Change point detection for subprime crisis in American banking: From the perspective of risk dependence

    Change Point Detection for Subprime Crisis in American Banking: from the Perspective of Risk Dependence Xiaoqian Zhu, Yongjia Xie, Jianping Li, Dengsheng Wu PII: DOI: Reference:

S1059-0560(15)00004-0 doi: 10.1016/j.iref.2014.12.011 REVECO 1028

To appear in:

International Review of Economics and Finance

Received date: Revised date: Accepted date:

13 February 2014 21 September 2014 30 December 2014

Please cite this article as: Zhu, X., Xie, Y., Li, J. & Wu, D., Change Point Detection for Subprime Crisis in American Banking: from the Perspective of Risk Dependence, International Review of Economics and Finance (2015), doi: 10.1016/j.iref.2014.12.011

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Change Point Detection for Subprime Crisis in American Banking:

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from the Perspective of Risk Dependence

Institute of Policy and Management, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100190, China

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Xiaoqian Zhua,b, Yongjia Xiea,b, Jianping Lia, Dengsheng Wua,*

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Change Point Detection for Subprime Crisis in American Banking:

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from the Perspective of Risk Dependence

Abstract: The subprime crisis has received great attention in academic research but there is no consensus on when the crisis started and when it ended. Previous researchers

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have only mentioned their subjective judgments in related papers and well accepted change point detection methods are not available. So the objective of this paper is to propose a multiple change point detection approach from the perspective of risk

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dependence by using copula function. Since the inter-dependence of different types of risks during crisis and non-crisis periods is significantly different, we monitor the change of dependence structure over time. The first step is to choose a proper copula that can

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accurately describe the dependence structure of the data. Thereafter, using the chosen

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copula to fit the data dynamically, a series of parameters are attained. Finally, the change points are identified by analyzing the trend of the fitted parameters. Based on the

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financial data of the top 100 American banks in Forbes’ list, we empirically detect the start point, end point and peak period of the subprime crisis in American banking. The results show that the crisis started in 2007Q4 and ended in 2011Q3, and the peak period

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of the crisis was from 2009Q3 to 2010Q2. Keywords: subprime crisis, risk dependence, copula, multiple change points, unbalanced panel data, American banks

I Introduction The U.S. subprime mortgage crisis was a series of events that led to a financial crisis and the subsequent recession. The crisis had severe, long-lasting consequences for the U.S. and world economies (Morales & Andreosso, 2014; Shiller, 2012). The core problems that caused the subprime crisis have been addressed in extensively in extant literature. One basic question is how to define the critical points of the event. A landmark incident is always used to define the start of a financial crisis; for example, the collapse of Leman Brothers on 15 September 2008 is always considered as the start of the US subprime 2

ACCEPTED MANUSCRIPT crisis. However, the end time has not been defined in a credible manner. During data collection and pre-processing stages in previous research works, the problem of dividing phases of the data has usually been encountered, which can affect the outcome to some

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degree (Dooley & Hutchison, 2009; Tsay & Ando, 2012; Wang et al., 2012). Thus, the exact

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start time and end time of the events are the premises of a rigorous comparative analysis, which requires a quantitative approach to detect the change points of subprime crisis in

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American banking.

Change point detection is a popular research area in statistics and has experienced rapid development in both theory and application in recent years. There are a lot of quantitative change point detection methods and classifications. According to the

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detection methods, the change point detection problem can be divided into three categories, parametric detection, semi-parametric detection and non-parametric

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detection (Chen & Gupta, 2012; Guan, 2004; Pettitt, 1979). According to the types of observations, the change point detection problem can also be divided into two categories: one is for time series data (Mensi etc., 2014) and the other is for the panel data (Éltető, et

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al., 2012; Wachter & Tzavalis, 2004). But all these methods are not valid for detecting the

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change points in financial crises. There are mainly two problems in the existing methods. Firstly, the current methods are mainly based on independent variables and only a few

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are proposed for dependent variables on the assumption of auto-aggression (Berkes, et al., 2004; Kokoszka & Leipus, 2000). But the experimental data in finance are apparently dependent and their correlations are complicated. Secondly, the financial data here are unbalanced panel data because some individual data are missing, which happens

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frequently in real situations. Currently, most studies on change-point detection for panel data focus on the balanced panel, which require high quality of the experimental data (Joseph & Wolfson, 1992; Horvath &Huskova, 2012). As for unbalanced panel data, Atak et al. (2011) construct an asymptotic framework to detect changes in the climate that allows a non-trivial fraction of the data to be missing. Therefore, a new approach that can be applied to unbalanced panel data, to detect the start and end point in financial crises is imperative. It is generally believed that the risk dependence of crisis and non-crisis periods is significantly different, that is financial crises ruin the stable relationship of risks (Syllignakis & Kouretas, 2011). So by observing the correlation of the two indicators of risks, the change point can be identified, which can translate the problem into tracing the change of risk correlation. Correlations among market risk, credit risk, operation risk and other kinds of risks have been examined extensively in extant research. The linear 3

ACCEPTED MANUSCRIPT correlation coefficient is commonly used for measuring the magnitude of dependency. However, the linear correlation coefficient is only valid when the relationships among variables are linear and the variance is finite (Grundke & Polle, 2012). A lot of the data

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that appear in the financial markets tend to submit to heavy-tailed distribution so that

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variance sometimes does not exist and the correlation coefficient cannot reflect the tail dependence (Boyer et al., 1999). Hence it is necessary to fully understand the stochastic

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dependence beyond linear correlation.

With copula function, the shortage of linear correlation mentioned above can be addressed. A copula can describe the dependence between a joint distribution and marginal distributions. Different copula functions reflect different dependence structures.

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The earliest theories put forward can be traced back to 1959, Sklar (Sklar, 1959). In the late 1990s, the theory got developed quickly and was first introduced to the financial field

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to analyze risk dependence (Embrechts et al., 2002). Bouye & Durrleman (2000) systematically expanded copula applications in finance, such as insurance, portfolio analysis and risk management. Copula has performed well in measuring the dependency

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structure. Patton (2001) constructed a bivariate copula model for returns of exchange

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rate and compared it with the BEKK model. Results showed that the copula model can better describe the dependence structure in financial markets. Hu (2006) put forward the

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concept of “Mixed Copula” where different copulas are linearly combined with weight parameters, so that several financial markets with different dependence structures can be described by one copula function. Zhu et al. (2014) investigates the dynamic dependence between crude oil prices and stock markets by using unconditional and conditional

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copula models and showed that time-varying copulas best capture the tail dependence. Therefore, allowing for the complexity of risk dependence, copula is a good option. For the change point detection issue and use of the copula function, previous researchers have proposed important theoretical basis and carried out empirical tests. Dias and Embrechts (2004) proposed a parametric copula model of dependence and discussed change point analysis question in this framework. The closest research to this paper is Caillault & Guégan (2010), in which a similar real problem is discussed. It focused on the time-varying parameters and sequence of copulas in computation of VaR and ES measures using a conditional copula and GOF tests. And the change points detected in empirical work can well reflect certain financial events (Guegan & Zhang, 2010). Developed from Guegan & Zhang, Ye et al. (2012) not only proved the existence but also measured the degree of crisis contagion using the tail dependence coefficient of copula function, of which the first step is to detect the change point of the subprime crisis. 4

ACCEPTED MANUSCRIPT Their core issue is similar to this paper. The objective of this manuscript is to propose a copula based change point detection approach which can quantitatively determine the start and end time of a financial crisis

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from the perspective of risk dependence. The proposed approach has two advantages.

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One is that it can determine the start and end point simultaneously. The other advantage is that it can make full use of the collected panel data, including balanced and unbalanced

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panel data, which happens very frequently in risk management because of inadequate data (Kuritzkes & Schuermann, 2006). In the experiment, this paper applies the proposed method in detecting the start point, end point and peak period of the subprime crisis in American banking, which is, to the best of our knowledge, the first paper to study this

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issue quantitatively.

The following sections are organized as follows. In Section 2, the rationale and steps of

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the proposed change point detection approach is introduced. In Section 3, the proposed approach is applied to detect the start point, end point and peak period of the subprime crisis in American banking based on the financial statements data. In Section 4, the

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conclusions and possible future directions are presented.

2 The proposed approach

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In this section, the proposed change point detection approach is presented in detail. The main idea of this approach is to find the change of the dependence structure of different types of risks, measured by the copula function. Firstly, the rationale is

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illustrated. The dependence of risks is varying all the time and will significantly change during specific periods of time. Then the steps of the approach are given. There are three steps to determine the exact change points of financial crisis.

2.1 The rationale of the proposed approach Financial institutions are always faced with different types of risks tightly associated with financial products, such as credit risk, market risk and operational risk. The total risk here is actually the integrated risk that a financial institution confronts, taking into account the inter-dependence among different types of risks. During periods of financial crises, the risk department in a financial institution is under great stress. It is generally believed that the inter-dependence of different types of risks is significantly different during crisis and non-crisis periods, that is, the financial crisis ruins the stable relationship of risks. At the beginning of a financial crisis, there is always an emergency 5

ACCEPTED MANUSCRIPT that becomes the spotlight, indicating the eruption of a crisis, which is called the risk trigger (Elmer & Seelig, 1999). As shown in Fig. 1, the risk trigger generates a risk that first affects the other kinds of risks and the total risk. The first affected risk generated by

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the risk trigger is called the “triggered risk” hereafter. Then the total risk is either directly

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affected by the triggered risk, or indirectly affected by the trigged risk by means of other risks and becomes larger. That is, during a financial crisis, the correlation of triggered risk

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and integrated risk become larger than in normal period. In other words, the triggered risk contributes more to the total risk during the crisis.

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Event Direct

Triggered risk

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Indirect

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Other risks

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Fig. 1 The rationale of this proposed approach

In this paper, we mainly consider the dependence between the triggered risk and the

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total risk. Since the dependence is always changing, we can detect the time point when the dependence structure changes significantly. As shown in Fig. 2, before a crisis, the dependence of trigger risk and total risk is stable. When a crisis starts, the dependence

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becomes larger and when the crisis ends, the dependence becomes stable again. Besides, in an economic cycle, there must exist a period during which the inter-dependence of trigger and total risk increased, peaks and decreases gradually after that. Therefore, by tracking the dependence of the trigger and total risk, the start time, the end time and the peak period of the financial crisis can be identified. So in this paper, the start time is defined as the time point when the dependence becomes larger and the peak period is defined as the period when the dependence reaches its peak and the end time is defined as the time point when the dependence becomes stable again.

End Dependence

Start

Before crisis

During crisis

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After crisis

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Then the question is how to describe the dependence at each time point. During a crisis, assume there are m banks involved. At a time point, each bank can produce a pair of

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triggered risk and total risk data. So all the pairs of data will constitute a m×n panel data.

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Panel data includes two categories, namely, balanced panel data and unbalanced panel

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data (Fig. 3). From Fig. 3 we can see that unbalanced panel data are different from balanced panel data because some sources don’t generate data at some time points. From the perspective of the assumed issue, each box in Fig. 3 denotes a pair of triggered risk

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and total risk.

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Fig. 3 The illustrative figure of panel data

The unbalanced panel data always appear in the situation of small data sources and some data are lost due to various reasons during data collection. In the proposed method,

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we are focusing on the dependence between triggered risk and total risk. Therefore, as long as the data appear by pairs, the dependence structure can be captured and measured.

2.2 The steps of the proposed approach We aim to observe the dependence of the triggered risk and the total risk here because dependence can describe the overall correlation between them, not only linear correlation. Copula function, put forward by Schweizer & Sklar (1983), has been gaining popularity in economics and finance due to its appealing features in modeling dependence between variables (Ye et al., 2012). Therefore, here we use copula to capture the dependence between triggered risk and total risk. There are many different copula functions exhibiting tail dependence, ranging from tail independence to tail dependence, symmetry to asymmetry (Rodriguez, 2007). Among all types of copulas, frequently-used copulas include Gaussian copula and t copula from elliptical copula family and Gumbel copula, 7

ACCEPTED MANUSCRIPT Clayton copula and Frank copula from Archimedean copula family (Cherubini et al., 2004; Nelsen, 1999). Besides, the survival Gumbel copula that has similar features to Clayton copula is also taken into consideration.

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The procedure of the proposed approach is shown in Fig. 4 and illustrated as follows at

Choose a proper copula

Three ways: 1. analysis 2. graph 3. goodness-of-fit test 1. Fit the data from begaining

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Fit copula to data dynamically

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Step 1

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End

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Final output: start, end and peak

Fig.4 The procedure of the proposed change point detection approach

Step 1: choose the proper copula The triggered risk and total risk might interact in different ways. So the task of the very first step is to select a proper copula to properly describe the interaction or dependence of the triggered risk and the total risk. There are three ways to choose a proper copula. The first is to analyze the dependence on the basis of previous literature. If the dependence structure of the two risks is well known, we just need to choose the corresponding copula. The second way is to draw a scatterplot of the triggered risk and total risk. From the scatter chart, we can preliminarily estimate what kind of tail dependence the two random variables are: whether it is tail symmetric; if it is asymmetric, whether it is upper tail dependence. Then we can judge the dependence subjectively, 8

ACCEPTED MANUSCRIPT according to the shape of the graph. The third way, also the usual way, is to test goodness-of-fit. The larger the P-values are, the better the copula can fit the dependence. The output of this step is a proper copula capable of describing the dependence best.

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Step 2: fit the copula to data dynamically

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After the proper copula is identified, we fit the copula to data dynamically. The output of this step is the fitted parameters of copula. The fitting process (Fig. 4) is carried out three

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times in different ways. Every time the data is fitted, n subsets are needed if there are n time points. Firstly, we fit copula to the data backward, adding new data one at a time from the beginning. Subset 1 that is fitted to the chosen copula contains the data at t1. And data at t2 is added to form Subset 2 to be fitted again. Then data at t3 is added to form Subset 3

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by the analogy. Finally we get a series of fitted parameters a: a1, a2, … , an. Secondly, we fit copula to the data forward adding new data one at a time from the ending. Subset 1 fitted

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to the chosen copula contains the data at tn. And data at tn-1 is added to form Subset 2 to be fitted again. Then data at tn-2 is added to form Subset 3 by the analogy. At last, we get a series of fitted parameters b: b1, b2, … , bn. Thirdly, we fit copula to the data with the time

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window=2 (a parameter set by requirement), Subset 1 fitted to the chosen copula

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contains the data at t1 and t2. Subset 2 contains the data t2 and t3 and it is fitted again. Then Subset 3 contains the data t3 and t4 by the analogy. Finally, we get a series of

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parameters p: p1, p2, … , pn-1.

Step 3: find the change points As the fitted parameters are given in Step 2, we can draw three lines a, b and p along with time, and then the start time, end time and peak time can be found out. The start

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time is the time point where the parameter becomes significantly larger and the end time is where the parameter becomes stable again. Besides, the peak is the time point where the parameter is the largest. The final output is the start, end and peak time of the crisis. This section presents all details of the proposed approach, both rationale and the steps. The main feature of the proposed approach is that the dependence structure of risks is fully taken into account by applying copula function to determine the change points in financial crisis.

3 Experiment In this section, based on the quarterly financial data of American banks, the proposed change point detection approach is applied to determine the period when the American banking was affected by the financial crisis. 9

ACCEPTED MANUSCRIPT 3.1 Experiment design The goal of this experiment is to detect the start time, end time and peak period of the subprime crisis in American banking. The subprime crisis was triggered by borrowers’

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failure to pay subprime loans. So we recognize the credit risk as the triggered risk and

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use copula to observe the dependence changes of credit risk and total risk along with time

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in this experiment. Since the definition and quantification of risk is not the focus of this study, we employ the definition of credit risk and integrated risk from Kuritzkes & Schuermann (2006). Therefore, the credit risk is defined as the potential for losses due to the failure of borrowers to repay and the total risk’s definition is the potential for

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deviation from expected results, that is, earnings volatility. According to the mapping relationship between risk and financial statements, we considered the provision as credit

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risk and pre-tax income as the total risk. The primary reason is that risk is defined as the uncertainty of profit or loss as mentioned above and the financial statements have recorded the final profits or losses. Therefore, the financial statements provide a simple

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and efficient way to measure risk. Based on the empirical data, we choose the proper copula function that can best describe the dependence between credit risk and total risk

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in American banking during the selected period. Then the dynamic fitting process is conducted to obtain three series of parameters. Finally, the start and end points and peak parameters.

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period of the subprime crisis are identified, after analyzing the trends of the three

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3.2 Experiment data

We adopted the Top 100 American banks in 2013 Forbes list (see Appendix). The sample banks are increased to 96, except Bank of New York Mellon (ranked 16th), Oriental Financial Group (ranked 63rd), Popular (ranked 85th) and Doral Financial (ranked 100th) because these four banks’ search codes in Reuters are not available within our capabilities. According to the mapping relationship between risk and financial statement, we extracted the data of provisions, pre-tax income and assets. The financial statements data of these 96 banks were downloaded from Reuter’s database from 2006Q1 to 2013Q1. Finally, 2513 pairs of quarterly data were used in the experiment. In order to make the data comparable among different banks, we pre-processed the provisions and pre-tax income by dividing their total assets. Therefore, we get the final data structure with the first dimension of the data being provision/asset and the second dimension of the data being pre-tax income/asset. 10

ACCEPTED MANUSCRIPT 3.3 Experiment result In this section, the proposed approach is applied to detect the start point, end point

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3.3.1 The detection of the period of subprime crisis

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and peak period of subprime crisis in American banking with the experiment data.

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As illustrated in Section 2.2, the first step is to find a proper copula function that can most accurately describe the dependence structure of credit risk and total risk. In the subprime crisis, the credit risk has greater effects on total risk under extreme economic circumstances than under normal periods of time. This kind of dependence results in

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significant effect in negative tail.

From the perspective of the dependence, it also proves the judgment above. The scatter

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plot between integrated risk and credit risk is shown in Fig. 5. Before the scatter plot is drawn, the experimental data are preprocessed. The two variables, credit risk and total risk, are calculated their empirical CDFs in [0, 1]. Therefore, in the scatter plot, the

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horizontal axis shows the first variable’s empirical CDF values and the vertical axis shows

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the second variable’s empirical CDF values. When the two variables are paired to be shown in the scatter plot, the denseness of dots can visualize the dependence structure of

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two variables. In Fig. 5, the dots at bottom left are densest, which means the dependence structure tends to be lower tail dependence. Therefore, Clayton Copula is known to be

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sensitive to the change of negative tail, which is theoretically suitable for this experiment.

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Fig.5 The scatter plot of integrated risk and credit risk

The goodness-of-fit test results of all candidate copulas are shown in Table 1. Several

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common copula functions, such as Gaussian copula, Student-t copula, Gumbel copula, Clayton copula, Frank copula and survival Gumbel copula are adopted to model the entire sample. A function named “BiCopGofKendall” in R package “CDVine” is used to test the goodness-of-fit of copulas. This copula goodness-of-fit test is based on Kendall's process

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as investigated by Genest and Rivest (1993) and Wang and Wells (2000). This function performs the goodness-of-fit test based on Kendall's process for bivariate copula data. It computes the Cramer-von Mises and Kolmogorov-Smirnov test statistics, respectively, as well as the according p-values using bootstrapping. The larger the P-values are, the more proper the copula is. It turns out that Clayton copula outperforms the other candidate copulas and best fits the data according to the significance level of their parameters. Three methods to choose the proper copula come to the same conclusion that is the Clayton copula is the best choice to fit the experiment data. Table 1 Goodness-of-fit results by maximum likelihood estimation

Gaussian

Student-t

Gumbel

Clayton

Frank

Survival

copula

copula

copula

copula

copula

Gumbel copula

Statistic.CvM

5.788646

4.121852

11.05378 12

0.2315973

4.576834

2.082201

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3.188443

5.131296

1.004552

3.540357

2.592167

P.value.CvM

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0

0.02

0

0

P.value.KS

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Statistic.KS

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The chosen Clayton copula is used to dynamically fit the data from 2006Q1 to 2013Q1 twice (Section 2.2). As all of the fitted parameters are significant, we draw two lines of

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the parameters separately to show the changes of the dependence over time (Figs. 6 & 7). We apply the first half of the time periods to the line graph instead of the whole periods. The main reason is to highlight the experiment results. As we conducted the fitting process dynamically, the number of fitting samples is growing larger, especially at the

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latter part of the experiment. The big amount data contain much more information compared with a newly-added quarterly data. So the change in dependence brought by

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newly added data is limited.

By analyzing the trend of the fitted parameters, the start and end point can be easily identified. Fig. 6 shows that during 2006Q2 to 2007Q3, values of the parameters

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experienced a relatively steady period, which means the dependence degree of credit risk

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and integrated risk is weak and nearly unchanged. Once the data of 2007Q4 is added, value of the fitted parameters obviously leaps. Therefore, 2007Q4 can be considered as

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the start of the subprime crisis.

Fig. 6 The fitted parameters of Clayton copula from the beginning

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upward tendency. Although the change point is not easy to identify like the start point,

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the two different trends indicate that the dependence degree of credit risk and integrated risk is undergoing a big change. Therefore, a quantitative analysis can help identifying the

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end point of subprime crisis.

Fig. 7 The fitted parameters of Clayton copula from the ending

The Quandt-Andrews test is used to quantitatively validate results concludes from

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Fig.6&7. Quandt-Andrews test is used to test the existence of the unknown breakpoints (Andrews, 1993). The maximum LR F-statistic appears at 2007Q4 with great significance, which meaning the most probable change point appears at 2007Q4. Similarly, the maximum LR F-statistic appears at 2011Q3 with great significance, which meaning the most probable change point appears at 2011Q3. Table 2 The detection results of subprime crisis in American banking

Quarter

Pre-crisis

Change point 1

During-crisis

Change point 2

After-crisis

2006Q1-

2007Q4

2007Q4-

2011Q3

2011Q3-

2007Q3

2011Q2

2013Q1

In conclusion, due to the two change points detected above, the subprime crisis can be divided into three periods, that is, pre-crisis period, during-crisis period and after-crisis 14

ACCEPTED MANUSCRIPT period. In the scope of the experimental data, the pre-crisis period is from 2006Q1 to 2007Q3. After the event erupted in 2007Q4, the subprime crisis remained till 2011Q2. And the world entered the post-crisis period after 2011Q3.

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3.3.2 Detection of the peak of the subprime crisis

In order to find the peak period of the subprime crisis, the data is fitted to the Clayton

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Copula with time window=4. The choice of the width of the time window directly affects two key points: one is the accuracies of the experiment results and the other is the significance of the copula fitting. During the process of the experiment, we must take into

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account both of the two contradictory aspects. We have tried the time windows from 1 to 5. Results turns out that when the time window =4 or 5, the significance of the fitted parameters is the strongest. However, if time window=5, the peak period contains 5

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quarters, covering nearly one-third of the entire subprime crisis. If the width of time window is very big, there will be little sense to detect the peak period. So taken these two aspects into consideration, we choose time window=4 to detect the peak period.

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The obtained parameters are drawn into a line graph (Fig. 8) from which it can clearly

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show the trend of values of the fitted parameters. Apparently, values of the parameters from 2009Q3 to 2010Q2 reached the peak during all the experimental periods. The

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dependence degrees of the periods either before it or after it are smaller. Therefore, we

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consider the 2009Q3 to 2010Q2 as the peak period.

Fig. 8 The fitted parameters of Clayton copula when window=4 Table 3 The peak period during subprime crisis Peak period Quarter

2009Q3-2010Q2 15

ACCEPTED MANUSCRIPT 3.4 Experiment analysis In terms of the start time and end time of the American subprime crisis, many researchers have mentioned different judgments in their papers. Wang et al. (2012)

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considered the period from Jun 2007 to May 2009. Tsay and Ando (2012) think the

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period is from August 2007 to September 2009. Duncan and Kabundi (2013) hold that

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the subprime starts on 1 July 2007 and ends on 31 October 2008. Claessens and Köse (2013) roughly give a time range from 2007-2009. Many other researchers have also given the start time of the subprime crisis but have not exactly mentioned the end time. For example, Acharya and Merrouche (2013) think it started on 9 August 2007. Most

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conclusions are qualitative analyses based on the scholars’ subjective judgments. As for the quantitative method, Ewing and Malik (2013) examine the volatility of gold and oil

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futures returns to find the structural breaks and results related to the subprime crisis turn out to be 22 March 2009 and 28 September 2009, respectively. And to the best of our knowledge, few have tried to identify the peak period of the subprime crisis.

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Our results show that the subprime crisis started in 2007Q4 and ended in 2011Q3 and the peak period of the crisis is from 2009Q3 to 2010Q2. Compared to the previous

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opinions reviewed above, the conclusions are not completely consistent. The start time is later than the scholars’ judgment, because the data in the financial sheet usually lag. As

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for the end time detection, it is when the relationship between triggered risk and total risk returns to a steady state, so it is a relatively conservative judgment. The results also first present the peak period during the whole subprime crisis. Corresponding to the

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event at that time, on 2 November 2007, the global stock markets plunged and threatened to push the global economy into deep turmoil. While in the first half of 2011, the Dow Jones Industrial Average recovered to pre-crisis levels. Therefore, the results in this experiment are confirmed to some degree.

4 Conclusion The approach proposed in this paper can quantitatively detect multiple change points and peak periods simultaneously with the unbalanced panel data. The main idea is to detect the change of risk dependence based on copula. During a certain period of time, such as when financial crises occur, the risk department in a financial institution is under great stress. It is generally believed that the dependence between different types of risks during crisis and non-crisis periods is significantly different, that is, the financial crisis ruins the stable relationship of risks. So we use copula function to fully describe the 16

ACCEPTED MANUSCRIPT dependence structure of the triggered risk and the total risk. In the empirical section, the financial statement data of the top 100 American banks in Forbes’ list from 2006Q1 to 2013Q1 is applied to the proposed approach to detect the start point, end point and peak

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period of the subprime crisis in American banking.

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The results show that the crisis started in 2007Q4 and ended in 2011Q3, and the peak period of the crisis was from 2009Q3 to 2010Q2. Previous researches had used

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qualitative analysis as listed in 3.4. Although the start time detected in this paper is a little later than the previous judgment, it is fairly confirmed by the researchers’ opinions if we take the lag of financial statements into account. However, the conclusion of end time shows a larger difference. The end time detected here is calculated when the relationship

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between triggered risk and total risk returns to a steady state in this paper, so it is a relatively conservative judgment.

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This research is only a preliminary attempt towards multiple change-points detection using unbalanced panel data. There still exist some limitations that need to be addressed. For example, if the crisis has a high fluctuation in a very short time, this approach can

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hardly work because of the low frequency of the collected data. Further research will be

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carried out considering more objective quantitative approach and advanced change points detection approach.

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Acknowledgement

This research has been supported by grants from the National Natural Science

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Foundation of China (71071148, 71301087), Key Research Program of Institute of Policy and Management, Chinese Academy of Sciences and Youth Innovation Promotion Association of the Chinese Academy of Sciences. We also extend our appreciation to Jianming Chen, Shuang Yang and Xiaolei Sun for their great help.

Appendix Table 1 The Top 100 American banks in 2013 Forbes list Rank

Bank

TA

ROAE (%)

1

State Street

204

9.9

2

Bank of Hawaii

13

3

Signature Bank

16

CCAR(%)

LR (%)

0.04

19.8

7.6

16.2

0.8

16.1

6.8

12

0.6

16.2

9.6

17

NPLR (%)

ACCEPTED MANUSCRIPT Prosperity Bancshares

14

10.1

0.1

14.4

6.9

5

First Republic Bank

33

13.7

0.2

13.6

9.3

6

East West Bancorp

22

12

1

15.3

9.7

7

Westamerica Bancorp

5

15.4

1.3

15

8.6

8

Citizens Republic Bancorp

10

33

1.3

9

Commerce Bancshares

21

12

0.7

10

BankUnited

13

11.7

1.3

11

Community Bank System

8

9.3

12

PacWest Bancorp

6

9.1

13

BancFirst

6

10.4

14

SVB Financial Group

22

9.6

15

National Penn Bancshares

8

16

Bank of New York Mellon

340

17

First Citizens BancShares

18

Cullen/Frost Bankers

19

BBCN Bancorp

20

CVB Financial

21

UMB Financial

22

BOK Financial

23

T

4

IP

15.1

9.7 10

34.3

12.9

0.8

14.9

8.3

1.1

14.9

10.3

0.7

13.4

8.3

0.5

13.1

8

7.9

1

17.5

12.8

6.8

0.8

15.3

5.6

21

7.5

1.4

15.1

9.7

22

10

1.2

14.1

8.6

5

9.1

1.2

15.2

13.2

6

10.4

2.1

18

11.2

13

10.2

0.5

11.7

7.2

27

11.9

1.1

13.2

9.3

Central Bancompany

10

8.9

1.3

15.8

10.8

24

NBT Bancorp

6

10

1

10.8

8.5

25

1st Source

4

9

1.2

15.1

12.1

26

City National

26

9.3

0.9

9.2

6.3

27

Texas Capital Bancshares

10

17.3

0.6

10.4

9.6

28

Umpqua Holdings

12

5.6

1.2

15.9

11.4

29

Independent Bank

5

9.1

0.8

10.7

8.7

30

Columbia Banking System

5

6.2

2

19.5

12.8

31

WesBanco

6

7.4

1

13.2

9.1

32

Capital One Financial

302

8.9

1.7

12.7

9.9

33

First Financial Bancorp

6

9.7

2.5

16.9

10.5

34

U.S. Bancorp

352

15.4

1.2

10.9

9.2

35

Trustmark

10

9.1

1.5

15.4

10.8

36

First National of Nebraska

15

8.6

1.6

13.1

10.6

37

BB&T

182

9.9

1.4

10.9

7.9

38

FirstMerit

15

7.9

1.1

11.4

8.3

NU

MA

D TE

CE P

AC

SC R

14.9

18

ACCEPTED MANUSCRIPT Heartland Financial USA

5

12.7

1.5

13.7

10.1

40

Cathay General Bancorp

11

7.5

1.3

17.1

13.6

41

Northern Trust

94

9

1

12.8

8.1

42

Citigroup

1931

4

2.3

13.9

7.4

43

Susquehanna Bancshares

18

4.8

1

44

New York Community Banc

44

8.9

1

45

Huntington Bancshares

56

10.8

1.3

46

Brookline Bancorp

5

5.6

47

Investors Bancorp

11

9.7

48

Pinnacle Financial Partners

5

5.3

49

Fifth Third Bancorp

117

11

50

Sterling Financial

10

51

Boston Private Financial

6

52

Taylor Capital Group

53

Webster Financial

54

JPMorgan Chase

55

Farmers & Merchants Bank

IP

11.4

9 9

11.9

10.3

0.8

10.4

9

1.3

11.6

8.1

1

12.1

10.5

1.6

10.9

10.1

37.7

2.6

17.6

12.6

9.4

1.5

13.1

9.2

5

31.5

1.7

12.3

9.4

20

8.7

1.4

11.9

8.2

2321

10.7

3.1

11.9

7.1

5

9.3

3.5

28.6

14.5

16

7.7

1.8

13.4

10.7

MA

NU

SC R

13.6

D

TE

of Long Beach

T

39

Fulton Financial

57

IBERIABANK

13

4.6

1.7

13.3

10

58

Hancock Holding

19

5.2

1.6

12.5

9.2

59

F.N.B.

12

7.9

1.3

10.6

8.2

60

Valley National Bancorp

16

9.1

1.4

10.9

8.1

61

First Citizens

8

7.3

2.6

16.9

8.7

AC

CE P

56

Bancorporation 62

Glacier Bancorp

8

7.9

3.3

18.7

11.5

63

Oriental Financial Group

6

4.3

9

36.3

10.9

64

KeyCorp

87

8.5

1.4

12.1

11.4

65

SunTrust Banks

173

8.2

1.4

10.6

8.5

66

MB Financial

9

6.6

2.8

15.8

10.6

67

Park National

7

10.2

3.7

13.1

9

68

First Interstate BancSystem

7

7

3.1

14.5

9.6

69

Associated Banc-Corp

23

6

1.8

13.6

10

70

Zions Bancorporation

53

5.6

1.9

13.5

11.1

71

Tompkins Financial

5

8.6

1.9

12

8.5

19

ACCEPTED MANUSCRIPT International Bancshares

12

7.4

2.3

21.8

12.8

73

United Community Banks

7

6.5

2.7

14.3

9.8

74

United Bankshares

8

8.5

1.5

12.5

10.6

75

Chemical Financial

6

8.4

2.2

12.4

9.4

76

Provident Financial Svcs

7

6.8

2.1

77

Beneficial Mutual Bancorp

5

2.7

3.3

78

CVB Financial

17

6.2

1.2

79

Old National Bancorp

9

8.6

80

Capital Bank Financial

6

4.8

81

PNC Financial Services

301

7.8

82

First Niagara Financial

36

3.5

83

National Bank Holdings

6

84

Wells Fargo

1375

85

Popular

86

Bank of America

87

M&T Bank

88

TFS Financial

89

BancorpSouth

90

First BanCorp

91

T

72

IP

13.3

9.1 9.7

12.2

10.2

3.4

12.9

8.8

6.5

23.1

15.9

2.5

11.7

10.4

1.5

9.5

6.8

-0.1

4.5

51.5

17.7

12.3

3.4

11.5

9.4

37

4.3

7.7

16.8

11.4

2168

2.3

3.9

13.6

7.8

81

9.3

2.1

10.2

9.8

11

0.8

1.8

20.9

13.3

13

5.9

2.4

13.6

10.2

13

0

10.2

16.2

12.7

First Commonwealth Fin’l

6

3.6

2.1

13.7

11.7

92

Western Alliance Bancorp

7

7.2

2.3

11

9.7

93

Comerica

63

7

1.6

10.4

10.7

94

Regions Financial

122

2

3.1

11.5

9.1

95

TCF Financial

18

-12.8

3.1

10.4

8.7

96

PrivateBancorp

13

5

2

12.2

11.2

97

First Horizon National

26

-1.4

2.6

13.2

10.6

98

Synovus Financial

26

4.7

3.6

13.2

11

99

First Midwest Bancorp

8

-2.8

3.9

9.9

8.1

100

Doral Financial

8

-2.4

10.9

11.9

9.3

AC

CE P

TE

D

MA

NU

SC R

19.2

Note: TA stands for Total Asset in billion dollars; ROAE stands for Return on Average Equity (%); NPLR stands for Non-performing Loan ratio (%); CCAR stands for Core Capital Adequacy Ratio (%); LR stands for Leverage Ratio (%).

20

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