This study investigates whether the COVID-19 pandemic has caused asset price bubbles in the stock and oil markets in the United States and Malaysia. More specifically, the study seeks to detect the onset and end of possible speculative bubbles and their causes in these markets. It also examines the existence of a contagion effect between the stock and oil markets during the Covid-19 pandemic. To achieve these objectives, the study used the Generalized SADF (GSADF) developed by Phillips et al. (2015) in order to check for existence of bubbles within the time frame from January 1, 2020, to April 24,2020. This technique allows one to look for the occurrence of multiple bubbles during the sample period with great precision. The findings showed that five out of the six equities, including the oil price indices had multiple bubbles. Evidence was also obtained which linked the explosive activity episodes between the crude oil market and the US stock markets from the start and end point of each bubble event. These findings add not only to the literature on the existence of bubbles in the financial and energy markets during the initial outbreak of COVID-19, but also to the significance of the negative impact of pandemics on bubble contagion effects under extreme market conditions.
The world is currently paralyzed by the COVID-19 outbreak that began in Wuhan, China in late December 2019. The stock markets plummeted across the world as investors fled the markets. As the novel COVID-19 virus scare continues to create havoc in the capital markets, it is important to investigate the market crashes due to the pandemic. Baker et al. (2020) provides several reasons why such a pandemic has a powerful impact on financial markets. They argue that the gravity of this pandemic, its high contagiousness and the large number of infections and deaths resulting from it, have all contributed to making the stock market shock critical.
To underscore the gravity of the pandemic, Figure 1 shows the COVID-19 cases for China and selected Euro nations. Countries which have been recording the highest number of daily cases are China, Germany, Italy, Spain, and the UK. Except for China, the peak for these countries was detected between 9 March to 14 April 2020. In Malaysia, the jump in the number of daily new cases was detected from 13 March 2020 until 14 April 2020 (see Figure 2). As for the US, it started to show a jump in the daily new cases on 17 March and continued until late April 2020. The US reported the highest market turbulence since the global financial crisis in December 2008 (Baker et al., 2020). This was following the US stock market crash by 20 percent on 11 and 12 March 2020 (Giglio et al., 2020).
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behavior episodes in selected US and Malaysia stock markets during the first four months of the COVID-19 outbreak, which was from 2 January 2020 to 24 April 2020. This period was selected because it covered the non-COVID-19 phase (2 January, 2020 – 16 January, 2020) and the early pandemic phase (17 January– 24 April 2020). Since the period of study also coincided with the Saudi-Russia oil price war in early March 2020, the study also investigated the bubble episodes in crude oil market and determined whether there existed a contagion effect between the bubbles in the equity markets and the oil markets.
As the epidemic wreaked havoc on world markets, determining the exact date when the initial bubbles occurred, how long each bubble lasted and the possible causes for each bubble became important for policy makers and investors alike. This is especially true given that the pandemic is still having an impact on the equities markets, any significant increase in COVID-19 cases is likely to spark future episodes of explosive behavior in the stock markets. Therefore, to test for bubbles in the stock markets and energy sector, the Supremum Augmented Dickey–Fuller (SADF) test introduced by Phillips et al. (2011) and the Generalized SADF (GSADF) by Phillips et al. (2015) were also used in the present study. While the SADF test detects a single bubble episode, the GSADF test overcomes this constraint by assessing the explosive behavior with multiple bubbles, resulting in more robust estimations for the present study.
The USA and Malaysia stock markets were selected for the following reasons. First, the USA has the largest stock markets in the world and one of the earliest markets to jolt following the coronavirus outbreak in February 20204. Second, the USA was also the largest oil producer in 2020, hence was directly affected by the oil price crash. Third, Malaysia was chosen due to its significant economic relations with the USA, with their bilateral trade volume surpassing RM100 billion in 2020, or 11.1 percent of the total Malaysian exports (MATRADE, 2020). Given that Malaysia was severely impacted by the GFC (Alp et al., 2012), which was precipitated by the collapse of the US subprime market, there was the anticipation that there would be a contagion effect from the asset price bubbles in the USA into the asset markets of Malaysia during the COVID-19 pandemic. The theorized contagion effect between these markets was seen as highly possible, given how strong the correlation between these markets was at the time of the study (evidence of this correlation is provided in the fourth section of the paper).
Besides the KLSE index, the present study also considered the Shariah Index for the asset price bubbles analysis of Malaysia. Given the differing rules and regulations that govern these two markets5, this study wanted to examine if there was a major difference (or differences) in how these two indices responded to the turbulent market conditions during the COVID-19 epidemic. This was because the Islamic stocks were hypothesized to be less vulnerable to shocks due to the lower leverage, smaller business size, and under-diversified market (Rizvi et al., 2015). It has also been found that the Islamic finance portfolio performed better than their conventional counterpart in the early wake of the 2008 GFC (Zhang et al., 2020). Therefore, findings from the analysis in the present study have important implications for investors who are looking for an alternative form of stock market that will be better in safeguarding investors’ rights from the numerous financial crises that have plagued the world over the past several decades.
This study reported in this paper contributes to the literature in several ways. First, it has shed light on the explosive behaviors and contagion effects in the US stock and crude oil markets during the initial phase of the COVID-19 outbreak from 2 January 2020 to 24 April 2020. Second, it has investigated bubbles episodes in the Malaysian stock market and its linkage with the implosion of COVID-19 cases from the US stock markets and crude oil markets. Third, it has compared the explosive behavior episodes between the conventional stock markets and the Islamic stock markets in the markets in Malaysia during the early wave of the COVID-19 pandemic. With this analysis, it was hoped that researchers can draw new insights about the reactions of investors and traders on these two markets during the initial pandemic outbreak.
The analyses have resulted in several significant outcomes. First, there was evidence of multiple bubbles in five out of the six of equity series. Second, there was evidence linking explosive behavior episodes between the crude oil market and the US stock markets from the date- stamp of the starting and ending points of each bubble incident. In contrast, the stock market in Malaysia exhibited explosive behavior more closely to the date-stamp of the COVID-19 implosion of local cases around the 13th of March, 2020. There was also evidence linking the bubbles episodes in the US and the crude oil market with the stock market in Malaysia. Finally, the empirical results showed that multiple bubbles episodes were detected in the Shariah stock market, as opposed to a single bubble in the KLSE index in Malaysia. Two bubbles episodes that were discovered only in the Shariah index corresponded to the initial spread of the COVID-19 virus in China, when the global market was still relatively unperturbed by the pandemic. The results of the present study seemed to suggest that the Islamic stock market could detect abnormal market behavior earlier, as compared to the conventional stock market. However, these results were limited to the Malaysian market experience, and more evidence is needed to generalize the findings to other Islamic stock markets.
The rest of the paper is organized as follows: Section 2 presents the literature review; Section 3 describes the method and data; Section 4 introduces the results and Section 5 concludes the paper.
LITERATURE REVIEW
The COVID-19 pandemic has been regarded as the most devastating worldwide health catastrophe since the Spanish flu in 1918. Nevertheless, there has been relatively little past research on how epidemics influence financial markets. As a result, the present study offers an important contribution to the literature in the field. According to Chen et al. (2009) and Loh (2006), the Severe Acute Respiratory Syndrome (SARS) outbreak had a detrimental impact on industries such as those in aviation, tourism, wholesale, and retail. Previous studies have also indicated that the severity of the pandemic might predict the likelihood of an equity market crash. Giglio et al. (2020) and Wen et al. (2019a, b), for example, demonstrated that short-run investor expectations might correlate with the risk of a stock market crash. However, previous research by Giglio et al. (2020, 2021) has also revealed that the likelihood of an equity market crash occurring before a crisis was lower since investors were more bullish about stock market returns.
Zhang et al. (2020) found that COVID-19’s rapid spread had a major impact on financial markets worldwide, resulting in huge increases in the global financial market risk and huge losses to investors for a short period of time. The study by Zeren and Hizarci (2020) which looked at the impact of COVID-19 shock on the stock markets of six countries found a co-integrating relationship between the daily total case and stock markets returns. Yilmazkuday (2020) analyzed the COVID-19 impact on the S&P 500 index and found a significant relationship between the two. Awadhi et al. (2020) also examined the effect of COVID-19 on stock market returns and revealed that there were sectoral differences in the market returns. According to Mazur et al. (2020), the March 2020 financial market meltdown was caused by government reactions. Their analysis also corroborated the findings of Mishkin and White (2002), who discovered that a stock market crash might result in a loss of 20 percent –25 percent in the United States (U.S.) equities index, compared to past crises (e.g., World War I, World War II, and so on) owing to a series of panic selling.
Several studies have looked at asset price bubbles when markets were in distress. A recent study by Chang et al. (2021) showed the existence of bubbles in the US stock market during the early months of the COVID-19 outbreak by using the GSADF test. Focusing on the gold and crude oil markets, Gharib et al. (2021) discovered that there existed mild explosive price behaviors in the WTI and gold markets from January 4, 2010, to May 4, 2020. The most notable finding was that a bilateral contagion impact of bubbles was found during the recent COVID-19 outbreak in the oil and gold markets. Zhao et al. (2020) examined the bilateral contagion effect of bubbles between oil price and the Chinese stock markets. They detected six bubbles in the oil and stock markets from September 1, 2004, to July 9, 2018. Another study by Sharma and Escobari (2018) have highlighted the existence of bubbles in the energy sector. They found that a strong spike in oil prices had caused the bubble to explode in 2015. Li et al. (2020), in their study of natural gas price bubbles in three major economies from 1996 to 2017, found that the 2008 global financial crisis (GFC) contributed to the rapid variations in natural gas prices in all three countries.
RESEARCH METHOD AND DATATheory and MethodA large set of studies in the existing literature have investigated
asset price bubbles using different asset-pricing models (Gürkaynak,
2008, De Long et al., 1990, Tirole, 1985). Price bubbles occur when
commodity prices deviate from their true values. Many researchers
have considered the model used by Gürkaynak (2008) as the most
reliable one from among the others highlighted in the aforementioned
studies. Three main assumptions formed the basis of the Gürkaynak
model. First, market information is assumed to be non-asymmetric,
such that uninformed traders are unable to influence asset price by
manipulating information about prices. Second, consumers are
assumed to be risk neutral with no premium on risk. This means that
fluctuations in prices are not caused by risk premiums that varies
𝑃𝑃𝑡𝑡 = (1 + 𝑟𝑟𝑓𝑓 ) 𝛦𝛦𝑡𝑡 (𝜕𝜕DETECTING
𝑡𝑡+1 + ℧ )
−1𝑡𝑡+1 ASSET PRICE BUBBLES DURING CO
𝑃𝑃𝑡𝑡 DETECTING
𝑃𝑃𝑡𝑡𝑃𝑃𝑡𝑡==(1(1+ASSET 𝛦𝛦𝑡𝑡𝛦𝛦PRICE
−1
+𝑟𝑟𝑓𝑓𝑟𝑟)𝑓𝑓 ) IMPLICATIONS:
(𝜕𝜕
𝑡𝑡 (𝜕𝜕 BUBBLES
𝑡𝑡+1++℧
𝑡𝑡+1 ℧
𝑡𝑡+1 DURING
) ) EVIDENCE
𝑡𝑡+1 COVID-19
FROM STOCK C
A
t t DETECTING IMPLICATIONS: ASSET EVIDENCE PRICE BUBBLES FROM
𝛦𝛦𝑡𝑡 DURING
STOCK AND COVID-19
OIL
The International 𝑃𝑃𝑡𝑡 Journal IMPLICATIONS: of Banking and Finance, Vol. 𝑃𝑃 17, = (1𝑃𝑃+
𝑡𝑡EVIDENCE
Number 𝑟𝑟𝑓𝑓 )−1
2𝑡𝑡 (July) 𝛦𝛦𝑡𝑡 (𝜕𝜕
FROM
2022, pp:𝑡𝑡+191–114 +STOCK
℧𝑡𝑡+1 ) AND OI
t DETECTING ASSET PRICE BUBBLES DURING COVID-19 CRISI
𝑃𝑃𝑡𝑡𝑃𝑃𝑡𝑡
𝛦𝛦𝑡𝑡 𝛦𝛦 𝑡𝑡 IMPLICATIONS: EVIDENCE FROM 𝜕𝜕𝑡𝑡+1 and STOCK ℧𝑡𝑡+1 AND OIL MAR
t
with timet and price variations. Third, the 𝑃𝑃𝑡𝑡 discount rate is assumed
to remain 𝛦𝛦𝑡𝑡 constant. In short, the general model which allows for the
𝜕𝜕𝑡𝑡+1 and ℧ tt 𝑃𝑃𝑡𝑡 = (1 + 𝑟𝑟𝑓𝑓 )−1 𝛦𝛦𝑡𝑡 (𝜕𝜕𝑡𝑡+1 + ℧𝑡𝑡+1 )t+1,
𝜕𝜕𝑡𝑡+1 and ℧
presence of𝑡𝑡+1 bubbles 𝑡𝑡+1 is given as follows: 𝛦𝛦𝑡𝑡
𝛦𝛦𝑡𝑡 −1
𝑃𝑃𝜕𝜕𝑡𝑡 = (1 + 𝑟𝑟𝑓𝑓 ) 𝛦𝛦 𝑡𝑡 (𝜕𝜕 𝑡𝑡+1 + ℧ 𝑡𝑡+1 ) t
𝑡𝑡+1 and ℧𝑡𝑡+1
t+1, 𝑃𝑃
t+1, 𝑡𝑡 = (1 + 𝑟𝑟𝑓𝑓𝑡𝑡)−1𝑃𝑃𝛦𝛦𝑡𝑡 (𝜕𝜕𝑡𝑡+1 + ℧𝑡𝑡+1 )
𝛦𝛦𝑡𝑡𝛦𝛦 ∞ [1]
𝑖𝑖
𝑃𝑃𝑡𝑡 = (1 + 𝑟𝑟𝑓𝑓 )−1 𝛦𝛦𝑡𝑡𝑡𝑡(𝜕𝜕𝑡𝑡+1 + ℧𝑡𝑡+1 ) 𝜕𝜕𝑡𝑡+1 and 𝑓𝑓 ℧
𝑃𝑃𝑡𝑡 =𝑡𝑡+1
𝜕𝜕𝑡𝑡+1 and ℧𝑡𝑡+1 ൬1+𝑟𝑟 ൰ 𝛦𝛦𝑡𝑡 (𝜕𝜕𝑡𝑡+1 + ℧
where 𝑃𝑃t+1, 𝑡𝑡 is the asset price in time t and 𝛦𝛦𝑡𝑡 signifies the expectation.
𝑓𝑓
∞ 𝑖𝑖=0
∞
The tand℧℧ 𝑖𝑖
𝑓𝑓 coefficients and represent thet+1, return and hidden
𝑃𝑃 𝑖𝑖 𝜕𝜕 and
𝑃𝑃𝑡𝑡𝑡𝑡𝑓𝑓 =൬ 1 𝜕𝜕𝑡𝑡+1 𝑡𝑡+1 1
𝑃𝑃 = t+1, ൰ 𝛦𝛦 ൬1+𝑟𝑟
(𝜕𝜕 ൰+𝑡𝑡+1𝛦𝛦℧𝑡𝑡𝑡𝑡+1
(𝜕𝜕 ), 𝑖𝑖 +
𝑡𝑡+𝑖𝑖𝑡𝑡+1 =For ℧ 𝑡𝑡+𝑖𝑖 ),…
0,1,2 𝑖𝑖 𝑛𝑛
= 0,1,2 … 𝑛𝑛
component for1+𝑟𝑟
𝑡𝑡 time 𝑓𝑓
∞ 𝑖𝑖=0 t+1, 𝑡𝑡 respectively.
𝑡𝑡+1 𝑓𝑓 the present study𝑓𝑓
Equation
[1] has𝑃𝑃tbeen 𝑓𝑓
𝑖𝑖=0
transformed 𝑖𝑖
1 into the following 𝜕𝜕𝑡𝑡+1 and ℧𝑡𝑡+1
Equation 𝑃𝑃𝑡𝑡
[2]:
𝑡𝑡 = ൬ ൰𝛦𝛦𝑡𝑡𝛦𝛦𝑡𝑡 (𝜕𝜕𝑡𝑡+1 + ℧𝑡𝑡+𝑖𝑖 ), 𝑖𝑖 = 0,1,2 … 𝑛𝑛 ∞
tt t+1, t+1,
∞ 1+𝑟𝑟𝑓𝑓
𝑖𝑖 𝑓𝑓 1
𝑖𝑖
𝑖𝑖=0
𝑓𝑓 𝑓𝑓
1 𝑃𝑃 = ൬1+𝑟𝑟 [2] ൰ 𝛦𝛦𝑡𝑡 (𝜕𝜕𝑡𝑡+1 + ℧𝑡𝑡+𝑖𝑖 ), 𝑖𝑖
𝑃𝑃𝑡𝑡 𝑃𝑃𝑡𝑡 = ൬ ൰ 𝛦𝛦𝑡𝑡 (𝜕𝜕𝑡𝑡+1 + ℧𝑡𝑡+𝑖𝑖 ), 𝑖𝑖 = 0,1,2 … 𝑡𝑡 𝑛𝑛
𝜕𝜕𝑡𝑡+1 𝑓𝑓
𝜕𝜕𝑡𝑡+1 and ℧𝑡𝑡+1 t+1,
1+𝑟𝑟𝑓𝑓
𝛦𝛦𝑡𝑡 𝑖𝑖=0
𝑖𝑖=0
𝛦𝛦𝑡𝑡 ∞∞
where 𝑃𝑃𝑡𝑡𝑓𝑓 𝛦𝛦 is𝑡𝑡 the underlying 𝑓𝑓 𝑓𝑓 stock market 1 1 (crude oil) index (price) and
𝑖𝑖 𝑖𝑖
𝜕𝜕𝑡𝑡+1 is the 𝑡𝑡+1 𝜕𝜕 stock market 𝑃𝑃 𝑃𝑃 = = (crude oil) ൬ ൬ ൰
return൰𝛦𝛦𝑡𝑡𝛦𝛦(𝜕𝜕𝑡𝑡in(𝜕𝜕
𝑡𝑡+1 +
period
𝑡𝑡+1 𝑓𝑓+ ℧℧𝜕𝜕𝑡𝑡+𝑖𝑖 ),),𝑖𝑖 𝑖𝑖==0,1,2
𝑡𝑡+1 . Equation
0,1,2……𝑛𝑛𝑛𝑛
𝑓𝑓
𝑡𝑡 𝑡𝑡
t+1,
1+𝑟𝑟1+𝑟𝑟
𝑓𝑓 𝑓𝑓 𝑃𝑃𝑡𝑡 ∞𝑡𝑡+𝑖𝑖
𝜕𝜕𝑡𝑡+1
[2] represents 𝑃𝑃 and ℧components 𝑖𝑖
𝜕𝜕𝑡𝑡 𝑡𝑡+1the and 𝑡𝑡+1 ℧𝑡𝑡+1 of𝑖𝑖=0
𝑖𝑖=0 the underlying
𝑃𝑃
𝑓𝑓
=
price without
1 bubbles.
൰ 𝛦𝛦𝑡𝑡 (𝜕𝜕 ),
𝜕𝜕
The asset-pricing 𝜕𝜕
𝑡𝑡+1 𝑡𝑡+1
𝜕𝜕𝑡𝑡+1 .
and ℧
model with the presence of bubbles is1+𝑟𝑟
𝑡𝑡+1 𝑡𝑡 depicted
𝑓𝑓𝑓𝑓 by𝑡𝑡+1 + ℧𝑡𝑡+𝑖𝑖 𝑖𝑖 =
𝜕𝜕𝑡𝑡+1 . 𝑖𝑖=0 𝑃𝑃𝑡𝑡 = 𝑃𝑃𝑡𝑡 + 𝐵𝐵𝑡𝑡
the following 𝜕𝜕𝑡𝑡+1
Equation 𝑓𝑓 𝑓𝑓
[3]: ∞
𝑖𝑖 𝜕𝜕𝑡𝑡+1
t+1, t+1, 𝑃𝑃 𝑃𝑃
𝑡𝑡 𝑡𝑡
𝑓𝑓 1
𝑃𝑃𝑡𝑡 = ൬1+𝑟𝑟 ൰ 𝛦𝛦𝑡𝑡 (𝜕𝜕𝑡𝑡+1 + ℧𝑡𝑡+𝑖𝑖 ), 𝑖𝑖 = 0,1,2 … 𝑛𝑛
𝜕𝜕𝑡𝑡+1 .
𝑓𝑓 t+1,
𝑓𝑓
𝑃𝑃𝑡𝑡 = 𝑃𝑃𝑡𝑡 +𝑃𝑃𝐵𝐵𝑡𝑡 𝑡𝑡= 𝑃𝑃𝑡𝑡 + 𝐵𝐵𝑡𝑡
𝑓𝑓
𝑖𝑖=0 𝑓𝑓 [3]
𝑃𝑃 𝜕𝜕𝑡𝑡+1 . 𝐵𝐵𝑡𝑡 ≠ 0,
𝜕𝜕𝑡𝑡+1 . ∞∞ 𝑡𝑡
𝑖𝑖
Equation 𝑓𝑓 =[3] 𝑓𝑓𝑃𝑃 𝑓𝑓 is 𝜕𝜕𝜕𝜕∞𝑡𝑡+1
+ 𝐵𝐵represented 1 𝑖𝑖
൬ 1 𝑓𝑓൰൰ 𝛦𝛦𝛦𝛦by two+ ℧ elements. The…first element
𝑃𝑃𝑃𝑃𝑡𝑡𝑡𝑡 𝑃𝑃 =𝑡𝑡 = 𝑡𝑡
𝑡𝑡 𝑡𝑡+1 ൬ 𝑃𝑃𝑡𝑡𝑓𝑓𝑓𝑓1 of𝑡𝑡𝑖𝑖𝑡𝑡(𝜕𝜕
1+𝑟𝑟
(𝜕𝜕𝑡𝑡+1
𝑡𝑡+1 + ℧𝑡𝑡+𝑖𝑖
), 𝑖𝑖 = 0,1,2
𝑡𝑡+𝑖𝑖 ), 𝑖𝑖 = 0,1,2 … 𝑛𝑛
𝑛𝑛
𝐵𝐵𝑡𝑡 ≠ 0, 𝐵𝐵𝑡𝑡𝑓𝑓the
represents ≠ 0, present value
1+𝑟𝑟 the projected capital return,
𝐻𝐻 𝑓𝑓 : 𝛣𝛣 = and
1 the
𝑃𝑃
𝑃𝑃𝑡𝑡𝑡𝑡 ==𝑃𝑃𝑡𝑡 + 𝐵𝐵𝑡𝑡 ൬the
𝑓𝑓 𝑖𝑖=0 𝑖𝑖=0 ൰ 𝛦𝛦𝑡𝑡 (𝜕𝜕𝑡𝑡+1 + 𝜕𝜕𝑡𝑡+1 ℧𝑡𝑡+𝑖𝑖 ),𝑃𝑃𝑡𝑡𝑖𝑖 = = 𝑃𝑃0,1,2
𝑡𝑡 + 𝐵𝐵…
𝑡𝑡 𝑛𝑛
second element captures 1+𝑟𝑟asset
𝑓𝑓 price bubbles. In Equation [3], price
bubbles𝐵𝐵𝑡𝑡are ≠ 0, treated𝜕𝜕as
𝐻𝐻𝑓𝑓0 : 𝛣𝛣 = 1
𝜕𝜕𝑖𝑖=0
𝑡𝑡+1 a. factor
𝑡𝑡+1 .
𝜕𝜕𝑡𝑡+1 in the pricing of an asset,𝑡𝑡 rather than a 𝑡𝑡−1
𝑦𝑦 = 𝑐𝑐𝑁𝑁 −𝜂𝜂 + 𝜃𝜃𝜃𝜃 + 𝜀𝜀𝑡𝑡
𝐻𝐻0 : 𝛣𝛣 = 1 𝑃𝑃
collective 𝑓𝑓 error
𝑡𝑡 by traders. Thus, when assuming 𝐵𝐵𝑡𝑡 ≠ 0, an interesting
𝑃𝑃𝑡𝑡 𝐵𝐵𝑡𝑡 ≠ 0, −𝜂𝜂 𝜕𝜕 .
premise
𝑦𝑦𝑡𝑡 = 𝑐𝑐𝑁𝑁𝐻𝐻0𝑃𝑃−𝜂𝜂 can
𝑦𝑦
: 𝛣𝛣 𝑓𝑓
𝑡𝑡+ = = be 𝑐𝑐𝑁𝑁articulated.
𝜃𝜃𝜃𝜃1𝑡𝑡−1 ++𝜀𝜀𝑡𝑡𝜃𝜃𝜃𝜃𝑡𝑡−1 Evans (1991) 𝑡𝑡+1
𝑓𝑓 𝑓𝑓 + 𝜀𝜀𝑡𝑡
opines thatNthe traditional
unit root tests 𝑡𝑡 𝑃𝑃𝑡𝑡𝑃𝑃𝑡𝑡==𝑃𝑃𝑡𝑡𝑃𝑃𝜕𝜕𝑡𝑡price
to ascertain ++𝐵𝐵
𝑡𝑡+1 . 𝐵𝐵𝑡𝑡 bubbles
𝑡𝑡 are ineffective when cyclical
𝜕𝜕𝑡𝑡+1 −𝜂𝜂 𝐻𝐻0 : 𝛣𝛣 = 1
unsustainable
𝑦𝑦𝑡𝑡 𝐻𝐻 =0 𝑐𝑐𝑁𝑁 : 𝛣𝛣behavior
= 1+ 𝜃𝜃𝜃𝜃𝑡𝑡−1 occurs + 𝜀𝜀𝑡𝑡 in the period. Therefore, 𝑓𝑓 Phillips and Yu
𝜕𝜕𝑡𝑡+1 N 𝑃𝑃 = 𝑃𝑃 + 𝐵𝐵
(2011) constructed the Supremum
N Augmented Dickey–Fuller
𝑡𝑡 𝑡𝑡 𝑡𝑡
𝑐𝑐
𝑦𝑦𝑡𝑡 = 𝑐𝑐𝑁𝑁 −𝜂𝜂 + 𝜃𝜃𝜃𝜃𝑡𝑡−1 + 𝜀𝜀𝑡𝑡
(SADF
𝜕𝜕𝑡𝑡+1 𝑐𝑐𝑁𝑁 −𝜂𝜂 𝐵𝐵𝐵𝐵 ≠𝜃𝜃𝜃𝜃
≠0, 𝑃𝑃0,
𝑡𝑡 =
𝑓𝑓
+𝑃𝑃𝜀𝜀𝑡𝑡behaviors.
+ 𝐵𝐵𝑡𝑡
hereafter)𝑦𝑦𝜕𝜕𝑡𝑡to =detect
𝑡𝑡+1 . +
𝑡𝑡explosive
𝑡𝑡 𝑡𝑡−1 𝑡𝑡 The SADF test carries out
the ADF N test repetitively on a forward recursive sample sequence
𝜕𝜕𝑡𝑡+1 𝑐𝑐 .
right-sided𝐵𝐵𝑡𝑡ADF ≠ 0,and
𝑐𝑐
that is also connected to𝐵𝐵 the N sup tests. When
N𝜕𝜕𝑡𝑡+1 . 𝑓𝑓 𝐻𝐻𝐻𝐻: 𝛣𝛣 ≠ 0,
0 𝑡𝑡0 : SADF
𝛣𝛣==11 test is shown to be more effective in
𝑡𝑡
the bubbles 𝑐𝑐
𝑃𝑃𝑡𝑡 =burst, 𝑃𝑃𝑡𝑡 + the 𝐵𝐵
detecting structural 𝑓𝑓 breaks than other tests (Homm 1 & Breitung,1 2012).
𝑃𝑃𝑡𝑡 = 𝑃𝑃𝑡𝑡 + 𝐵𝐵 𝑦𝑦 𝑦𝑦
𝑡𝑡 = = 𝑐𝑐𝑁𝑁 𝑐𝑐𝑁𝑁 −𝜂𝜂 −𝜂𝜂
+ + 𝜃𝜃𝜃𝜃
𝜃𝜃𝜃𝜃 + 𝐻𝐻
+ 𝜀𝜀 𝜀𝜀: 𝛣𝛣 = 𝑐𝑐1
The null 𝑐𝑐hypothesis 𝑓𝑓 𝑡𝑡 𝑡𝑡of 𝐻𝐻0 : 𝛣𝛣 = 1 𝑡𝑡−1 is𝑡𝑡−1
rejected 𝑡𝑡 𝑡𝑡 when the largest right-
𝑃𝑃
𝐵𝐵 = ≠ 0, 𝑃𝑃𝑡𝑡 + 𝐵𝐵𝑡𝑡 1 is evidence of
tailed ADF𝑡𝑡𝑡𝑡 is bigger than the critical value, i.e. there −𝜂𝜂
𝑦𝑦𝑡𝑡 = 𝑐𝑐𝑁𝑁 −𝜂𝜂 𝑦𝑦𝑡𝑡 +
+ estimation
𝜃𝜃𝜃𝜃𝑡𝑡−1 =𝜀𝜀𝑐𝑐𝑁𝑁 + 𝜃𝜃𝜃𝜃𝑡𝑡−1 + 𝜀𝜀𝑡𝑡
an asset price bubble. Details of the 𝑡𝑡 process are given as
𝐵𝐵𝑡𝑡 ≠ 0, N N 1
follows in𝐻𝐻Equation [4]:
𝐵𝐵0 :≠
𝑡𝑡
𝛣𝛣 = 0, 1
N N
𝑦𝑦𝑡𝑡 = 𝑐𝑐𝑁𝑁 −𝜂𝜂 + 𝜃𝜃𝜃𝜃𝑡𝑡−1 + 𝜀𝜀𝑡𝑡
[4]
𝐻𝐻0 : 𝛣𝛣 = 1 𝑐𝑐 𝑐𝑐
𝐻𝐻0 : 𝛣𝛣 = 1
98 𝑐𝑐
= 𝑐𝑐𝑁𝑁 −𝜂𝜂 + 𝜃𝜃𝜃𝜃𝑡𝑡−1 + 𝜀𝜀𝑡𝑡
𝑦𝑦𝑡𝑡 N 𝑐𝑐
𝑦𝑦𝑡𝑡 = 𝑐𝑐𝑁𝑁 −𝜂𝜂 + 𝜃𝜃𝜃𝜃𝑡𝑡−1 + 𝜀𝜀𝑡𝑡 11
2 𝑤𝑤 𝑦𝑦𝑡𝑡 =2),𝑐𝑐𝑁𝑁
𝜂𝜂>1/2, 𝜀𝜀𝑡𝑡 ~ NID(0,𝜃𝜃
𝜃𝜃=1.
+ 𝜃𝜃𝜃𝜃𝑡𝑡−1 + 𝜀𝜀𝑡𝑡 𝜂𝜂>1/2, 𝜀𝜀𝑡𝑡2~ NID(0,𝜃𝜃 𝑤𝑤 2
),
𝑘𝑘 𝑖𝑖
∆𝑦𝑦𝑡𝑡 = 𝛽𝛽𝑟𝑟1,𝑟𝑟2 + 𝛽𝛽𝑟𝑟1,𝑟𝑟2 𝑆𝑆𝑆𝑆𝑡𝑡−1∆𝑦𝑦 σ𝑖𝑖=1 = 𝜑𝜑𝛽𝛽 ∆𝑦𝑦+ 𝛽𝛽 +
𝑡𝑡−𝑖𝑖𝜀𝜀𝑟𝑟1,𝑟𝑟2 𝜇𝜇𝑆𝑆𝑆𝑆
𝑡𝑡
𝑡𝑡 ~ NI
𝑟𝑟2 while 𝑟𝑟𝑤𝑤𝑘𝑘 𝑘𝑘𝑘𝑘 𝑖𝑖 𝑖𝑖𝑖𝑖𝜃𝜃=1. 𝑡𝑡 𝑟𝑟1,𝑟𝑟2𝜂𝜂>1/2,
𝑟𝑟1,𝑟𝑟2
𝑟𝑟 =
2 ∆𝑦𝑦
The International 𝑟𝑟 1 ∆𝑦𝑦+
∆𝑦𝑦 = 𝑟𝑟 =
𝛽𝛽
Journal
𝑤𝑤 = . 𝛽𝛽
2𝛽𝛽 of + 𝛽𝛽
Banking + + N 𝛽𝛽𝛽𝛽and 𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆
𝑆𝑆𝑆𝑆
Finance, σ Vol. σ σ 17, 𝜑𝜑 𝜑𝜑
Number𝜑𝜑 ∆𝑦𝑦 2 ∆𝑦𝑦
∆𝑦𝑦
(July) + 2022, +
𝜇𝜇+ 𝜇𝜇
𝜇𝜇
pp: 𝑟𝑟
91–114
2 = 𝑟𝑟1 + 𝑟𝑟
𝑤𝑤 .
𝜂𝜂>1/2, 𝜀𝜀𝑡𝑡 ~𝑟𝑟2NID(0,𝜃𝜃 while
𝑡𝑡 𝑡𝑡
𝑡𝑡 ),2 𝑟𝑟1,𝑟𝑟2
𝑟𝑟𝑤𝑤𝑟𝑟1,𝑟𝑟2𝑟𝑟1,𝑟𝑟2
𝑟𝑟1,𝑟𝑟2 𝑟𝑟1,𝑟𝑟2
𝑟𝑟1,𝑟𝑟2 𝑡𝑡−1
𝑡𝑡−1
𝑡𝑡−1 𝑖𝑖=1 𝑟𝑟1,𝑟𝑟2 𝑖𝑖=1
𝑖𝑖=1 𝑟𝑟1,𝑟𝑟2
𝑟𝑟1,𝑟𝑟2 𝑡𝑡−𝑖𝑖 𝑡𝑡−𝑖𝑖
𝑡𝑡−𝑖𝑖
𝜃𝜃=1.
𝑡𝑡 𝑡𝑡𝑡𝑡
𝜂𝜂>1/2, 𝜀𝜀𝜃𝜃=1. 𝑡𝑡 ~ NID(0,𝜃𝜃 ),
𝑟𝑟2 = 𝑟𝑟1 + 𝑟𝑟𝑤𝑤 . 𝑟𝑟2 while 𝑟𝑟𝑤𝑤 𝜃𝜃=1.
𝑟𝑟2 = 𝑟𝑟1 + 𝑟𝑟𝑤𝑤 .
where ∆𝑦𝑦 N𝑡𝑡 is= the
𝛽𝛽 sample
𝑟𝑟1,𝑟𝑟2 + 𝛽𝛽size,
𝑟𝑟1,𝑟𝑟2 is 𝑆𝑆𝑆𝑆
𝑐𝑐 𝑡𝑡−1 σ𝑘𝑘𝑖𝑖=1 𝜑𝜑𝑟𝑟1,𝑟𝑟2
akconstant, 𝑖𝑖
𝜂𝜂>1/2, ∆𝑦𝑦𝑡𝑡−𝑖𝑖𝜀𝜀𝑡𝑡+ ~ 𝜇𝜇NID(0,𝜃𝜃
𝑡𝑡𝑟𝑟 while k 2),∆𝑦𝑦 and
𝑡𝑡 = 𝛽𝛽𝑟𝑟1,𝑟𝑟2 + 𝛽𝛽𝑟𝑟1
𝜃𝜃=1. 𝑟𝑟2 while 𝑟𝑟𝑤𝑤 2 𝑟𝑟𝑤𝑤
𝜃𝜃=1. k kk
Assuming ∆𝑦𝑦that the regression ∆𝑦𝑦 sample
𝑡𝑡𝜇𝜇σ 𝑡𝑡=~NID(0,𝜑𝜑
𝛽𝛽𝑟𝑟1,𝑟𝑟2𝑖𝑖begins + 𝑖𝑖 from
𝛽𝛽𝑡𝑡−𝑖𝑖 𝑟𝑟+2) 𝜇𝜇= 𝑡𝑡 𝑟𝑟
𝑆𝑆𝑆𝑆 1+ and σ𝑟𝑟𝑤𝑤𝑘𝑘ends 𝑖𝑖 at 𝑟𝑟 while
𝜇𝜇.𝑡𝑡 ~NID(0,𝜑𝜑
𝜑𝜑𝑟𝑟1,𝑟𝑟2 ∆𝑦𝑦 𝑖𝑖 +𝑟𝑟𝜇𝜇
1 )𝑤𝑤𝑡𝑡
𝑘𝑘
𝑡𝑡 = 𝛽𝛽𝑟𝑟1,𝑟𝑟2 + 𝛽𝛽𝑟𝑟1,𝑟𝑟2 𝑆𝑆𝑆𝑆 𝑡𝑡−1 𝑖𝑖=1 𝜑𝜑𝑟𝑟1,𝑟𝑟2 ∆𝑦𝑦𝑟𝑟1,𝑟𝑟2𝑟𝑟1,𝑟𝑟2 𝑡𝑡−1 𝑖𝑖=1 2 𝑡𝑡−𝑖𝑖 𝑟𝑟1,𝑟𝑟2
while 𝑟𝑟𝑤𝑤 𝑟𝑟is
𝑟𝑟2 while the
=𝜇𝜇
𝜇𝜇
𝜇𝜇2𝑡𝑡 ~NID(0,𝜑𝜑 window
~NID(0,𝜑𝜑
𝑟𝑟~NID(0,𝜑𝜑
𝑡𝑡𝑡𝑡 1 + 𝑟𝑟 𝑖𝑖 size of
. 𝑖𝑖𝑖𝑖
)
𝑟𝑟1,𝑟𝑟2
𝑟𝑟1,𝑟𝑟2 ) ) the regression, 𝜃𝜃=1. such that 𝑟𝑟2 = 𝑟𝑟1 + 𝑟𝑟
𝑤𝑤 . ,
2 while
𝑟𝑟then 𝑤𝑤 𝑟𝑟1,𝑟𝑟2
k 𝑟𝑟𝑤𝑤𝜂𝜂>1/2,[4]
Equation 𝜀𝜀𝑡𝑡 ~can NID(0,𝜃𝜃be written 2
), 𝑟𝑟𝑤𝑤as Equation [5]: 𝑟𝑟𝑤𝑤 k 𝑟𝑟 = 𝑟𝑟 +𝑘𝑘𝑟𝑟 .
𝑤𝑤 𝑖𝑖
𝑟𝑟1,𝑟𝑟2 + 𝛽𝛽𝑟𝑟1,𝑟𝑟2 𝑆𝑆𝑆𝑆𝑡𝑡−1 σ𝑖𝑖=1 𝜑𝜑𝑟𝑟
∆𝑦𝑦 = 𝛽𝛽 2 1
𝑟𝑟2 = 𝑟𝑟 + 𝑟𝑟 𝑟𝑟k𝑤𝑤 . 𝑟𝑟𝑟𝑟𝑤𝑤𝑤𝑤 𝑟𝑟 while 𝑡𝑡
𝑟𝑟
𝑟𝑟
1 𝜇𝜇 ~NID(0,𝜑𝜑
2 = 𝑟𝑟1
𝑤𝑤
𝑡𝑡 +∆𝑦𝑦 𝑟𝑟𝑤𝑤𝑡𝑡 .= 𝛽𝛽𝑟𝑟1,𝑟𝑟2
𝑖𝑖
𝑟𝑟1,𝑟𝑟2 )+ 𝛽𝛽𝑟𝑟1,𝑟𝑟2 k 𝑆𝑆𝑆𝑆 𝑟𝑟0𝑡𝑡−1 σ𝑘𝑘𝑖𝑖=12𝜑𝜑𝑟𝑟1,𝑟𝑟2 𝑖𝑖 𝑤𝑤
∆𝑦𝑦𝑡𝑡−𝑖𝑖 + ∆𝑦𝑦𝜇𝜇𝑡𝑡𝑡𝑡𝑟𝑟= [5]+ 𝛽𝛽𝑟𝑟1,𝑟𝑟2𝑟𝑟1,𝑟𝑟2
𝛽𝛽 𝜇𝜇𝑡𝑡 ~NID(0,𝜑𝜑
0 𝑟𝑟1,𝑟𝑟2
𝑖𝑖𝑆𝑆𝑆𝑆
𝑡𝑡−1)
𝜃𝜃=1.
𝜇𝜇
𝑟𝑟0𝑡𝑡 00 𝑟𝑟
𝑟𝑟
~NID(0,𝜑𝜑 𝑖𝑖
𝑟𝑟1,𝑟𝑟2 )
∆𝑦𝑦 = 𝛽𝛽𝑟𝑟1,𝑟𝑟2 +
where 𝑟𝑟𝑤𝑤k is the lag order and𝑘𝑘𝜇𝜇𝑡𝑡 ~NID(0,𝜑𝜑 𝑟𝑟0𝑖𝑖 𝑖𝑖 The window𝑟𝑟size
2 =)𝑟𝑟1 + 𝑟𝑟𝑤𝑤 .
𝑟𝑟𝑟𝑟1,𝑟𝑟2 𝑟𝑟𝑤𝑤lies 𝑡𝑡
∆𝑦𝑦𝑡𝑡between
= 𝛽𝛽𝑟𝑟1,𝑟𝑟2𝑟𝑟2and + while 𝛽𝛽 𝑆𝑆𝑆𝑆 σ 𝜑𝜑 ∆𝑦𝑦 + 𝜇𝜇 0
+k𝜇𝜇(2015),
𝑟𝑟𝑤𝑤As suggested
𝑟𝑟𝑤𝑤0 𝑟𝑟𝑟𝑟00+1.𝛽𝛽𝑟𝑟1,𝑟𝑟2 σ𝑘𝑘𝑖𝑖=1by 𝑖𝑖Phillips
𝑟𝑟1,𝑟𝑟2 𝑡𝑡−1 𝑟𝑟1,𝑟𝑟2 𝑡𝑡−𝑖𝑖 et al. 𝑡𝑡 the window
∆𝑦𝑦𝑡𝑡 = 𝛽𝛽𝑟𝑟1,𝑟𝑟2 𝑆𝑆𝑆𝑆𝑡𝑡−1𝑖𝑖=1 𝜑𝜑𝑟𝑟1,𝑟𝑟2 ∆𝑦𝑦𝑡𝑡−𝑖𝑖 𝑡𝑡
size 𝑟𝑟0 fork sample size N is determined 𝑟𝑟𝑤𝑤 as follows, using k Equation 𝑟𝑟0[6]:
𝑟𝑟0 𝑖𝑖
𝑟𝑟2 = 𝑟𝑟1 + 𝑟𝑟𝑤𝑤 . 𝑟𝑟0 = 0.01∆𝑦𝑦 +𝑡𝑡1.8. =𝜇𝜇𝛽𝛽𝑡𝑡ξ𝑁𝑁 ~NID(0,𝜑𝜑
𝑟𝑟1,𝑟𝑟2 + 𝛽𝛽𝑟𝑟1,𝑟𝑟2 𝑟𝑟𝑟𝑟1,𝑟𝑟2
0 = 𝑆𝑆𝑆𝑆)0.01
𝑡𝑡−1kσ
𝑘𝑘
+𝑖𝑖=11.8. 𝑖𝑖
𝜑𝜑𝑟𝑟1,𝑟𝑟2
ξ𝑁𝑁 ∆𝑦𝑦
𝑟𝑟0
𝜇𝜇𝑟𝑟𝑟𝑟00𝑡𝑡 ~NID(0,𝜑𝜑
𝑟𝑟𝑟𝑟000.01
= = = 0.01 0.01 𝑖𝑖
+𝑟𝑟1,𝑟𝑟2 ++1.8.
1.8. )ξ𝑁𝑁
1.8. 𝑟𝑟ξ𝑁𝑁
0ξ𝑁𝑁 𝜇𝜇 𝑡𝑡 ~NID(0,𝜑𝜑 𝑟𝑟 [6]𝑖𝑖
0 𝑟𝑟1,𝑟𝑟2 )
k
k 𝑘𝑘 𝑖𝑖 𝑟𝑟 𝑤𝑤 𝑖𝑖
For example,
∆𝑦𝑦𝑡𝑡 = 𝛽𝛽𝑟𝑟1,𝑟𝑟2 + 𝛽𝛽𝑟𝑟1,𝑟𝑟2 𝑆𝑆𝑆𝑆𝑡𝑡−1 σ𝑖𝑖=1 𝜑𝜑𝑟𝑟1,𝑟𝑟2 ∆𝑦𝑦𝑡𝑡−𝑖𝑖 + 𝜇𝜇𝑡𝑡
a sample size of 𝑁𝑁 = 79 requires a window size of 16.𝜇𝜇𝑡𝑡 ~NID(0,𝜑𝜑𝑟𝑟1,𝑟𝑟
𝑟𝑟𝑤𝑤𝑖𝑖 𝑟𝑟 0 𝑟𝑟𝑤𝑤 𝑁𝑁 = 79
𝜇𝜇𝑡𝑡 ~NID(0,𝜑𝜑 𝑟𝑟𝑟𝑟1,𝑟𝑟2 =𝑖𝑖 0.01 ) + et 1.8.al. ξ𝑁𝑁(2015), the lag order
𝜇𝜇Following
𝑟𝑟0 =𝑁𝑁
𝑡𝑡 ~NID(0,𝜑𝜑
00.01 Phillips
𝑁𝑁
=𝑁𝑁
𝑟𝑟1,𝑟𝑟2
=
79 =+)79 791.8. ξ𝑁𝑁 k is
𝑟𝑟0 set to zero using 𝑟𝑟0 the
= 0.01 + 1.8. ξ
BIC information criterion. The critical values are calculated using 𝑟𝑟𝑤𝑤
𝑟𝑟 𝑟𝑟0
𝑟𝑟𝑤𝑤 the Montek0 Carlo simulation with 𝑐𝑐 and 1000 𝜂𝜂 +replications. 𝑟𝑟0 𝑖𝑖 In addition, 𝑐𝑐 and 𝜂𝜂 the
𝑟𝑟𝑤𝑤 𝑁𝑁 = 79 𝑟𝑟0 = 0.01 𝑡𝑡 ~NID(0,𝜑𝜑
𝜇𝜇1.8. ξ𝑁𝑁 𝑟𝑟1,𝑟𝑟2 )
parameters
𝑁𝑁 =𝑟𝑟𝑐𝑐079 and
𝑐𝑐𝑐𝑐 and
and 𝜂𝜂 𝜂𝜂𝜂𝜂𝑖𝑖 ) of the data generating process (whereby 𝑟𝑟0 c 𝑁𝑁 =𝑟𝑟079
is the
𝑟𝑟0 constant drift 𝜇𝜇 ~NID(0,𝜑𝜑
𝑟𝑟0
𝑡𝑡
factor𝑟𝑟1,𝑟𝑟2 and η is a coefficient𝑟𝑟𝑤𝑤that controls the magnitude
of the drift) 𝑐𝑐 and 𝜂𝜂
𝑟𝑟𝑤𝑤 is set according 𝑁𝑁to = 𝑐𝑐 = 79𝜂𝜂 =et1 al. (2015)
Phillips 𝑟𝑟0 = where 0.01 +𝑐𝑐1.8. = 𝜂𝜂ξ𝑁𝑁 =𝑟𝑟01
𝑟𝑟0 The asymptotic
𝑟𝑟0 𝑐𝑐 and 𝑟𝑟𝑐𝑐0 𝜂𝜂= =𝑐𝑐𝑐𝑐𝜂𝜂= ==distribution
0.01 𝜂𝜂𝜂𝜂1= =+111.8. ξ𝑁𝑁of the SADF𝑟𝑟statistic 0
may be = 0.01𝑐𝑐 +
𝑟𝑟0 summarized and 1.8. 𝜂𝜂 ξ𝑁𝑁
as follows 𝑟𝑟𝑐𝑐0= in𝜂𝜂Equation
=1 [7]: 𝐿𝐿
𝑟𝑟 1
𝑟𝑟𝑤𝑤 ቂ∫0 𝑤𝑤 𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟𝑤𝑤𝐿𝐿ቃ−−∫0 𝑤𝑤 𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟
𝑟𝑟
𝑟𝑟+ )𝑟𝑟
𝑟𝑟 = 0.01 𝑤𝑤 ቂ∫1.8
𝑤𝑤
𝑐𝑐 and 𝐴𝐴𝐴𝐴𝐴𝐴
𝑟𝑟𝑤𝑤 𝜂𝜂 𝑤𝑤→ 1𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠
𝑟𝑟𝑟𝑟𝑤𝑤 11 𝑁𝑁 𝑟𝑟ቐ
𝑤𝑤 = 79
𝑟𝑟𝑟𝑟𝑤𝑤
𝑤𝑤 𝐴𝐴𝐴𝐴𝐴𝐴
→ 0 𝑠𝑠𝑠𝑠𝑠𝑠 ቐ 1 Τ
2 ቑ
𝑟𝑟0 = 0.01 𝐿𝐿 𝐿𝐿𝐿𝐿 𝑟𝑟𝑟𝑟𝑤𝑤𝑤𝑤ቂ∫
ቂ∫00 𝑊𝑊𝑊𝑊𝑊𝑊− 𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟0 ቃ−−∫ 𝑟𝑟𝑟𝑟𝑤𝑤𝑤𝑤ቃ−−∫
ቃ−−∫𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟
0Τ2𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟
𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟 )) 2
𝑐𝑐 =+𝑁𝑁 𝑟𝑟𝜂𝜂01.8. =𝐴𝐴𝐴𝐴𝐴𝐴
𝐴𝐴𝐴𝐴𝐴𝐴 1ξ𝑁𝑁 → → 𝑠𝑠𝑠𝑠𝑠𝑠
𝑠𝑠𝑠𝑠𝑠𝑠
𝑟𝑟𝑟𝑟𝑤𝑤 ቂ∫
ቐቐ 0 𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
2 𝑟𝑟𝑤𝑤𝑤𝑤22 0 ,1ሿ0 𝑟𝑟01 ൜𝑟𝑟 𝑁𝑁 ∫ =
𝑤𝑤𝑤𝑤) 𝑤𝑤
𝑟𝑟
𝑊𝑊
𝑤𝑤2
ቑቑ 𝑐𝑐
𝑑𝑑𝑑𝑑−ቂ∫= [7]
𝑟𝑟𝑤𝑤
𝜂𝜂
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 = 01,1ሿ 𝑟𝑟 1ൠΤ2 ൜𝑟𝑟
𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ
𝑟𝑟0 = 0.01𝐴𝐴𝐴𝐴𝐴𝐴 +=1.8. 𝐿𝐿→ ξ𝑁𝑁
79 𝑠𝑠𝑠𝑠𝑠𝑠 𝑤𝑤ቐ 0 ቑ
𝑤𝑤 1 𝑤𝑤
𝑟𝑟 ቂ∫ 𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟 ቃ−−∫ 𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟 𝑤𝑤) 0 11ΤΤ22 0 𝑤𝑤
2 𝑤𝑤 𝑟𝑟𝑟𝑟 0 𝑤𝑤
2 1Τ22
𝐴𝐴𝐴𝐴𝐴𝐴 → 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
ቐ 0,1ሿ
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
0 ,1ሿ0 ,1ሿ1Τ2𝑟𝑟𝑟𝑟11ΤΤ22൜𝑟𝑟
൜𝑟𝑟𝑟𝑟𝑤𝑤
𝑤𝑤
∫∫𝑊𝑊𝑤𝑤
𝑤𝑤 22 𝑟𝑟𝑤𝑤 1𝑤𝑤 2ቑ
Τ𝑤𝑤
𝑟𝑟𝑟𝑟
00 𝑟𝑟𝑊𝑊
𝑊𝑊 𝑑𝑑𝑑𝑑−ቂ∫
𝑑𝑑𝑑𝑑−ቂ∫ 00 𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ
𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ
ൠ ൠൠ
1Τ2𝑟𝑟
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿ ൜𝑟𝑟
𝑤𝑤 𝑤𝑤 ∫ 𝑤𝑤
𝑟𝑟𝑤𝑤 𝑤𝑤 ∫ 𝑤𝑤 𝑑𝑑𝑑𝑑−ቂ∫
𝑤𝑤 2𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ 𝑟𝑟𝑐𝑐
𝑤𝑤 = 𝜂𝜂 = 1 1
𝑟𝑟𝑤𝑤 ൜𝑟𝑟 20 0
𝑤𝑤 0 𝑊𝑊 𝑑𝑑𝑑𝑑−ቂ∫0 𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ ൠ
𝑟𝑟𝑤𝑤 𝑐𝑐 and 𝜂𝜂
𝑁𝑁 = 79
𝑟𝑟0 𝐿𝐿= 0.01 + 1.8.𝑟𝑟ξ𝑁𝑁
𝑤𝑤 ቂ∫ 𝑟𝑟0 =
𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟𝑤𝑤 ቃ−−∫ 0.01 + 𝑤𝑤1.8.
𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟 ) ξ𝑁𝑁 𝐿𝐿𝑟𝑟 𝑟𝑟
𝑁𝑁 = 79𝐴𝐴𝐴𝐴𝐴𝐴 0 0
and
where
= 79W𝑐𝑐 → is the𝜂𝜂𝑠𝑠𝑠𝑠𝑠𝑠
standard ቐ Wiener process. = 𝑟𝑟 Despite the 1Τ2𝑐𝑐 superiority𝜂𝜂 𝐴𝐴𝐴𝐴𝐴𝐴 of
𝑁𝑁 and 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝑠𝑠𝑠𝑠𝑠𝑠 ൛𝑠𝑠𝑠𝑠𝑠𝑠 2
ቑ
𝑟𝑟 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
∈ሾ0,1−𝑟𝑟 ሿ 𝐴𝐴𝐴𝐴𝐴𝐴
= →𝑟𝑟
𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠
ൟ
𝑠𝑠𝑠𝑠𝑠𝑠 ൛𝑠𝑠𝑠𝑠𝑠𝑠ቐ
𝑟𝑟 1 𝑤𝑤 1
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿ 𝑟𝑟 1Τ2 ൜𝑟𝑟 ∫ 𝑤𝑤 𝑊𝑊 2 𝑑𝑑𝑑𝑑−ቂ∫ 𝑟𝑟𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿ 𝑟𝑟
the SADF over
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 traditional
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
= ==𝑠𝑠𝑠𝑠𝑠𝑠
𝑠𝑠𝑠𝑠𝑠𝑠 𝑤𝑤𝑠𝑠𝑠𝑠𝑠𝑠 unit
𝑠𝑠𝑠𝑠𝑠𝑠
൛𝑠𝑠𝑠𝑠𝑠𝑠 root
𝑤𝑤 0൛𝑠𝑠𝑠𝑠𝑠𝑠
൛𝑠𝑠𝑠𝑠𝑠𝑠 tests 𝐴𝐴𝐴𝐴𝐴𝐴 in𝑤𝑤𝑤𝑤ൟ𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ
distinguishing
0𝑟𝑟
𝐴𝐴𝐴𝐴𝐴𝐴
𝐴𝐴𝐴𝐴𝐴𝐴 𝑤𝑤 𝑟𝑟𝑟𝑟𝑟𝑟
,1ሿ
ൟ𝑟𝑟1𝑤𝑤1ൠ 𝑊𝑊𝑊𝑊𝑊𝑊−
𝑤𝑤𝑤𝑤
ൟൟ bubbles, 𝑟𝑟 it 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿ
𝑁𝑁 = 79 = ൛𝑠𝑠𝑠𝑠𝑠𝑠 𝑟𝑟01ሿ 𝐴𝐴𝐴𝐴𝐴𝐴
𝑟𝑟1 ∈ሾ0,1−𝑟𝑟 𝑟𝑟𝑤𝑤𝑟𝑟1ሿ1∈ሾ0,1−𝑟𝑟
𝑟𝑟1𝐿𝐿∈ሾ0,1−𝑟𝑟 ∈ሾ0,1−𝑟𝑟 𝑤𝑤ሿሿ 𝑟𝑟𝑤𝑤
𝑤𝑤 𝑟𝑟1ቂ∫0𝑟𝑟 𝑟𝑟𝑤𝑤 ቃ−−∫0𝑐𝑐𝑤𝑤and 𝜂𝜂 𝑤𝑤)
𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟
lacks the ability to detect 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 ,1ሿ
0several
0𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 ,1ሿ0𝐴𝐴𝐴𝐴𝐴𝐴
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 ,1ሿ bubbles
0,1ሿ → 𝑠𝑠𝑠𝑠𝑠𝑠 𝑤𝑤
𝑁𝑁 across
= ቐ79 𝑐𝑐 an= 𝜂𝜂extended =1 2
horizon. ቑ
𝑐𝑐 and 𝜂𝜂 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 ,1ሿ 1Τ2 𝑐𝑐 =
𝑟𝑟𝑤𝑤 2
𝜂𝜂 = 1 𝑟𝑟𝑤𝑤 2 1Τ2
As shown by
𝑐𝑐 and 𝜂𝜂 𝑐𝑐 = 𝜂𝜂 = 1 Phillips et al. (2015), the SADF
0 𝑟𝑟𝑤𝑤 is ൜𝑟𝑟unable
𝑤𝑤 0 ∫ 𝑊𝑊 to pick
𝑑𝑑𝑑𝑑−ቂ∫ 0 up𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ ൠ
𝑟𝑟𝑤𝑤
multiple 𝑐𝑐 occurrences
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺and= 𝜂𝜂 𝑠𝑠𝑠𝑠𝑠𝑠 of bubbles when 𝐴𝐴𝐴𝐴𝐴𝐴 the period is prolonged. 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝑟𝑟𝑤𝑤 =
൛𝑠𝑠𝑠𝑠𝑠𝑠 ൟ 1𝑠𝑠𝑠𝑠𝑠𝑠
𝑟𝑟1 ∈ሾ0,1−𝑟𝑟𝑤𝑤 ሿ 𝑟𝑟1 𝐿𝐿 𝑟𝑟𝑤𝑤𝑐𝑐ቂ∫= 0 𝜂𝜂 =𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟 ቃ−
2 𝑤𝑤
Correspondingly, Phillips 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 0 ,1ሿ et al. 𝑟𝑟(2015) has
1 𝑐𝑐 introduced
and 𝐴𝐴𝐴𝐴𝐴𝐴
𝜂𝜂𝑟𝑟 → the 𝑠𝑠𝑠𝑠𝑠𝑠
Generalized ቐ 𝑟𝑟𝑤𝑤 0 ,1
𝑐𝑐 = 𝜂𝜂 = 1 𝐿𝐿
𝑤𝑤
𝑟𝑟𝑤𝑤 ቂ∫0 𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟𝑤𝑤 ቃ−−∫0 𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟 𝑤𝑤
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 𝐿𝐿
𝑤𝑤 ) 0 ,1ሿ 1Τ𝑟𝑟2 𝑟𝑟𝑤𝑤 02 𝑊𝑊𝑊𝑊
𝑟𝑟 𝑤𝑤 ቂ∫
𝑐𝑐sup
= 𝜂𝜂ADF = 𝐴𝐴𝐴𝐴𝐴𝐴
1 (GSADF)
𝑐𝑐 = 𝜂𝜂 = 1 → 𝑠𝑠𝑠𝑠𝑠𝑠 by adjusting
ቐ 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 the = beginning
𝑠𝑠𝑠𝑠𝑠𝑠 ൛𝑠𝑠𝑠𝑠𝑠𝑠 and 𝑟𝑟
ending
𝐴𝐴𝐴𝐴𝐴𝐴
∈ሾ0,1−𝑟𝑟
1 2 1 Τ 2 ቑ points,
→ሿ 𝐴𝐴𝐴𝐴𝐴𝐴 𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠
𝑤𝑤 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
𝑤𝑤 𝑤𝑤൜𝑟𝑟
𝑟𝑟1 ,1ሿൟ 𝑤𝑤 ∫ ቐ0 𝑊𝑊 𝑑𝑑𝑑𝑑−ቂ
1Τ2 𝑟𝑟𝑤𝑤
thus allowing for a more
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 0 ,1ሿ 𝑟𝑟 1flexible
𝑤𝑤 ൜𝑟𝑟𝑤𝑤 ∫0 window
Τ2 𝑟𝑟𝑤𝑤 2
𝑊𝑊 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
𝑑𝑑𝑑𝑑−ቂ∫ 0 ,1ሿsize in the estimation
𝑟𝑟𝑤𝑤
0 𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ ൠ
0 𝐿𝐿𝑟𝑟𝑤𝑤 ൜𝑟𝑟𝑤𝑤 ∫0
𝑟𝑟𝑤𝑤 1 𝑟𝑟𝑤𝑤 𝐴𝐴𝐴𝐴𝐴𝐴 → 𝑠𝑠𝑠𝑠𝑠𝑠
process.
𝐿𝐿 The GSADF 𝑟𝑟𝑤𝑤 ቂ∫0 𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟is useful 𝑟𝑟2𝑤𝑤 𝑤𝑤1 when
𝑟𝑟 ቃ−−∫10 analyzing 𝑟𝑟𝑤𝑤𝑐𝑐𝑟𝑟𝑤𝑤= 𝜂𝜂𝑤𝑤=
𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟 ) the 1 volatile behavior 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1
𝐴𝐴𝐴𝐴𝐴𝐴 → 𝐿𝐿 𝑠𝑠𝑠𝑠𝑠𝑠 ቐ 𝐿𝐿 𝑟𝑟𝑤𝑤 ቂ∫0 𝑤𝑤𝑟𝑟𝑊𝑊𝑊𝑊𝑊𝑊−
𝑤𝑤 ቂ∫0 𝑊𝑊𝑊𝑊𝑊𝑊− 𝑟𝑟𝑤𝑤 ቃ−−∫ 𝑟𝑟 ቃ−−∫𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟
0 𝑊𝑊𝑊𝑊𝑊𝑊.𝑊𝑊(𝑟𝑟 ቑ𝑤𝑤𝑤𝑤) )
of an
𝐴𝐴𝐴𝐴𝐴𝐴 → asset price
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟𝑠𝑠𝑠𝑠𝑠𝑠
𝐴𝐴𝐴𝐴𝐴𝐴 →
0 ,1ሿ 𝑟𝑟 1
with
ቐ 2 ൜𝑟𝑟 ∫ 𝑤𝑤1𝑊𝑊
Τ𝑠𝑠𝑠𝑠𝑠𝑠 several
ቐ𝑟𝑟 2
2parts 𝑟𝑟
2 𝑤𝑤 0
𝑤𝑤 in a long 2 1 Τ period
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
ቑ ቑ which
= includes
𝑠𝑠𝑠𝑠𝑠𝑠 ൛𝑠𝑠𝑠𝑠𝑠𝑠 𝑟𝑟1 ∈ሾ0,1−𝑟𝑟𝑤𝑤 ሿ 𝐴𝐴
𝑤𝑤 Τ𝑟𝑟2 𝑑𝑑𝑑𝑑−ቂ∫ 𝑟𝑟𝑤𝑤 2 𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ ൠ 2 21Τ12Τ2
𝑟𝑟𝑤𝑤 𝑟𝑟𝑤𝑤 𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ
several subsamples
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿ𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
𝑤𝑤 1
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 =𝑟𝑟𝑤𝑤 𝑠𝑠𝑠𝑠𝑠𝑠
Τ0for
2,1ሿ 0this𝑟𝑟 𝑤𝑤 procedure.
൜𝑟𝑟 ∫2 𝑊𝑊 0 𝑑𝑑𝑑𝑑−ቂ∫
The GSADF 𝑟𝑟ൠ is represented
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0𝑟𝑟,1ሿ in 1
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
𝑟𝑟𝑤𝑤 ቂ∫0= 𝑠𝑠𝑠𝑠𝑠𝑠 𝑟𝑟𝑤𝑤൛𝑠𝑠𝑠𝑠𝑠𝑠
𝑟𝑟1 ∈ሾ0,1−𝑟𝑟𝑤𝑤 ሿ 𝐴𝐴𝐴𝐴𝐴𝐴
൜𝑟𝑟𝑤𝑤 ∫0 ൛𝑠𝑠𝑠𝑠𝑠𝑠
𝐿𝐿𝑟𝑟1 ൟ
𝑤𝑤 𝑊𝑊 𝑑𝑑𝑑𝑑−ቂ∫0 𝑊𝑊(𝑟𝑟)𝑑𝑑𝑑𝑑ቃ
𝑤𝑤 0 0 ൠ 𝑤𝑤 𝑤𝑤 𝑟𝑟
𝑊𝑊𝑊𝑊𝑊𝑊− ቃ−−∫0𝑟𝑟1𝑤𝑤∈ሾ 𝑊𝑊
the Equation [8]: 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿ 𝐴𝐴𝐴𝐴𝐴𝐴 → 𝑠𝑠𝑠𝑠𝑠𝑠 ቐ 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 0 2,1ሿ 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿ 𝑟𝑟 1Τ2 ൜𝑟𝑟 ∫ 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝑟𝑟 𝑤𝑤 2 = 𝑤𝑤 𝑊𝑊(
𝑟𝑟 𝑠𝑠𝑠𝑠
𝑟𝑟𝑤𝑤𝐴𝐴𝐴𝐴𝐴𝐴 𝑟𝑟𝑤𝑤 ൟ 2 𝑤𝑤 𝑤𝑤 0 𝑊𝑊 𝑑𝑑𝑑𝑑−ቂ∫0𝑟𝑟𝑟𝑟∈ሾ
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
𝑠𝑠𝑠𝑠𝑠𝑠 = ൛𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠𝑠𝑠 ൛𝑠𝑠𝑠𝑠𝑠𝑠 𝐴𝐴𝐴𝐴𝐴𝐴 ሿ ൟ
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟
= 𝑠𝑠𝑠𝑠𝑠𝑠
𝑟𝑟1 ∈ሾ0,1−𝑟𝑟 𝑟𝑟1 ∈ሾ0,1−𝑟𝑟
𝑤𝑤 ሿ 𝑟𝑟1 𝑟𝑟𝑤𝑤
𝑤𝑤 𝑟𝑟 1
[8]
൛𝑠𝑠𝑠𝑠𝑠𝑠 𝑟𝑟1 ∈ሾ0,1−𝑟𝑟𝑤𝑤 ሿ 𝐴𝐴𝐴𝐴𝐴𝐴𝑟𝑟1 ൟ 2
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟 0 ,1ሿ
0 ,1ሿ
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿ
2 22 99 𝑟𝑟
𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 = 𝑠𝑠𝑠𝑠𝑠𝑠 ൛𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟1 ∈ሾ0,1−𝑟𝑟𝑤𝑤ሿ 𝐴𝐴𝐴𝐴𝐴𝐴𝑟𝑟1𝑤𝑤 ൟ
𝑟𝑟𝑟𝑟∈ሾ𝑟𝑟0 ,1ሿData
The present study has examined the asset price bubbles due to the COVID-19 outbreak during the period from January 2, 2020, to April 24, 2020. Four daily equities, namely two American stock markets indices (S&P 500 (^GSPC) and Dow Jones (^DJI)), two Malaysian stock market indices (FTSE Bursa Malaysia (^KLSE?P=^KLSE) and Emas Shariah (FBMS.FGI)) were sourced from Yahoo Finance (https://finance.yahoo.com). For oil prices, two indices (WTI Spot Crude and NYMEX Futures) were sourced from the US Energy Information Administration (https://www.eia.gov/petroleum/data. php).
Summary Statistics on Stock Market Indices and Oil Markets
Dow
S&P 500
KLSE
Shariah
NYMEX
Spot
Jones
Crude
Mean
25801.71
2985.72
1461.79
11043.17
40.56
40.36
Maximum
29551.42
3386.15
1611.38
12104.30
63.27
63.27
Minimum
18591.93
2237.40
1219.72
9120.49
-37.63
-36.98
Std. Dev.
3344.77
340.71
109.40
865.65
17.66
17.99
Skewness
-0.42
-0.46
-0.39
-0.51
-1.25
-1.20
Kurtosis
1.73
1.83
1.86
2.00
6.01
5.50
Jarque-Bera
7.61***
7.33***
6.41***
6.87***
48.41**
37.96**
Observations
79
79
81
81
76
76
Note. Dow Jones denotes Dow Jones Industrial Average, KLSE denotes FTSE Bursa Malaysia KLCI, Shariah denotes FTSE Bursa Malaysia EMAS Sharia, NYMEX denotes NYMEX Futures, Spot Crude denotes WTI Spot Price. Note. **, *** Indicates statistical significance at 5% and 1% level, respectively.
Table 1 shows the summary statistics for the four stock market indices and two oil price measures. The range of stock indices in percentage terms was 37 percent for the Dow Jones and 34 percent for the S&P 500. For the KLSE and the Shariah, the variation was 24 percent each from January,1 to April,24 2020. Fluctuations in crude oil prices were more severe, accounting for 158 percent during the sample period, largely attributed to the unprecedented drop to negative value for first time in history on 20th April 2020 at USD37. These large and drastic fluctuations of prices were the result of the severity of the impact of COVID-19 on these markets, and which increased the likelihood of causing bubbles. Furthermore, all prices exhibited negative skewness, with similar skewness value for the four stock market indices and two oil price measures. According to the Jacque-Bera test of normality, none of the prices were normally distributed. The present study then tested the pairwise correlation between prices of different equity pairs, as is shown in Table 2. The observed correlations between the markets in the USA and Malaysia were high and significant at the 1 percent level of significance. Similar positively significant correlations between the crude oil prices and the stock markets in the USA and Malaysia were recorded. The significant correlations between the six equities seemed to suggest the possibility of contagion effects from speculative bubbles in the stock and oil markets during the epidemic.
Correlation Statistics of Stock Market Indices and Oil Price
Correlation
Dow Jones
NYMEX
S&P 500
KLSE
SHARIAH
Spot Crude
Dow Jones
NA
NYMEX
0.78719*
NA
S&P 500
0.99049*
0.76149*
NA
KLSE
0.87351*
0.90628*
0.84603* NA
Shariah
0.88625*
0.89566*
0.86239* 0.98933*
NA
Spot Crude
0.79376*
0.99045*
0.76916* 0.91176*
0.90353* NA
Note. Dow Jones denotes Dow Jones Industrial Average, KLSE denotes FTSE Bursa Malaysia KLCI, Shariah denotes FTSE Bursa Malaysia EMAS Sharia, NYMEX denotes NYMEX Futures, Spot Crude denotes WTI Spot Price. Note. * denotes significant at 1% level of significance.
RESULTS
Table 3 presents the SADF and the GSADF test results for the six asset prices. Each test was achieved by performing 1000 replications of the Monte Carlo simulations. Both the statistical values of the GSADF and the SADF for Dow Jones and the S&P 500 indices were > 99 percent of the threshold level. For example, in the case of the S&P 500, the statistical value of the GSADF test and the SADF test were 2.19 and 1.93, respectively and exceeded the 99 percent threshold level of 1.57 for the GSADF and 1.08 for the SADF. For the KLSE and Shariah markets, the GSADF tests values exceeded 95 percent, while the SADF statistical values were > 99 percent of the threshold. Regarding oil prices, the statistical values of the GSADF were 1.92 for the NYMEX and 1.12 for the Spot Crude, respectively. The former exceeded the 99 percent threshold, whilst the latter exceeded the 95 percent threshold. The statistically significant results indicate the presence of bubbles between January 2020 and April 2020, hence supporting the notion of COVID-19 induced uncertainty shocks in these markets.
Results of the GSADF and SADF tests for Stock Market Indices and
Oil Market
Market
Test method
Statistical value
Critical level
99%
95%
90%
Dow Jones
GSADF SADF
1.71*** 1.71***
GSADF Threshold SADF Threshold
1.57 1.08
1.13 0.45
0.93 0.23
S&P 500
GSADF SADF
2.19*** 1.93***
GSADF Threshold SADF Threshold
1.57 1.08
1.13 0.44
0.93 0.22
KLSE
GSADF SADF
1.43** 1.38***
GSADF Threshold SADF Threshold
1.54 0.9
1.09 0.39
0.86 0.15
Shariah
GSADF SADF
1.14** 1.09***
GSADF Threshold SADF Threshold
1.55 0.9
1.12 0.4
0.89 0.18
NYMEX
GSADF SADF
1.92*** 0.56**
GSADF Threshold SADF Threshold
1.55 0.94
1.1 0.45
0.91 0.2
Spot Crude
GSADF SADF
1.12** -0.04
GSADF Threshold SADF Threshold
1.55 0.94
1.1 0.45
0.91 0.2
Note. Dow Jones denotes Dow Jones Industrial Average, KLSE denotes FTSE Bursa Malaysia KLCI, Shariah denotes FTSE Bursa Malaysia EMAS Sharia, NYMEX denotes NYMEX Futures, Spot Crude denotes WTI Spot Price. Note. **, *** Indicates statistical significance at 5% and 1% level, respectively.
Since the GSADF test is prone to succeeding bubbles and results from the GSADF tests have shown that each price index has surpassed the threshold values at 95 percent level, the analysis in the present study had proceeded by comparing the GSADF statistical values with the GSADF sequence threshold’s critical value for the four stock markets and two oil price indices at the 95 percent sequence. The graphical assessment in Figures 3-8 shows the date-stamp for the beginning and end of each bubble. For the purpose of comparison, the original data series were plotted in the same figures and corresponds to the right vertical axis.
Figures 3-4 plot the corresponding GSADF statistics (the brown line) against the corresponding 95 percent threshold (the green line) for the Dow Jones and the S&P500 during the COVID-19 pandemic. The date
of beginning is identified as the first point when the GSADF statistics
surpassed the critical value. The end date is equivalent to the point
after the date of origin when the GSADF statistics dropped below the
threshold value. These periods are represented by the shaded areas as
identified by Equation (8). Figure 3 and Figure 4 show evidence of
two statistically significant bubbles, with the first bubble detected on
the 21st of February and lasted around seven days. During this short
span of The
sequence. time, both
graphical indices
assessment suffered
in Figures 3-8 shows a 12 percentfordrop
the date-stamp in theandindex.
the beginning end of
The second
each bubble. For successive bubble was
the purpose of comparison, detected
the original onwere
data series theplotted
12th of March
in the and
same figures
and corresponds to the right vertical axis.
lasted for two days, when both markets registered a 10 percent drop
inFigures
the index.
3-4 plot theThese resultsGSADF
corresponding are in line with
statistics theline)
(the brown findings in corresponding
against the Chang et al. 95
percent threshold (the green line) for the Dow Jones and the S&P500 during the COVID-19
(2021). These two consecutive short bubbles in these two US stock
pandemic. The date of beginning is identified as the first point when the GSADF statistics surpassed
markets coincided
the critical value. The end with
date is the declining
equivalent to the point global
after theoil
datedemand, as the
of origin when reflected
GSADF
by the persistent
statistics dropped belowdrop in crude
the threshold value. oil
Theseprices
periodsas are much
represented as by22thepercent from
shaded areas as
identified by Equation (8). th Figure 3 and Figure 4 show evidence of two statistically significant
20 th
February to 13 March 2020. This was largely due to the
bubbles, with the first bubble detected on the 21st of February and lasted around seven days. During
weakened
global
this shortoil
spandemand,
of time, bothasindices
China shuta its
suffered economy
12 percent drop intothecurb
index. the spread
The second of the
successive
virus
bubble and manyoncountries
was detected began
the 12th of March to limit
and lasted for twoor suspend
days, when bothair travels
markets to and
registered a 10
percent drop in the index. These results are in line with the findings in Chang et al. (2021). These two
from China. The explosive behavior episodes of the
consecutive short bubbles in these two US stock markets coincided with the declining global oil
Dow Jones and
the S&P500
demand, were
as reflected a manifestation
by the of the
persistent drop in crude adverse
oil prices as mucheconomic
as 22 percentimpact
from 20ofththe Coronavirus outbreak and this was made worse after the Russia–
February to 13th
March 2020. This was largely due to the weakened global oil demand, as China shut
its economy to curb the spread of the virus and many countries began to th limit or suspend air travels to
Saudi
and from Arabia
China. Theoil explosive
price war which
behavior started
episodes of theon theJones
Dow 8 of andMarch
the S&P500 2020.were a
manifestation of the adverse economic impact of the Coronavirus outbreak and this was made worse
after the Russia–Saudi Arabia oil price war which started on the 8 of March 2020. th
Bubbles Date Stamp for Dow Jones Index Bubbles Date Stamp for Dow Jones Index
explosive behavior on the 24th of February might be rooted in the continuous drop in oil prices after hovering around 53 US dollars for several weeks and has never recovered since. The second bubble on February 24 might also be due to the political upheaval in Malaysia when the Pakatan Harapan coalition government collapsed on 24 February 2020 (Ho Wah Foon), it created a power vacuum when the then Prime Minister, Tun Dr. Mahathir Mohamad resigned. The stock market slid down further and subsequently led to the third bubble in the Shariah index following the sudden surge in COVID-19 cases and the imposition of a nationwide partial lockdown by the Malaysian government.
Bubbles Date Stamp for the KLSE Index
Note. GSADF denotes the GSADF sequence; GSADF_CV denotes the 95% critical value sequence.
104Bubbles Date Stamp for the Sharia Gold Index
For the KLSE, only one episode of bubble was detected between March,13 and March 21, 2020. The inception date of the bubble for this period corresponded to the sudden spike in COVID-19 cases in Malaysia, from the daily confirmed cases of 9 on 13th March to 190 confirmed cases on 16th March 2020 (see Figure 2). Although the Shariah index experienced an explosive episode around the same starting date as that of the KLSE, the GDASF test picked up two prior bubbles for the Shariah as is shown in Figure 6. There was a double sudden explosion in prices on 31st January and 24th February 2020. The implosion on 31st January was likely a response to the surge in the COVID-19 confirmed cases in China by 10,000 percent between 20th January-31st January 2020 (see Figure 1), while the explosive behavior on the 24th of February might be rooted in the continuous drop in oil prices after hovering around 53 US dollars for several weeks and has never recovered since. The second bubble on February 24 might also be due to the political upheaval in Malaysia when the Pakatan Harapan coalition government collapsed on 24 February 2020 (Ho Wah Foon), it created a power vacuum when the then Prime Minister, Tun Dr. Mahathir Mohamad resigned. The stock market slid down further and subsequently led to the third bubble in the Shariah index following the sudden surge in COVID-19 cases and the imposition of a nationwide partial lockdown by the Malaysian government. Note. GSADF denotes the GSADF sequence; GSADF_CV denotes the 95% critical value sequence.
Figure 6 Bubbles Date Stamp for the Sharia Gold Index Bubbles Date Stamp for the Sharia Gold Index
the study by Gharib et al. (2021), as it also identified several explosive episodes in the West Texas Light (WTI) oil price during the corresponding period.
sequence. Bubbles Date Stamp for the Spot Crude Oil Price
According to the IEA6, the coronavirus outbreak had caused the first quarterly contraction of oil demand by 435 kb/d in 10 years. The oil market took another major blow in March when Saudi Arabia engaged in a price war with Russia on March 8, 2020. A movement of panic ensued in the energy market, triggering yet another implosion of the oil price bubble, highlighted in the12green-shaded area in Figure 7 and
The GSADF test picked up another short-lived bubble for the
NYMEX index on March 18, 2020 (purple-shaded area in Figure 7), as crude oil price tumbled by another 23 percent during an overnight trade. As oil traders began to accommodate the free fall of the crude oil price over the next four weeks, in the midst of the ongoing price war between Russia and Saudi Arabia reached a stalemate, there was no episode of explosive behavior in the energy market until the 20th of April when oil price turned negative for the first time in history. Both the NYMEX and the Spot Crude prices were traded at 37 US dollars a day before the May futures contracts expired on April 20th. The GSADF statistics for the NYMEX and the Spot Crude were above its critical values for one day on the 20th of April 2020, as is shown in the red-shaded region in Figure 6 and Figure 7, consistent with the monthlong crisis in the oil markets. Overall, findings for the crude oil markets are consistent with the results in the study by Gharib et al. (2021), as it also identified several explosive episodes in the West Texas Light (WTI) oil price during the corresponding period.
The present study has focused on how the COVID-19 pandemic has caused the asset bubbles in the stocks markets of the USA and Malaysia, CONCLUSION as well as in the global oil markets. It found that the increase in COVID-19 cases has exacerbated the volatility in the financial markets, as well as in the oil market. The GSADF has provided a powerful explanatory story by documenting a series of bubble The in thepresent study period between Januaryhas focused 1, 2020, on2020, to April 24, how theequity for six COVID-19 pandemic series. The findings showed has that caused the asset the major bubble episodesbubbles in the in the US stock stocks markets markets byofthethe were precipitated USA slump in theand oil market, and to a lesser extent was due to COVID-19. This finding is consistent with the position of Malaysia, the USA as aas well major oil as in thestriding producer globalpastoil markets. Saudi Arabia andItRussia found that since theTheincrease 2017. analysis carried out in this study revealed that the volatility of the US stock markets was more influenced by oil price movements rather than the shock created by COVID-19. This is in line with the findings of Gao et al. (2021). Investors can utilize this information as a prediction and diagnostic tool for future stock prices, as well as an early warning system for economic instability, notably in the US stock market.
in COVID-19 cases has exacerbated the volatility in the financial markets, as well as in the oil market. The GSADF has provided a powerful explanatory story by documenting a series of bubble in the period between January 1, 2020, to April 24, 2020, for six equity series. The findings showed that the major bubble episodes in the US stock markets were precipitated by the slump in the oil market, and to a lesser extent was due to COVID-19. This finding is consistent with the position of the USA as a major oil producer striding past Saudi Arabia and Russia since 2017. The analysis carried out in this study revealed that the volatility of the US stock markets was more influenced by oil price movements rather than the shock created by COVID-19. This is in line with the findings of Gao et al. (2021). Investors can utilize this information as a prediction and diagnostic tool for future stock prices, as well as an early warning system for economic instability, notably in the US stock market.
In contrast, the stock market in Malaysia exhibited explosive behaviour episodes that correlated strongly with COVID-19 cases. This is not surprising because of the high degree of market integration between Malaysia and its major trading partners, particularly China and the USA. There was also evidence of bilateral contagion effects of bubbles between the oil stock market and the Shariah stock market in Malaysia. Multiple bubble episodes began simultaneously in the crude oil markets (NYMEX and Spot Crude) following the oil price slump in February 2020 and then spread to the Shariah market in Malaysia in March 2020. The explosive behaviour of the stock markets in Malaysia is also explained by the political instability as a result of the collapse of the Pakatan Harapan government in March 2020. These findings have important implications for Government leaders, the single most important was the need to create a stable climate for the financial system so that all other sectors of the economy could thrive, especially during the pandemic.
In comparing the conventional (KLSE) market and the Islamic
(Shariah) market, it was found that there was a single bubble episode in the KLSE index, but multiple bubbles in the Shariah index. In particular, the GSADF test was able to pick up two bubble episodes earlier in the Shariah index, both of which coincided with the initial implosion of the COVID-19 outbreak in China, and the impending collapse of the crude oil price. The noteworthy differences of bubble episodes in Malaysia has revealed the sensitivity of the Shariah index to market shocks when the asset prices were heated. The early detection of a bubble episode due to the COVID-19 pandemic from the Shariah index could help protect investors and fund managers in both markets (conventional and Islamic) from impending market crashes. This may be attributed to the rigorous screening of Shariah compliant companies included in the Islamic index, which has excluded gambling and riba (unproportionate interest). Furthermore, the strict benchmarking criteria, namely the nature of asset (Jakhura & Mangera, 2010) and the low debt-to-equity ratio (Zandi et al., 2014) for Shariah-compliant companies would ensure that investors’ rights were protected. Consequently, this will make investment in Islamic financial markets a viable feature for diversification. However, since the present findings have been limited to the market in Malaysia, it is suggested that further investigation be carried out on the Shariah index of other countries to confirm the current results.
ENDNOTES
Oil Market Report. (April 2020). by the International Energy Agency.https://www.iea.org/reports/oil-market-report- april-2020. Jacobs, Trent. OPEC+ moves to end price war with 10 million B/D cut. pubs.spe.org. Journal of Petroleum Technology. Archived from the original on 10 April 2020. Retrieved from https://www.oecd.org/coronavirus/policy- responses/global-financial-markets-policy-responses-to-covid- 19-2d98c7e0/ (24 February 2020). The Dow Jones Industrial Average and FTSE 100 dropped more than 3%. Retrieved from “Global stock markets plunge on coronavirus fears”. BBC News. 24 February 2020. The KLSE index is derived based on Shariah and non-Shariah (conventional) compliant stocks; the Shariah index is calculated solely on stocks which are Shariah compliant. Information on the listing criteria can be found in https://www.bursamalaysia. com. Retrieved from https://www.iea.org/reports/oil-market-report- february-2020
ACKNOWLEDGMENT
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
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