NCoVaR Granger Causality

Transcription

1 NCoVaR Granger Causality Cees Diks 1 Marcin Wolski 2 1 Universiteit van Amsterdam 2 European Investment Bank Bank of Italy Rome, 26 January 2018 The opinions expressed herein are those of the authors and do not necessarily reflect those of the EIB.

2 Introduction Complexity within the financial system emphasised again by the crisis: Linkages and risk exposure again proved to be significant for transmitting distress across the system Both directly through mutual risk exposures and indirectly via price effects/liquidity spirals Tail co-movements during times of financial distress Strong negative adverse effects on real economy: Individual risk exposures risk of disrupting entire financial system (systemic risk) important to measure and monitor systemic risk + individual + mutual risk exposures

3 Introduction (cntd) Majority of approaches focus on co-risk measures (intuition: externalities of excessive risk taking + leverage) Growing body of literature on CoVaR (Adrian and Brunnermeier, 2016) using (linear) quantile regression Linear approach susceptible to model misspecification E.g. presence of nonlinearities Contribution of institution to systemic risk linear in default probability but nonlinear w.r.t. institution size and asset correlation (Huang et al., 2010) Also supported by empirical evidence (He, 2012) Xiao and Hua (2012) designed an EWS based on neural networks, exploiting nonlinearity

4 Introduction (cntd) CoVaR, when based on linear quantile regression: is essentially linear (correlation-type dependence) does not distinghuish between direct causal effects and common factor effects Causality important for network analysis (contagion, spillover effects, cascades, shock propagation) We extend CoVaR in two directions Take a nonparametric approach based on (conditional) tail probabilities rather than quantiles (U-statistics) Focus on causal links X Y (higher-variate problem) between one institution and another or between an institution and the system

5 Unconditional VaR CoVaR P ( Y VaR Y q ) = q Conditional VaR (CoVaR) ( ) P Y CoVaR Y C(X) q C(X) = q Dependence measure based on CoVaR CoVaR Y q = CoVaR Y X=VaRX q q CoVaR Y X=VaRX 0.5 q Obtained using quantile regression a tail dependence measure, but essentially linear in the Gaussian setting equivalent to covariance/correlation

6 Nonparametric CoVaR (NCoVaR) Idea related to CoVaR, but focus on (tail) event probabilities rather than quantiles Let A denote an extreme event for Y, and C and D be events for X near the γ and 0.5 quantiles of X, respectively, e.g. A = [y γ µ, y γ + µ] (or (, y γ ]), Y near (or below) tail quantile C = [x γ µ, x γ + µ] (or (, x γ ]) X near (or below) tail quantile D = [x 0.5 µ, x µ], X near median Then study NCoVaR = P(Y A X C) P(Y A X D)

7 Nonparametric CoVaR (NCoVaR) mechanics Y (st. returns) X (st. returns) X D Y A 0.1 X C µ-region around y γ µ-region around x γ NCoVaR =P(Y A X C) P(Y A X D)

8 Nonparametric CoVaR (NCoVaR) mechanics Y (st. returns) X (st. returns) X D Y A 0.1 X C µ-region around y γ µ-region around x 0.5 NCoVaR = P(Y A X C) P(Y A X D)

9 Rewrite Nonparametric CoVaR (NCoVaR) cntd. NCoVaR = P (Y A X C) P (Y A X D) = P (Y A, X C) P (Y A, X D) P (X C) P (X D) The null hypothesis of no (instantaneous) NCoVaR relation from variable X onto Y ( NCoVaR = 0), implies q P (Y A, X C) P (X D) P (Y A, X D) P (X C) = 0 Or equivalently E [I A C (Y 1, X 1 )I D (X 2 ) I A D (X 1, Y 1 )I C (X 2 )] = 0 A natural estimator of, for instance P(X 2 D) = E[I D (X 2 )] is by counting the frequency of events X k D among the observed X 1,..., X n.

10 NCoVaR estimator Consider a sample of a random process {(X t, Y t )}, t = 1,..., n One can estimate q as 1 ˆq n = n(n 1) [I A C (Y k, X k )I D (X l ) I A D (Y k, X k )I C (X l )] l k To develop asymptotics, define W k = (X k, Y k ) and write the estimator as a weighted average of a symmetric kernel function (as a U-statistic) where ˆq n = 1 n(n 1) K(W k, W l ), k l K(W k, W l ) = 1 2 [I A C(Y k, X k )I D (X l ) I A D (Y k, X k )I C (X l ) +l k]

11 Asymptotic normality Consider a strictly stationary bivariate random process {(X t, Y t )} with t Z Then for fixed events A, C and D, nˆq n q ˆσ(ˆq n ) d N(0, 1), where ˆσ 2 (ˆq n ) is a consistent (HAC) estimator of the asymptotic variance of n(ˆq n q)

12 Granger non-causality Definition (Granger non-causality (bivariate)) For a strictly stationary bivariate time series process {(X t, Y t )}, t Z, {X t } does not Granger cause {Y t } if, for all k 1, (Y t+1,..., Y t+k ) (F X,t, F Y,t ) (Y t+1,..., Y t+k ) F Y,t. F X,t and F Y,t : info contained in X s, s t and Y s, s t, respectively We focus on k = 1, so the condition simplifies to Y t+1 (F X,t, F Y,t ) Y t+1 F Y,t. H 0 : {X t } {Y t }, for one lag, let (X, Y, Z) (X t, Y t, Y t+1 ), then H 0 becomes that X and Z are conditionally independent given Y.

13 NCoVaR Granger causality (X, Y, Z) (X t, Y t, Y t+1 ), H 0 : X and Z are cond. indep. given Y. A = [z γ µ, z γ + µ] (or (, y γ ]), Z near (or below) tail quantile C = [x γ µ, x γ + µ] (or (, x γ ]) X near (or below) tail quantile D = [x 0.5 µ, x µ], X near median Implies P(Z A, X C Y = y) P(X C Y = y) = So one might want to test whether P(Z A, X D Y = y) P(X D Y = y) A, C, D, y P(Z A, X C Y = y )P(X D Y = y ) P(Z A, X D Y = y )P(X C Y = y ) = 0 for a given past Y t value y (e.g. some unconditional Y-quantile) P(X D Y = y ), for instance, is now estimated by counting the frequency of events X D, among the vectors close to y.

14 NCoVaR Granger causality mechanics Y (st. returns) X (st. returns) X t D Y t = y y Y t+1 A 0.1 X t C µ-region around y γ µ-region around x γ P(Y t+1 A, X t C Y t = y ) = P(Y t+1 A, X t D Y t = y ) P(X t C Y t = y ) P(X t D Y t = y ) A, C, D, y

15 NCoVaR Granger causality mechanics Y (st. returns) X (st. returns) X t D Y t = y y Y t+1 A 0.1 X t C µ-region around y γ µ-region around x 0.5 P(Y t+1 A, X t C Y t = y ) P(X t C Y t = y ) = P(Y t+1 A, X t D Y t = y ) P(X t D Y t = y ) A, C, D, y

16 Estimation considerations A natural estimator for P(X D Y = y ) is the Nadaraya-Watson nonparametric regression function estimator P(X D Y = y ) = 1 n n k=1 I D(Z k )K h (y Y k ) 1 n n k=1 K h(y Y k ) where we take K h ( ) to be the Gaussian kernel K h (s) = 1 2πs exp( s 2 /(2h 2 )), But the denominator is just a kernel density estimate of f Y (y )! We can get rid of the denominator and obtain simple U-statistics estimates if we multiply the probabilities by f Y (y )

17 Estimation considerations (cntd.) Therefore, for a given quantile y of Y we define q(y ) = fy 2(y ) (P(Z A, X C Y = y )P(X D Y = y ) P(Z A, X D Y = y )P(X C Y = y )) By construction, q(y ) = 0 under H 0 Now the term e.g. can be simply estimated as f Y (y )P(X D Y = y ) 1 n n I D (Z k )K h (y Y k ) k=1

18 U-statistic kernel for estimating NCoVaR Granger causality We use a Gaussian kernel for estimating the marginal density f Y (y ) where y is taken to be either the unconditional median y 0.5 of Y, or the unconditional quantile y γ in the left tail of Y Then the U-statistic kernel used for estimation of q(y ) is K(W k, W l ; h) = 1 2 [I A(Z k )I C (X k )K h (y Y k )I D (X l )K h (y Y l ) I A (Z k )I D (X k )K h (y Y k )I C (X l )K h (y Y l ) +k l] (1)

19 Asymptotic normality Consider a strictly stationary bivariate random process {(X t, Y t )} with t Z Then for given the events A, C and D, and Y-quantile y, for a kernel density bandwidth parameter tending to zero at an appropriate rate nˆq n (y ) q(y ) ˆσ(ˆq n (y )) d N(0, 1), where ˆσ 2 (ˆq n (y )) is a consistent (HAC) estimator of the asymptotic variance of n(ˆq n (y ) q(y ))

20 DGPs VAR model context X t = ax t a 2 ε 1,t, Y t = ax t τ + 1 a 2 ε 2,t ARCH model context (to mimick volatility spillover) The lag τ is either set to X t N(0, 1), Y t N(0, 1 + axt τ 2 ), τ = 0 for instantaneous Granger causality (for simulating size/power for CoVaR and NCoVaR), or τ = 1 for Granger causality (for simulating size/power for NCoVaR Granger causality)

21 Bandwidth selection Minimise the MSE of ˆq n (y ) MSE optimal bandwidth for the Nadaraya-Watson estimator for f Y (y ) Bias/variance trade-off VAR model context h AR(1) opt = n 1/3. ARCH model context (bias of lower order in kernel bandwidth) h ARCH(1) opt = n 1/5. (after standardizing the data to zero mean and unit variance)

22 Simulated size/power for data from the VAR(1) model Rejection rates n=100 n=200 n=500 n=1000 Rejection rates n=100 n=200 n=500 n=1000 Size Size Rejection rates n=100 n=200 n=500 n=1000 Rejection rates n=100 n=200 n=500 n=1000 Size Size CoVaR NCoVaR a = 0.4, x γ = x 0.05, y = y 0.5, z γ = z 0.05 (1,000 replications)

23 Simulated size/power for data from the VAR(1) model Rejection rates n=100 n=200 n=500 n=1000 Rejection rates n=100 n=200 n=500 n=1000 Size Size Rejection rates n=100 n=200 n=500 n=1000 Rejection rates n=100 n=200 n=500 n=1000 Size Size CoVaR GC NCoVaR GC a = 0.4, x γ = x 0.05, y = y 0.5, z γ = z 0.05 (1,000 replications)

24 Simulated size/power for data from the ARCH(1) model Rejection rates n=100 n=200 n=500 n=1000 Rejection rates n=100 n=200 n=500 n=1000 Size Size Rejection rates n=100 n=200 n=500 n=1000 Rejection rates n=100 n=200 n=500 n=1000 Size Size CoVaR NCoVaR a = 0.4, x γ = x 0.05, y = y 0.5, z γ = z 0.05 (1,000 replications)

25 Simulated size/power for data from the ARCH(1) model Rejection rates n=100 n=200 n=500 n=1000 Rejection rates n=100 n=200 n=500 n=1000 Size Size Rejection rates n=100 n=200 n=500 n=1000 Rejection rates n=100 n=200 n=500 n=1000 Size Size CoVaR GC NCoVaR GC a = 0.4, x γ = x 0.05, y = y 0.5, z γ = z 0.05 (1,000 replications)

26 Multivariate problems Assume we have an extra conditioning variable, call it Q, satysfying the regularity conditions. Then for all q NCoVaR (becomes quasi NCoVaR GC) P(Y A X C, Q = q ) = P(Y A X D, Q = q ) P(Y t+1 A, X t C Q = q ) P(X t C Q = q ) = P(Y t+1 A, X t D Q = q ) P(X t D Q = q ) NCoVaR Granger causality (non-trivial asymptotics) P(Y t+1 A, X t C Y t = y, Q = q ) P(X t C Y t = y, Q = q ) = P(Yt+1 A, Xt D Yt = y, Q = q ) P(X t D Y t = y, Q = q )

27 Non-trivial asymptotics explained Estimator of P(X D Y = y, Q = q ) is the Nadaraya-Watson estimator P(X D Y = y, Q = q ) = Asymptotics is driven by bandwidths 1 n n k=1 I D(Z k )K h (y Y k )K h (q Q k ) 1 n n k=1 K h(y Y k )K h (q Q k ) take h = Cn β n β and r being bias order of kernel K h, i.e. f Eˆf h r + o(h r ) consistency of the estmator achieved only under 1 2r < β < 1 2(d Y +d Q ) increased dimensionality has to be supported by higher precision of the estimates curse of dimensionality

28 Data empirical application Euro area Member States: Germany and France Five vulnerable Member States: Spain, Portugal, Italy, Ireland and Greece. The Sovereign Price Index (SPI) is calculated from the price-yield relation of a 1-year zero-coupon bond, based on a generic 1-year sovereign bond yield for each country The Banking Price Index (BPI) is taken as the FTSE banking price index for each country SPI obtained from Bloomberg, and BPI obtained from Datastream. Time spans in range 03/01/ /10/2016 ( observations) Time series I(1) without cointegration analysed in log-differences

29 Time series (log returns) DE ES IT GR Dlog_sov_DE Dlog_sov_ES Dlog_sov_IT Dlog_sov_GR Time Time Time Time Dlog_bank_DE Time FR Dlog_sov_FR Time Dlog_bank_FR Dlog_bank_ES Time PT Dlog_sov_PT Time Dlog_bank_PT Dlog_bank_IT Time IE Dlog_sov_IE Time Dlog_bank_IE Dlog_bank_GR Time Time Time Time

30 Scatter plots (log returns) Dlog_sov_DE Dlog_bank_DE DE Dlog_sov_ES Dlog_bank_ES ES Dlog_sov_PT Dlog_bank_PT PT Dlog_sov_FR Dlog_bank_FR FR Dlog_sov_IT Dlog_bank_IT IT Dlog_sov_IE Dlog_bank_IE IE Dlog_sov_GR Dlog_bank_GR GR

31 Empirical results Table: Bank-sovereign feedback loops in selected euro area countries. BPI and SPI denote the Banking Price Index and Sovereign Price Index, respectively. ***, **, * denote 1%, 5% and 10% significance levels. For nonparametric NCoVaR and NCoVaR Granger causality tests we set µ = 0.8. Risky quantiles are estimated at γ = CoVaR NCoVaR CoVaR Gc NCoVaR Gc X Y X Y Y X X Y Y X X Y Y X X Y Y X Germany BPI SPI *** France BPI SPI *** Spain BPI SPI *** ** *** *** ** *** ** Italy BPI SPI *** *** ** *** *** * Portugal BPI SPI *** *** *** *** Ireland BPI SPI *** *** Greece BPI SPI *** ** ***

32 Empirical results - financial crisis Table: Bank-sovereign feedback loops in selected euro area countries for the financial crisis sample between February 27, 2007 and April 13, ***, **, * denote 1%, 5% and 10% significance levels. For nonparametric NCoVaR and NCoVaR Granger causality tests we set µ = 0.8. Risky quantiles are estimated at γ = CoVaR NCoVaR CoVaR Gc NCoVaR Gc X Y X Y Y X X Y Y X X Y Y X X Y Y X Germany BPI SPI ** France BPI SPI ** ** * Spain BPI SPI * ** Italy BPI SPI *** Portugal BPI SPI *** *** ** Ireland BPI SPI * *** Greece BPI SPI *** * *** **

33 Empirical results - sovereign debt crisis Table: Bank-sovereign feedback loops in selected euro area countries for the sovereign debt crisis sample between October 4, 2009 and May 30, ***, **, * denote 1%, 5% and 10% significance levels. For nonparametric NCoVaR and NCoVaR Granger causality tests we set µ = 0.8. Risky quantiles are estimated at γ = CoVaR NCoVaR CoVaR Gc NCoVaR Gc X Y X Y Y X X Y Y X X Y Y X X Y Y X Germany BPI SPI France BPI SPI Spain BPI SPI *** *** *** *** * * Italy BPI SPI *** *** *** *** ** Portugal BPI SPI *** *** * * * *** Ireland BPI SPI ** ** * Greece BPI SPI * **

34 Summary/Conclusions We extended CoVaR in directions of nonparametric risk dependence measure (NCoVaR) involving lagged variables (NCoVaR Granger causality) NCoVaR and NCoVaR Granger causality lead to different size/power and empirical results/interpretations Future work includes fine-tuning the exact events to focus on (e.g. near/below which quantiles) data-driven optimal kernel density bandwidth selection shed more light on multivariate applications