ Vˆ V ] [ = μ 3 μ 3μ μ μ 8 μ σ ] σ 8 μ + 1μ μ μ 6 μ μ σ 6 σ + O as 0 LEMMA S.3: Under Assumpton H, condtonally on σ, ES = 0 ES = 1 ES 3 = B 1 σ 6 ES = 3 + B σ 8 ES U = A 1 σ 6 ES U = A σ 8 and as 0, ES 3 U = A 3 σ 6 + O ES U = [D 1 σ 8 + D σ ]+O 6 3/

4 S. GONÇALVES AND N. MEDDAHI ES U = O 1/ ES 3 U = O 1/ ES U =[C 1σ 8 + C σ 6 ]+O ES U =[E 1σ 8 + E σ 6 ]+O Te constants A 1 A B 1, B, and C 1 areasnteorema.1, and A 3 = 3A 1, C = A 1, D 1 = 6A D = A 1 B 1, E 1 = 3C 1, and E = 1A 1 REMARK 1: Te WB analogue of Lemma S. replaces σ q wt R q and μ q wt μ q = E η q Te WB analogue of Lemma S.3 replaces σ qp wt R qp and μ q wt μ = q E η q, yeldng for example, E S 3 = B R 1 6, were B = 1 μ 6 3μ μ + μ 3 /μ μ 3/ Lemma S.7 below s te..d. bootstrap analogue of Lemma S.3. Te next results are auxlary n provng Lemma S.7. LEMMA S.: Let r..d. from {r : = 11/}. Under Assumpton H, condtonally on σ, for any q>0 and for any = 11/, a1 a a3 a a5 a6 a7 a8 a9 a10 a11 a1 a13 E r q = q/ R q and E R q = R q = O P 1 E [r R ]= R R E [r R 3 ]= 3 R 6 3R R + R 3 E [r R ]= R 8 R 6 R + 6R R 3R E [r R 5 ]= 5 R 10 5R 8 R + 10R 6 R 10R R 3 + R5 E [r R 6 ]= 6 R 1 6R 10 R + 15R 8 R 0R 6 R R R 5R6 E [r R q ]=O P q for any q 7 as 0 E [r R ]= R 8 R E [r E [r E [r R r R r R 3 r R ] R ]= 3 R 6 R R R ]= R 8 R R 6R + R R = 5 R 10 R R 6 3R 8 R + 3R R + 3R 6 R 3R R 3 E [r R r R ] = 6 R1 R R 8 R 10 R + R R 6 R + 6R 8 R 6R R R 6R 3 + R R E r R r R = 5 R 10 R R 6 R 8 R + R R

5 BOOTSTRAPPING REALIZED VOLATILITY 5 a1 for any q p > 0 E [r R q r R p ] = O P q+p as 0 LEMMA S.5: Let r..d. from {r : = 11/}. Under Assumpton H, condtonally on σ, a5 a1 V Var 1/ R = R R a Vˆ V = R R [R R + R R R ] a3 E [R R 3 ]= R 6 3R R + R 3 a E [R ]= [3R R ] + 3 R 8 R 6 R + 1R R 6R 3R E [R R 5 ]= 3 [10R 6 3R R + R 3 R R ] + O P as 0 a10 a6 E [R R 6 ]= 3 [15R R 3 ]+O P as 0 a7 E [R R q ]=O P for q = 7 8 as 0 a8 E [R R R R ]=R 6 R R a9 E [R R R R ]= R 8 R R 6R + R R E [R 3 R R ]=3 R 6 R R R R + O P 3 as 0 a11 a1 a13 a1 a15 a16 E [R R R R ] [ R = 3 6 3R R + R 3R ] 6 R R + 6R R R 8 R R 6R + R R + O P as 0 E [R R 5 R R ]= 3 [15R R R 6 R R ] + O P as 0 E [R R 6 R R ]=O P as 0 E [R R R R ]= R 10 R R 6 R 8 R + R R E [R R R R ] = [R R R 8 R + R 6 R R ]+O P 3 as 0 E [R R 3 R R ]=O P 3 as 0

6 6 S. GONÇALVES AND N. MEDDAHI a17 [ E [R R R R 3R ]= 3 R R 8 R ] + 1R 6 R R R R + O P as 0 LEMMA S.6: Let r..d. from {r : = 11/}. Under Assumpton H, condtonally on σ, as 0, a1 a a3 a a5 a6 a7 a8 E [R R ˆ V V ]=R 6 3R R + R 3 + O P E [R R ˆ V V ] = [ R8 R R 6R + R R 3R R R R 6 3R R + R 3 ] + O P 3 E [R R 3 Vˆ V ]= [3R R R 6 3R R + R 3 ] + O P 3 E [R R Vˆ V ] = 3 [ R 6 3R R + R 3 R 6 R R + 6R R R 8 R R 6R + R R 3 [15R R 3 ] 3 [0R R 6 3R R + R 3 R R ]+O P E [R R Vˆ V ]=O P E [R R Vˆ V ] R R R 8 R + R 6 R R = 1R 6 R R R R R + O P 3 + R [3R R ] E [R R 3 Vˆ V ]=O P 3 E [R R Vˆ V ] = 3[ 3R R R 8 R + 1R 6 R R R R ] 3 [60R R R 6 R R R ] + 3 [60R R 3 R ]+O P ] LEMMA S.7: Let r..d. from {r : = 11/}. Under Assumpton H, condtonally on σ, E S = 0

7 E S = 1 E S 3 = B 1 E S = 3 + B and as 0 BOOTSTRAPPING REALIZED VOLATILITY 7 E S U = Ã1 + O P E S U = Ã + O P 3/ E S 3 U = Ã3 + O P E S U = D + O P 3/ E S U = O P E S U = C + O P 1/ E S 3 U = O P 1/ E S U = Ẽ + O P Te bootstrap constants Ã1 Ã B C D, and Ẽ are as n Teorem A., and Ã3 and B 1 are suc tat Ã 3 = 3Ã1 and B 1 = Ã1 PROOFS OF LEMMAS S.1 S.7 PROOF OF LEMMA S.1: For S1, note tat 1/ 1/ 1/ 1/ σ q σ p j = σ q = 1+q/ j=1 σ p j 1/ 1 q/ σ q 1+q+p/ σ q+p 1+p/ 1/ 1 q+p/ = +q+p/ σ q σ p σ q+p For S, note tat σ q σ p j σ s = 1/ k k σ q+p 1/ 1/ σ q σ p j σ s k j=1 k=1 1/ 1 p/ σ p j 1/ j=1 σ q+p+s σ q+p σ s j σ q+s σ p j σ q σ p+s j

8 8 S. GONÇALVES AND N. MEDDAHI and ten proceed as for S1. Q.E.D. PROOF OF LEMMA S.: a1 Equalty a1 follows from r = σ u,wereu..d. N01. a Note tat R = 1/ r,werer s condtonal on σ ndependent wt Varr = Er Er = μ σ μ σ = μ μ σ,wtσ σ. To prove te remanng results, we use te multnomal formula to compute te coeffcents n te expansons. In partcular, we ave tat a 1 + a + +a d n = n! n 1!n! n d! an 1 1 an an d d a3 Wrte I 1 E [ R μ σ 3] = E [ 1/ n 1 n n d 0 n 1 + +n d =n 1/ 1/ r μ σ r μ j σ j r μ k σ k ] j=1 k=1 Te only nonzero contrbuton to I 1 s wen = j = k, n wc case we get E[r μ σ 3 ]=μ 6 3μ μ + μ 3 σ 6 and I 1 = μ 6 3μ μ + μ 3 σ 6 provng a3. a Usng te ndependence and zero mean property of {r μ σ },weave tat E [ R μ σ ] 1/ 1/ = E[r μ σ ]+3 E[r μ σ ]E[r μ j σ j ] 1/ = E[u μ ] σ E[u μ ] 1/ σ σ j = μ 8 3μ + 6μμ μ μ 6 3 σ 8 + 3μ μ [ σ ] σ 8 = 3 μ μ σ + 3 μ 8 μ μ 6 + 1μ μ 6μ 3μ σ 8 were we ave made use of Lemma S.1 and of te results E[u μ ]=E[u 8 u μ3 + 6u μ u6 μ + μ ]

9 BOOTSTRAPPING REALIZED VOLATILITY 9 = μ 8 3μ + 6μμ μ μ 6 E[u μ ]=E[u uμ + μ ]=μ μ a5 We ave a6 We ave E [ R μ σ Vˆ V ] = μ μ E R μ σ ˆ V V = μ μ μ 1/ 1 μ 1 1/ Er 6 r μ σ μ σ r + μ μ σ 6 = μ μ 1 μ 6 μ μ σ 6 μ = μ μ μ 6 μ μ σ 6 μ E[r μ σ r μ σ ] = μ μ μ 8 μ μ μ 6 + μ μ σ 8 μ a7 Wrte ER μ σ 3 ˆ V V = μ μ /μ 1 I were by te ndependence and mean zero property of r q μ σ q 1/ I = E[r μ σ 3 r μ σ ] 1/ + 3 E [ r μ σ r μ j σ j r μ j σ ] j = M 1 1/ σ 10 1/ + 3E[u μ ]E[u μ j u μ j ] σ σ 6 j = 3 3 μ μ μ 6 μ μ σ σ 6 + O

10 10 S. GONÇALVES AND N. MEDDAHI gven Lemma S.1, te fact tat σ 10 = O1 under our assumptons, and were M 1 = E[u μ 3 u μ ] s a constant, and E[u μ ]=μ μ and E[u j μ u μ ]=μ 6 μ μ a8 Wrte ER μ σ ˆ V V = μ μ /μ 1 I 3 were by te ndependence and mean zero property of r q μ σ q and Lemma S.1, 1/ I 3 = E[r μ σ r μ σ ] 1/ + E[r μ σ 3 ]E[r μ j σ j r μ j σ ] j 1/ + 6 E[r μ σ ]E[r μ j σ j r μ j σ ] j = M 1 5 σ 1 + M σ 6 5 σ 1 = [ M σ 6 + 6M 3 σ σ 8 ] + O 5 + 6M3 σ σ 8 5 σ 1 were M 1 E[u μ u μ ]M E[u μ 3 ]E[u μ j u μ j ]= μ 6 3μ μ + μ 3μ 6 μ μ,andm 3 E[u μ ]E[u μ u μ ]=μ 8 μ μ μ 6 + μ μ μ μ and gven te fact tat σ q = O1 under our assumptons. a9 Wrte ER μ σ Vˆ V = μ μ /μ 1/ E[r μ σ r μ σ ]=O. a10 Wrte ER μ σ Vˆ V = μ μ /μ I were by te ndependence and mean zero property of r q μ σ q, 1/ I = E[r μ σ r μ σ ] 1/ + E[r μ σ ]E[r μ j σ j ] 1/ + E[r μ σ r μ σ ]E[r μ j σ j r μ j σ ] j = D 1 5 σ 1 + D σ σ 8 5 σ 1 + D3 σ 6 = D σ σ 8 + D 3 σ 6 + O 5 5 σ 1

11 BOOTSTRAPPING REALIZED VOLATILITY 11 gven Lemma S.1, and were D 1 = E[u μ u μ ], D = E[u μ ]E[u μ j ]=μ μ μ 8 μ,andd 3 =[Eu μ u μ ] = μ 6 μ μ a11 Wrte ER μ σ 3 Vˆ V = μ μ /μ I 5 wt 1/ I 5 = E[r μ σ 3 r μ σ ] 1/ + E[r μ σ 3 ]E[r μ j σ j ] 1/ + 3 E[r μ σ ]E[r μ j σ j r μ j σ j ] 1/ + 6 E[r μ σ r μ σ ]E[r μ j σ j r μ j σ ] j 5 σ 6 σ 8 6 σ 1 = K 1 6 σ 1 + K + 6K 5 σ 8 σ 6 6 σ 1 = 5 K σ 6 σ 8 + 3K 3σ σ K3 5 σ σ 10 6 σ 1 + 6K σ 8 σ K 1 K 3K 3 6K σ 1 were we ave used te ndependence and mean zero property of r q μ σ q Lemma S.1, and were K 1 troug K are constants tat depend on μ q.snce σ q = O1, te result follows. a1 Wrte ER μ σ Vˆ V = μ μ /μ I 6 wt 1/ I 6 = E[r μ σ r μ σ ] 1/ + E[r μ σ ]E[r μ j σ j ] 1/ + 8 E[r μ σ 3 r μ σ ]E[r μ j σ j r μ j σ ] j

12 1 S. GONÇALVES AND N. MEDDAHI 1/ + 6 E[r μ σ r μ σ ]E[r μ j σ j ] 1/ + E[r μ σ 3 ]E[r μ j σ j r μ j σ j ] 1/ + 6 E[r μ σ r μ σ ]E[r μ j σ j r μ j σ ] j / k 1/ k E[r μ σ ]E[r j μ σ j ]E[r k μ σ k ] E[r μ σ ]E[r j μ σ j r j μ σ j ] E[r μ k σ k r μ k σ ] k = 5[ 3J 7 σ σ 8 + 1J 8 σ 6 σ] + O 6 + O 7 gven te ndependence and mean zero property of r q μ σ q Lemma S.1, and were J 7 = E[u μ ] E[u μ ]=μ μ μ 8 μ and J 8 = E[u μ ]E[u μ u μ ] = μ μ μ 6 μ μ Q.E.D. PROOF OF LEMMA S.3: Te frst two results are obvous gven S.Teremanng results follow from te defnton of S and Lemma S.. Q.E.D. PROOF OF LEMMA S.: Part a1 follows from te propertes of te..d. bootstrap. Te remanng results follow from a1, gven te bnomal expansons. Note n partcular tat snce R q = O P 1, t follows tat E [r R q ]=O P q For nstance, for a, E [r R ]=E r r R + R = R R Te oter results follow smlarly. Q.E.D. PROOF OF LEMMA S.5: For a1, snce r are..d. from {r : = 11/}, t follows tat 1/ 1/ V = 1 Var r = 1 Var r = Var r 1 But Var r = 1 E r 1 E r 1 = R R Tus, V = R R Part a follows because V = R R and ˆ V = R R For te remanng of

13 BOOTSTRAPPING REALIZED VOLATILITY 13 te proof, note tat 1/ 1 = 1, k 1 = and 1/ k m 1 = In addton, note tat 1/ 1/ R R = r R and R R = 1 r R were for any, q>0, { r q q/ R q } are condtonally on te sample..d. wt zero mean and R q = O P 1. Usng ts ndependence property, we evaluate te bootstrap expectatons of te sums of products and cross-products of r q q/ R q by relyng on Lemma S. to compute te approprate bootstrap moments of products and cross-products of r q q/ R q We proceed as n te proof of Lemma S. and use te multnomal expansons to compute te number of coeffcents n eac sum. Q.E.D. PROOF OF LEMMA S.6: Usng part a of Lemma S.5, for q = 1 we can wrte S3 E [R R q Vˆ V ] = E [R R q R R ] E [R R +q ] R E [R R 1+q ] I q 1 Iq Iq 3 Smlarly, for q = 1 note tat E [R R q Vˆ V ] = E [R R q R R ] E [R R +q R R ] R E [R R 1+q R R ]+E [R R +q ] + R E [R R 3+q ]+R E [R R +q ] For a1, set q = 1 n S3. We ave tat I 1 = 1 E [R R R R ]=R 6 R R I 1 = E [R R 3 ]= R 6 3R R + R 3 I 1 = R 3 E [R R ]=R [R R ] by Lemma S.5 a8, a3 and a1, respectvely. Tus E [R R Vˆ V ] = [R 6 R R R R R ] R 6 3R R + R 3 = R 6 3R R + R 3 + O P

14 1 S. GONÇALVES AND N. MEDDAHI Te remanng results follow smlarly. Q.E.D. PROOF OFLEMMA S.7: Te prooffollowste proofoflemmas.3, gvente defnton of V and gven Lemmas S.5 and S.6. Q.E.D. PROOFS OF PROPOSITIONS. AND.3C IN SECTION PROOF OF PROPOSITION.: Wen gz = log z we ave tat q 1log x = 3 σ 6 σ 3/ x σ 1/ g σ log g σ log = 1 [ σ 1/ σ x 6 σ σ 3 σ wereas for gz = z, g z = 0and q 1 x = 3 σ 6 σ 3/ x + 1 = 1 σ 1/ σ [ x 3 σ 6 σ σ + 3 σ 6 σ σ σ 6 σ σ Snce σ σ 6 σ by te Caucy Scwarz nequalty, t follows tat 1 3 σ 6 σ /σ 1 < σ 6 σ /σ, wc mples tat q 3 3 1log x < q 1 x for x fxed and nonzero. Wen x = 0, t follows trvally tat q 1log 0 = q 1 0 Q.E.D. PROOF OF PROPOSITION.3C: Defne C = σ 6 / σ 3/ and C = 15σ 6 9σ σ + σ 3 /3σ σ 3/ and note tat C>0. It suffces to prove tat C C C, wc n turn s equvalent to provng 0 C C Next we sow tat C 0. Te Jensen nequalty mples tat σ σ,and snce σ > 0 t follows tat te denomnator of C s postve. For te numerator of C, note tat we can wrte 15σ 6 9σ σ + σ 3 15σ 6 9 σ 3/ + σ 3 ] ] 9 σ 3/ σ 3/ + 6σ 6 + σ 3 usng σ σ Snce te functon ψx = x 3/ for x>0 s convex, we ave tat σ 3/ σ 3/ 0 wc mples 15σ 6 9σ σ + σ 3 6σ 6 +

15 BOOTSTRAPPING REALIZED VOLATILITY 15 σ 3 > 0, provng tat te numerator of C s also postve. Next we prove C /C. We can wrte C C = 15σ 6 9σ σ + σ 3 σ 3/ 8σ 6 3σ σ C 3/ 1 C We sow tat C 1 andc 1 Frst, note tat 15σ 6 9σ σ + σ 3 8σ 6 15σ 6 9σ σ + σ 3 16σ 6 0 σ 6 + 7σ σ + σ σ σ wc proves te result snce σ σ 0and0 σ 6 + 7σ σ Fnally, we ave tat σ 3/ 3σ σ 3/ 1 8 σ 3 3σ σ 3 8 σ 3 σ + σ σ 3 wc olds true snce σ σ 0. Q.E.D. Département de Scences Économques, CIREQ and CIRANO, Unversté de Montréal, C.P. 618, Succ. Centre-Vlle, Montréal, QC, H3C 3J7, Canada; slva.goncalves@umontreal.ca and Fnance and Accountng Group, Tanaka Busness Scool, Imperal College London, Exbton Road, London SW7 AZ, U.K.; n.medda@mperal.ac.uk. Manuscrpt receved July, 005; fnal revson receved June, 008.

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