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

See e.g., Upper and Worms (2004), Nier et al. (2007) and Gai et al. (2011).

See Haldane and May (2011) and Albert et al. (2000).

See Albert et al. (2000).

This is not to say these papers do not yield very useful phenomenological information about the pertinent network structures. The point here is rather that there seems to exist a convention in the network literature to produce a power-law graph without a deeper statistical analysis of this issue and also often in spite of a relatively small sample of observations that additionally impedes such a strong inference on the underlying distribution.

Since we cannot easily observe the state of a hypothesized network of interbank links at a given point in time, some data aggregation is necessary. Usually, for time-aggregated data, a link is assumed to exist between two banks, if there has been a trade at any time during the aggregation period.

This is of course only true when taking banks as consolidated entities.

Directed means that \(d_{i,j}\ne d_{j,i}\) in general. Sparse means that at any point in time, the number of links is only a small fraction of the \(N(N-1)\) possible links. Valued means that interbank claims are reported in monetary values as opposed to 1 or 0 for the presence or absence of a claim, respectively.

The vast majority of trades (roughly 95 %) is conducted in Euro.

See also the e-MID website http://​www.​e-mid.​it/​.

Note that the mean in- and out-degree are identical by definition.

Interestingly, after standardizing the degrees, we find structural breaks in all three time series close to quarter 39, i.e., around the GFC.

Note that the first subsample roughly coincides with the dataset used by De Masi et al. (2006).

In Appendix 4, we present a similar analysis for the distribution of transaction volumes of individual institutions.

This is important, since we cannot replicate the large number of zero values that we observe in the empirical data based on these distributions. Ignoring zeros reduces the number of quarterly observations to 1,742, 3,271, and 788 for the in-variables, and 1450, 2733, and 663 for the out-variables, respectively. For the daily data, this leaves 70,584, 133,280, and 28,093 for the in-variables, and 39,619, 83,723, and 17,961 for the out-variables, respectively. The number of observations for the total degree and ntrans variables remains unaffected, since only active banks are in the sample.

There exist a number of alternative approaches in statistical extreme value theory for determining the optimal tail size. The approaches by Danielsson et al. (2001) and Drees and Kaufmann (1998) yielded results very similar to those reported in the text. We also checked certain fixed thresholds for identifying the tail region. The results remain qualitatively the same as long as the chosen upper quantile is reasonably large.

In principle, we could also use likelihood-based criteria, e.g., AIC or BIC. However, Clauset et al. (2009) provide some evidence that the KS statistic is preferable as it is more robust to statistical fluctuations.

See Clauset et al. (2009) and Stumpf et al. (2005) for similar approaches.

De Masi et al. (2006) report power-law exponents between 2 and 3 for total degree, in-degree, and out-degree for daily data of our period 1 of the e-MID overnight record. Soramäki et al. (2007) also report values in this range for daily interbank payments within the U.S. Fedwire system. Boss et al. (2004) fit two power-laws to the more central and the extremal region of the in-degree distribution of monthly Austrian interbank liabilities with the extremal region exhibiting slopes of 1.73 for in-degrees and 2.01 for out-degree. This visual illustration could, however, be as well interpreted as indicating an overall exponential shape. Bech and Atalay (2010) study daily federal fund credits in the U.S. Their study constitutes a rare example of a comparative fit of a number of candidate distributions. While the overall shape of the distribution does not quite show a straight linear shape in a log–log plot, they report that the power-law gave the best fit among the candidates for out-degrees while the negative Binomial did best fit the in-degree distribution.

We have set 7 as the upper bound of the power-law parameter in our numerical ML implementation. For larger values, the evaluation of the zeta function appearing in the discrete Pareto law, cf. Appendix 2, is not accurate enough to obtain reliable estimates. The fact that the estimated values hit the upper bound quite frequently indicates that the estimated values may become even larger when increasing the upper bound.

We also generated synthetic power-law distributed random draws and estimated their scaling parameters based on the algorithm for the selection of the tail region detailed above (not reported). For the small sample sizes of the typical daily data, the tail parameter of these synthetic data is highly volatile as well, even though the very large values observed for the actual data are very rare. As usual, however, increasing the number of observations (say more than 500), typically yields estimates very close to the true parameters.

This result is driven by the higher noise level in the tail data due to a smaller number of observations compared to the complete distributions.

The stability under aggregation of power-laws characterizing the tails of iid random variables is one of the basic tenets of the statistical theory of extremes, cf. Reiss and Thomas (2007). In this sense, summing up daily power-law networks should preserve the tail index for different frequencies.

Clauset et al. (2007) show that this is necessary for datasets from the social sciences, where the maximum value is usually only a few orders of magnitude larger than the minimum, i.e., the tail is heavy but rather short. In such cases, the estimated exponents can be biased severely when using the continuous approximation.

Using a quadratic approximation of the log-likelihood at its maximum, Clauset et al. (2009) also derive an approximate closed-form solution for the estimate of \(\alpha \simeq 1 + n / \left( \sum \nolimits _{i=1}^{n} \ln \left[ \frac{x_i}{x_m - .5} \right] \right) \). This can be seen as an adjusted Hill-estimator, see Hill (1975). While we always report the exact ML estimator, we checked that the approximation is typically not too bad.

Clauset et al. (2009) also derive an (approximate) estimator for the standard error based on discrete data, which is, however, much harder to evaluate as it involves derivatives of the generalized zeta function.

See Clauset et al. (2007, 2009) for an extensive discussion.

Note that the maximum daily (quarterly) transaction volumes were 3.75 bn (113.46 bn) Euros for in-tvol, 4.96 bn (111.93 bn) Euros for out-tvol, and 5.32 bn (146.06 bn) Euros for total tvol, respectively. For such huge numbers, the estimation procedure, in the numerical optimization for the power-law parameters, tends to take a very long computation time. Therefore, the results in this section should be treated with care, since the rescaling might affect our statistical analysis.