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¨ osch FOM University of Applied Sciences, Munich, Germany Erhan Uluceviz Istanbul Bilgi University, Istanbul, Turkey Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 1 of 25
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¨ osch / Erhan Uluceviz 2 of 25
Introduction Measuring spillovers using fevds Shock propagation Speed of shock digestion Empirical findings Conclusions The series of daily returns dji 10 0 −10 ftse 10 0 −10 sx5e 10 0 −10 n225 10 0 −10 ssec 10 0 −10 1998 2000 2002 2004 2006 2008 2010 2012 Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 3 of 25
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¨ osch / Erhan Uluceviz 4 of 25
Introduction Measuring spillovers using fevds Fevds Shock propagation Spillover table & spillover index Speed of shock digestion Example Empirical findings The need for summary Conclusions Forecast error variance decomposition — four markets A , B , C , D . Forecast error variance (market A ; Φ = an irf): ǫ A , t + n − i n − 1 ǫ B , t + n − i � � � Φ A A ( i ) , Φ B A ( i ) , Φ C A ( i ) , Φ D var A ( i ) × ǫ C , t + n − i i =0 ǫ D , t + n − i This expression equals σ 2 A · � n − 1 i =0 (Φ A A ) 2 ( i ) σ 2 B · � n − 1 i =0 (Φ B A ) 2 ( i ) + + C · � n − 1 D · � n − 1 σ 2 i =0 (Φ C A ) 2 ( i ) σ 2 i =0 (Φ D A ) 2 ( i ) + Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 5 of 25
Introduction Measuring spillovers using fevds Fevds Shock propagation Spillover table & spillover index Speed of shock digestion Example Empirical findings The need for summary Conclusions 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¨ osch / Erhan Uluceviz 6 of 25
Introduction Measuring spillovers using fevds Fevds Shock propagation Spillover table & spillover index Speed of shock digestion Example Empirical findings The need for summary Conclusions Example: dji, ftse, sx5e, n225, ssec Spillover index series: spillover index 60 50 40 30 1998 2000 2002 2004 2006 2008 2010 2012 Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 7 of 25
Introduction Measuring spillovers using fevds Fevds Shock propagation Spillover table & spillover index Speed of shock digestion Example Empirical findings The need for summary Conclusions 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 . 8 0 . 1 0 . 1 0 . 1 0 . 6 0 . 3 0 . 4 0 . 5 0 . 1 0 . 1 0 . 3 0 . 6 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¨ osch / Erhan Uluceviz 8 of 25
Introduction Measuring spillovers using fevds A hypothetical shock hitting the network Shock propagation The size of a shock Speed of shock digestion The location of a shock Empirical findings Relative market entropy Conclusions 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 : 0 . . . 0 n 0 = 1 ← i -th component 0 . . . 0 What happens when such a shock hits the network? Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 9 of 25
Introduction Measuring spillovers using fevds A hypothetical shock hitting the network Shock propagation The size of a shock Speed of shock digestion The location of a shock Empirical findings Relative market entropy Conclusions The size of a shock Assumptions Given: A spillover matrix M t for day t Propagation of a shock within next day: initial shock size: n 0 (a unit vector) shock propagation in short time interval according to n s +1 = M t · n s , s = 0 , 1 , . . . Question: What happens to shock size n s as s → ∞ ? Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 10 of 25
Introduction Measuring spillovers using fevds A hypothetical shock hitting the network Shock propagation The size of a shock Speed of shock digestion The location of a shock Empirical findings Relative market entropy Conclusions 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: 0 . 6 0 . 2 0 . 2 0 . 8 0 . 1 0 . 1 (1 , 2 , 2) 0 . 1 0 . 6 0 . 3 (1 , 0 . 30 , 0 . 26) 0 . 4 0 . 5 0 . 1 0 . 1 0 . 3 0 . 6 0 . 3 0 . 2 0 . 5 I II Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 11 of 25
Introduction Measuring spillovers using fevds A hypothetical shock hitting the network Shock propagation The size of a shock Speed of shock digestion The location of a shock Empirical findings Relative market entropy Conclusions The location of a shock Can we use the spillover table as a Markov transition matrix? M t is row-stochastic: If p (column vector) is a probability distribution, then: M t · p need not be a distribution p ′ · M t is a distribution However, a Markov chain with p ′ s +1 = p ′ s · M t is running backward in time (relative to the setup of M t ). p : distribution of shock location; p ′ · M t : distribution of shock origin Time needs to be reversed. Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 12 of 25
Introduction Measuring spillovers using fevds A hypothetical shock hitting the network Shock propagation The size of a shock Speed of shock digestion The location of a shock Empirical findings Relative market entropy Conclusions The location of a shock: time reversal A Markov chain running forward in time can be defined for strongly connected networks. Transformation of M t into a forward Markov transition matrix (using the eigenvalue structure): V − 1 · M ′ t · V t t (Tuljapurkar, 1982) s · V − 1 A Markov chain with p ′ s +1 = p ′ · M ′ t · V t is running forward in t time. Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 13 of 25
Introduction Measuring spillovers using fevds A hypothetical shock hitting the network Shock propagation The size of a shock Speed of shock digestion The location of a shock Empirical findings Relative market entropy Conclusions 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 M t . Dual interpretation of propagation values! Market connectedness. . . relative market entropy — Harald Schmidbauer / Angi R¨ osch / Erhan Uluceviz 14 of 25
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