复杂网络，变滞后传递熵，FDA

Question

1.VaR、CoVaR、Delta CoVaR 和 MES具体是如何被拟合到y(t)曲线里的，是在已知基函数∮（t）里吗？能否给出具体的公式推导过程

2.1）这篇文献提到的用DFSRI曲线是如何构建部门内部的风险传染复杂网络的？
2）如何利用转移熵因果推断得出三部门之间系统性风险传染复杂网络？请给出具体公式推导过程，谢谢！

Answer 1

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Let's break down the complex questions related to financial risk networks, time-varying transfer entropy, and FDA.

1. VaR, CoVaR, Delta CoVaR, and MES in Functional Data Analysis (FDA)

These risk metrics are not directly "fitted" into a y(t) curve in the same way you fit a regression line. Instead, FDA treats the entire time series of returns (or another relevant variable) as a single function, y(t). The risk metrics are then calculated from this functional data object. The basis functions φ(t) are used to represent y(t) in a functional space, enabling various operations.

Here's a breakdown:

• Basis Functions (φ(t)): These are a set of functions (e.g., B-splines, Fourier basis) that are used to represent the observed data y(t) as a linear combination: y(t) ≈ Σ cᵢφᵢ(t), where cᵢ are the coefficients. The basis functions allow us to smooth the data and perform calculations in the functional domain.

• Fitting the curve: The "fitting" process involves finding the optimal coefficients (cᵢ) that best represent the observed data y(t) using the chosen basis functions. This is typically done using least squares regression.

• Calculating Risk Metrics: Once you have the smooth functional representation of y(t), you can calculate the risk metrics:

  • VaR(α): The Value-at-Risk at level α is the α-quantile of the distribution of y(t) (e.g., returns). In the FDA context, you'd find the value y*(t) such that the probability of y(t) being less than or equal to y*(t) is α.

  • CoVaR(α|Y=y): The Conditional Value-at-Risk measures the VaR of institution A conditional on institution B's return being at a specific level y. In FDA, you'd model the joint distribution of the returns of institutions A and B as functions and then calculate the conditional quantile.

  • ΔCoVaR(α): This is the difference between CoVaR(α|Y=VaR(α)) and CoVaR(α|Y=median(Y)). It captures the change in risk for institution A when institution B moves from its median state to its distressed state (VaR).

  • MES (Marginal Expected Shortfall): This measures the expected loss of an institution given that the system is in a distressed state. It requires modeling the joint distribution of all institutions in the system.

2. DFSRI and Transfer Entropy for Risk Contagion Networks

2.1 DFSRI-based Network Construction:

The Dynamic Financial Stress Risk Index (DFSRI) likely captures various aspects of financial stress (e.g., market volatility, credit spreads). The paper you're referencing probably uses the correlation or cross-correlation between the DFSRI series of different sectors to build the adjacency matrix of the risk contagion network.

1. Calculate DFSRI: Compute the DFSRI time series for each sector.

2. Correlation/Cross-correlation: Calculate the pairwise correlation or cross-correlation between the DFSRI series of different sectors. Cross-correlation can capture lead-lag relationships.

3. Adjacency Matrix: Use the correlation or cross-correlation values to define the connections (edges) between sectors (nodes) in the network. For example, you could set a threshold – if the correlation between two sectors exceeds the threshold, a link is established in the adjacency matrix. This creates the risk contagion network.

2.2 Transfer Entropy-based Network Construction:

Transfer entropy (TE) quantifies the directed information flow between two time series. It's used to infer causal relationships.

1. Calculate Transfer Entropy: For each pair of sectors (X and Y), calculate the transfer entropy TE(X→Y) and TE(Y→X). The formula for transfer entropy is:

   TE(X→Y) = Σ P(yₜ₊₁ , yₜ⁽ᵏ⁾, xₜ) log₂ [P(yₜ₊₁ | yₜ⁽ᵏ⁾, xₜ) / P(yₜ₊₁ | yₜ⁽ᵏ⁾)]

   where:

   • yₜ₊₁ is the future state of Y.
   • yₜ⁽ᵏ⁾ represents the past k states of Y.
   • xₜ is the current state of X.
   • P(.) represents probabilities.

2. Adjacency Matrix: Create the adjacency matrix of the network. The (i, j)-th entry represents the directed edge from sector i to sector j. You can use different approaches:

   • Thresholding: If TE(X→Y) exceeds a certain threshold, set the corresponding entry in the adjacency matrix to 1 (representing a directed link from X to Y). Otherwise, set it to 0.

   • Weighted Network: Use the TE values directly as the weights of the directed edges in the network. A higher TE value indicates a stronger causal influence.

Code Example (Conceptual - Transfer Entropy):

import numpy as np

def transfer_entropy(x, y, k=1):
  """Calculates transfer entropy from x to y."""
  # Implement the transfer entropy calculation here.  This is a simplified example
  # and requires a proper probability estimation method (e.g., kernel density estimation).
  # This is just to illustrate the general idea.
  # ... (complex probability estimation and summation) ...
  return te

# Example usage (assuming you have time series data for sectors)
sector_data = { 'SectorA': np.random.rand(100), 'SectorB': np.random.rand(100), 'SectorC': np.random.rand(100)}

adjacency_matrix = np.zeros((3, 3))

for i, sector1 in enumerate(['SectorA', 'SectorB', 'SectorC']):
  for j, sector2 in enumerate(['SectorA', 'SectorB', 'SectorC']):
    if i != j:
      te = transfer_entropy(sector_data[sector1], sector_data[sector2])
      adjacency_matrix[i, j] = te

print(adjacency_matrix)

This is a simplified example. Calculating transfer entropy accurately requires sophisticated probability estimation techniques and careful consideration of parameters like k (the history length). Libraries like JIDT can be helpful for computing transfer entropy. Also, the DFSRI-based network construction would involve calculating correlations and applying a threshold. Remember to adapt the code and methodology to your specific data and research question.