ONLINE SUPPLEMENT FOR EVALUATING FACTOR PRICING MODELS USING HIGH FREQUENCY PANELS

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1 ONLINE SUPPLEMEN FOR EVALUAING FACOR PRICING MODELS USING HIGH FREQUENCY PANELS YOOSOON CHANG, YONGOK CHOI, HWAGYUN KIM AND JOON Y. PARK hi online upplement contain ome ueful lemma and their proof and the proof of the theorem preented in the main paper. Ueful Lemma and heir Proof Let be an Ito proce given by d = f t dt+g t db t, where B t i a Brownian motion with repect to a filtration F t, to which f t and g t are adapted. We aume Aumption A For all 0 t, a t t g2 udu b t, where a and b are ome contant depending only upon. Aumption A2 up t 0 f t = O p. Aumption A3 inf t 0 > 0 and up 0 t = O p c, with c depending only on. In the ubequent development of our theory, we aume that the Ito proce A atifie Aumption A-A3. Lemma A We have up A = O p δ /2 ε b /2 c t δ for any ε > 0, uniformly in 0,t. Correponding author. Addre correpondence to Yoooon Chang, Department of Economic, Indiana Univerity, Wylie Hall Rm 05, 00 S. Woodlawn, Bloomington IN , or to yoooon@indiana.edu. Department of Financial Policy, Korea Development Intitute Department of Finance, May Buine School, exa A&M Univerity Department of Economic, Indiana Univerity and Sungkyunkwan Univerity

2 Proof of Lemma A Write A = t f u A u du+ t g u A u db u for 0 t. We may eaily deduce that t t f u A u du f u du = O p δc up 0 t uniformly in 0,t, due to Aumption A2 and A3. Moreover, if we let C t = t 0 g A db, then C i a continuou martingale with t [C] t [C] = gua 2 2 udu t up A 2 t 0 t gudu 2 = O p δb c 2 2 uniformly in 0,t. Since C i a continuou martingale, we may repreent it a C t = D [C] t 3 with the DDS Brownian motion D of C, due to the celebrated theorem by Dambi, Dubin and Schwarz in e.g., Revuz and Yor 993, heorem 5..6, p73. Now we may deduce from 3, together with the modulu of continuity of Brownian motion and 2, that up C t C up t δ t δ up t δ D [C]t D [C] [C]t [C] /2 ε = Op δ /2 ε b /2 c for any ε > 0, uniformly in 0,t. Upon noticing that c δ = o δ /2 ε b c for any ε > 0, the tated reult follow immediately from and 4. he proof i therefore complete. 4 Lemma A2 We have m M m δ d A mδ A m δ = O pδ ε b c A m δ for any ε > 0. Proof of Lemma A2 Define R m = = m δ m δ d A mδ A m δ A m δ A m δ dat = m δ A m δ A m δ f t dt+g t db t. 5 2

3 We have m δ A m δ f t dt mδ up A A m δ inf t A m δ t m δ t mδ = O p δ 3/2 ε b /2 c = O p δ ε b c m δ f t dt 6 uniformly in m =,,M, due in particular to Lemma A. Moreover, t m δ A A m δ g db A m δ i a continuou martingale, whoe increment in quadratic variation over interval [m δ,mδ] i bounded by At A m δ gt 2 m δ A dt mδ m δ inf t A 2 up A m δ 2 gt 2 dt t m δ t mδ m δ Conequently, we may how that = O p δ 2 ε b 2 c2. m δ A m δ A m δ g t db t = O p δ ε b c 7 uniformly in m =,,M, uing the ame argument a in the proof of Lemma A2. he tated reult now follow immediately from 5, 6 and 7. Subequently, we let and define df t = d and dg t = g t db t, Amδ A m δ 2 [F] δ t = A m δ [G] δ t = G mδ G m δ 2. Lemma A3 We have up 0 t [G] δ t [G] t = Op δ /2 b. Proof of Lemma A3 Under Aumption A, the tated reult follow immediately from Lemma A3. of Park

4 Lemma A4 We have for any ε > 0. up 0 t Proof of Lemma A4 Define [F] δ t [G] t = O p δ /2 ε b 3/2 c2 [F δ ] t = F mδ F m δ 2 = m δ d 2, and note that [F] δ t [G]δ t [F] δ + [F t [Fδ ] δ t ] t [G] δ t. 8 We may readily deduce from Lemma A and A2 that [F] δ Amδ A 2 t [Fδ ] t = m δ 2 inf t M A m δ m δ d 2 A mδ A m δ m M d A mδ A m δ m M m δ =/δo p δ /2 ε b /2 c for all 0 t. Moreover, it follow that [F δ ] t = [G] δ t +2 m δ f t dt A m δ O p δ ε b c = Op δ /2 ε b 3/2 c2 G mδ G m δ + m δ f t dt 2, 9 where we have and m δ f t dt 2 MO p δ 2 = O p δ mδ f t dt G mδ G m δ m δ 2 /2 mδ f t dt G mδ G m δ m δ = O p δ /2 O p b /2 = O p δ /2 b /2 /2 4

5 uniformly in 0 t. Note that δ = o δ /2 b /2 [F δ ] t [G] δ t = O p δ /2 b /2. Conequently, we have 0 uniformly in 0 t. he tated reult follow from Lemma A3, and 8, 9 and 0. Note that δ /2 b /2,δ/2 b = o δ /2 ε b 3/2 c2, and therefore, the term we conider in Lemma A3 and 0 become negligible. In what follow, we let { H t = inf [G] > t } >0 and analogouly define for 0 t [G]. Lemma A5 We have for any ε > 0. up 0 t [G] { } Ht δ = inf [F] δ > t >0 Ht δ H t = Op δ /2 ε a b3/2 c2 Proof of Lemma A5 he proof i virtually identical to that of Corollary 3.3 of Park 2009, and therefore, it i omitted. In the following lemma, we define M n by δm n = Hn δ for n =,...N. Lemma A6 We have Hn d H n for any ε > 0. Proof of Lemma A6 We let R n = M n m=m n + Hn H n d A mδ A m δ A m δ M n m=m n + = O p δ /4 ε /2 a /2 b 5/4 c A mδ A m δ A m δ, and write R n R a n + R b n, 5

6 where Hn Rn a = da H δ t n H n R b n = H δ n H δ n d H δ n M n m=m n + d A mδ A m δ A m δ. Moreover, we define I n = min H n,hn δ and J n = H n,hn δ for n =,...,N. We have Hn Rn a f t dt H n he firt term i bounded by H δ n H δ n Jn 2 f t dt 2 I n Hn f t dt + g t db t H n up f t 0 t H δ n H δ n H n Hn δ g t db t. for all n =,...,N, and the quadratic variation of the econd term i bounded by Jn 2 g 2 tdt 2b H n H δ n I n for all n =,...,N. Clearly, the firt term i of order maller than that of the econd term. herefore, it follow from Lemma A5 that Rn a = O p δ /4 ε /2 a /2 b 5/4 c, 2 uniformly in n =,...,N. Furthermore, we have R b n M n m=m n + d A mδ A m δ m δ A m δ Hn δ d Hδ n A mδ A m δ m M m δ for all n =,...,N. However, we may readily deduce that Hn δ Hδ n H n H n +2 6 A m δ H n Hn δ,

7 and Hn H n = O p a. a Conequently, it follow from Lemma A2 that Rn b = O p δ ε a b c 3 for any ε > 0, uniformly in n =,...,N. Note that H n Hn δ = O p δ /2 ε a b3/2 c2 = o p a, due to Lemma A5. he tated reult now follow immediately from, 2 and 3. Note that Rn b i of order maller than that of the firt term of Rn. a he Proof of heorem Proof of heorem 3. hroughout the proof, we et n = H n, where H i introduced above Lemma A5. Note that b /2 = ON /2, ince N b and i contant. he reult for c n may eaily be obtained if we let X = A and apply Lemma A5. It follow that c δ n c n = 2 n δ n δ n n H n Hn δ = O p δ /2 ε a b3/2 c2 = o p b /2 = o p N /2. Similarly, we may imply apply Lemma A6 with X j = A, and note that δ /4 ε /2 b 5/4 c a /2 = o /2 b /2 to deduce the tated reult for x nj. he proof for u in i lightly more involved. Note that u δ ni u ni 2 However, we have U i δ n U in = whoe quadratic variation i bounded by n δ n = δ n 0 = on /2. U i δ n U in. 4 n ω it dz it ω it dz it, 0 H δ n H n = Op δ /2 ε a b3/2 c2 7

8 uniformly in n =,...,N, due to Lemma A5. It follow that U i δ n U in = Op δ /2 ε a b3/2 c2 /2, 5 and therefore, u δ ni u ni = on /2, due to 4 and 5, and δ /2 ε a b3/2 c2 /2 = ob /2 = on /2. o finih the proof, we note that yni δ y ni α i c δ n c n+ J β ij x δ nj x nj+ u δ ni u ni, j= uniformly in i =,...,I, from which and our previou reult we may eaily deduce the tated reult for y ni. Proof of Corollary 3.2 We may readily deduce the tated reult for ˆΣ from N N n= û δ nûδ n = N = N N n= u δ n uδ n +O pn /2 N u n u n +O p N /2, n= due to the well known regreion aymptotic and heorem 3.. For the proof of our reult for Σ, we aume that I = J = and uppre the ubcript i and j for notational implicity. he proof for the general cae i eentially the ame and can eaily be etablihed a in the imple cae we conider here. We write Û mδ Ûm δ = U mδ U m δ R mδ with o that R mδ = ˆα αδ +ˆβ β X mδ X m δ X m δ, 2 Ûmδ Ûm δ = Umδ U m δ 2 2U mδ U m δ R mδ 6 for m =,...,M. However, we have N M m= R 2 mδ 2ˆα α2δ2 M N +2ˆβ β 2 N M Xmδ X m δ m= X m δ = on 2 +ON = ON, 7 2 8

9 and M U mδ U N m δ R mδ m= [ ] /2[ M U mδ U N m δ 2 N m= /2 M Rmδ] 2 = ON /2. 8 m= Now it follow immediately from 6, 7 and 8 that Σ δ = Σ + O p N /2, and the proof i complete. Proof of Corollary 3.3 Let τα and τβ j be the continuou time verion of the Wald tatitic τ δ α and τ δ β j introduced in 29 and 3 of the main paper, i.e., τα = c c c XX X X c ˆα Σ ˆα τβ j = x jx j x jx j X jx j X jx j ˆβ j Σ ˆβj where c,x, ˆα, x j,x j, ˆβ j and Σ are defined from regreion 2 of the main paper correpondingly a c δ,x δ, ˆα δ, x δ j,xδ j, ˆβ δ j and Σ δ that are defined from regreion 24 of the main paper. Furthermore, let ˆγ = ˆα, ˆβ, where ˆα and ˆβ = ˆβ,..., ˆβ J, which are repectively I and IJ-dimenional, are the OLS etimator of α and β = β,...,β J. Define Z = c,x, and let R be an I IJ + -dimenional matrix given by R = I I,0 I IJ o that we may repreent the null hypothei H 0 : α = = α I = 0 a Rγ = 0 with γ = α,β. hen we may write τα = ˆγ γ R R [ Z Z Σ ] R Rˆγ γ. 9 However, due to Aumption 3., we have Z Z/N p Λ and Nˆγ γ d N 0,Λ Σ, and therefore, it follow that τα = a N. Now, we write [ R ] Nˆγ γ R [ Z Z N τ δ α = ˆγ δ γ R R Σ]R [ R ] Nˆγ γ d χ 2 I 20 [ Z δ Z δ Σ δ] R Rˆγ δ γ analogouly a in 9, where ˆγ δ i defined imilarly a ˆγ from ˆα δ and ˆβ δ = ˆβ δ,..., ˆβ δ J, and Z δ = c δ,x δ. herefore, we may eaily deduce from heorem 3., Corollarie 3.2 and 3.3 that τ δ α = τα+o p, 9

10 from which and 20 it follow that τ δ α d χ 2 I a N. hi wa to be hown. he proof for τ δ β j i entirely analogou and omitted to ave pace. Reference Park, J.Y Martingale regreion for conditional mean model in continuou time, mimeographed. Revuz, D. and M. Yor 994. Continuou Martingale and Brownian Motion, 2nd ed., Springer-Verlag: New York, New York. 0