A Proofs of Main Results

We use C, C1; C₂; : : : to denote generic positive constants without further clari…cation.

In what follows, we assume that A is a compact subset of (0; 1)², while the results can be easily extend to the case for which A is the union of a …nite number of disjoint compact subsets of (0; 1)².

We de…ne an empirical processes indexed by u = (u1; u2)^| 2 R² as

V^T;k(u) = T ¹⁼² XT t=k+1

f1[x^1t u₁; x_{2t k} u₂] G_k(u)g ;

for k 2 f1; : : : ; pg. The empirical processes based on the stationary bootstrap is denoted by

VT;k(u) = T ¹⁼² XT t=k+1

1[x_1t u₁; x_{2t k} u₂] 1[x_1t u₁; x_{2t k} u₂] :

We also consider a sequential empirical process over (s; u) 2 [!; 1] R² for some

! 2 (0; 1), given by

V^T;k(s; u) = T ¹⁼²

[T s]

X

t=k+1

f1[x^1t u₁; x_{2t k} u₂] G_k(u)g :

The below lemma shows the weak convergence of the empirical process de…ned above.

We de…ne a set of mean-zero Gaussian process fV1;k(u) : u2 R²g^pk=1 with covariance

matrix function given by

kk⁰(u; u⁰) E [V1;k(u)V1;k⁰(u⁰)] = X1 l= 1

cov ('_l(u; k); '₀(u⁰; k⁰)) ;

for k; k⁰ 2 f1; : : : ; pg and u; u⁰ 2 R², where '_t(u; k) 1[x_1t u₁; x_{2t k} u₂] G_k(u).

We also de…ne a set of mean-zero Gaussian process fV1;k(s; u) : (s; u)2 [0; 1] R²g^pk=1

with covariance matrix function given by

E[V1;k(s; u)V1;k⁰(s⁰; u⁰)] = minfs; s⁰g ^kk0(u; u⁰)

for k; k⁰ 2 f1; : : : ; pg and (s; u); (s⁰; u⁰) 2 [0; 1] R². Notice that kk⁰(u; u⁰) is equiv-alent to the (1; 1)th element of the matrix kk⁰( ; ⁰) de…ned in the main text, when (u; u⁰) = (q ; q 0).

Lemma A.1 Under Assumption A1-A4,

(a) [V^T;1( ); : : : ; V^T;p( )]^| =) [V1;1( ); : : : ; V1;p( )]^|.

(b) [VT;1( ); : : : ; VT;p( )]^| =) [V1;1( ); : : : ; V1;p( )]^| in probability.

(c) [V^T;1( ; ); : : : ; V^T;p( ; )]^| =) [V1;1( ; ); : : : ; V1;p( ; )]^|.

Proof. (a) is proved by the similar argument used in Theorem 7.3 of Rio (2000). (b) is shown by a slight modi…cation of Theorem 3.1 of Politis and Romano (1994). (c) can be shown by using the argument in Bucher (2013) with (a).

Also, we de…ne a vector of the one-parameter empirical process:

W^T;k(u) = W⁽¹⁾T;k(u₁); W⁽²⁾T;k(u₂)

|

;

where W⁽¹⁾T;k(u₁) = lim_u2!1V^T;k(u) and W⁽²⁾T;k(u₂) = lim_u1!1V^T;k(u). Likewise, the bootstrap and sequential versions are denoted as follows:

WT;k(u) = W⁽¹⁾T;k(u₁); W⁽²⁾T;k(u₂)

|

; and W^T;k(u; s) = W⁽¹⁾T;k(u₁; s); W⁽²⁾T;k(u₂; s)

|

;

respectively. In the following lemma, we shall obtain an asymptotic linear approxi-mation for the cross-quantilogram.

Lemma A.2 Suppose that Assumption A1-A4 hold. Then, for each k 2 f1; : : : ; pg, pT (^ (k) (k)) = V^T;k(q ) +rG^k(q )^|p

T (^q q ) p

1(1 ₁) ₂(1 ₂) + o_p(1);

uniformly in 2 A.

Proof. Let ^ _k(q) = T ¹PT

t=k+1 1(x_1t q₁)

2(x_{2t k} q₂)and _k(q) = E[

1(x_1t q₁) ₂(x_{2t k} q₂)] for q = (q1; q₂)^| 2 R². Using a similar argument used to show Lemma 2.1 of Arcones (1998), we can show

sup

i2Ai

T ¹⁼² XT

t=1

i(x_it q^_i( _i)) = o_p(1); (A.1) for i = 1; 2. It follows that, uniformly in 2 A,

^ _k(^q ) = T ¹ XT t=k+1

1[x_1t q^₁( ₁); x_{2t k} q^₂( ₂)] _{1 2}+ o_p(T ¹⁼²);

and T ¹PT t=k+1

2

i(x_it q^_i( _i)) = _i(1 _i) + o_p(1) for i = 1; 2: Thus, we have pT f^ (k) (k)g =

ppTf^ k(^q ) _k(q )g

1(1 1) 2(1 2) + o_p(1);

uniformly in 2 A.

The remaining task is to obtain a uniform asymptotic approximation ofp

Tf^ k(^q )

k(q )g. Since k(q ) = G_k(q ) _{1 2}, we have pT f^ k(^q ) _k(q )g = V^T;k(^q ) +p

TfG^k(^q ) G_k(q )g :

Let E^M denote the event that sup _2Ak^q q k M T ¹⁼² for some M > 0 and EM^c

be the complement of E^M. Given an arbitrary constants > 0 and > 0, we have

P sup

2AjV^T;k(^q ) V^T;k(q )j > and E^M P sup

U( ) V^T;k(u⁽²⁾) V^T;k(u⁽¹⁾) >

!

; (A.2) for a su¢ ciently large T , where U( ) = f(u⁽²⁾; u⁽¹⁾) 2 R² : ku⁽²⁾ u⁽¹⁾k g. From Lemma A.1, V^T;k( ) is stochastically equicontinuous and thus the right-hand side of (A.2) converges to zero for some as T ! 1. An application of Lemma A.1 and A.5 shows that lim_{T !1}P (EM^c ) = 0 for some M > 0, which implies

T !1lim P sup

2AjV^T;k(^q ) V^T;k(q )j > = 0:

Because Gk( )is di¤erentiable from Assumption A4, the mean value expansion yields pT fG^k(^q ) G_k(q )g = rG^k(q )^|p

T (^q q ) ;

uniformly in 2 A, where q = (q¹( ₁); q₂( ₂))^| with qi( _i) being between ^q_i( _i) and qi( _i) for i = 1; 2. It follows that

pT f^ k(^q ) _k(q )g = V^T;k(q ) +rG^k(q )^|p

T (^q q ) + o_p(1);

uniformly in 2 A.

Proof of Theorem 1. From the uniform Bahadur representation, we have pT f^q_i( _i) q_i( _i)g = 1

f_i(q_i( _i))W⁽ⁱ⁾T;k(q_i( _i)) + o_p(1); (A.3) uniformly in i 2 Aⁱ, for i = 1; 2. We de…ne an empirical process indexed by 2 A as

T;k( ) = V^T;k(q ); W^T;k(q )^{| |}:

Then, Lemma A.2 together with (A.3) implies that, for every k 2 f1; : : : ; pg, pT (^ (k) (k)) = ^|_{k T;k}( ) + o_p(1);

uniformly in 2 A, where ^k is de…ned in (4).

We …rst prove

sup

2A

^^{(p) (p)} = o_p(1):

Using the Cauchy-Schwarz inequality, we have j ^|k T;k( )j k ^kkk ^T;k( )k. From Assumption A2 and A4, k ^kk is bounded uniformly in 2 A. Furthermore, Lemma A:1 implies that sup _2Ak ^T;k( )k = O^p(1), which together with the cr-inequality leads to that sup _2Ak^ (k) (k)k Pp

k=1sup _2Aj^^{(p) (p)}j = o^p(1).

To show (3), let ^(p) = diag( ^|₁; : : : ; ^|_p) and ^(p)_T ( ) = [ _T;1( ); : : : ; _T;p( )]^|. Then, we have

pT ^^{(p) (p)} = ^{(p) (p)}_T ( ) + o_p(1);

uniformly in 2 A. From Lemma A.1, we can establish the the …nite dimensional distributions convergence of ^{(p) (p)}_T ( ) over 2 A and it su¢ ces to show the sto-chastic equicontinuity of ^{(p) (p)}_T ( ). An application of the cr-inequality with the uniform boundedness of k over 2 A yields that, for any ; 2 A,

(p) (p)

T ( ) ^{(p) (p)}_T ( )

Xp k=1

fC k ^{T k}( ) _{T k}( )k + k ^{k k}kk ^{T k}( )kg :

Let ; 2 A with k k for some > 0. From Assumption A2, inf _i2Aif_i(q( _i)) C₁ for some C1 > 0. It follows that kq q k ~ C₁ ¹ , because i i = Rqi( i)

qi( _i) f_i(v)dv for each i = 1; 2, by de…nition. It follows that sup

; 2A:k k k ^T;k( ) _T;k( )k sup

U( ~)

V^T;k(u⁽²⁾) V^T;k(u⁽¹⁾) + sup

U( ~)

W^T;k(u⁽²⁾) W^T;k(u⁽¹⁾) ;

where U(~) = f(u⁽²⁾; u⁽¹⁾)2 R² :ku⁽²⁾ u⁽¹⁾k ~g. Lemma A.1 implies the stochastic equicontinuity of V^T;k( )and W^T;k( ), which yields that, for arbitrarily small positive constants and , there exist an > 0 such that

T !1lim P sup

; 2A:k k k ^T;k( ) _T;k( )k >

!

< :

From Lemma A.1, sup _2Ak ^{T k}( )k = O^p(1) and the uniform continuity of k un-der Assumption A2 and A4 implies sup _{; 2A:k k} k ^{k k}k = o(1) for some

> 0. Collecting the results given the …nite lag order, we establish the stochastic equicontinuity, from which (3) follows.

Lemma A.3 Suppose that Assumption A1-A5 hold. Then, pT (^ (k) ^ (k)) = VT;k(q ) +rG^k(q )^|p

T (^q q )^ p

1(1 ₁) ₂(1 ₂) + r_3;T( ) where P (sup _2Ajr3;T( )j > ) !^p 0 for every > 0.

Proof. This proof is similar to the proof of Lemma A.2 and thus we outline the key steps. First, we consider ^ _k(^q ) ^ _k(^q ), where ^ _k(q) = PT

t=k+1 1(x_1t q₁) ₂(x_{2t k} q₂) for q = (q1; q₂)^| 2 R². It can be shown that, for i = 1; 2,

P sup

i2Ai

T ¹⁼² XT

t=1

i(x_it q^_i( i)) >

!

= op(1): (A.4)

for any > 0, which together with (A.1) implies

pT f^ k(^q ) ^ _k(^q )g = VT;k(^q ) + V^T;k(^q ) V^T;k(^q ) + G_k(^q ) G_k(^q ) + r_1;T( );

where P (sup _2AjrT( )j > ) = o^p(1) for any > 0.

From Theorem 2 of Doss and Gill (1992), P (sup _2Ak^q q^ k > M¹T ¹⁼²) = o_p(1)

for some M1 > 0. Since sup _2Ak^q q k = O^P(T ¹⁼²), we have P sup _2Ak^q q k M₂T ¹⁼² =

o_p(1)for some M2 > 0. Lemma A.1(b) implies the stochastic equicontinuity of VT;k( ) conditional on the original sample, thereby yielding that, for any > 0,

P sup

2AjVT k(^q ) VT k(q )j > = o_p(1):

Similarly, Lemma A.1 implies that, for any > 0,

P sup

2AjV^{T k}(^q ) V^{T k}(^q )j > = op(1):

Thus, it follows from the mean-value expansion that pTf^ k(^q ) ^ _k(^q )g = V k(q ) +rG^k(q )^|p

T (^q q ) + r^ _2;T( );

where P (sup _2Ajr2;T( )j > ) = o^p(1) for any > 0.

Next, we consider the denominator. It can be shown that

sup

12A1

T ¹ XT t=k+1

2

1(x_1t q^₁( ₁)) ₁(1 ₁) sup

12A1

T ¹ XT

t=1

i(x_it q^_i( _i)) where the right-hand side converges to zero in probability, conditional on the orig-inal sample from (A.4). A similar argument can apply for the other term of the denominator and thus we obtain desired result.

Proof of Theorem 2. As in the proof of Theorem 2 of Doss and Gill (1992), we have the uniform Bahadur representation

pTf^q_i( i) q^i( i)g = 1

f_i(^q_i( _i))W⁽ⁱ⁾T;k(^qi( i)) + r_{T 1}( );

for i = 1; 2, where P (sup _i2AijrT 1( _i)j > ) = o^p(1) for any > 0. Since p Tf^q q g = O^p(1)uniformly in 2 A, Lemma A.1(b) and A.3 imply implies that, for every k 2 f1; : : : ; pg,

pT (^ (k) ^ (k)) = ^|_{k T;k}( ) + r_{T 2}( );

where P (sup _2AjrT 2( )j > ) = o^p(1) for any > 0, k is de…ned in (4), and

T;k( ) = [VT;k(q ); WT;k(q )^|]^|. We de…ne ^(p) = diag( ^|₁; : : : ; ^|_p) and ^(p)_T ( ) = [ _T;1( ); : : : ; _T;p( )]^|. Then, we have

pT ^^{(p) (p)} = ^{(p) (p)}_T ( ) + r_{T 3}( );

where P (sup _2AjrT 3( )j > ) = o^p(1) for any > 0. The cross-quantilogram based on the SB resample is linearly depend on the empirical process based on the SB resample. Since Lemma A.1(b) shows the validity of SB method for the empirical distribution function and the similar argument used to show Theorem 1 can apply for the rest of the proof.

Proof of Theorem 3. Under both …xed and local alternatives, Lemma A.2 implies pT ^^{(p) (p)} = ^{(p) (p)}_T ( ) + o_p(1):

uniformly in 2 A, and it follows from Theorem 1 that

(p) (p)

T ( ) = OP(1);

uniformly in 2 A.

(a) Under the …xed alternative, there is some 2 A such that is some non-zero constant and then

pT ^^(p) = ^{(p) (p)}_T ( ) +p

T ^(p)+ o_p(1):

This implies that, under the …xed alternative,

sup

2A

Q^^(p) = T sup

2Ak ^(p)k²(1 + o_p(1)) :

Thus, sup _2AQ^^(p) !^p 1 under the …xed alternative, whereas the critical value cQ; is bounded in probability from Theorem 2. Therefore, lim_{T !1}P (sup _2AQ^^(p) > c_Q; ) =

1. Therefore, our test is shown to be consistent under the …xed alternative.

(b) Under the local alternative, we can write ^(p) = ^(p)=p

T, where ^(p) is a p-dimensional constant vector, at least one of elements is non-zero. Thus, we have

Q^^(p) =k ^{(p) (p)}T ( ) + ^(p)k²+ o_P(1);

uniformly in 2 A. From Theorem 1 and the continuous mapping theorem,

sup

2A

Q^^(p) =) sup

2Ak ^(p)B^(p)( ) + ^(p)k²: Also, Theorem 2 implies

sup

2A

Q^^(p) =) sup

2Ak ^(p)B^(p)( )k²; in probability. Thus, the desired result follows.

Lemma A.4 Suppose that Assumption A1-A4 hold. Then, for each k 2 f1; : : : ; pg and for each 2 A,

pT ^ _{;[T r]}(k) (k) = r ¹V^k(q ; r) +rG^k(q )^|p

T ^q _{;[T r]} q p

1(1 ₁) ₂(1 ₂) + o_p(1);

uniformly in r 2 [!; 1].

Proof. The proof follows the line of Lemma A.2, using Lemma A.1(c) and Lemma A.5. We omit the details.

Proof of Theorem 4. Lemma A.5 shows that, for i = 1; 2, pT q^_{i[T r]}( _i) q_i( _i) = 1

rf_i(q_i( _i))W⁽ⁱ⁾T;k(q_i( _i); r) + o_p(1); (A.5)

uniformly in r 2 [!; 1]. We de…ne an empirical process indexed by r 2 [!; 1],

T;k; (r) = V^T;k(q ; r); W^T;k(q ; r)^{| |};

and let ^(p)_T; ( ) = [ _T;1; ( ); : : : ; _T;p; ( )]^|. Then, Lemma A.4 together with (A.5) implies

pT ^^(p)_{;[T r]} ^(p) = r ^{1 (p) (p)}_T; (1) + o_p(1);

uniformly in r 2 [!; 1]. It follows that [T r]

pT ^^(p)_{;[T r]} ^^(p)_;T = ^(p)n _(p)

T; (r) r ^(p)_T; (1)o

+ o_p(1);

uniformly in r 2 [!; 1]. From Lemma A.1(c), f ^(p)T; (r) r ^(p)_T; (1) : r 2 [!; 1]g weakly converges to the Brownian bridge process f ^(p)(B^(p)(r) rB^(p)(1)) : r 2 [!; 1]g with

(p)( ^(p))^{0 (p)}( ; ), and thus it follows from the continuous mapping theorem that

A^ _p =) ^{(p) (p)} A^{p (p) (p) |}: Since p

T ^^(p)_;T =) ( ^{(p) (p)})B^(p)(1), we have

S^^(p) =) B^(p)(1)^|Ap¹B^(p)(1):

We complete the proof.

Proof of Theorem 5. Under both …xed and local alternative, the argument used in Theorem 4 gives

pT ^^(p)_{;[T r]} ^(p) = r ^{1 (p) (p)}_T; (r) + o_p(1);

thereby yielding ^A _p =) ( ^{(p) (p)})A^p( ^{(p) (p)})^|.

(a) Under the …xed alternative, we have pT ^^(p)_;T = ^{(p) (p)}_T; (1) +p

T ^(p)+ o_p(1);

where the right-hand side diverges in probability as T ! 1. Since the critical value we use is …nite in probability from Theorem 4, we obtain the desired result.

(b) Under the local alternative,

pT ^^(p)_;T = ^{(p) (p)}_T; (1) + ^(p)+ o_p(1):

Proof of Theorem 6. Consider (a). Two l-dimensional vectors, _z1 and _z2, and a l l matrices z and their sample analogues, ^_z1 and ^_z2, satisfy the following partitioned matrices:

which leads to that ^p _k;12 = p _k;12+ o_p(1) and that pT (^p _k;12 p _k;12) =p

T (^r _k;12 r _k;12) ^|_{z1 z}¹p

T (^_{z2 z2}) + o_p(1):

Applying arguments used to prove Lemma A.2, we can show pT (^p _k;12 p _k;12) = ⁽¹⁾_k;z^|T ¹⁼²

XT t=k+1

~1t( ; k) + ⁽²⁾_k;z^|p

In document The Cross-Quantilogram: Measuring Quantile Dependence and Testing Directional Predictability between Time Series (Page 40-51)