Critical Slowing Down as Early Warning Signals for Financial Crises? Cees Diks, Cars Hommes and Juanxi Wang Centre for Nonlinear Dynamics in Economics and Finance (CeNDEF) University of Amsterdam Tinbergen Institute Conference May 18-20, 2015
Crises are of all times Tulip mania 1637 Stock market crash, 1929 Financial crisis, 2008 Greek debt crisis, 2010 2 / 37
Jean-Claude Trichet: In the face of the crisis we felt abandoned by conventional tools. 3 / 37
4 / 37 Motivation FED coservatively estimated the cost of the financial crisis to be $ 14 trillion approx. one year of US yearly output (GDP)! about $ 2,000 per person on the planet! Aim to prevent/reduce such costs resulting from future crises Even if we lack insight necessary to build structural models, we may try to use time series methods to obtain early warning signals (EWS) for upcoming critical transitions
5 / 37 Motivation (continued) Market instability may develop endogenously by a self-organizing market process (Sornette et al., 1996, 1997,... ) Leverage cycles (Geanakoplos, 2009) This time is different (Reinhart & Rogoff, 2009) EWS based on critical slowing down have been applied successfully in other fields (physics, ecology, biology) to predict critical transitions Based on model-free (normal form) approach
6 / 37 Example of critical transitions in ecology 1 Caribbean coral reef collapse ( from Scheffer, 2009): Sudden collapse number of sea urchins Sudden increase of algal cover
7 / 37 Example of critical transitions in ecology 2 Greenhouse-icehouse transition: CaCO3(%) 0 20 40 60 80 4.0e+07 3.8e+07 3.6e+07 3.4e+07 3.2e+07 years before present Sudden increase of the percentage of Calcium Carbonate
Critical slowing down 8 / 37
9 / 37 Example (Dakos et al., 2008) CaCO3 (%) AR(1) coeff residuals 0.80 0.85 0.90 0 20 40 60 80 A B end of greenhouse Earth N=482 Kendall τ=0.83 (P<10-4 ) x10 6 40 38 36 34 32
10 / 37 Critical slowing down approach to EWS applicable to financial time series? How universal is the approach based on critical slowing down? Does it apply to financial systems? Main question: Can we use early warning signals based on detection of critical slowing down to detect critical transitions or market crashes in finance? Certain agent-based models with boundedly rational agents suggest the answer is yes (they show saddle-node bifurcations)
Experimental economics 11 / 37
12 / 37 Predicted and realised prices 65 Group 2 90 Group 4 65 Group 1 70 Price 55 Price 50 30 Price 55 Predictions 45 65 55 45 35 2 0-2 Predictions 10 90 70 50 30 10 30 0-30 Predictions 45 65 55 45 35 5 0-5 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 65 Group 5 75 Group 7 65 Group 6 Price 55 Price 65 55 Price 55 Predictions 45 65 55 45 35 2 0-2 Predictions 45 75 65 55 45 10 0-10 Predictions 45 65 55 45 35 5 0-5 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
13 / 37 Example: Agent-based model for housing market Estimated deviation from fundamental price, assuming a fundamental price-rent ratio price index 100 120 140 160 180 200 real house price fundamental real house price X 0.1 0.0 0.1 0.2 1970 1980 1990 2000 2010 Time 1970 1980 1990 2000 2010 Time Price index and estimated fundamental price (left) and relative difference (right)
14 / 37 Two types of agents Relatieve difference between price and fundamental price X t = H t /Q t 1. Twee rules of thumb : X e,1 X e,2 t+1 = φ 1 X t 1, 0 < φ 1 < 1(fund. revert.) t+1 = φ 2 X t 1, φ 2 > 1(fund. divert.) Fractions n 1,t and n 2,t, updated based on past performance π h,t 1 (realised profits) Price equation (in terms of deviations from fundamental price) X t = n 1,tφ 1 + n 2,t φ 2 X t 1 R + ᾱ
Estimated time dependent fractions (US) ts.union(yt, n1, ar1) ar1 0.90 0.95 1.00 1.05 1.10 n1 0.2 0.4 0.6 0.8 yt 0.1 0.0 0.1 0.2 1970 1980 1990 2000 2010 Time 15 / 37
Fancharts US 16 / 37
17 / 37 Saddle node bifurcations Bifurcation diagram of the model estimated for The Netherlands the interest rate being the control parameter NLD X 0.25 0.2 0.15 0.1 0.05 0 1.018 1.02 1.022 1.024 1.026 1.028 1.03 R Change in interest rate can induce critical transitions
18 / 37 Dynamics close to Critical Transition and Critical Slowing Down Continuous time dx = f (X, ρ)dt + g(x, ρ)dw Locally near fixed point Ornstein-Ühlenbeck process Fixed point loses stability if real part eigenvalue Jacobian matrix crosses the imaginary axis Discrete time X t+1 = h(x t, ρ) + s(x t, ρ)ε t+1 Locally near fixed point AR(1) process Fixed point loses stability if eigenvalue Jacobian approaches 1
19 / 37 Leading indicators of critical slowing down in literature autoregressive coefficient of AR(1) model: Held and Kleinen (2004) Measure the slowest relaxation time of the system and estimate its trend autocorrelation increases (but not necessarily to 1) variance increases as system approaches critical transition (Carpenter and Brock, 2006) Other indicators: Detrended fluctuation analysis indicator (Livina and Lenton 2007) Skewness (Guttal and Jayaprakash 2008, 2009) We follow Dakos, Scheffer,... (PNAS, 2008)
20 / 37 Methodology: Detrending Time Series To achieve stationarity filter out long term trends; we subtract a slow moving average from the data (z t = log p t ) Gaussian kernel (bandwith σ) Moving average G(x) = 1 2πσ e x2 2σ 2, MA j = N i=1 G(i j)z i N, i=1 G(i j) Residual Time Series y j = z j MA j.
Illustration methodology: 1987 Black Monday (S&P 500) 1987 Black Monday (S&P500) 5.4 5.5 5.6 5.7 5.8 Sliding Window 0.04 0.00 0.04 1987 1988 Residuals 0.0145 0.0155 0.0165 0.74 0.78 0.82 0.86 0.75 0.80 0.85 1987 1988 kendall tau = 0.718 (p= 0 ) 1987 1988 kendall tau = 0.685 (p= 0 ) 1987 1988 kendall tau = 0.259 (p= 0 ) 1987 1988 AR(1) ACF(1) Standard Deviation 21 / 37
22 / 37 Methodology: Leading indicators Autocorrelation at lag 1: AR(1) AR(1) model: Assume residuals can be modelled by linear stochastic system y j+1 = λy j + θη j+1 θη j+1 is a vector of random disturbances of std.dev. θ. OLS estimation AR(1) coefficient using sliding window w (half sample size N)
23 / 37 Methodology: Leading indicators Autocorrelation at lag-1: ACF first order sample autocorrelation coefficient (from Yule-Walker equations) ρ 1 = E[(y t µ)(y t+1 µ)] σ 2 y Increase of variance measured by standard deviation St.Dev. = 1 N (y t µ) N 1 2 t=1
24 / 37 Methodology: establishing trends Kendall tau: For each indicator observed over time, we quantify the trend using the nonparametric Kendall rank correlation coefficient τ (Dakos et al.,2008) τ represents the degree of concordance between two ordinal variables: τ = C D N, where C is the number of concordant pairs, D is the number of discordant pairs, N = n(n 1)/2 Range: 1.0 τ 1.0 (we expect τ > 0)
25 / 37 Financial time series Four financial crises: Black Monday, Asian Crises, Dot.com Bubble burst and 2008 Financial crisis. Crisis Year Data Sample Size (N) Black Monday 1987 S&P 500 200 Asian Crisis 1997 Hangseng 500 Dot.com 2000 NASDAQ 400 2008 Financial Crisis 2008 S&P 500 398 2008 Financial Crisis 2008 TED spread 280 2008 Financial Crisis 2008 VIX 80
26 / 37 1987 Black Monday (S&P 500) 1987 Black Monday (S&P500) 5.4 5.5 5.6 5.7 5.8 Sliding Window 0.04 0.00 0.04 1987 1988 Residuals 0.0145 0.0155 0.0165 0.74 0.78 0.82 0.86 0.75 0.80 0.85 1987 1988 kendall tau = 0.718 (p= 0 ) 1987 1988 kendall tau = 0.685 (p= 0 ) 1987 1988 kendall tau = 0.259 (p= 0 ) 1987 1988 Time (Days) AR(1) ACF(1) Standard Deviation
27 / 37 1997 Asia Crises (Hangseng) 1997 Asia Crises (Hangseng) 0.76 0.80 0.84 0.88 0.10 0.00 0.05 0.10 9.0 9.2 9.4 9.6 Sliding Window 1996 1997 1998 1996 1997 1998 kendall tau = 0.385 (p= 0 ) Residuals AR(1) 0.74 0.76 0.78 0.80 0.82 0.014 0.016 0.018 0.020 1996 1997 1998 kendall tau = 0.334 (p= 0 ) 1996 1997 1998 kendall tau = 0.462 (p= 0 ) ACF(1) Standard Deviation 1996 1997 1998 Time (Days)
28 / 37 2000 Dot.com Bubble Burst (NASDAQ) 2000 Dot.com Bubble Burst (NASDAQ) 0.10 0.00 0.05 0.10 7.0 7.4 7.8 8.2 0.65 0.70 0.75 Sliding Window 1999 2000 1999 2000 kendall tau = 0.501 (p= 0 ) Residuals AR(1) 0.65 0.70 0.75 1999 2000 kendall tau = 0.478 (p= 0 ) ACF(1) 0.025 0.027 0.029 1999 2000 kendall tau = 0.343 (p= 0 ) 1999 2000 Time (Days) Standard Deviation
29 / 37 2008 Lehman Brothers collapse (S&P 500) 2008 Lehman Brothers collapse (S&P 500) 6.6 6.8 7.0 7.2 Sliding Window 0.013 0.015 0.017 0.64 0.68 0.72 0.64 0.68 0.72 0.76 0.04 0.00 0.04 kendall tau = 0.503 (p= 0 ) kendall tau = 0.47 (p= 0 ) kendall tau = 0.354 (p= 0 ) 2008 2009 2008 2009 2008 2009 2008 2009 2008 2009 Time (Days) Residuals AR(1) ACF(1) Standard Deviation
30 / 37 2008 Lehman Brothers collapse (TED spread) 0.0 0.5 1.0 1.5 2008 Lehman Brothers collapse (TED spread) Sliding Window 0.10 0.12 0.14 0.65 0.75 0.85 0.65 0.75 0.85 0.3 0.1 0.1 0.3 kendall tau = 0.699 (p= 0 ) kendall tau = 0.705 (p= 0 ) kendall tau = 0.441 (p= 0 ) 2008 2009 2008 2009 2008 2009 2008 2009 Residuals AR(1) ACF(1) Standard Deviation 2008 2009 Time (Days)
2008 Lehman Brothers collapse (VIX) 2008 Lehman Brothers collapse (VIX) 6.6 6.8 7.0 7.2 15 log S&P 500 3.0 3.5 4.0 Sliding Window log VIX 0.15 0.00 0.10 Residuals 0.45 0.50 0.55 0.60 0.65 0.45 0.55 0.65 kendall tau = 0.463 (p= 0 ) kendall tau = 0.446 (p= 0 ) AR(1) ACF(1) 0.060 0.070 kendall tau = 0.149 (p= 0.175 ) Standard Deviation May Jul Sep Nov Jan Time (Days) 31 / 37
32 / 37 Naive results Possible early detection of critical transition for Black Monday, Asian Crisis and Dot.com Mixed results for 2008 financial crisis
33 / 37 Bootstrapped time series Generate surrogate time series to approximate the distribution. Bootstrap I: bootstrap residuals after detrending Bootstrap II: bootstrap log-returns log(p t /p t 1 ) = z t z t 1 of the original time series Bootstrap III: generate surrogate time series by fitting GARCH(1,1) log-returns y t = σ t ɛ t, σt 2 = ω + αy 2 t 1 + βσ2 t 1, t = 1,..., T where ɛ t is a white noise process with ɛ t N(0, 1)
GARCH Bootstrap Results GARCH(1,1) to S&P500 log-returns around Black Monday 1987 Histogram of Kendall s τ coefficients and of p-values of AR(1) of 1000 bootstrapped time series. The red lines indicate the trend statistic of the original residual records, τ, (N = 200) Histogram of p Frequency 0 200 600 0.0 0.2 0.4 0.6 0.8 1.0 p Histogram of kendall Frequency 0 10 20 30 40 1.0 0.5 0.0 0.5 1.0 kendall 34 / 37
35 / 37 Bootstrapped p-values (1000 bootstrap replications) Observed probabilities that the surrogate estimated trend is at least as high as the trend estimated for the original data (based on τ for AR(1))
36 / 37 Robustness of parameters 1987 Black Monday. Left: contour plots of τ as a function of the rolling window size and bandwidth Right: Histogram of τ-values Indicator AR1 Rolling Window Size 60 80 100 120 140 1.0 0.5 0.0 0.5 1.0 occurence 0 1000 2000 3000 50 100 150 200 Filtering Bandwidth 1.0 0.5 0.0 0.5 1.0 Kendall tau estimate
37 / 37 Concluding Remarks We find positive trends in autocorrelation and variance before Black Monday, the Asian Crisis and the Dot.com bubble burst, but not for the recent Financial Crisis Bootstrap methods show that the results for the Asian Crisis and Dot.com are not significant, but those for Black Monday 1987 remain significant (at 10% significance level) Financial crises and crashes may be harder to forecast than critical transitions in other systems Differences: The equilibrium price follows a process close to a unit root rather than a smooth ODE with some noise Saddle node shown in the ABM was for model in terms of deviations from fundamental price (not price itself) Multivariate dependence measures may be useful for assessing financial stress