Diagnosis and Prediction of IIGPS’ Countries Bubble Crashes during BREXIT

We herein employ an alternative approach to model the financial bubbles prior to crashes and fit a log-periodic power law (LPPL) to IIGPS countries (Italy, Ireland, Greece, Portugal, and Spain) during Brexit. These countries represent the five financially troubled economies of the Eurozone that have suffered the most during the Brexit referendum. It was found that all 77 crashes across the five IIGPS nations from 19 January 2015 until 17 February 2020 strictly followed a log-periodic power law or other LPPL signature. They all had a speculative bubble phase (following the power law growth) that was then followed by a sudden crash immediately after reaching a critical point. Furthermore, their pattern coefficients were similar as well. This study would surely assist policymakers around the Eurozone to predict future crashes with the help of these parameters.


Introduction
Financial crashes have weakened the economy of various countries. A financial bubble is often outlined as a positive acceleration of asset prices that does not reflect an increase in their true value. With the trend of speculation amongst market players constantly on the rise, stock markets have become circumvented. Conceiving and predicting financial bubbles is of paramount importance in the stock markets in order to stabilize the economy.
Numerous controversies are based upon understanding the concepts behind the formation of bubbles. Studies have shown that major crashes also occur due to the consistent collective approach that several investors follow, thus further exemplifying an intermittent positive movement [1]. The issue of the financial crisis in the late 2000s was a matter of great concern that prompted both researchers and experts to conduct research for which the LPPL model was implemented. In terms of effective bubble detection, the log-periodic power law model was developed [2][3][4][5][6][7]. The aim of the log-periodic power law model is to investigate whether the LPPL patterns in the development of credit default swaps (CDS) can be conducive to default classification. Hence, this entire approach facilitates a computable analysis behind bank runs. Traces of LPPL patterns were observed in CDS spread variations during global economic collapses [8]. Researchers point to the fact that major crashes in stock markets are parallel to critical points studied in logarithmic science with log-periodic correction to scaling. They analyze the presence of log-frequency shift over time in the log-periodic oscillations prior to a crash by performing tests on two of the largest crashes, viz, October 1929 and October 1987 [9].
Over the past decade, the LPPL model has been widely used for detecting bubbles and crashes in various markets [10][11][12][13][14][15][16][17][18][19][20]. Hence, a model was created that would enable a reduction in demand with an increasing number of obstacles, causing stock prices to fall as a result of the power law. The model was applied to the Japanese Nikkei stock index from 1990 and the gold future prices after 1980, both after their peak levels, and it was subjected to various parametric and nonparametric analyses [21]. One of the studies revealed the significance of the log-periodic power law in the collapse of the Mont Blanc glacier in Italy, in which case the incident was predicted in advance, enabling officials to take the appropriate precautions. At the same time, the retrospective analysis of this incident was further used to establish a potential early warning system [22]. While acknowledging the challenges of fluctuations in the housing prices of Wuhan, China, especially so in the real estate market and the economy, the LPPL strategy was applied with the implementation of a multi-population genetic algorithm.
It was observed that this LPPL model conquers the challenges related to multivariate and univariate techniques. It will assist governments in formulating better policies for the real estate market and at the same time protect the interests of purchasers [23]. Empirical studies have revealed the fact that through the reformulated version of the LPPL by Filimonov and Sornette [24], it is possible to predict the embedded risk of future enigmatic events in the Indian stock market [25]. Researchers suggested that certain parameters such as time asymmetry, robust alliance between market players, definite rationality, and a probabilistic elucidation are essential in building a model of stock price variations. In addition, previous models based on log-periodic behavior were compared prior to the crash and accordingly, it was discovered that the model that adhered to the above specifications resulted in greater authenticity [26].
In this study, we adopt the LPPL methodology to detect the positive and negative bubbles in the IIGPS countries (Italy, Ireland, Greece, Portugal, and Spain) using the daily data of five stock indices (Irish Stock Exchange, Italian FTSE MIB Index, Athens Stock Exchange, Portuguese Stock Index, and Madrid Stock Exchange). IIGPS countries are usually referred to as PIIGS or GIPSI [27][28][29][30][31][32]. Since the above acronyms are derogatory, we herein use IIGPS as a new acronym for the five financially troubled economies of the Eurozone and cover the time period from 19 January 2015 to 17 February 2020, when the Brexit referendum occurred, in order to identify LPPL traces. This study is the first of its kind that identifies the existence of bubbles in the stock markets of the IIGPS countries (Italy, Ireland, Greece, Portugal, and Spain) with the advanced bubble detection methodology of the LPPLS confidence indicator. Thus motivated by previous research on the characteristics of bubbles, we aim to examine the characteristics of the recent bubbles that occurred from 2016-2020 (Brexit impact) in the IIGPS countries (Italy, Ireland, Greece, Portugal, and Spain), which, as mentioned above, suffered the most during the Brexit referendum.
The research questions that this paper attempts to answer are the following: (a) Is there a common pattern for all the IIGPS countries during Brexit? (b) Are Italy, Ireland, Greece, Portugal, and Spain, as financially troubled economies of the Eurozone, interconnected with each other (from the perspectives of the respective stock markets during the Brexit)?
The contribution of our research to the existing literature is threefold. Firstly, this is the first attempt to search for a common thread across the financial crashes of the IIGPS countries during Brexit using Filimonov and Sornette's [24] modified LPPL. Secondly, we differentiate from previous studies and test the robustness of the LPPL following the reformulated version of the LPPL calibrations proposed by Filimonov and Sornette [24]. Thirdly, the IIGPS or PIIGS countries are relatively vulnerable to financial crashes when compared to their Western EU counterparts, especially during structural events such as Brexit. If the LPPL model fits the past crashes of these countries without any glitches, that ensures future crash predictions for those countries by regulators well in advance (especially in the advent of a possible structural event as big as Brexit). The LPPL model fitted pretty well in our study and paves the way for future crash predictions well in advance for the IIGPS countries, before any large structural events. Market stability instruments such as circuit filters can be used to control volatility (downwards) well in advance.
The rest of the paper is organized as follows: Sections 2 and 3 describe the data and present the results. Section 4 concludes the paper.

Data and Methodology
The study analyzed the daily closing prices of five stock indices (the Irish Stock Exchange, the Italian FTSE MIB Index, the Athens Stock Exchange, the Portuguese Stock Index, and the Madrid Stock Exchange). The time period was selected from the data that was collected for the period from 19 January 2015 to 17 February 2020 in order to identify LPPL traces. Each index had 1240 observations; thus, the total numbers of observations was 6200.
The study complied with the LPPL approach proposed by Filimonov and Sornette to support the identification of LPPL traces [24]). We began with the standard Johansen-Ledoit-Sornette (JLS) log-periodic power law algorithm: where t c denotes the most plausible time of the market crash, β signifies the exponential growth, y t is the price index at time t (y t > 0), ω regulates the magnitude of the oscillations, and t is any time in the bubble preceding (t < t c ). A, B, C, and Φ are merely units with minor details. A signifies the expected value of yt when the end of the bubble is reached at t c (A > 0). B denotes the fall in y t over the time period before the crash if C is near zero (B < 0). C is the proportional magnitude of the oscillations around the exponential growth (|C| < 1).
Fitting the LPPL algorithm into financial data, the bubble behavior is apprehended by log-periodic oscillations and proceeds with the stock index at the critical time of the crash (t c ). Rapid acceleration in asset or equity prices is the prime indication followed by periodic fluctuations at a low magnitude, when t comes nearer to critical time (t c ). Fundamentally, t = t c is considered to be the most plausible time of the crash. However, the JLS algorithm required a reconstruction so as to avoid technical hitches. Filimonov and Sornette [24] have since amended the classical model into an advanced one: where C 1 = CCos∅ and C 2 = CSin∅.
The advanced version of the LPPL algorithm has four linear variables (A, B, C 1 , C 2 ) and three non-linear variables (t c , ω, β). The four linear parameters (A, B, C 1 , C 2 ) are based on the "standard slaving" principle. The subordination procedure is used to propagate non-linear parameters (ω, β). This amendment enables the examination of the bubble formation followed by the anticipation of future crashes [24].
This latest version of the LPPL is used on a daily basis with various cryptocurrencies and commodities by Sornette at the Financial Crisis Observatory, ETH Zurich (https: //er.ethz.ch/financial-crisis-observatory.html accessed on 28 January 2021). The original JLS model (2001) proposed three linear variables (A, B, C) and four non-linear variables (t c , ω, β, ∅). This was difficult to calibrate. Moreover, it was also not robust from the error (RMSE) perspective.
In conclusion, the advantages of the methodology used are that firstly, the LPPL has no competitor. In fact, Sornette runs a financial crisis observatory. However, they started with the JLS model [5]) back in 2001, which had three linear (A, B, C) and four non-linear parameters (∅, t c , ω, β), which was difficult to calibrate and was not reliable in empirical data crunching. Filimonov-Sornette [24] (the model we used) has four linear (A, B, C 1 , C 2 ) and three non-linear parameters (tc, ω, β), making it a robust calibration.

Empirical Results
The LPPL framework is subject to certain restrictions. For instance, the drawdown threshold or all the stock indices were taken as ≥7%. Primarily, drawdown is defined as an accretive fall from one local maximum value to the next proximal minimum value. Conjointly, Python codes were operated on the Anaconda 3 platform in order to generate empirical results. It is a surprising fact that Greece alone exhibited 22 events with cogent LPPL signatures. The prominent underlying facts, which validate the low economic growth of Greece, include the turmoil of the Great Recession, which led to budget deficits and a higher unemployment rate. Table 1 represents the constraints on the LPPL variables associated with its literature. Tables 2-6 represent the coefficients of the LPPL framework providing the drawdown >7% in Ireland, Italy, Greece, Portugal, and Spain, respectively. The LPPL signatures were observed on 77 occasions. C 2 (Sine function) Filimonov and Sornette [24] t c (t to ∞) Kuropka and Korzeniowski [33] β (0.1 to 0.9) Lin et al. [34] ω (4.8 to 13) Johansen [35] Note: The table above presents the constraints on the LPPL parameters of Equation (2) used in our empirical analysis. These stylized facts were found consistent in most previous literature.  (2) in the text. DD, or drawdown, is the gap between the local minima to the next local maxima and it is ≥7%. RMSE is the root mean square error. All estimates were generated by the authors using Python language in the Anaconda program with a project Jupyter notebook (please see Figure A1 in the Appendix A).  (2) in the text. DD, or drawdown, is the gap between the local minima to the next local maxima and it is ≥7%. RMSE is the root mean square error. All estimates were generated by the authors using Python language in the Anaconda program with a project Jupyter notebook (please see Figure A2 in the Appendix A).  (2) in the text. DD, or drawdown, is the gap between the local minima to the next local maxima and it is ≥7%. RMSE is the root mean square error. All estimates were generated by the authors using Python language in the Anaconda program with a project Jupyter notebook (please see Figure A3 in the Appendix A).  (2) in the text. DD, or drawdown, is the gap between the local minima to the next local maxima and it is ≥7%. RMSE is the root mean square error. All estimates were generated by the authors using Python language in the Anaconda program with a project Jupyter notebook (please see Figure A4 in the Appendix A).  (2) in the text. DD, or drawdown, is the gap between the local minima to the next local maxima and it is ≥7%. RMSE is the root mean square error. All estimates were generated by the authors using Python language in the Anaconda program with a project Jupyter notebook (please see Figure A5 in the Appendix A).
It is worth mentioning here that the LPPL is a generalized method as it was conceived by Sornette in the late 1990s. It indicates a common behavior of a group of interconnected underlying time series. Here, in this study, the power law coefficient β value range mentioned above signifies the extent of power law growth of a speculative bubble in these stock exchanges during Brexit. Since it is on the higher side (refer to Table 1), the pattern of the speculative bubble in the IIGPS countries would be steep in nature. Angular frequency, or ω, clocked closer to the highest range (refer to Table 1), signifying an enormous increase in volatility during the bubble build-up phase in all of the IIGPS countries' stock exchanges during Brexit. The most important of is the "drawdown," or the bubble buildup phase measurement just before the crash. Our study found that all of the IIGPS countries' stock markets under a specified time period witnessed a total of 77 crashes. The drawdown before all those crashes was the same.
Furthermore, although the LPPL looks non-stationary, it is a stationary process due to the increments of the process. Hence, persistence (though a weaker one) would manifest. During the bubble build-up phase and even during the crash phase it will exhibit persistency due to the increments of its process.
The robustness of the LPPL model was supported through root mean square errors (RMSE) in all 77 cases. The accuracy was indicated by extremely lower values, which was exhibited in all the cases.
The predicted crashes were validated by analyzing the actual events that occurred in the Eurozone countries. Here is a brief synopsis of the events that triggered a crash during different critical points. Table 7 represents the crashes that had a drawdown point exceeding -10%, termed "large crashes" (double-digit drawdowns).

Conclusions
We presented a model for market bubbles or crashes, termed the "log-periodic power law," or LPPL signature, in order to describe and diagnose situations when excessive expectations of future market price increases cause prices in the IIGPS countries (Italy, Ireland, Greece, Portugal and Spain) to be temporarily elevated and vice versa. We covered the time period from 19 January 2015 to 17 February 2020, a time which the Brexit referendum occurred, in order to identify LPPL traces.
The LPPL model for pre-crash bubbles on stock markets has important consequences. Our analysis led us to the following conclusions: We found a profound LPPL signature with common pattern parameters for all the IIGPS countries across all 77 bubble-crash cases. The speculative bubble followed by a crash remained a part of IIGPS during the Brexit period (extended).
LPPL-based models are apparently non-stationary and more of mean-shifting process. However, in detailed observation they are mean-reverting due to their increments in the process. It is not a strong stationarity, though. However, this stationarity further indicates persistence. Interestingly, a milder policy shock stays for longer in cases that border non-stationarity, according to a Portuguese research group [36]). We can draw a parallel and suggest transitory policy shocks for the IIGPS countries' stock markets based on our findings, which were similar.
We believe that these findings are useful to policymakers as a vision for appropriate policymaking, for entrepreneurs in financial companies of all sizes to develop their competitive strategies, and for investors to adjust their investment strategies.
Another economic implication is that most stock markets deviate from the efficient market theory (EMH) in practice. Hence, an alternate prediction mechanism would assist policymakers to control unnecessary volatility and safeguard investor wealth.
This study contributes to the existing literature as it is the first attempt to search for a common thread across the financial crashes of the IIGPS countries during Brexit using the Filimonov-Sornette modified LPPL. In addition, we differentiated from previous studies and tested the robustness of the LPPL following the reformulated version of the LPPL calibrations proposed by Filimonov and Sornette [24].
The IIGPS or PIIGS countries are relatively vulnerable to financial crashes compared to their Western EU counterparts, especially during structural events such as Brexit. If the LPPL model fits their past crashes without any glitches, it ensures future crash prediction for those countries by regulators well in advance (especially in the advent of a possible structural event as big as Brexit). The LPPL model fitted pretty well in our study and paves the way for future crash predictions for the IIGPS countries well in advance, before any large structural events. Market stability instruments such as circuit filters can be used to control volatility (downwards) well in advance.
The limitations of the study concerning the limitations of the research methodology (LPPL) are twofold. Firstly, it tries to generalize, which may not work sometimes, and secondly, as drawdowns are case specific and empirically found, this cannot be generalized for a very long period.
Transitory policies are required for all the IIGPS countries. The debt-led growth policy probably needs to be revised or reconsidered. Future crashes can be forecasted well in advance (for the IIGPS countries) using the specific stylized facts found during our study. Thus, volatility curbing measures could prevent further crashes and safeguard the public wealth invested in the respective stock markets.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The