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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions Market connectedness: spillovers, information flow, and relative market entropy Ko c University, March 26, 2014

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

Market connectedness: spillovers, information flow, and relative market entropy

Ko¸ c University, March 26, 2014

Harald Schmidbauer

Istanbul Bilgi University, Istanbul, Turkey

Angi R¨

  • sch

FOM University of Applied Sciences, Munich, Germany

Erhan Uluceviz

Istanbul Bilgi University, Istanbul, Turkey Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

  • sch / Erhan Uluceviz

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

Market connectedness and fevd

Assessing the degree of connectedness of equity markets Diebold & Yilmaz, 2005–2014: VAR model for return series forecast error variance decomposition (fevd) spillover table collapsed into the spillover index Our contribution: characterization of dynamic behavior a Markov chain perspective application of entropy measures

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

The series of daily returns

−10 10

dji

−10 10

ftse

−10 10

sx5e

−10 10

n225

1998 2000 2002 2004 2006 2008 2010 2012 −10 10

ssec

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

Scatterplots of daily returns

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions Fevds Spillover table & spillover index Example The need for summary

Forecast error variance decomposition — four markets A, B, C, D.

Forecast error variance (market A; Φ = an irf):

var       

n−1

  • i=0
  • ΦA

A(i), ΦB A(i), ΦC A(i), ΦD A(i)

  • ×

    ǫA ,t+n−i ǫB ,t+n−i ǫC ,t+n−i ǫD ,t+n−i           

This expression equals

σ2

A · n−1 i=0 (ΦA A)2(i)

+ σ2

B · n−1 i=0 (ΦB A)2(i)

+ σ2

C · n−1 i=0 (ΦC A)2(i)

+ σ2

D · n−1 i=0 (ΦD A)2(i) Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

  • sch / Erhan Uluceviz

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions Fevds Spillover table & spillover index Example The need for summary

Spillover table & spillover index

Spillover table, schematically: from (time t) A B C D A

  • B
  • to (time t + n)

C

  • D
  • Spillover index =

+ (Diebold & Yilmaz, 2005)

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

  • sch / Erhan Uluceviz

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions Fevds Spillover table & spillover index Example The need for summary

Example: dji, ftse, sx5e, n225, ssec

Spillover index series:

30 40 50 60 1998 2000 2002 2004 2006 2008 2010 2012 spillover index

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions Fevds Spillover table & spillover index Example The need for summary

The need for summary

For each day: the procedure yields an n × n table. Spillover index: summary measure. If spillover index = 40%, what is the spillover table? (3 markets) . . . this? . . . or this?   0.6 0.2 0.2 0.1 0.6 0.3 0.1 0.3 0.6     0.8 0.1 0.1 0.4 0.5 0.1 0.3 0.2 0.5   I II “Average” spillover of shocks to other markets: 40%!

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

  • sch / Erhan Uluceviz

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

A hypothetical shock hitting the network

Spillover matrix: most recent information available for a day defines weights in a network Hypothetical shock to node (or market) i on day t:

n0 =             . . . 1 . . .             ← i-th component

What happens when such a shock hits the network?

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

  • sch / Erhan Uluceviz

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

The size of a shock

Assumptions Given: A spillover matrix Mt for day t Propagation of a shock within next day:

initial shock size: n0 (a unit vector) shock propagation in short time interval according to ns+1 = Mt · ns, s = 0, 1, . . .

Question: What happens to shock size ns as s → ∞?

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

The size of a shock

It holds that: The relative size of a shock, as s → ∞, is determined by the left eigenvector of the spillover matrix. left eigenvector: value of a shock to which the market is exposed as seed for future variability or risk (“propagation value”) Example:

(1, 2, 2)   0.6 0.2 0.2 0.1 0.6 0.3 0.1 0.3 0.6   (1, 0.30, 0.26)   0.8 0.1 0.1 0.4 0.5 0.1 0.3 0.2 0.5   I II

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

The location of a shock

Can we use the spillover table as a Markov transition matrix? Mt is row-stochastic: If p (column vector) is a probability distribution, then: Mt · p need not be a distribution p′ · Mt is a distribution However, a Markov chain with p′

s+1 = p′ s · Mt is running backward

in time (relative to the setup of Mt). p: distribution of shock location; p′ · Mt: distribution of shock origin Time needs to be reversed.

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

The location of a shock: time reversal

A Markov chain running forward in time can be defined for strongly connected networks. Transformation of Mt into a forward Markov transition matrix (using the eigenvalue structure): V−1

t

· M′

t · Vt

(Tuljapurkar, 1982) A Markov chain with p′

s+1 = p′ s · V−1 t

· M′

t · Vt is running forward in

time.

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

The location of a shock: distributional characteristics

If a shock hits a node (market, asset) of the network on day t: Where will the (hypothetical) shock be settling? Share of time spent in each node of the network? Stationary distribution of the Markov chain? It can be shown that: The stationary probability distribution of the Markov chain running forward in time equals the (normed) left eigenvector of Mt. Dual interpretation of propagation values!

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

Markov chain interpretation: two perspectives

Single-day perspective: Information flow according to a single Mt? Emphasizes what happens on a day in isolation. Sensitive to any change between days. Day-to-day perspective: Shock in the (remote) past. (Mt)t=0,1,... defines a non-homogeneous Markov chain. Emphasizes what happens from day to day. Sensitive to abrupt changes only.

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

Relative market entropy

Several probability distributions are associated with each day: initial shock distribution (a unit vector) stationary distribution of shock position non-stationary distribution of shock position Informational distance between distributions? Relative market entropy: Kullback-Leibler distance, defined as KLIC =

  • i

πa(i) · log2 πa(i) πb(i)

  • with (for example)

πa = today’s (non-) stationary distribution, πb = yesterday’s (non-) stationary distribution

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

Entropy & KLIC

Given: random variable X with distribution P random variable Y with distribution Q Entropy of X: H(X) = −

  • x

P(x) · log2 P(x) Kullback-Leibler divergence (KLIC) of (false) Q from (true) P: DKL(P Q) =

  • x

P(x) · log2 P(x) Q(x)

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

Example: entropy

Given: random variable X and its distribution P A B C D X, P 1/2 1/4 1/8 1/8 Suppose you know the support {A, B, C, D} and P. Average number of (“clever”) guesses required to identify a realization: 1 · 1 2 + 2 · 1 4 + 3 · 1 8 + 3 · 1 8 = 1.75 This equals the entropy of X: H(X) = −1 2 · log2 1 2 − 1 4 · log2 1 4 − 1 8 · log2 1 8 − 1 8 · log2 1 8 Case of a uniform distribution: H(X) = 2.

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions A hypothetical shock hitting the network The size of a shock The location of a shock Relative market entropy

Example: KLIC

Given: random variables X, Y , their respective distributions P, Q A B C D X, P 1/2 1/4 1/8 1/8 Y , Q 1/8 1/8 1/4 1/2 Suppose you know the support {A, B, C, D} and Q, but not P. Average number of guesses required to identify a realization of X when using the coding for Y : 1 · 1 8 + 2 · 1 8 + 3 · 1 4 + 3 · 1 2 = 2.625 > 1.75 The difference equals the Kullback-Leibler divergence of (false) Q from (true) P: DKL(P Q) = 1 2 · 2 + 1 4 · 1 − 1 8 · 1 − 1 8 · 2 = 0.875

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions Kolmogorov-Sinai entropy Time-varying characteristics

Kolmogorov-Sinai entropy

Idea: The system will converge to equilibrium after being hit by a shock. How fast will it converge? Appropriate measure: Kolmogorov-Sinai entropy, KS = −

  • i,j

π(i) · log2

  • ppij

ij

  • ,

where π = stationary distribution, pij = entries of the transition matrix. Can be seen as a measure of network stability.

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions Kolmogorov-Sinai entropy Time-varying characteristics

Time-varying characteristics

Algorithm: Given a spillover matrix, find a VAR process with this spillover matrix. Connect processes with different properties together. Tool to study joint spillover / speed of convergence behavior. Does an increase in spillover imply an increase in speed of convergence? — No!

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions Kolmogorov-Sinai entropy Time-varying characteristics

Time-varying characteristics

spillover index (percent) KS entropy KLIC

200 400 600 800 1000 40 42 44 46 48 50 200 400 600 800 1000 1.20 1.25 1.30 1.35 1.40 1.45 1.50 200 400 600 800 1000 0.00000 0.00010 0.00020 0.00030

scenario 1 spillover index (percent) KS entropy KLIC

200 400 600 800 1000 40 42 44 46 48 50 200 400 600 800 1000 1.20 1.25 1.30 1.35 1.40 1.45 1.50 200 400 600 800 1000 0.00000 0.00010 0.00020 0.00030

scenario 2 Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

Progagation values

0.0 0.1 0.2 0.3 0.4 0.5 0.6 1998 2000 2002 2004 2006 2008 2010 2012 dji ftse sx5e n225 ssec propagation values

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

Relative entropy, different shock origins (hypothetical)

2 4 6 8 10 1998 2000 2002 2004 2006 2008 2010 2012 dji ftse sx5e n225 ssec

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

Relative entropy, stationary distributions (actual)

0.00 0.05 0.10 0.15 0.20 0.25 1998 2000 2002 2004 2006 2008 2010 2012

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

Relative entropy, non-stationary distributions (actual)

0.000 0.005 0.010 0.015 0.020 0.025 1998 2000 2002 2004 2006 2008 2010 2012

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

Kolmogorov-Sinai entropy

0.8 1.0 1.2 1.4 1.6 1.8 2.0 KS entropy 1998 2000 2002 2004 2006 2008 2010 2012

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Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions

The spillover approach lends itself to further network-related techniques. New tools can reveal further details in information flow. An increase in spillover index does not necessarily imply an increase in network stability.

Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨

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