UvA-DARE ( Digital Academic Repository ) Critical slowing down as an early warning signal for financial crises

The global impact of the recent financial crisis has once more stressed the urgency of new approaches to designing early warning signals (EWS) for financial crises. In the recent literature on constructing EWS through identifying characteristics of critical slowdown on the basis of time series observations, finance has repeatedly been coined as an important potential application area. On the one hand, this appealing idea is supported by the fact that there is ample empirical and experimental evidence to suggest that nonlinearities play a role in the expectations feedback governing market dynamics. On the other hand, financial markets differ from many natural complex systems, for which evidence of critical slowing down has been reported, in that market dynamics are not necessarily captured well by an ordinary differential equation, the fixed point of which may lose stability through a saddle-node bifurcation, as is the case for the cusp catastrophe. Also, financial time series exhibit persistent near unit root behaviour. In this paper we consider a number of historical financial crises, to investigate whether there is indeed evidence for critical slowing down prior to market collapses. The four events considered are Black Monday 1987, the 1997 Asian Crisis, the 2000 Dot.com bubble burst, and the 2008 Financial Crisis. Our analysis shows evidence for critical slowing down before Black Monday 1987, while the results are mixed and insignificant for the other financial crises. ∗J.Wang@uva.nl, CeNDEF, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam.


Introduction
Several researchers have recently applied complexity tools, and already demonstrated their value in economics and finance (?, ?; ?, ?). Applying complexity approaches to financial systems can help deepen our understanding on the dynamic behaviour of financial markets and (parts of) the economy. In this paper, we will apply concepts from complexity theory to real financial data related to historical financial crises. We will also investigate, for the first time, whether the crises considered can be linked to critical transitions of the corresponding financial system. In particular, we will investigate whether in principle it is possible to develop an early warning signal based on a number of indicators for different types of historical financial crisis.
The exploration of indicators of critical transitions in complex systems has been quite fruitful in other disciplines. Important signals that have been suggested in the complexity literature as an early warning indicator are measures of "Critical slowing down". This approach is based on the slowing down of the dynamics of a complex system approaching a critical point. Several authors developed methods to extract signals of critical slowing down from time series data. ? observed that power spectral properties changed as the earth system moved closer to a bifurcation point in a hemispheric thermohaline circulation (THC) model. ? used a trend in the decay rate of climate sub-systems as an indicator of critical slowing down. The first-order autoregressive coefficient obtained from time series was used as a measure of the decay rate of the system. This method was applied to the North Atlantic THC model, providing an early warning signal for the climate system. ? developed another way of detecting critical slowing down by using detrended fluctuation analysis (DFA). This analysis was originally developed by ? to detect DNA sequences' long-range correlations. ? found an early warning signal for an upcoming critical transition in the North Atlantic THC system by investigating model output, as well as Greenland ice core paleotemperature data. The subsequent work by ? was the first to study critical slowing down in empirical time series. The results showed increased autocorrelation in eight climate time series prior to critical transitions. ? contributed to the early warning signal literature by offering some general guidelines for choosing the parameter settings of the analysis.
They improved the robustness of ACF and DFA techniques and gave additional examples showing evidence of critical slowing down in both palaeodata and climate model output. Recently, ? expanded the theory of critical slowing down to a broader class of situations where a system becomes increasingly sensitive to perturbations even without catastrophic transitions. They showed that critical slowing down could even be used in a more general sense as an early warning signal.
The possibility of applying early warning tools to financial data was suggested by ? (?, ?). Encouraged by the successes of early warning signals in many complex systems, some efforts have been made to explore the possibility of contructing early warnings from financial time series. For instance, by using information dissipation length (IDL) as an indicator, ? detected early warning signals prior to Lehman Brothers collapse in both USD and EUR interest rate swaps (IRS). They suggested that the IDL may be used as an early warning signal for critical transitions. ? observed critical slowing down in the U.S. housing market. They detected strong early warning signals associated with a sequence of coupled regime shifts during the period of subprime mortgage loans transition and the sub-prime crisis. They also found weaker signals during the Asian financial crisis and technology bubble crisis. However, up until now, no evidence of early warning signals has been found in time series data of stock markets. To fill this gap, this paper is, to our best knowledge, the first attempt to apply complexity theory based on "critical slowing down" to real financial data. Four financial crises are analysed: Black Monday 1987, the 1997 Asian Crisis, the Dot-com Bubble in 2000 and the 2008 Financial Crisis. The sample lag-1 autocorrelation and the variance are considered as early warning indicators to examine whether financial systems slow down before the critical point is reached. This paper is organized as follows. In Section 2 we provide some theoretical background of nonlinear dynamical systems and bifurcations underlying critical transitions.
We describe the data and EWS methodology in Section 3 and 4, respectively. Subsequently, we analyse how well such early warning indicators perform when applied to financial time series. The results of our analyses are presented and discussed in Section 5. Section 7 provides a summary and conclusions.

Theory
The mechanism driving critical transitions in complex systems mathematically is a bifurcation, which is an abrupt qualitative change in the dynamical systems when one or more control parameters change. ? reviewed different types of bifurcations in dissipative dynamical systems. Three categories are classified based on their continuous or discontinuous dependence on the control parameters. These categories consist of safe bifurcations, with continuous growth of a new stable attractor, explosive bifurcations, with a discontinuous growth to a newly enlarged attractor along with itself, and dangerous bifurcations, in which the current attractor simply disappears, forcing the system to jump in a fast dynamic transient to a remote and entirely new attractor.  Figure 2 provides an example of a saddle-node bifurcation. With the increase of a single control parameter, a critical point is approached. Even a small perturbation would lead to a large qualitative change of the dynamics when the system is very close to this critical point. Once this threshold is exceeded, the whole system transits toward a different attractor. Even if the control parameter is reversed, the response will typically remain close to the new attractor. The dynamics will not automatically return to the original attractor. This highlights the irreversibility of critical transitions. ? (?, ?) proposed that this bifurcation can also be used to describe the dynamical behaviour of financial systems, systematic crashes in stock markets for instance. This suggests the possible use of applying complexity theory to financial systems 1 .
To develop tools to forecast critical transitions based on time series, it is necessary to assume that the observed time series is generated by a rather general nonlinear dynamical system, driven by white noise, with control parameter ρ: where X t is the state of the system, F(X t , ρ) specifies the deterministic part of the system, while G(X t , ρ)ε t is the stochastic part with ε t a white noise process. The equilibrium of the undisturbed system is stable in all directions while approaching the critical value.
By varying the control parameter ρ, the system reaches a threshold. When a real eigenvalue of the Jacobian matrix DF ρ (X) of the steady state finally crosses +1 (with the other real eigenvalue smaller than 1 in absolute value), a saddle-node bifurcation occurs. This bifurcation is corresponding with a critical transition in a time series. This scenario is descried in detail in ?, ? and ?. It offers ways to provide early warnings before the critical transition actually happens. As long as we understand the statistical properties of the system approaching a critical transition, we may predict the time of the transition in advance, up to some estimation uncertainty.
Several earlier attempts have been made to develop an early warning system to monitor the risk of financial systems, such as binomial/multinomial logit/probit models (?; ?; ?), multivariate probability models (?), Markov switching models (?; ?), binomial tree approaches (?), and so on. However, their failure to give warnings ahead of the financial crisis in 2008 makes their predictive ability questionable. At the same time, the exploration of indicators of critical transitions in complex systems has been quite successful in other disciplines. Important signals suggested in the literature as early warning indicators are related to critical slowing down; when a dynamical system approaches a critical point, we can expect it to become increasingly slow in recovery from small perturbations. This is characterised by the linear decay rate decreasing to zero.
The theory of critical slowing down used to describe this phenomenon, is illustrated in Figure 2. Panels a, b and c of Fig. 2 show the behaviour of a dynamical system approaching a saddle-node bifurcation. The local minima of the potential well represent stable attractors and the ball shows the present state of the system. While approaching the bifurcation point, the local minimum on the right becomes shallower, and the recovery of the ball in response to small perturbations is increasingly slowing down. When this local minimum finally disappears, the ball quickly rolls into the minimum on the left. This implies that the system transits into a different steady state. The mechanisms behind this behaviour can be explained in mathematical terms. Approaching a saddle node bifurcation, the maximum real part of the eigenvalue of the Jacobian matrix tends towards 1. It indicates a slower recovery from perturbations (?, ? and ?, ?). Therefore, as long as we are able to detect a signal of slowing down, it would be possible in principle to predict future critical transitions with some accuracy. In what follows, by exploring critical slowing down prior to some extreme events in financial history, we would be able to assess whether such early warning signals might be possible for financial time series.

Data
For the analyses time series from subsystems of the economy which clearly indicate critical transitions from one state to another are needed. Since stock market prices rather than returns often show sharp transitions, the analysis focuses on (log) prices rather than the more commonly analysed (log) returns. Moreover, stock prices often are nearunit root processes, similar to a saddle-node bifurcation with eigenvalue +1, while returns typically have close to 0 autocorrelations. Therefore, time series of daily stock market (log) prices are taken as the dynamical subsystem to be analysed in our work.
As long as we find early warning signals in the stock market price system, we are able to detect critical transitions. In addition with the S&P 500, we analysed the TED spread, since it is an indicator of perceived credit risk. Moreover, as the volatility index of the S&P500, the VIX index is also analysed. Detailed information on data is presented in Table 1.
Daily time series data of stock prices, such as the S&P 500 index, the NASDAQ com- Since the random growth in stock prices data is in percentage term and not in absolute term, we take logarithms of data. Doing so also linearises the exponential growth in original series and stabilises the variance of the analysed residuals.

Detrending
In order to achieve a stationary stochastic process, the first step is to remove the trend pattern from the original time series. The residuals after detrending are further analysed by using linear and nonlinear time series techniques. Subtracting a moving average is the most commonly used technique in detrending. In this paper, we introduce the weighting scheme and use gaussian kernel smoothing. This allows data near the given time point to receive larger weights.
In the analysis of many other complex systems, additional interpolation is needed to have evenly spaced time series. This may produce spurious results (?). Because stock market system can be already considered as a time discrete dynamical system, which is with fixed time-step ∆t = 1 trading days, we skip the interpolation and therefore avoid its possible adverse effects. We detrend time series using a gaussian kernel, based on which the moving average is given by where z is the logarithm of original price index with fixed time-step ∆t = 1. The bandwidth is σ. It is also the standard deviation of the Gaussian kernel distribution. In kernel-based nonparametric regression, the choice of bandwidth is important, due to the bias-variance trade-off. A large bandwidth would lead to oversmoothed, biased, estimates in which details of the trend are washed out and, the magnitude of peaks and troughs is underestimated. Conversely, if a too small bandwidth is applied, this will result in estimation based on only a few local data points, which will lead to large variance of the detrended signal. So the aim is to choose the bandwidth in such a way that we filter out the slower trends from the data while keeping the details of the fluctuations around the local equilibrium value. ? had obtained a considerable number of significant results by setting the bandwidths around 10 to 25. In this paper, we use bandwidth σ = 10 and later doublecheck the results over a range of σ-values in a robustness check. By subtracting moving average from the logarithm of the original time series, the residuals time series is then given by which fluctuates around 0.

Leading Indicators
An increase in autocorrelation and variance is expected for critical slowing down. Whether this is the case can be estimated using a moving estimation window. Three early warning indicators are considered: AR(1), ACF(1) and Variance.

AR(1) indicator
When approaching a critical transition, consecutive observations in the state of the system become increasingly similar to each other. The lag-1 autocorrelation is considered as the leading indicator to measure critical slowing down. It can be estimated from an autoregressive model of lag-1: which is known as an AR(1) model. f (t j ) is the residual y j in Equation (4). ∆t = t j+1 − t j and η j is a zero mean innovation. κ indicates the magnitude of the recovery rate. λ = e −κ∆t is the AR(1) coefficient, which is the autoregressive coefficient at lag1. In a saddlenode bifurcation scenario, κ vanishes on the way to the bifurcation point. As κ goes to 0, λ goes to 1.
The random disturbances are assumed to be white noise with zero mean and variance equal to 1. Therefore, the first order autocorrelation coefficient λ is approximated as constant in a local time window of length w. We estimate λ by an ordinary leastsquare (OLS) fitting method of the regression with u k white noise, over the set of indices k = j − w + 1, . . . , j. The local window slides from left to right and traces a series of AR(1) coefficients varying with respect to index.
This new series can be interpreted as the time-varying AR(1) coefficient. If it increases, this indicates that the system is driven gradually closer to a bifurcation.
Apart from the bandwidth, the window size is also a very important parameter. A smaller window size allows us to track short term changes in autocorrelation. However, taking a too small window size with very few observations will make the estimation of autocorrelation less reliable. Following ?, we use half the size of the analysed time series as the sliding window size.

ACF(1) indicator
An alternative and more straightforward way to estimate autocorrelation at lag-1 is by using the first value of the autocorrelation function (ACF) where µ is the mean of y t in the window considered, and σ 2 y the variance. Like the AR(1) indicator, the moving window produces a proxy series of ACF(1). It also serves as an indicator to detect critical slowing down prior to a critical transition.
Variance indicator An increased slowing down also induces an increased amplitude on the way of approaching threshold. This amplitude corresponds with the variance and is measured by standard deviation: This proxy series of variance produced by moving window also serves as one of the early warning indicators preceding a critical transition. Carpenter and Brock (2006) and Biggs et al. (2009) have argued that variance measures seem to be most robust as EWS and also seem much easier to generalize to multivariate cases.

Estabishing Trends
For each indicator observed across time, i.e. λ, ρ, or the standard deviation, we test the trend over time for significance using the noparametric Kendall rank correlation τ between the indicator and time variable (?). It is a statistic tool used to measure the degree of concordance between two pairs of ordinal variables: where C is the number of concordant pairs, D is the number of discordant pairs, and N = n(n − 1)/2 is the total number of different pair combinations. The quantity τ is in the range of [−1, 1]. If Kendall's τ is close to 1, the agreement between the two rankings is perfect. A high Kendall's τ suggests a strong trend. In the presence of critical slowing down, one expects to find a significant upward trend as indicated by a significantly positive value of Kendall's τ.

Results
The evidence for early warning signals is evaluated in two steps. Firstly, we observe the early warning signals before real critical transitions. By using six time series, we examine four well-known extreme financial events in history -Black Monday 1987, the Asian Crisis, the Dot-com Bubble and the 2008 Financial Crisis. Secondly, we examine the likelihood of spurious early warnings. The probability of obtaining similar or more extreme early warning signals by chance is estimated using bootstrap time series. In the end, we perform an extensive analysis to examine the robustness of the results with respect to the choice of user-set parameters.

Financial Time Series
First of all, we examine whether there is evidence of critical slowing down in time series data of stock prices. Four financial crisis are investigated.            In order to obtain more evidence, we also analyze the volatility index (VIX) in Figure   7. The VIX index is a commonly used estimated time series of the implied volatility of S&P 500 in the next 30 days. Figure 7    and lasts for a long time. The whole process coincides with the depression of the economy. It also shows that the early warnings on the critical transitions in volatility index can be considered as early warnings on the crash in S&P500 index.
The same early warning methodology is applied to VIX time series. The analysis shows significant upward trends in AR(1) and ACF(1) indicators preceding the critical transition in VIX time series. It demonstrates increased slowing down and early warnings before the critical transition in the volatility index. However, the variance indicator is just horizontal with a slight downward trend. Moreover, the p-value of the estimated Kendall's tau indicates insignificance of the trend. Therefore, no trend is found for the variance indicator.
All the above results and the parameters used in the analysis are summarized in Table 2. The symbols "(+)" indicate that significant early warning signals are detected, while the symbols "(-)" indicate that the transitions are not preceded by indicators. As shown in Table 2, the window size we choose is half the sample size in each example following ?. The choice of bandwidth is 10 under the condition that we do not overly smooth the data but still have a stationary time series.  The summary of the results in Table 2 suggests early warnings indicated by the ex-

Bootstrapped Time series
In order to test the likelihood of having the trend statistic estimation of Kendall's τ by chance, we apply the same early warning methodology to surrogate time series.
We also calculate the probability of the trend statistics in surrogate time series being at least as high as for the original records. The surrogate time series are generated by bootstrapping the financial time series of the S&P500 index, the Hangseng index, the NASDAQ composite, TED spread and the VIX index in three different ways.
Bootstrapping Residuals Firstly, we bootstrap the residuals after detrending following the test in ?. By resampling the order, we generate surrogate time series with similar means and variances.
Bootstrapping Log-returns Secondly, instead of residuals, the log-returns of the original time series are bootstrapped. Similar with the first method, we bootstrap the time series by randomly picking data with replacement. Moreover, we take the cumulative sum before detrending.

Random Segments Bootstrap
Spurious early warnings can occur in the fluctuations in a single regime without transiting to a different one. In order to test this possibility, we analyse the early warnings on random segments in financial time series. Figure 8 shows an example of the test of the early warning indicators for random segments in Hangseng time series. It follows the same format of figures as in the analysis of financial time series. The analysis is based on a randomly picked segment from a long period of Hangseng time series. However, the sample size is kept the same as it in the case of the Asian Crisis. As shown in Figure 8 based on our random segments bootstrap, there is no evidence of early warnings in this case with respect to the early warnings of AR(1),

ACF(1) and Variance indicators.
Many other random segments of the financial time series are tested in the same way.
However, the trends of the early warning indicators are diverse. Some of them show no early warnings while the others do. The early warning indicators obtained from 1000 random segments are analysed. Figure 9

Random Walk Bootstrap
To compare with financial time series, we perform the analysis on realisations of a random walk process. A simple random walk is presented as: where t is white noise.
We generate 1000 random walk process realisations and caculate the early warning indicators based on them. Figure 10 shows the histograms of Kendall's τ and p-values of AR(1) and Variance indicators. Due to gaussian detrending in the methodology, −1 < τ < 1 is expected as shown in the figure. Moreover, if it is the methodology which creates spurious trends in Kendall's τ, the distribution would left skewed and clustered near 1. However, Figure 10 shows all the Kendall's τ are distributed evenly between -1 and 1, which rejects the hypotheses above.   Table 3.

Robustness of parameters
The early warning analysis in this paper is influenced by two key parameters: bandwidth and moving window size. Bandwidth size is a very important parameter when filtering out long term trends from the original time series. There is a trade-off when making the choice. A too narrow bandwidth would not only remove the long run trends but also the short run fluctuations we intend to study; a too wide bandwidth would not remove enough long run trends. There would be still some slow trends left which may lead to spurious trends of the indicators. A similar trade-off also affects the window size. A smaller window size is good to track short run changes, but a too small window size with too few sample points would make the estimations less reliable.
In order to check the robustness of the parameters in our analysis, we perform an additional analysis by using rolling window and rolling bandwidth. The contour plots in Figure  Kendall's tau distributions in Figure 12 and 13(b) confirm the strong positive trends of Kendall's tau in the contour plots. This robustness analysis indicates that the results are quite robust with respect to the parameters chosen. It also shows that even more significant trends could be obtained by moving parameters in the direction of the black dot, which in turn confirms the robustness of the results with respect to changes in the parameters in this paper. With the increase of the complexity of the nonlinear dynamics, the prediction of bifurcations may become harder. ? tried to extend the techniques using nonlinear features.
However, they did not find discernible trends in the nonlinear case. The methodology used in this paper is also based on a first order linear approximation of the dynamics.
Therefore, due to the complexity of financial dynamic systems, false early warnings can be expected. Secondly, the transitions in complicated financial systems may happen far from local bifurcations and do not necessarily correspond to cusp catastrophe transitions. For instance, there may be an early escape from a stable equilibrium due to exogenous shocks -a so-called noise induced transition (?). In particular, the emergence of new technology and financial instruments nowadays makes the financial markets more complex. This could explain the failure to detect the 2008 financial crisis, despite their use of advanced economic and financial models. Thirdly, the catastrophe theory approach is based on one dimensional systems with only one control variable, while the situation in financial systems is far more complicated. Perhaps a multivariate approach is required to capture the sophisticated dynamic behaviours in financial systems. It is known, for instance, that in periods of market stress returns on individual stocks are more correlated than in more tranquil periods. Fourthly, asset pricing models are usually based on the assumption that the fundamental price follows a geometric random walk process. Bottom-up heterogeneous agent models where agents are boundedly rational typically describe endogenous fluctuations around this fundamental price, where the latter is typically allowed to be a unit-root process (?). This means that (log) prices are naturally characterised by having an eigenvalue close to 1, so that a trend of the dominating eigenvalue from below one towards one may not be easy to observe in near unit root (log) price time series data. Finally, a fifth reason why the standard critical slowing down approach may not be suitable for financial/economic time series data, or more generally social systems, is that these systems differ from most natural systems due to the presence of smart agents ("atoms that can think") whose main activities are based on their expectations on future price developments. Agents learn and adapt their behaviour and may respond to news announcements, and in particular to early warning signals of a crisis. Their expectations and adaptive behaviour in response to the risk of an upcoming market devaluation may actually trigger or accelerate a selffulfilling crisis. This touches upon the issue whether being able to detect an upcoming financial crisis could cause a panick reaction of the agents and thus trigger an upcoming crisis even earlier. These nonlinear expectations feedback mechanisms and adaptive behaviour may make detection of early warning signals for social and economic systems much harder than for complex natural systems.
In this paper we have deliberately focused on EWSs based on critical slowing down using a narrow set of techniques that have been successfully applied in other fields. We therefore have not addressed the issue of optimising the algorithm in any way by finetuning the test parameters, sample size, or data frequency. This is considered to be out of the scope of this preliminary study, and left for future research. Neither did we discuss which alternative techniques might be used to extract EWSs from financial time series data. In recognition that economic/financial systems may behave differently than many natural systems, and that critical slowing down may not be a typical phenomenon prior to market collapses, we conclude with a number of remarks regarding possible future approaches towards developing EWSs for financial crises. There are many possibilities still open here. For instance forecasting techniques based on pattern recognition might be exploited to extract signals from financial time series data. Such signals might be based, for example, on machine learning algorithms for forecasting prices. One could also imagine the use of not one but a number of different machines, the forecasts of which could be combined into a single signal. In view of the increased correlations between stock returns in periods of market strees, also multivariate approaches, for instance based on increased cross-sectional correlations between individual stocks, are promising for future EWS studies based on financial time series data. Finally, the use of complex networks techniques to monitor the evolving structure of financial-economic networks also has potential as early warning indicators for crises.