On the distribution of links in the interbank network: evidence from the e-MID overnight money market

Abstract

Previous literature on statistical properties of interbank networks has reported various power-laws, particularly for the degree distribution (i.e., the distribution of credit links between institutions). In this paper, we revisit data for the Italian interbank network based on overnight loans recorded on the e-MID trading platform during the period 1999–2010 using both daily and quarterly aggregates. In contrast to previous reports, we find no evidence in favor of a power-law characterizing the degree distribution. Rather, the data are best described by negative Binomial distributions. For quarterly data, Weibull, Gamma, and Exponential distributions tend to provide comparable fits. We find similar results when investigating the distribution of the number of transactions, even though in this case, the tails of the quarterly variables are much fatter. The absence of power-law behavior casts doubts on previous claims that these interbank data fall into the category of scale-free networks.

Appendices

Appendix 1: Truncated distributions and maximum likelihood

The distribution fitting approach described in the main text involves fitting a set of candidate distributions with possibly differing support. For example, some distributions have support at zero, while others do not. Similarly, when focusing on the tail observations, we have to get rid of the probability mass below the cutoff point in order to accurately calculate the statistics. Therefore, we describe the use of truncated distributions and ML fitting in this Appendix in more detail.

1.1 Normalization

When working with truncated variables, we need to make sure to use the correct pdfs and cdfs, since the ML estimation and the evaluation of the fit (KS statistic) depend on them. In order to illustrate this issue, let variable $x$ have the pdf $p(x)$ with support $[0,\infty ]$. As usual, the cdf is defined as

$$\begin{aligned} P(a)=P(X \le a) = \int _0^a p(x) \mathrm{d}x. \end{aligned}$$

(6)

Now, suppose the data are (left-)truncated at some value $x_m$, i.e., the variable $\tilde{x}$ follows the same distribution as $x$, but the pdf has limited support $[x_m,\infty ]$ with minimum value $x_m >0$. For our purposes, it is therefore useful to define the quantity

$$\begin{aligned} P_<(a)=P(X < a) = 1 - \int _a^{\infty } p(x) \mathrm{d}x, \end{aligned}$$

(7)

or more compactly

$$\begin{aligned} P_<(a)= P(a)-p(a). \end{aligned}$$

(8)

We can properly construct the pdf of $\tilde{x}$, say $\tilde{p}$, as

$$\begin{aligned} \tilde{p}(x)={\left\{ \begin{array}{ll} \frac{p(x)}{1-P_<(x_m)}, &{}\quad \text {if }\,\, x \ge x_m\\ 0, &{}\quad \text {else.} \end{array}\right. } \end{aligned}$$

(9)

where the denominator distributes the probability mass of $p(x)$ among the support of $\tilde{x}$.

For the calculation of the KS statistics, we also need the adjusted cdf. For the supported values of $\tilde{x}$ it takes the form

$$\begin{aligned} \tilde{P}(x)=\int _{x_m}^x \frac{p(x)}{1-P_<(x_m)} \mathrm{d}x = \frac{1}{1-P_<(x_m)} \int _{x_m}^x p(x) \mathrm{d}x, \end{aligned}$$

(10)

or

$$\begin{aligned} \tilde{P}(x)=\frac{P(x)-P_<(x_m)}{1-P_<(x_m)}, \end{aligned}$$

(11)

which can be easily evaluated.

1.2 Maximum likelihood for truncated variables

Using the previous definitions, we can show that the ML estimator for left-truncated variables does not coincide with the standard estimator. The standard ML estimator, i.e., using a sample of $n$ observations of $x$ and denoting by $\theta$ the vector of parameters, can be written as

$$\begin{aligned} L(\theta |x_1,\ldots ,x_n)=p(x_1,\ldots ,x_n|\theta )=\prod _i^n p(x_i | \theta ), \end{aligned}$$

(12)

or in logs

$$\begin{aligned} \ln (L)=\sum _i^n \ln [p(x_i | \theta )]. \end{aligned}$$

(13)

Using the definitions from above, we can show that the ML estimator for left-truncated variables differs from the one in Eq. (13). Using Eq. (9), we can write the likelihood as

$$\begin{aligned} L=\prod _i^{\tilde{n}} \tilde{p}(x_i | \theta ) = \prod _i^{\tilde{n}} \frac{p(x|\theta )}{1-P_<(x_m | \theta )}, \end{aligned}$$

(14)

where $x$ ignores those observations smaller than $x_m$ and the total number of observations is $\tilde{n}$ instead of $n$. Taking logarithms we obtain

$$\begin{aligned} \ln (L)=\sum _i^{\tilde{n}} \ln \left[ \frac{p(x_i | \theta )}{1-P_<(x_m | \theta )}\right] = \sum _i^{\tilde{n}} \ln [p(x_i | \theta )] - \sum _i^{\tilde{n}} \ln [1-P_<(x_m | \theta )], \end{aligned}$$

(15)

which can be written as

$$\begin{aligned} \ln (L)= -\tilde{n} \ln [1-P_<(x_m | \theta )] + \sum _i^{\tilde{n}} \ln [p(x_i | \theta )]. \end{aligned}$$

(16)

The second part of this equation looks familiar, as it corresponds to Eq. (13) for the $\tilde{n}$ observations with values $\ge x_m$. However, the normalization term on the left does not vanish (as it depends on the parameter vector) and affects the location of the ML estimator. Therefore, we need to find the $\theta$ that maximizes Eq. (16). The standard ML estimator would not be efficient.

Appendix 2: Discrete power-laws and parameter estimation

This presentation is mostly based on Clauset et al. (2009).

1.1 Discrete power-laws

A power-law distributed variable $x$ obeys the pdf

$$\begin{aligned} p(x) \propto x^{-\alpha }, \end{aligned}$$

(17)

where $\alpha >0$ is the tail exponent with ‘typical’ interesting values in the range between 1 and 3. In many cases, however, the power-law only applies for some (upper) tail region, defined by the minimum value $x_m$. While it is common to approximate discrete power-laws by the (simpler) continuous version, for our (integer-valued) data, we employ the more accurate discrete version in the paper.²³

In the discrete case, the cdf of the power-law can be written as

$$\begin{aligned} P(x)=\frac{\zeta (\alpha ,x)}{\zeta (\alpha ,x_m)}, \end{aligned}$$

(18)

where

$$\begin{aligned} \zeta (\alpha ,x_m)=\sum _{n=0}^\infty (n+x_m)^{-\alpha } \end{aligned}$$

(19)

is the generalized or Hurwitz zeta function.

1.2 Estimation of $\alpha$ and $x_m$

For a given lower bound $x_m$, the ML estimator of $\alpha$ can be found by direct numerical maximization of the log-likelihood function

$$\begin{aligned} \mathcal {L}(\alpha )=-n \ln [\zeta (\alpha ,x_m)] - \alpha \sum _{i=1}^n \ln [x_i], \end{aligned}$$

(20)

where $n$ is the number of observations.²⁴ For simplicity, we approximate the standard error of the estimated $\hat{\alpha }$ (for $\hat{\alpha }>1$) using the closed-form solution based on continuous data.²⁵ Neglecting higher-order terms, this can be calculated as

$$\begin{aligned} \sigma =\frac{\hat{\alpha }-1}{\sqrt{n}}. \end{aligned}$$

(21)

However, the equations assume that $x_m$ is known in order to obtain an accurate estimate of $\alpha$.²⁶ When the data span only a few orders of magnitude, as usual in many social or complex systems, an underpopulated tail would come along with little statistical power. Therefore, we employ the numerical method proposed by Clauset et al. (2007) for selecting the $x_m$ that yields the best power-law model for the data. To be precise, for each $x_m$ over some reasonable range, we first estimate the scaling parameter using Eq. (20) and calculate the corresponding KS statistic between the fitted data and the theoretical distribution with the estimated parameters. The reported $x_m$ and $\alpha$ are those that minimize the KS statistic, i.e., minimize the distance between the observed and fitted probability distribution. According to Clauset et al. (2007, 2009), minimizing the KS statistic is generally superior to other distance measures, e.g., likelihood-based measures such as AIC or BIC.

Appendix 3: Goodness-of-fit test for the estimated distributions

Since the distribution of the KS statistics is unknown for the comparison between an empirical subsample and a hypothetical distribution with estimated parameters, we carry out a Monte Carlo approach. We sample synthetic datasets from the estimated distribution, compute the distribution of KS statistics, and compare the results to the observed value for the original dataset. If the KS statistic of the empirical dataset is beyond the $\alpha$ percent quantile of the Monte Carlo distribution of KS values, we reject the pertinent distribution at the $1-\alpha$ level of significance. In our results, we indicate significant fits at the 5 % confidence level using asterisks. We should stress that we carry out this (very time-consuming) GOF experiment only for the distribution with the minimum KS statistic for each sample and variable, respectively. This can be justified by the fact that, even though other candidate distributions may not be rejected as well, they are clearly inferior to the optimal distribution in terms of the KS statistic.

Fig. 14

Quarterly data, tvol. Complementary cumulative distribution functions (ccdf) in-tvol (top), out-tvol (center), and total tvol (bottom) for all time periods on a log–log scale

Fig. 15

Daily data, tvol. Complementary cumulative distribution functions (ccdf) in-tvol (top), out-tvol (center), and total tvol (bottom) for all time periods on a log–log scale

Appendix 4: Distributional properties of transaction volumes

Here, we report the results using another important measure for interbank networks, namely the transaction volumes (tvol). We use the same distribution fitting approach as before, differentiating between in-tvol, out-tvol, and their sum (total tvol), respectively. Figures 14 and 15 show the ccdfs for the quarterly and daily variables on a log–log scale. We should stress that the minimum trade size on the e-MID market is 50,000 Euros. In order to run our estimation procedure in a reasonable amount of time, we rescale the tvol variables by a factor of $10^{-6}$ such that a transaction size of 50,000 is represented by a value of .05.²⁷ We then round the tvol variable toward the nearest integer (otherwise the discrete candidate distributions could not be accurately evaluated), again ignoring zero values. In this way, we restrict our samples to relatively large transaction volumes with at least 500,000 Euros, represented by positive integer values. Note that, besides the upward bias of the data and the fact that the data now span several orders of magnitude, it is again hard to visually detect linear decay over several orders of magnitude in the ccdfs. We should also stress that we did not perform the GOF exercise for the tvol variables, since it is too time-consuming in this case.

1.1 Daily data

Tables 9, 10, and 11 show the results for the daily data. The complete distributions are now usually fitted best by Log-normal distributions, whereas the fit of the power-law is very poor in general. The power-law parameters are again very small, with typical values around 1.22, cf. Table 10 (top, complete). For the tail observations, the best fit again is always provided by Log-normal distributions, cf. Table 11. Interestingly, the tail exponents of the daily data are within the typical range of meaningful power-laws, cf. Table 10 (top, tail), but the power-law is still not the best description of the data. In the end, for the transaction volumes, we find no evidence in favor of power-laws.

Table 9 Daily data, tvol. KS statistic for the candidate distributions (complete)

Table 10 Power-law parameters and standard errors, tvol. Values obtained via numerical maximization of the log-likelihood for discrete data

Table 11 Daily data, tvol. KS statistic for the candidate distributions (tail)

Table 12 Quarterly data, tvol. KS statistic for the candidate distributions (complete data)

Table 13 Quarterly data, tvol. KS statistic for the candidate distributions (tail)

1.2 Quarterly data

Tables 12 and 13 show the results for the quarterly data. The complete in-, out-, and total degree distributions are now fit best by Weibull, negative Binomial, and Log-normal distributions, respectively. In many cases, these distributions yield comparable KS statistics, but the clear advantage of the Log-normal distribution for the daily data does not carry over to the quarterly level in all cases. Similar to the daily estimates, the power-law parameters are within the usual range of empirical power-laws. As before, however, the tails are best described by Log-normal distributions. Therefore, while the tails of the tvol variables are somewhat fatter compared to the degree and ntrans variables, the power-law remains a poor description of the data.