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Bank characteristics and the interbank money market: a distributional approach

Giulia Iori, Burcu Kapar and Jose Olmo

Abstract

This paper studies the relationship between bank characteristics, such as size, nationality, operating currency and sovereign debt in the parent country, and the distribution of funding spreads observed in the e-MID interbank money market during the Great financial crisis. Our setup is a pseudo-panel with a random number of international banks acting in the interbank market in each period. We develop new econometric tools for panel data with random effects and discrete covariates, such as a nonparametric kernel estimator of the distribution function of the response variable conditional on a set of covariates and a consistent test of first order stochastic dominance. Our empirical results, based on these tests, shed light on the survivorship bias in the e-Mid market, and reveal the existence of a risk premium on small banks, banks with currencies different from the Euro, and banks based on countries under sovereign debt distress in the periphery of the European Union. Finally we assess the impact of policy intervention in the aftermath of the financial crisis.

JEL Classification: C12; C14; G01; G20

Corresponding author: Jose Olmo, School of Social Sciences, Economics Division, University of Southampton, Room 3015, Bld 58 (Murray Bld), Highfield Campus, Southampton, SO17 1BJ, UK, e-mail:

Acknowledgments

The research leading to this paper has received funding from the European Union, Seventh Framework Programme FP7/2007–2013 under grant agreement FOC “Forecasting Crisis”, No. 255987, and to a lesser extent from grant agreement CRISIS “Complexity Research Initiative for Systemic Instabilities”, No. 288501.

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)=zZ1j(z,x)[Fz(y)μ(z)Fx(y)μ(x)]/μ(x) with Z the support of the covariate vector Ij,it and 1j(z,x)=1(|zx|=1)j=1k1(|zx|=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 N1/2h→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{Iis=x, Iit=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)=T1s,t=1tsTP{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)(1Fx(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{Iit=xIis=x}=1 with s<t. The asymptotic variance Σx(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˜)=n1/2i=1Nt=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˜)=n1/2i=1Nt=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˜)=T1s,t=1tsTP{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 supyΩ|sit(y;x,x˜)| is bounded by construction of dit(y) and the kernel functions wh(Ij,it; xj) in Wh(Iit; 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 cα denote the (1–α) quantile of the distribution of supyΩ|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 supyΩ|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 supyΩ|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 HA, note that Tn(x,x˜) as N→∞, implying that

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

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Published in Print: 2015-06-01

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