Bank characteristics and the interbank money market: a distributional approach

Appendix: Proofs

Proof of Theorem 1: For the sake of clarity in the presentation, we assume throughout the proof a balanced panel. There is no loss of generality in doing so.

1. The proof of the consistency of (5) follows from the proof of Theorem 2.1 in Li and Racine (2008) for the cross-sectional case. More specifically, let Bj(x, y)=∑z∈Z1j(z, x)[Fz(y)μ(z)−Fx(y)μ(x)]/μ(x) with Z the support of the covariate vector I_j,it and 1j(z, x)=1(|z−x|=1)∏j=1k1(|z−x|=0); let |h|=∑j=1khs. Our assumptions A.1–A.6 contain conditions (C1)-(C3) in Li and Racine (2008). These authors show that under these conditions

   F ^ x ( y ) = F x ( y ) + ∑ j = 1 k h j B j ( x ,   y ) + o ( | h | 2 ) ,

   for any fixed pair (x, y)∈Ω˜. Now, under assumption A.6 the leading bias term converges to zero as N→∞ and the asymptotic consistency of the estimator follows.

2. To derive the asymptotic distribution of the standardized estimator we note that E[n1/2(F^x(y)−Fx(y))] converges to zero in probability. This is so by assumption A.6 that imposes that N^1/2h→0 as with T fixed. Under this assumption the bias of the nonparametric estimator given by ∑j=1khjBj(x, y) converges to zero as N increases.

To complete the proof we need to find the limiting variance of n1/2(F^x(y)−Fx(y)) as N→∞, and apply a Liapunov Central Limit Theorem. First, note that n1/2(F^x(y)−Fx(y))=n1/2(F^x(y)−Fx(y))μ^(x)/μ(x)+oP(1). Thus, V(n1/2(F^x(y)−Fx(y)))=V(n1/2(F^x(y)−Fx(y))μ^(x))/μ2(x)+oP(1). Then

(13) V ( n 1 / 2 ( F ^ x ( y ) − F x ( y ) ) μ ^ ( x ) ) = n − 1 ∑ i = 1 N ∑ t = 1 T V ( ( d i t ( y ) − F x ( y ) ) W h ( I i t ;   x ) ) + n − 1 ∑ i = 1 N ∑ s , t = 1 t ≠ s T C o v ( d i s ( y ) − F x ( y ) ) W h ( I i s ;   x ) ,   ( d i t ( y ) − F x ( y ) ) W h ( I i t ;   x ) ) .  (13)

The first term in (13) is such that

V ( ( d i t ( y ) − F x ( y ) ) W h ( I i t ;   x ) ) = E [ ( d i t ( y ) − F x ( y ) ) 2 W h 2 ( I i t ;   x ) ] + O ( | h | ) = E [ ( η i 2 + F z ( y ) − 2 F z ( y ) F x ( y ) + F x 2 ( y ) ) W h 2 ( I i t ;   x ) ] + O ( | h | ) = ( σ i 2 + F x ( y ) ( 1 − F x ( y ) ) ) μ ( x ) + O ( | h | ) . ]

The second term in (13) is

C o v ( ( d i s ( y ) − F x ( y ) ) W h ( I i s ;   x ) ,   ( d i t ( y ) − F x ( y ) ) W h ( I i t ;   x ) ) = E [ V ( μ i ) W h ( I i s ;   x ) W h ( I i t ;   x ) ] + O ( | h | ) = σ i 2 P { I i s = x , I i t = x } + O ( | h | ) .

Then, assuming that P{I_is=x, I_it=x} is the same across individuals and n=NT, expression (13) takes the following form:

V ( n 1 / 2 ( F ^ x ( y ) − F x ( y ) ) μ ^ ( x ) ) = F x ( y ) ( 1 − F x ( y ) ) μ ( x ) + N − 1 ∑ i = 1 N σ i 2 ( μ ( x ) + λ ( x ) ) + O ( | h | ) ,

with λ(x)=T−1∑s,t=1t≠sTP{Iis=x, Iit=x}. Applying the law of large numbers to the iid cross section we obtain that

l i m N → ∞ V ( n 1 / 2 ( F ^ x ( y ) − F x ( y ) ) μ ^ ( x ) ) = F x ( y ) ( 1 − F x ( y ) ) μ ( x ) + E [ σ i 2 ] ( μ ( x ) + λ ( x ) ) + O ( | h | ) ,

Using a Liapunov Central Limit Theorem, it follows that

n 1 / 2 ( F ^ x ( y ) − F x ( y ) ) → d N ( 0,   Σ x ( y ) ) ,

with Σx(y)=(Fx(y)(1−Fx(y))/μ(x)+E[σi2](μ(x)+λ(x))/μ2(x).

Proof of Corollary 1:

1. The presence of a persistent estimator is defined in this case as P{I_it=x∣I_is=x}=1 with s<t. The asymptotic variance Σ_x(y) in Theorem 1 becomes

   Σ x ( y ) = ( F x ( y ) ( 1 − F x ( y ) ) + T σ 2 ) / μ ( x ) .

2. If the covariates are serially independent the asymptotic variance is

   Σ x ( y ) = ( F x ( y ) ( 1 − F x ( y ) ) + σ 2 ) / μ ( x ) + ( T − 1 ) σ 2 .

Proof of Theorem 2: The function Sn(y; x, x˜)=n1/2((F^x(y)−Fx(y))−(F^x˜(y)−Fx˜(y))) can be expressed as Sn(y; x, x˜)=n−1/2∑i=1N∑t=1Tsit(y; x, x˜) with

s i t ( y ;   x ,   x ˜ ) = ( d i t ( y ) − F x ( y ) ) W h ( I i t ;   x ) / μ ^ ( x ) − ( d i t ( y ) − F x ˜ ( y ) ) W h ( I i t ;   x ˜ ) / μ ^ ( x ˜ ) .

Its empirical covariance function is Kn(y1, y2; x, x˜) that is defined as

K n ( y 1 ,   y 2 ;   x ,   x ˜ ) = n − 1 ∑ i , j = 1 N ∑ s , t = 1 T s i s ( y 1 ;   x ,   x ˜ ) s j t ( y 2 ;   x ,   x ˜ ) .

Under A.1–A.8, for each y∈Ω and fixed pair (x, x˜)∈Ω*,sit(y; x, x˜) is a square integrable stationary sequence with an asymptotic mean equal to zero as N→∞, to which the pointwise central limit theorem applies. Furthermore, note that sit(y; x, x˜)=s˜it(y; x, x˜)+oP(1), with

s ˜ i t ( y ;   x ,   x ˜ ) = ( d i t ( y ) − F x ( y ) ) W h ( I i t ;   x ) / μ ( x ) − ( d i t ( y ) − F x ˜ ( y ) ) W h ( I i t ;   x ˜ ) / μ ( x ˜ ) ,

and the covariance of Sn(y; x, x˜), defined as E[Sn(y1; x, x˜)Sn(y2; x, x˜)], satisfies that

E [ S n ( y 1 ;   x ,   x ˜ ) S n ( y 2 ;   x ,   x ˜ ) ] = E [ S ˜ n ( y 1 ;   x ,   x ˜ ) S ˜ n ( y 2 ;   x ,   x ˜ ) ] + o P ( 1 ) ,

with S˜n(y; x, x˜)=n−1/2∑i=1N∑t=1Ts˜it(y; x, x˜). The cross-sectional independence imposed on A.2 implies that the covariance kernel for Sn(y|x, x˜) can be approximated by

E [ S ˜ n ( y 1 ;   x ,   x ˜ ) S ˜ n ( y 2 ;   x ,   x ˜ ) ] = n − 1 ∑ i = 1 N ∑ t = 1 T E [ s ˜ i t ( y 1 ;   x ,   x ˜ ) s ˜ i t ( y 2 ;   x ,   x ˜ ) ] + n − 1 ∑ i = 1 N ∑ s , t = 1 t ≠ s T E [ s ˜ i s ( y 1 ;   x ,   x ˜ ) s ˜ i t ( y 2 ;   x ,   x ˜ ) ] ,

where

E [ s ˜ i t ( y 1 ;   x ,   x ˜ ) s ˜ i t ( y 2 ;   x ,   x ˜ ) ] = ( σ i 2 + F x ( m i n ( y 1 ,   y 2 ) ) − F x ( y 1 ) F x ( y 2 ) ) / μ 2 ( x ) + ( σ i 2 + F x ˜ ( m i n ( y 1 ,   y 2 ) ) − F x ˜ ( y 1 ) F x ˜ ( y 2 ) ) / μ 2 ( x ˜ ) + O ( | h | ) ,

and

E [ s ˜ i s ( y 1 ;   x ,   x ˜ ) s ˜ i t ( y 2 ;   x ,   x ˜ ) ] = σ i 2 P { I i s = x ,   I i t = x ˜ } / ( μ ( x ) μ ( x ˜ ) ) + O ( | h | ) .

Then,

E [ S ˜ n ( y 1 ;   x ,   x ˜ ) S ˜ n ( y 2 ;   x ,   x ˜ ) ] = ( F x ( m i n ( y 1 ,   y 2 ) ) − F x ( y 1 ) F x ( y 2 ) ) / μ 2 ( x ) + ( F x ˜ ( m i n ( y 1 ,   y 2 ) ) − F x ˜ ( y 1 ) F x ˜ ( y 2 ) ) / μ 2 ( x ˜ ) + N − 1 ∑ i = 1 N σ i 2 ( 1 / μ 2 ( x ) + 1 / μ 2 ( x ˜ ) ) + N − 1 ∑ i = 1 N σ i 2 λ ( x ,   x ˜ ) / ( μ ( x ) μ ( x ˜ ) ) + O ( | h | ) ,

with λ(x, x˜)=T−1∑s,t=1t≠sTP{Iis=x, Iit=x˜}.

The asymptotic covariance kernel of the limiting Gaussian process S∞(y; x, x˜) is

K ∞ ( y 1 ,   y 2 ;   x ,   x ˜ ) = ( F x ( m i n ( y 1 ,   y 2 ) ) − F x ( y 1 ) F x ( y 2 ) ) / μ 2 ( x ) + ( F x ˜ ( m i n ( y 1 ,   y 2 ) ) − F x ˜ ( y 1 ) F x ˜ ( y 2 ) ) / μ 2 ( x ˜ ) + E c [ σ i 2 ] ( μ 2 ( x ) + μ 2 ( x ˜ ) + λ ( x ,   x ˜ ) μ ( x ) μ ( x ˜ ) ) / ( μ 2 ( x ) μ 2 ( x ˜ ) ) .

The multivariate central limit theorem establishes the finite dimensional distributional convergence. To establish stochastic equicontinuity, we appeal to Theorem 1 of Doukhan, Massart, and Rio (1995). By assumption A.2, the summands sit(y; x, x˜) satisfy the necessary absolute regularity mixing decay rate. Further, the envelope function supy∈Ω|sit(y; x, x˜)| is bounded by construction of d_it(y) and the kernel functions w_h(I_j,it; x_j) in W_h(I_it; x). The rest of the proof follows from the proof of Theorem 1 in Hansen (1996).

Proof of Corollary 2: The proof is analogous to the proof for the cross-sectional case presented in Barrett and Donald (2003). Under A.1–A.8, the application of Theorem 2 and the continuous mapping theorem imply that

s u p y ∈ Ω | S n ( y ;   x ,   x ˜ ) | → d s u p y ∈ Ω | S ∞ ( y ;   x ,   x ˜ ) | .

Let c_α denote the (1–α) quantile of the distribution of supy∈Ω|S∞(y; x, x˜)|. This implies that

l i m N → ∞ P { s u p y ∈ Ω | S n ( y ;   x ,   x ˜ ) | > c α } = α .

Further, it is not difficult to see that under the null hypothesis H₀, the statistic Tn(x, x˜) is majorized by supy∈Ω|Sn(y; x, x˜)|, hence,

l i m N → ∞ P { T n ( x ,   x ˜ ) > c α } ≤ l i m N → ∞ P { s u p y ∈ Ω | S n ( y ;   x ,   x ˜ ) | > c α } = α .

Under the least favourable case, it follows that Tn(x, x˜) and supy∈Ω|Sn(y; x, x˜)| are the same process, hence the above condition holds with equality for all y∈Ω.

Finally, to derive the consistency of the test under H_A, note that Tn(x, x˜)→∞ as N→∞, implying that

l i m N → ∞ P { T n ( x ,   x ˜ ) > c α } = 1.