RDP 2018-01: A Density-based Estimator of Core/Periphery Network Structures: Analysing the Australian Interbank Market Appendix A: The Maximum Likelihood Estimator
February 2018
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Partition the N × N adjacency matrix (A) into four blocks: the core lends to core block, the core lends to periphery block, the periphery lends to core block, and the periphery lends to periphery block. Assuming the links in each block come from an Erdős-Rényi random network, the probability mass function for a link in block i is a Bernoulli distribution (with support k ∈ {0,1} and probability p_{i}):
Since each link in an Erdős-Rényi network is independent, the joint probability function of all the links in the adjacency matrix is the product of N(N − 1) Bernoulli distributions:
where s_{i} is the number of possible links in block i and λ_{i} is the number of actual links in block i. With each bank's designation as core or periphery determining the composition of the four blocks, the joint probability distribution is defined by N + 4 parameters (the designations of the N banks, and the four p_{i} parameters).
Both s_{i} and λ_{i} depend on the matrix A and on the core/periphery partition, but not on the probability p_{i}. Therefore, we can determine the maximum likelihood estimator of p_{i} conditional on the data and the core/periphery partition in order to produce a concentrated log-likelihood function where the only unknown parameters are whether each node is in the core or the periphery.
The conditional maximum likelihood estimator of p_{i} is λ_{i}/s_{i}. However, for our analysis it is handy to concentrate-out the λ_{i} parameter so that the concentrated log-likelihood function becomes:
where p_{i} is now a function of the data and the core/periphery partition via the relationship λ_{i} = s_{i}p_{i}.
For a given core/periphery partition, the log-likelihood has the same value if the designation of every bank is switched (i.e. every core bank becomes a periphery bank and every periphery bank becomes a core bank). Therefore, the maximum likelihood estimator requires the identifying restriction p_{CC} ≥ p_{PP} (where i = CC is the block of core lends to core links and i = PP is the block of periphery lends to periphery links).
Since p_{i} ∈[0,1], each component of the sum in Equation (A1) is bounded above by zero.^{[32]} This upper bound occurs when either p_{i} = 0 or p_{i} = 1. Since p_{i} = λ_{i}/s_{i}, the upper bound for each component is reached when either no links exist in the corresponding block or all the links in the block exist. Therefore, if we ignore the components of Equation (A1) that correspond to the off-diagonal blocks, the true CP split of an ideal CP network produces the largest possible value of the likelihood function.
Footnote
Evaluating xlnx when x = 0 as its limit value as x → 0 (to prevent the log-likelihood from being undefined at the bounds of p_{i}). [32]