Bubble measurement and its contagion models in financial markets
محورهای موضوعی : Mathematical Methods in Financial, Physical, Biological, Social, and Behavioral Sciences.
Vahid Mohammadi
1
(Department of financial management,Central Tehran branch,Islamic azad univercity,Tehran, Iran)
Mir feiz Fallah shams
2
(Department of Financial Management, Tehran Azad University, Center, Tehran, Iran)
Gholamreza Zomorodian
3
(Department of Financial Management, Central Tehran Branch, Islamic Azad University, Tehran, Iran)
کلید واژه: RADF, BEKK GARCH, SADF, DCC-GARCH, GSADF,
چکیده مقاله :
The aim is to investigate three methods to measure the bubble and categorize the methods of its contagion .The price bubbles of the capital market were tested with three methods (RADF), (SADF) and (GSADF) and the dates of their formation and collapse were determined. Two models of contagion using DCC-GARCH and BEKK GARCH methods were expressed and compared.The results indicated four bubble periods as follows 2015:11:17-2016:02:09, 2017:06:13-2017:07:18, 2017:07:25-2018:01:30 and 2018:03:20-2020:12:16. The results showed that in all three methods, the existence of a price bubble in stock exchange companies was confirmed
The aim is to investigate three methods to measure the bubble and categorize the methods of its contagion .The price bubbles of the capital market were tested with three methods (RADF), (SADF) and (GSADF) and the dates of their formation and collapse were determined. Two models of contagion using DCC-GARCH and BEKK GARCH methods were expressed and compared.The results indicated four bubble periods as follows 2015:11:17-2016:02:09, 2017:06:13-2017:07:18, 2017:07:25-2018:01:30 and 2018:03:20-2020:12:16. The results showed that in all three methods, the existence of a price bubble in stock exchange companies was confirmed
Bubble measurement and its contagion models in financial markets | |||
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Article Info Article history:
Keywords: DCC-GARCH, BEKK GARCH, GSADF, SADF, RADF, |
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Abstract | |
The aim is to investigate three methods to measure the bubble and categorize the methods of its contagion .The price bubbles of the capital market were tested with three methods (RADF), (SADF) and (GSADF) and the dates of their formation and collapse were determined.Two models of contagion using DCC-GARCH and BEKK GARCH methods were expressed and compared.The results indicatedfour bubble periods as follows 2015:11:17-2016:02:09, 2017:06:13-2017:07:18, 2017:07:25-2018:01:30 and 2018:03:20-2020:12:16. The results showed that in all three methods, the existence of a price bubble in stock exchange companies was confirmed.
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1 Introduction
A bubble usually occurs when there is a successive increase and a sudden fall in prices in a certain period [1].A bubble is an increase in the price of an asset in a continuous process, in which the initial increase in price leads to expectation of future price increase and leads to the attraction of new buyers. According to the most accepted definition of a bubble, if the price of an asset deviates from its base price, it can be said that there is a bubble [2]. But usually after some time, this increase and deviation of the price is associated with opposite expectations and as a result a sudden decrease in the price, which often leads to financial crises [3].
The emergence of a bubble crisis in the financial markets can lead to the bankruptcy of large financial organizations, the panic of depositors, the bankruptcy of institutions, credit defaults, etc[18].The research conducted in this field shows that there has been a lot of research in developed countries regarding the bubble and its contagion, but in developing countries like Iran there is a research gap in the field of bubble detection with new methods. Also, most of the studies conducted in Iran have only investigated the existence or non-existence of the bubble, and few researches have been done in the field of bubble dating and determining the beginning and end of bubble periods. In this research, we tried to introduce the latest bubble measurement methods and use them to measure the Tehran Stock Exchange bubble.We will also determine the start and end periods of the bubble in Tehran Stock Exchange by using the provided methods. Another advantage is the estimation of three bubble measurement models in determining the optimal method for bubble detection. By comparing the three bubble measurement methods, we will show which one of the methods is operationally more capable of detecting the bubble.
Financial markets are influenced by each other and affect each other's efficiency. In other words, inefficiency and creating a crisis like a price bubble in a market as the stock exchange affects other markets. The existence of this strong relationship in markets has important applications for economic policy makers and capital budgeting decisions at the international level [4]
In due course, several definitions have been proposed for financial contagion, and exhaustive literature has developed. As a starting point, the term “contagion” refers to a “shift” in the interrelationship between financial markets during times of crisis and non-crisis. Following that, the transmission mechanism of the crisis is explained as investors’ irrational behavior. Additionally, contagion is defined as“ any channel connecting countries and causing markets to move in unison“ [5 6 7].
In the last decade, the occurrence of financial and economic crises in the world has had an impact on economic development and the principles of evaluating the severity of crises. Financial analysts and market participants were worried that the contagion of crises might cause instability in global financial markets .However , there is no consensus on how to measure the contagion of crises and their effects. The different models presented to investigate the contagion of crises and the dispersion of these models in different researches at the global level indicate that there is a gap in the field of investigating contagion models and comparing them. For the first time, this research has tried to provide a comprehensive view of bubble contagion models by comparing DCC-GARCH model with The VAR-BEKK-GARCH model. In this article, dynamic conditional correlation (DCC) estimators are proposed that have the flexibility of univariate GARCH but not the complexity of conventional multivariate GARCH. These models, which parameterize the conditional correlations directly, are naturally estimated in two steps a series of univariate GARCH estimates and the correlation estimate. These methods have clear computational advantages over multivariate GARCH models in that the number of parameters to be estimated in the correlation process is independent of the number of series to be correlated. Thus potentially very large correlation matrices can be estimated.
Several methods are reported in the literature to examine contagion. A method proposed for the measurement of cross-market turbulence is the correlation technique. To estimate varying time correlations among the markets DCC-GARCH model proposed by Engle [13] is deployed in studies(Corsetti et al. [8]; Chiang et al. [9]; Celık [10]; Syllignakis and Kouretas [11]; Ahmad et al. [12]; Corbet et al. [7])..DCC measures a non-linear correlation structure; additionally, the model’s parameters are not dependent on the selected variables. A measurement bias in contagion tests is heteroskedasticity. Utilizing the DCC-GARCH model, the issue is resolved as the residuals generated by the modelling technique are uniform[6].
Another important model that is frequently used for contagion in financial markets is The VAR-BEKK-GARCH. this model,is a multivariate GARCH model proposed by Engle and Kroner (1995), estimates the conditional mean function and the conditional volatility function of high dimensional relationships, which we use to test volatility spillovers between multi markets[14].
2 Preliminaries
In general, the main methods of investigating the existence of bubbles in asset prices that have been used in various studies can be classified into four categories:The first category is tests based on the Variance Bound , which was one of the first methods to evaluate the existence of bubbles in asset prices. In this test, it is stated that assuming the existence of rational expectations, the difference between actual and expected profit is not predictable and has zero mean. Also, the price variance is naturally Bounded; Because the expected increases in prices are uncorrelated with the information available in the market; Therefore, if the data crosses the variance boundary, it can be a sign that the asset price does not follow a bubble-free fundamental pricing equation Among the studies that have used this method, Shiller[19] and Akdeniz et al [20] can be mentioned However, due to the criticism of this method, it is not used in recent studies related to the investigation of bubbles[21].
Another category of methods for checking bubbles is known as West's two-step test. In this method, the presence of bubbles is directly included in the opposite hypothesis (hypothesis one). This method also faced criticism. Dezbakhsh and Demirguc-Kunt [22]stated that this test in Samples with few observations do not provide valid results. Also, Flood et al[23] criticized the method of estimating the bubble in this way.
Another category of bubble detection methods was introduced by Wu[24]. In this method, the bubble is considered as a deviation from the present value. One of the criticisms of this method is based on the results of this method. In the studies that have used this method, the bubble has been considered as a deviation from the fundamental prices, which has been obtained negative most of the time. This has not been confirmed in the framework of theoretical foundations [25].
The fourth category of bubble investigation methods is based on the concept of Cointegration. This method was introduced by Diba and Grossman[26], which was used in many studies. The most important criticism of this method was raised by Evans[27]. In his study, he showed that these tests cannot detect the collapse or bursting of the bubble; Because the bubbles in the collapses show a calm and continuous behavior rather than their behavior resembling an explosive process.Therefore, not rejecting the H0 hypothesis through this test cannot confirm the absence of bubbles in the time series of observations.
In order to overcome the criticism of Evans and some other criticisms of conventional methods, Phillips et al[15]. introduced the SADF method. In this method, it is possible to detect the increase in asset prices during inflationary periods. After that, due to the possibility of several bubbles occurring in a time series, Phillips et al[16]. introduced the GSADF method to check the existence of multiple bubbles. The main feature of this test is that it allows considering non-linear dynamics and structural break at the same time as examining multiple bubbles in the time series.
Subsequently,researchers such as Hu and Oxley [28] also used this method for the Japanese market. They used RTADF tests and it was interested by the researchers due to the advantages of dating bubble. Onyibor and safakli,[29] investigated the Turkish capital market bubble, they also used methods GSADF, SADF, RADF and ADF and indicated that there have been bubbles in the Turkish stock market and these bubbles reached their peak in the 2008 global crisis
Yan et al.[30] employed the method GSADF for examining the bubble in Bitcoin and periods of bubble and found four periods for the Bitcoin price bubble. They stated that the emergence and bursting of bubbles were more shaped in the time of occurrence of special events and news in each period and the Bitcoin market has not yet reached maturity.
The studies conducted in Iran show that most of the contagion models are based on different GARCH models
Hosseinyoun et al. [31] studied the volatility spillover in the stock and gold markers using VAR-MGARCH. Their results showed that there is a two-way shock transmission between the currency and gold markets and between gold and stocks. There is also a one-way shock from the stock market to the currency market.
Ranjbar and Sefidbakht [32] investigated the fluctuations and spillover in the oil, currency, gold and stock markets using the models BEEK and VAR and showed that there is a significant effect between the exchange rate and the stock index. But when structural failure is used in transactions, the results will be different.
Rostami and Farahmandi[33] investigated the effects of OPEC crude oil and West Texas crude oil yield contagion using the BEEK and VEC model.
2.1 right-tail augmented Dickey-Fuller (RTADF)
Phillips et al. [15] proposed a dating strategy based on the ADF statistic. In order to identify the time of emergence and collapse of bubbles or explosive growth, they suggested that the ADFn0 sequence test statistic to be compared with the right critical value of the standard (conventional) ADF sequence. The first observation whose ADF statistic became larger than the critical value is determined as the starting time and the start of estimation. They showed that SADF is able to identify a bubble in the desired time series. However, due to the possibility of more than one bubble, Phillips et al [16] introduced another strategy called Augmented Dickey-Fuller generalized supremum (GSADF). They showed that the GSADF was superior to the SADF in various aspects. In this research, the right unit root test based on forward regression is used to evaluate the repetitive behavior of the unit root with respect to explosive prices. Hence, the ADF and Phillips-Perron tests provide the basis for stationary tests with an explosive alternative as follows.
∆yt=α+βyt-1 +
+ εtεt῀ NID(0, σ 2) (1)
Whereyt is the total index of the stock exchange, α is the constant value, L is the number of delays, εt is the error and NID represents the independent normal distribution.
SADF(r0) = sup {ADFr2}
r2€[r0,1] (2)
Deciding on the existence of an explosive process in a time series is done by the SADF test based on the Supremum value of the ADF test sequence compared to the right critical value of the sequence with its limited distribution. Under the null hypothesis, if the ADF statistic for the sub-period is specified as (0, r2), then the corresponding ADF statistic supremum is estimated as follows:
SADF(r0) = sup {ADF r1r2} (3)
r1€[0,r2-r0]
r2€[r0,1]
where
)4)
where W ~ and W are the standard Brownian process
) 5)
The GSADF statistic can be defined as the largest available limited ADF statistic of r1 and r2. The statistic of the GSADF test is:
(6)
The advantage of SADF and GSADF tests is that they test the non-stationary behavior of time series against the presence of mild explosive behavior periodically. Mild explosive behavior can be modeled through an autoregressive process with a root greater than one and close to one. Therefore, if there is an explosive root (if the assumption against mild explosive behavior is not rejected), the SADF and GSADF tests will provide a means to determine the behavior of the bubble and the date of its occurrence and destruction.
2.2Dynamic Conditional Correlation, GARCH (DCC-GARCH) Model
The estimation of the DCC-GARCH model involves a two-step process. GARCH parameters are calculated followed by time-varying conditional correlations. The following equations must be solved to obtain the results of the analysis.
In the first equation, Xt-1 demonstrates lagged returns, and εitis the error term with conditional variance hit , whereas 
represents a vector of residuals
Xt= a + cXt-1+ εit (7)
Ht=DtRtDt
In Equation (3), time-varying conditional correlations are estimated [12]. Dtis a diagonal (s ˟s) matrix of conditional standard deviations from univariateGARCH. Rtis (s ˟s) dynamic correlation matrix, whereas Htrepresents multivariateconditional variance.
(8)
Rt =ɚt*-1ɚtɚt*-1
ɚt*-1is the diagonal matrix consisting the square root of diagonal element in ɚt
(9)
Here, ɚt= (qij,t) is(s ˟s)positive definite matrix of εitand ɚt should meet a condition
(10)
Conditional correlation is computed with estimates of the univariate GARCH (1,1) model, wherein αand βare the two parameters of model [13]
(11)
ɚt =(1-a-b)
t +a1 εt-1
t-1+b ɚt-1
The multivariate DCC parameters aand b are non-negative. Additionally, a+bshould lie between zero and one.
Dt in Equation (4) it can be estimated as 
The likelihood function for estimating the two-step DCC-GARCH model of Engle is described as:
(12)
2.3BEKK GARCH model
The full BEKK-GARCH model imposes positive definiteness restrictions.
It specifies Ht as:
(13)
Where
and Ht denotes the conditional variance-covariance matrix.By multiplying the matrices, the following equation is obtained:
(14)
(15)
(16)
The elements of matrix A indicate the coefficients of the ARCH term, which capture, (a) the effects of past own shock on the conditional volatility of the same series, and (b) the effect of past shocks in one series on the conditional volatility of other series. The elements of matrix B represent the coefficients of the GARCH term, which captures, (a) the effects of past own volatility on the conditional volatility of the same series, and (b) the effect of past volatility in one series on the conditional volatility of other series.
Compare the difference between the two models
The key difference between DCC-GARCH and BEKK (a popular multivariate GARCH) is that BEKK assumes constant conditional correlation between assets, i.e. the change in the covariance between two assets with time is due to the changes in the two variances (but the conditional correlation is constant.
The DCC-GARCH model is a forecasting model that quantifies volatility persistence for a sample period, while also providing dynamic correlation coefficients for the same period.This model has an advantage over the BEKK GARCH model as there is no dimensionality issue in the DCC-GARCH model, and it can include many variables. Another competing model, the constant conditional correlation (CCC) model, is unsuitable for our study as it does not capture variables’ dynamic interaction. Several studies, such as Yousaf et al. (2021)[17], deployed DCC-GARCH to examine financial contagion
Empirical example for bubble test
Statistical population: The statistics of this research included all stock exchange companies that have been examined in a six year period (2015-2020). Various tests have been used including sequence, skewness, kurtosis, co-linearity, fractional accumulation and unit root in domestic studies to investigate the bubble, however, these tests are not able to determine the date of the bubble occurrence. The relevant tests are only able to check the presence or absence of bubbles. It is necessary to use tests based on Augmented Dickey-Fuller right-tailed test RTADF to determine the date of the bubble occurrence
Table 2: The results of right-tailed unit root tests for the stock market
Index | Test | p-value | Statistic | Critical value (1%) | Critical value (5%) | Critical value (10%) |
Total stock price index | ADF | 0.0060 | 0.7393 | 0.5546 | -0.0976 | -0.4130 |
| RADF | 0.0000 | 7.1316 | 0.7083 | -0.0146 | -0.03840 |
| SADF | 0.0000 | 19.3622 | 2.0830 | 1.4667 | 1.1800 |
| GSADF | 0.0000 | 19.3622 | 2.8841 | 2.1383 | 1.9381 |
Table 2 shows the results of RTADF tests of the stock market. It can be seen that the ADF, RADF and SADF tests indicate the rejection of the null hypothesis of the existence of a unit root in the 99% confidence interval. In other words, the results depict the existence of a bubble in the stock price index. In addition, the findings of the GSADF test do not reject the explosive behavior and the existence of multiple bubbles during the period of 2015-2020 in the Tehran Stock Exchange. In other words, in this period of time, formation and then collapse of the bubble in the Iranian stock market can be observed several times.
Figure 1: GSADF test for total stock price index
Source: Researcher's findings
Figure 1 shows price bubble periods in Tehran Stock Exchange based on the GSADF test for the total price index. In this figure, the green (upper) curve indicates the total index, the red (middle) curve depicts the critical values at the 95% level, and the blue (lower) curve shows the GSADF test statistic. As you can see in the figure 1, whenever the blue line that indicates the GSADF test statistic is higher than the red line that indicates the critical values, it means that there is a bubble. In other words, whenever the blue line crosses the red line upwards, the intersection point indicates the start time of the bubble, and whenever the blue line crosses the red line downwards, the intersection point indicates the end time of the bubble. Also, the comparison of the green line with the blue line shows how the situation of the total index was compared to the bubble periods. Accordingly, the GSADF unit root test for the total stock price index determines four bubble periods, which correspond to the periods 2015:11:17-2016:02:09, 2017:06:13-2017:07:18, 2017:07:25-2018:01:30 and 2018:03:20-2020:12:16. This evidence shows that from the end of the first half of 2017 to the end of 2020, except for short periods, the price in the Tehran Stock Exchange was accompanied by a bubble. The most important advantage of using RADF, SADF and GSADF methods compared to other bubble identification methods is to simplify the process of showing the bubble. so that the process of comparison can be understood. Comparing time series with critical values makes it possible to identify bubble periods at any moment. Bubbles can be caused by various reasons. Among the reasons that can play a role in creating a bubble are changes in the exchange rate in Iran, changes in the central bank's policies, the creation of economic crises, sanctions, etc.
Next, in order to experimentally compare the RADF, SADF and GSADF tests in the detection of bubbles, the results of the RADF and SADF tests are presented in figures 2 and 3. These tests have lower accuracy and power in determining bubble periods than GSADF. As Phillips and Shi and Yu [16] also indicated, the GSADF test significantly improves the ability to detect multiple bubbles.
Figure 2: SADF test for total stock price index
Source: Researcher's findings
Figure 3: RADF test for total stock price index
Source: Researcher's findings
In this article, we have practically shown that the best bubble test method in Tehran Stock Exchange is the GSADF method. Therefore, investors, legislators, and researchers in bubble estimation can use it to investigate the market bubble based on the advantages that this method creates over other bubble estimation methods. Also, in this research, we showed in a practical way how the start and end time of the bubble in the stock exchange is determined by this method. As a result, it can be concluded that new methods that allow determining the time of the beginning and end of the bubble create a significant advantage for researchers and investors. Providing the possibility of comparing different variables during the time period of the bubble is a very important feature that is one of the capabilities of new methods. legislators can measure the effects of policies and changes during the bubble period and use it to prevent future crises. Due to the gap in the field of contagion models in Iran, in this research, two bubble contagion models were introduced and compared for the Iranian market. The use of these models can help investors and researchers to discover the origin and reasons of the bubble. and understand how the crisis is transmitted between financial markets.
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