**Debt-Growth Nexus in the MENA Region: Evidence from a Panel Threshold Analysis**

#### **Mohammed Daher Alshammary, Zulkefly Abdul Karim \*, Norlin Khalid and Riayati Ahmad**

Center for Sustainable and Inclusive Development Studies (SID), Faculty of Economics and Management, Universiti Kebangsaan Malaysia (UKM), Bangi 43600, Selangor, Malaysia; mhdd1986@gmail.com (M.D.A.); nrlin@ukm.edu.my (N.K.); riayati@ukm.edu.my (R.A.)

**\*** Correspondence: zak1972@ukm.edu.my

Received: 8 October 2020; Accepted: 12 November 2020; Published: 20 November 2020

**Abstract:** This study examines whether a debt-to-GDP threshold exists in the public debt and economic growth relationship for 20 Middle East and North Africa (MENA) countries from 1990 to 2016 using the threshold estimation technique. The empirical results reveal that there is a threshold effect in the public debt and economic growth relationship. The MENA region's debt-to-GDP threshold value as a developing region is lower than the debt threshold computed by earlier studies for developing countries. We found that the effect of public debt on economic growth is significant and positive only below the threshold value of debt-to-GDP. More precisely, debt has a promoting influence on economic growth when the debt is less than 58% of the GDP. This finding indicates that the relationship between public debt and economic growth is contingent on the debt-to-GDP ratio. Importantly, policymakers need to be more prudent when establishing a policy regarding debt issues.

**Keywords:** public debt; economic growth; threshold effects; MENA region

**JEL Classification:** E62; H63; N17; C24

#### **1. Introduction**

The substantially growing threat to the economic growth of debt accumulation has been one of the most controversial and discussed issues among economists and policymakers in both developed and developing countries. Recently, economic challenges, such as chronic fiscal balance deficits, which may affect the public debt ratio, have gained significance. In the aftermath of the financial crisis in 2007–2008, many studies have been conducted to examine the relationship between public debt and economic growth (Egbetunde 2012; Al-Zeaud 2014; Zouhaier and Fatma 2014; Akram 2015; Spilioti and Vamvoukas 2015; Fincke and Greiner 2015; Muye et al. 2017). By and large, the previous studies have confirmed that there is a closer link between public debt accumulation and economic growth, whether this linkage is positive or negative for any country.

In the 1980s, many countries in the Middle East and North Africa (MENA) region undertook reforms in the financial sector. These reforms were a part of the structural adjustment programs (SAPs) often adopted with the support of the IMF and the WB, particularly in low-income countries1. The main goal of SAPs is to promote the long-term economic growth of a developing economy by decreasing borrowing in the country's fiscal imbalances in both the short and medium terms. These reforms led to considerable economic growth by the late 1990s. Nonetheless, the economic growth rate in the MENA region has underperformed and fluctuated compared to that in other developing regions, such as

<sup>1</sup> For example, Jordan, Morocco and Tunisia.

ASEAN-5 and Sub-Saharan Africa. Over the last decade, the average economic growth rates were 3.4% in MENA, 5% in ASEAN-5, and 4.7% in Sub-Saharan Africa2. The MENA region is still facing various economic challenges, such as the fiscal deficit, current account deficit, and high debt accumulation (see Saeed and Somaye 2012; Asghari et al. 2014; Samadi 2006). Some countries of the MENA region, particularly non-oil countries, were affected by the global financial crisis in 2007–2008 because of their high interdependence on the US's financial assistance, which put pressure on them when this assistance was lost3. The MENA region witnessed significant development in its infrastructure accompanied by expansions in government expenditures, government size, and debt accumulation. As a developing region, MENA's debt-to-GDP ratio is high and troubling compared to that in other developing regions, such as ASEAN-5 and Sub-Saharan Africa. Over the last few decades, the debt-to-GDP ratio has held an average of approximately 43.7%, which is 2.1% higher than the average for ASEAN-5 and 9.9% higher than that for Sub-Saharan Africa4. Researchers studying the MENA region consider that the economic challenges are related to the broad interventions and the low quality of fiscal policies adopted by MENA governments, such as the larger government size than that in other developing regions.

These stylized facts about public debt and economic growth raise serious concerns regarding the debt-to-GDP ratio's threshold value at which economic growth can be sustained in the MENA region. This study aimed to address this issue. Reinhart and Rogoff (2010) showed that both developed and developing countries are highly concerned with their debt level. The authors suggested that a suitable debt-to-GDP threshold rate is 60% for developing countries and that exceeding this threshold will lead to negative consequences for developing countries' economies by increasing the burden on their economies through debt service and repayment.

This study examines whether there is a debt-to-GDP threshold in the public debt and economic growth relationship for 20 MENA countries. The MENA region is selected because many MENA countries undertook reforms in the institutional and financial sector by decreasing borrowing to reduce their fiscal imbalances in both the short and medium terms and attained economic growth through the gradual removal of trade barriers, which led to strengthened trade relationships. Data from only 20 MENA countries during the period from 1990 to 2016 are included because the data are bounded. This study contributes to the existing literature in three aspects. First, in terms of the policy, this study can directly help fiscal policymakers in the MENA region. Overall, in the MENA region, there is no fixed threshold for the debt-to-GDP ratio; it is dependent on a country's condition. The region must address many crucial issues to avoid potential bankruptcy. Therefore, determining the optimum debt-to-GDP level can prevent the adverse effects of overborrowing in the MENA region by providing signals for policymakers in managing debt accumulation. Second, while previous studies have identified the debt-to-GDP threshold values for some countries and regions, for example, the debt-to-GDP threshold of 90% for OECD countries estimated by Reinhart and Rogoff (2010), the MENA region may have a different debt-to-GDP ratio threshold value because of its different economic structure. The secure level of debt-to-GDP for the MENA region has not previously been investigated to the best of our knowledge. Finally, this study also sheds light on other explanatory variables and explores their influence on the MENA region's economic growth.

This study is organized as follows. Section 2 reviews the literature on the relationship between public debt and economic growth. Section 3 discusses the methodology and describes the data. Section 4 is devoted to discussing the empirical results. Finally, the last section presents a summary and conclusion of the findings.

<sup>2</sup> Source: world Bank.

<sup>3</sup> In 1961, congress passed the Foreign Assistance Act, reorganizing U.S. foreign assistance programs and separating military and non-military aid. Egypt, Jordan, Iraq, Israel and Lebanon are the recipient countries in MENA.

<sup>4</sup> Source: world Bank.

#### **2. Literature Review**

From a theoretical perspective, two streams of thought have been proposed to distinguish the relationship between public debt and economic growth. First, in the classical theory, Ricardian equivalence states that if a government borrows today, then it has to repay this borrowing in the future by raising taxes above the normal level; thus, the impact of debt accumulation on growth will be neutralized (Ricardo 1817). The Great Recession of the 1930s was the key catalyst for the development of the modern theory of public debt. According to Keynes, foreign assistance or foreign investment is required to fill the savings-investment gap. Furthermore, Keynes argued that fiscal policy is the best policy for growth in any economy because it meets the general public's needs. Second, the neo-classical theory argues that public debt directly influences economic growth contingent on the amount borrowed. If a government exploits and takes advantage of debt in inefficient ways, the amount of investment is anticipated to increase in the long run through spending on hospitals, schools, sanitation, and infrastructure as a strategy to counteract the negative effects of debt. As long as countries use the borrowed funds for productive investment and do not suffer from macroeconomic turbulence, policies that distort the economic motivation should be implemented to stimulate appropriate debt services and repayment.

In the neo-classical view, Diamond (1965) formally proposed public debt as a variable demonstrating economic growth. Diamond postulated that internal debt decreases the attainable capital stock due to the replacement of public debt for physical capital. In addition, Diamond contended that the decrease in capital stock arises from the internal borrowing of the government to finance the deficit. Fiscal and monetary policies play an essential role in promoting economic growth in the endogenous growth model. Consequently, through public debt as a fiscal policy tool, technical progress can be reached and result in a boost in economic growth. Nevertheless, Saint-Paul (1992) noted that using endogenous growth models noted that economic growth slowdowns arise from debt accumulation.

Furthermore, Krugman (1988) proposed the concept of "Debt Overhang" where the government's ability to repay external debt decreases below the actual debt value. Modern growth theories have recently demonstrated that the government debt-growth nexus is related to governance (see Zak and Knack 2001; Acemoglu and Robinson 2006). Against this backdrop, the main channel where public debt can affect economic growth is the long-term interest rate. The government's overborrowing to finance the fiscal balance deficit will result in a higher long-term interest rate. Consequently, the higher long-term interest rate tends to decelerate economic growth via the crowd-out of private investments.

Empirically, several macroeconomic studies have provided inconsistent or even contradictory results about the relationship between public debt and economic growth in developed and developing countries. By and large, empirical studies have demonstrated either a positive or negative association between the two variables. The preponderance of evidence has indicated that there is a significant influence of public debt on economic growth. Recently, research has shifted towards estimating the threshold value of debt-to-GDP that can maintain economic growth, rather than investigating the effect of public debt on economic growth itself. Nevertheless, many growth theories have admitted that government expansion is necessary for low-income countries to stimulate economic growth (see Levine 2005; Carlin and Mayer 2003). At the other extreme, most of the recent economic studies on the relationship between the public debt and growth relationship have contended that there is a non-linear (inverted U-shape) relationship between the accumulation of public debt and economic growth, where the effect of public debt on economic growth may change based on the level of economic development (See Pattillo et al. 2003; Kumar and Woo 2010; Cordella et al. 2010). As a consequence, there should be a level of secured debt that can boost economic growth. If a country borrows more without limitations, then economic growth will decrease even further.

After the financial crisis of 2008–2009, which was followed by the European debt crisis in countries including the PIIGS (Portugal, Italy, Ireland, Greece, and Spain), the relationship between public debt and economic growth for both developing and developed countries gained increased attention among scholars. As government size increases over time, overborrowing to finance many unproductive projects may dampen long-term economic growth. El-Mahdy and Torayeh (2009); Bal and Rath (2014); Puente-Ajovín and Sanso-Navarro (2015); Zouhaier and Fatma (2014); Eberhardt and Presbitero (2015); and Mitze and Matz (2015) also contended that there is a negative relationship between public debt and economic growth. Conversely, several studies have employed recent econometric techniques to evaluate the debt-growth nexus. For example, Al-Zeaud (2014); Spilioti and Vamvoukas (2015); Fincke and Greiner (2015); Owusu-Nantwi and Erickson (2016); and Muye et al. (2017) maintained that public debt affects economic growth in a positive and significant manner.

In a major break with the existing macroeconomic literature, for 20 developed countries, Lof and Malinen (2014) commented that public debt has no real effect on economic growth even at higher public debt rates. In the second set of estimations, which considers more than two countries, as in our case, the results are different depending on the sample, and methodological tools are shown in Table 1.


**Table 1.** The main studies which estimate the debt-growth threshold level with more than two countries.

Given this backdrop, this present study fills the gaps in the following ways. First, while Khanfir(2019) and Omrane et al. (2017) focused on a few MENA countries using the panel smooth transition model (PSTR), this study differs in terms of the countries considered and methodology used. More precisely, this study conducts an empirical estimation of the entire sample of MENA countries using a different method (two regimes model) developed by Hansen (1996) during the most recent years (1990–2016) to determine the debt-growth nexus in the MENA region. Second, this study provides paramount evidence highlighting the role of the debt-to-GDP threshold in sustaining economic growth. In particular, we estimate whether there exists an optimum debt-to-GDP threshold value for the public debt-growth nexus. The outcomes of the study may have significant implications for economic policy. Therefore, if there is clear evidence that debt accumulation significantly restrains economic growth, then policymakers need to establish an effective policy that sustains and boosts economic growth. In summary, while most studies conducted in the MENA region have focused on a small group of countries, the present study adopts a different sample and methodology.

#### **3. Methodology**

#### *3.1. Model Specification*

The log-linearized Cobb-Douglas production function has empirically estimated the debt role in economic growth by controlling other explanatory variables. The model can be written as follows:

$$Y\_{i,t} = \alpha\_t + \beta\_1 K\_{i,t} + \beta\_2 Pop\_{i,t} + \beta\_3 Debt\_{i,t} + \beta\_4 X\_{i,t} + \mu\_{i,t} \tag{1}$$

where *Yi*,*<sup>t</sup>* denotes real GDP per capita growth. α*<sup>t</sup>* denotes time-fixed effect, *Ki*,*<sup>t</sup>* denotes physical capital and *Popi*,*<sup>t</sup>* denotes the population growth. *Debti*,*<sup>t</sup>* denotes general government debt. *Xi*,*<sup>t</sup>* is a vector of explanatory variables government expenditure (*GEi*,*t*), which captures government policy, trade openness (*Openi*,*t*) to measure the effect of trade policies; inflation rate (*in fi*,*t*); and money supply (*M*2*i*,*t*) as a percentage of GDP, and the initial income (*Initiali*,*t*). Initial income is the lagged-dependent variable of the real GDP per capita growth to consider the convergence effect of the model.

#### *3.2. Data and Variable Description*

This study employs data for 20 MENA countries in estimating the growth model in Equation (1). The data adopt an annual frequency from 1990 to 2016. All the datasets are collected World Bank (WB) databases, except the data for Syria for the period 2010–2016, which has been obtained from the central bank and Central Bureau of Statistics of Syria. The real GDP per capita growth (*Y*) is a measure of economic growth, and (*K*) is the total investment as a percentage of GDP to proxy for gross capital formation. The population growth rate (*Pop*) has been used as a proxy for the labor force. Data for the public debt (*Debt*), the general government debt as a percentage of GDP, were collected from the IMF. Government expenditure (*GE*) is the total government expenditure as a percentage of GDP and captures government policy. The trade openness (*Open*) is the sum of exports plus imports as a percentage of GDP and is used as a proxy to capture trade openness. Money supply (*M*2) as a percentage of GDP used as a proxy for financial depth, as in Baharumshah et al. (2017). Inflation rate (*Inf*) is the annual percentage of average consumer prices. Finally, the initial income is the GDP per capita at constant prices (2010).

#### *3.3. Methodology Selection*

Previous studies have provided inconclusive results regarding the impact of debt accumulation on economic growth. The earlier studies show that debt accumulation effects may have nonlinear characteristics because of the potential presence of threshold in debt-growth nexus (see Pattillo et al. 2003; Kumar and Woo 2010; Cordella et al. 2010). Given the inconsistent results reported by earlier literature, this study estimates threshold effects in the debt-growth nexus for the case of MENA regions. The classical static panel data methods, such as fixed or random effect, have some limitations, showing only the heterogeneity in intercepts and linear relationship among the variables. In contrast, the panel threshold method's main advantages are the shifting character or structural break in the association between dependent and explanatory variables (Lee and Wang 2015), explaining the possible nonlinear relationship between the threshold and dependent variables. The threshold model is considered an essential model for estimating many economic issues and has recently been utilized in many macroeconomic studies (Lee and Wang 2015). Consequently, to assess the nonlinear behavior of the debt-to-GDP ratio in the relationship with economic growth, this study employs the threshold regression approach suggested by Hansen (1996). The model in Equation (1) can be written based on the fixed effect panel threshold regression as follows:

$$\text{Trgdpc}\_{i,t} = \mu\_i + \beta\_1 X\_{i,t} I(\text{Debt}\_{i,t} \le \lambda) + \beta\_2 X\_{i,t} I(\text{Debt}\_{i,t} \ge \lambda) + \text{ e}\_{i,t} \tag{2}$$

where *rgdpcgi*,*<sup>t</sup>* is the real GDP per capita growth, *Debti*,*<sup>t</sup>* is the public debt, which is a threshold variable that divides the sample into upper and lower regimes; λ is the unknown threshold parameter; *I*(.) is the indicator function, which takes the value 1 if the argument in the indicator function is valid, and 0 otherwise; μ*<sup>i</sup>* is the individual effect; and e*it* is the disturbance. *Xi* is a vector of the control variables (investment, population growth rate, government expenditure, trade openness, inflation, money supply, and initial income). Equation (2) can be rewritten as two equations, Equation (3) represents the lower regime threshold and Equation (5) represents the upper regime threshold:

$$r\text{gdpc}\emptyset\_{i,t} = \mu\_{i,t} + \beta\_1 X\_{i,t} \, I(\text{Debt}\_{i,t} \le \lambda) \tag{3}$$

$$
\log \text{bpc} \text{g}\_{i,t} = \; \mu\_{i,t} + \; \beta\_2 \text{X}\_{i,t} \; I(\text{Debt}\_{i,t} > \lambda) \tag{4}
$$

$$
\log \text{pcg}\_{i,t} = \; \mu\_{i,t} + \; \beta\_1 X\_{i,t} \; I(\text{Debt}\_{i,t} \ge \lambda) \tag{5}
$$

$$
\log \text{pcg}\_{i,t} = \mu\_{i,t} + \beta\_1 X\_{i,t} \, I(\text{Debt}\_{i,t} < \lambda) \tag{6}
$$

Therefore, this methodology allows for the examination of differentiating effects of public debt on economic growth in the lower and upper regimes depending on whether the threshold variable is smaller or higher than the threshold value γ. Coefficient β<sup>1</sup> and β<sup>2</sup> indicate the considered effects in the lower and higher regimes, respectively.

To carry out the panel threshold regression, we have to test the null hypothesis of linearity against the threshold model in Equation (2), where the null hypothesis is *H*0: β<sup>1</sup> = β2. According to Hansen (1996), there is a problem executing the LM and Wald test statistics under the null hypothesis because the λ parameter is not specified; therefore, inferences are implemented by calculating an LM or Wald statistic for each potential value of λ and depend on the least upper bound of the Wald or LM for all potential λs (Law et al. 2013). Consequently, the inferences are conducted via bootstrapping in a model whose validity and properties were developed by Hansen (1996) because tabulations are not possible. Once the value of λ is obtained, the slope parameters βˆ(λˆ) and γˆ(λˆ) can also be obtained.

#### **4. Empirical Results**

Table 2 shows the descriptive statistics for each variable employed in the model. The dataset is free from any extreme values, which may affect the estimated results' significance by affecting the mean, standard deviation, minimum, and maximum of each variable of the entire sample. Table 3 shows the correlation matrix of the variables employed in the analysis. Table 4 reports the panel threshold estimation results and presents the number of thresholds in terms of the debt-to-GDP ratio. According to the bootstrap *p*-values, the corresponding statistics *F*1, *F*<sup>2</sup> and *F*<sup>3</sup> suggest the number of thresholds. The test for the single threshold *F*<sup>1</sup> is highly significant, with a bootstrap *p*-value of 0.05, while *F*<sup>2</sup> and *F*<sup>3</sup> are nonsignificant, with bootstrap *p*-values of 0.16 and 0.68, respectively. In addition, the F statistic is highly significant. Therefore, we reject the linear model. The statistical significance of the threshold estimate is evaluated by the *p*-value calculated utilizing the bootstrap method with 1000 replications and a 0.1% trimming percentage. Consequently, the sample is split into two regimes. The point

estimate of the debt-to-GDP threshold value is 58.51%, with a corresponding 95% confidence interval of (52.3465, 58.9060) for the full sample model. The MENA region's debt-to-GDP threshold value as a developing region is close to 60%, as computed by Reinhart and Rogoff (2010), and 64% as computed by Grennes et al. (2010) for developing countries. This finding is also close to the debt-to-GDP threshold calculated by Sanusi et al. (2019), which is 57% for Southern African countries, and higher than the debt-to-GDP threshold value computed by Khanfir (2019); Omrane et al. (2017) and Boukhatem and Kaabi (2015). The estimated debt-to-GDP threshold is different from that found in previous studies conducted on the MENA countries because of the different sample and methodology employed.

**Table 2.** Descriptive and Summary Statistics for 20 Middle East and North Africa (MENA) Countries, 1990–2016.


**Table 3.** Correlations between Variables for 20 MENA Countries, 1990–2016.


**Notes:** Rgdpcg = real GDP per capita; INV = total investment (% of GDP); Popgr = population growth; Exp = government expenditure (% of GDP); Debt = public debt (% of GDP); Trade openness = openness in trade policy (% of GDP); Inflation = inflation rate; M2 = money supply (% of GDP) and Initial = GDP per capita in constant prices (2010).

**Table 4.** Tests for Threshold Effects for 20 MENA Countries, 1990–2016.


**Note:** A total of 1000 bootstrap replications were used for each of the three bootstrap tests. \*\*\* denotes significant at 1% level.

To estimate how the debt-to-GDP threshold affects the economy under the upper and lower threshold values of the debt-to-GDP ratio, Table 5 shows the panel threshold model's empirical results for the debt-growth relationship from Equation (2). The coefficient estimates of the explanatory variables are significant for promoting growth. First, public debt turns out to be a significant positive determinant of economic growth in the lower regimes. Still, in the upper regime, public debt has a nonsignificant positive effect promoting growth. In other words, debt at a level below 58% of GDP positively influences economic growth. The result is ambiguous when debt exceeds 58% of GDP because the coefficient is insignificant. This implies that if the debt-to-GDP threshold level is beyond 58%, it tends to have positive or negative effects on the MENA region's economic growth. Arguably, this finding is consistent with the Laffer curve and the theory that debt accumulation tends to dampen economic growth through the burden of higher debt servicing in the long term.


**Table 5.** Regression Results using Public Debt (Debt) as a Threshold Variable. Dependent Variable: Real GDP per Capita.

**Note:** \*\*\*, and \*\* denotes significant at 1% and 5% level respectively. The results correspond to a trimming percentage of 0.1%. F test that all ui = 0: F(19, 511) = 1.99. Prob > F = 0.0008.

Turning to physical capital, it positively affects economic growth, which is consistent with theory. Moreover, the labor force has a significant negative effect on economic growth, which reflects the labor market policies in the MENA region. Furthermore, government expenditure behavior is conflicting for the anticipations, since government expenditure is negatively and significantly associated with economic growth. This result shows the significance of fiscal policy for the MENA region. At 58% of debt-to-GDP ratio, government expenditure hinders economic growth. A plausible explanation is that ineffective governments in MENA countries depress private investment, so public investment may substitute private investment. Consequently, the MENA reign needs to allocate government expenditures prudently to curb adverse influences and take full advantage of beneficial economic growth. However, trade openness is in line with expectations. It has a significant positive influence on economic growth, possibly because of trade barriers removal.

The coefficient estimates of inflation are significant and negatively associated with economic growth. Besides which, the model shows that the money supply plays a significant negative role in promoting economic growth in the studied period. Although monetary policy has a vital role in enhancing the economic growth of any country, there has been a long debate in economics about the role of money in the economy. At 58% of the debt-to-GDP ratio, the money supply increases over time, coupled with a decrease in the interest rate. As a consequence, consumption, lending and borrowing will be increased. Hence, expansionary money policy leads to crowd-out investment, often resulting in stagnation, which finally hinders economic growth. Finally, the initial income coefficients are significant

and have a positive effect on economic growth. These findings are consistent with the theory. Overall, the ambiguity regarding the MENA region's estimations may be because of the development of the financial sector and indicate that the MENA region has not yet reached a sustainable level of debt.

To smooth short-run fluctuations, we split the MENA region's full sample into three periods over nine years for all variables. This leaves us with three intervals: 1990–1998, 1999–2007 and 2008–2016. Table 6 shows the threshold estimates for the periods.


**Table 6.** Threshold Estimates.

With the two estimated thresholds categorizing the countries into three levels based on their debt-to-GDP ratio, the significant estimate of each level proves the existence of nonlinear effects on the debt-to-GDP ratio. Table 7 shows the countries in each level of debt-to-GDP in the observed periods. In 1990, two countries, namely Bahrain and the UAE, had low levels of debt-to-GDP. On the other hand, 13 countries had a high debt-to-GDP threshold. By 1999, the number of countries with a medium level of debt-to-GDP had increased from five to seven, and the number of countries with a high level had decreased to nine. By 2008, Comoros, Djibouti, Ethiopia, and Tunisia had notably decreased their debt accumulation level and joined the countries with a medium level of debt-to-GDP, while Morocco and Jordan still had high levels of debt-to-GDP.


**Table 7.** 20 MENA Countries with Different Debt-to-GDP Threshold Levels, 1990–2016.

#### **5. Summary and Conclusions**

This study examined whether there is a debt-to-GDP threshold value in the public debt-economic growth nexus for 20 MENA countries over 1990–2016. This study's novel contribution is the adoption of the regression model proposed by Hansen (1996) to provide a reliable optimum debt-to-GDP threshold, which has not previously been addressed. The results reveal that debt promotes economic growth only below the debt-to-GDP threshold. The empirical findings also demonstrate that the estimated threshold percentage value for the developing MENA region is lower than the debt threshold computed by Reinhart and Reinhart and Rogoff (2010) and Grennes et al. (2010) for developing countries. This study concludes that debt has a positive effect on growth below the threshold value. In contrast, the impact of debt above the threshold value is ambiguous because the coefficient is nonsignificant.

This study's policy implications are that accumulating debt to boost economic growth is not a wise policy choice for countries in the MENA region. Instead, reducing the debt-to-GDP ratio seems to enhance these countries' economic performance; the average debt-to-GDP ratio of approximately 52% over the full sample period seems to support this view. Arguably, the relationship between public debt and economic growth depends on the debt-to-GDP ratio. This study finds that government debt can promote economic growth if used for productive projects and limited to optimal. Therefore, policymakers may eventually enhance growth by reducing the debt-to-GDP ratio, efficiently allocating financial sources, reducing sterile government-funded programs, and using timely austerity measures to curb shocks' effects financial crisis.

**Author Contributions:** M.D.A. carried out the experiment, wrote, and revised the manuscript with support from Z.A.K., N.K. and R.A. The central idea of this research, is given by M.D.A. and Z.A.K. The earliest manuscript is verified by Z.A.K., N.K. and R.A. Z.A.K. and N.K. have also verified the analytical method and the interpretation of the results of this article. Z.A.K. supervises the revised version of this article as a correspondence author. All authors have contributed significantly from the earlier draft until the final stage of the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

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

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## *Article* **Applying Quantum Mechanics for Extreme Value Prediction of VaR and ES in the ASEAN Stock Exchange †**

**Chukiat Chaiboonsri \* and Satawat Wannapan**

Modern Quantitative Economic Research Center (MQERC), Faculty of Economics, Chiang Mai University, Chiang Mai 50200, Thailand; lionz1988@gmail.com

**\*** Correspondence: chukiat1973@gmail.com

† This research work was partially supported by Chiang Mai University.

**Abstract:** The advantage of quantum mechanics to shift up the ability to econometrically understand extreme tail losses in financial data has become more desirable, especially in cases of Value at Risk (VaR) and Expected Shortfall (ES) predictions. Behind the non-novel quantum mechanism, it does interestingly connect with the distributional signals of humans' brainstorms. The highlighted purpose of this article is to devise a quantum-wave distribution methodically to analyze better risks and returns for stock markets in The Association of Southeast Asian Nations (ASEAN) countries, including Thailand (SET), Singapore (STI), Malaysia (FTSE), Philippines (PSEI), and Indonesia (PCI). Data samples were observed as quarterly trends between 1994 and 2019. Bayesian statistics and simulations were applied to present estimations' outputs. Empirically, quantum distributions are remarkable for providing "real distributions", which computationally conform to Bayesian inferences and crucially contribute to the higher level of extreme data analyses in financial economics.

**Keywords:** quantum mechanics; wave function; extreme value analysis; Bayesian inference; stock market; Value at Risk (VaR); Expected Shortfall (ES); prediction

#### **1. Introduction**

Physicists' interest in the social sciences is not novel. The word "econophysics" is the perspective applied to economic computational models and concepts associated with the "physics" of systematical complexity—e.g., statistical mechanics (quantum mechanics), self-organized criticality, microsimulation, etc. (Hooker 2011). Fundamentally, most econophysicists have in mind that the approach—computational physics for econometrics—seeks to structure physically realistic models and theories of economic phenomena from the actually observed features of economic systems. Practically, econophysicists and economists are connected by analyzing financial markets, but the problem is that they are trained in "different schools" (Ausloos et al. 2016). The exploration of the tangible of computational physics and financial econometrics is the nexus of this paper.

Not surprisingly, physics students have been trained and have known that the frontier of modern physics uses plain language—quantum physics and relativity (Bowles and Carlin 2020). The power of quantum physics substantially existed in the chaotic period of World War II—reasonably called the "beginning" of modern physics—by Werner Heisenberg, who was a founder of quantum mechanics and a significant contributor to the physics of fluids and elementary particles (Saperstein 2010). With this great exploration, modern quantum physics has brought people to distinctly shift the standard of living through numerous inventions such as microwaves, fiber optic telecommunications, super computers, etc.

In economics, "Marshall plus Keynes" neoclassical synthesis is still teachable for the non-specialist future citizen since the "visible hands" stated by Adam Smith has been elusive, and the story has therefore never been all that easy to see—as a perusal of his original text demonstrations (Persky 1989). However, human decision-making processes are significantly from their "power" explained by integrating psychological aspects and

**Citation:** Chaiboonsri, Chukiat, and Satawat Wannapan. 2021. Applying Quantum Mechanics for Extreme Value Prediction of VaR and ES in the ASEAN Stock Exchange . *Economies* 9: 13. https://doi.org/10.3390/ economies9010013

Academic Editor: Helmi Hamdi Received: 16 November 2020 Accepted: 23 December 2020 Published: 3 February 2021

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individual social-economic ideas (Sijabat 2018). From this perspective, quantum physics can potentially affect a merger with computational economic models through the concept of the power behind a decision. More expressly, Figure 1 displays the diverse iceberg for economic movements. Along with fluctuations in the trend, traditionally computational economics restrictedly captures only the observable zone (the top of the iceberg). However, numerous amounts of information exist underneath the water, which potentially motivate human perspectives to conclude a final decision, are intentionally neglected by the assumption called "normality". With this strongly theoretical supposition, traditional econometrics has been trustworthy for more than a hundred years.

**Figure 1.** Diverse economies iceberg.

Unfortunately, there have been many economic collapses after the industrial revolution in the mid-18th century. Economists and econometricians are blemished with their predictive foresight's computational failures, especially forecasting in financial investments. The deep root of the problem is about their fundamental thoughts. Thinking as a traditional econometrician is to model observed information by a random walks model, the easiest way to imitate rational human aspects. Box 1 represented in Figure 2 displays the concept that the random walk model is the logic to reach B from A. This principle's systematic idea is to sample only a static spot when the arrow is tangible B. However, the critical query is that this fundamental cannot seemingly support the existence of human thinking.

**Figure 2.** Two different concepts between Random walks and wave function (Quantum walks).

Human decisions are sourced from electromagnetic waves and particles since cell-tocell communications occur through a process known as "synaptic transmission", where chemical signals are passed between cells generating electrical spikes in the receiving cell. To think like modern econophysicists (Quantum Mechanics: Wave-particle duality), it is reasonable to state the arrow from E to F displayed in box 2 is not a random walk model (Quantum walk process). The observation at the point F exists when it passes the process of dynamic wave movements. In other words, the reason to decide to select F is the origin of numerous latencies, which are attitudes, perspectives, morals, etc. Interestingly, this fundamental can potentially raise a novel horizon for observing information using modern econometric (Quantum econometrics) estimations and predictions.

Financial markets have several of the properties that characterize complex systems and interact nonlinearly in the presence of estimations (Mantegna and Stanley 2000). One of the interesting areas in finance is the pricing of derivative instruments. The graphical trend displayed in Figure 3 shows the example of dynamic stock exchanges in Thailand between 2000 and 2020—investors have a variety of reasons and decide the process of brainstorms. The graph implies that it is identical to a wave transmission. Hence, it is time to seek an alternative tool to compute this kind of complex data econometrically.

**Figure 3.** The dynamic movement of the Thailand stock exchange between 2000 and 2020. Source: Stock Exchange of Thailand (SET)

This research tries to fill the research gaps between the traditional econometrics method and modern econophysicists (Quantum Econometrics), which apply in financial markets, especially the extreme value prediction in the ASEAN stock exchange. However, this research is organized, as follows, by explaining the conceptual framework between traditional econometrics and Quantum Econometrics (Modern Econophysicist). The second part is how to apply this conceptual framework in extreme value prediction, especially the extreme value of Value at Risk (VaR) and Expected Shortfall (ES) of five stock markets in ASEAN countries. The last part of this research is an exclusive summary for comparison between two methods to forecast the extreme value of Value at Risk (VaR) and Expected Shortfall (ES) in five stock markets of ASEAN countries based on Risk management analysis.

#### **2. Literature and Critical Thinking**

It is not simple to exactly explain and picture humans' decisive believes or faiths. In terms of a mathematical expression, deductive logic was the ideal invention trying to reach a conclusion. However, thoughts are not logically controllable as similar to a robotic mechanism. In econometrics, this is defined as "extreme distribution". To make sense of it, the root of distributional generating is interesting to reconsider.

#### *2.1. The Origin of Quantum Distributions*

Inside of the area of traditional econometrics, the original process of a random walk based on the scaling limit, which generalizes the so-called iterated Brownian motion, is useable and acceptable academically. This theoretical concept of the random walk graphically displayed in Figure 4 was considerably generalized and extended by the Polish physicist Marjan Smoluchowsk (Kac 1947). It is continuously modified to be functional in modern quantitative research. Jung and Markowsky (Jung and Markowsky 2013) showed the random walk's advantage at random times to be considered the "alternating random", which rewards the schematic scale to indicate fractional stable motions. Although the theory of random-walk processes is continuously acceptable, the theory has started to be criticized. The weakness of random walk algorithms is stated by (Saghiri et al. 2019). The non-intelligent random walk models may not be a problem-solving method in real-world

problems since some complex systems such as biological networks or social networks work as a "learning mechanism". It seems the performance of random-walk processes is low when used to explain mechanical information about the practical problems' nature.

**Figure 4.** The example of random walk patterns based on Brownian motion. The data visualize two log-return sets of quarterly stock indexes in Thailand (SET) and Singapore (STI) between 1995Q1 and 2019Q1. The fluctuation of index trends is the difference in pitch between positive and negative values, but it inclines stationary (close to 0 and unchangeable) in the long-term consideration.

Modernly, there are many attempts to make a reconsideration in the impetus of humans' decision calls. As Adam Smith said, invisible hands are behind the scenes. This undetectable power is deliberately linked to quantum behaviors. Quantum mechanics based on Newton's motion laws are good enough to predict how behavioral complexities are inspired. Newton's laws are seen to be consequences of the fundamental way the quantum world works (Ogborn and Taylor 2005). However, in accounting for small occurrences, Newtonian physics' failure is evident at the atomic level. This implies that the lack of precise calculations cannot be simply captured by Newtonian quantum computing. It turns out that the "Hamiltonian formulation" is the formalism that most readily generalizes to quantum mechanics via the Schrodinger equation (Piziak and Mitchell 2001), which is the crucial fundament for exploring quantum key distributions

One of the highlighted obstacles in quantum key distributions is capturing and pointing signals of quantum mechanism exactly. As stated in the contribution by (Bruß and Lütkenhaus 2000), the problem of cross-polarized cryptography between the two polarization modes and a random (classical) rotation of the polarization along the propagation direction is informationally detected by using Ekert's privacy amplification (Ekert et al. 1994). Interestingly, the quantum distribution is being a truly evolutional data analysis for post-modern econometrics.

#### *2.2. Quantum Computing in Financial Econometrics*

The Hamiltonian formulation for the time-independent Schrödinger equation composes a quantum evolutional enlargement of the classical harmonic oscillator approaches to economics' business cycle dynamics. As the literature on Piotrowski and Sładkowski (2001); Choustova (2007); Gonçalves and Gonçalves (2008); Choustova (2009); Gonçalves (2013); and Gonçalves (2015); Chaiboonsri and Wannapan (2021), a quantum application to extreme financial optimizations, therefore, contributes to the novel discussion within forecasting financial economics and raises a criticism to the empirical validity of the geometric Brownian motion and geometric random walk models of price dynamics, which is commonly employed in financial economics as mathematical tools for solving pricing problems, especially risks and returns analyses.

#### **3. The Objective and Scope of Research**

The risk and return of financial markets are the main investigations of the paper. Quarterly data from 1995 to 2019 were observed. Five major stock exchanges in five ASEAN countries such as Thailand (SET), Singapore (STI), Malaysia (FTSE), Philippines (PSEI), and Indonesia (PCI) were processed in three sections of the research framework, including data visualization, risks management (Value at Risk: VaR), and returns forecasting (Expected Shortfall: ES). Technically, parametric estimations' main statistical tool is the subjective method called "Bayesian inference", and observations are deliberately focused on the extreme tail loss of distributional portraits, theoretically known as "extreme value".

The scope of the research processes is displayed in Figure 5. Expressly, the observations are processed to the section of data visualization (screening data). Descriptive explanation, stationary testing, and normality checking are the main consideration, and then the raw data is modified by two critical concepts—a random walk (Gaussian) distributional set and quantum-wave distribution. The next step is to insert two distinguished data into the function of the Generalized Pareto Distribution (GPD) extreme value analysis. Heavy loss tails are clarified and analyzed by setting the prior density for parameters at the Bayesian estimation threshold. The most precise prediction between two distributions is validated by computing the Deviation Information Criteria (DIC).

**Figure 5.** The scope of research.

#### **4. Methodology**

#### *4.1. Quantum Mechanics and Wave Function for Time Series Movement*

Since 1926, Erwin Schrödinger developed the wave function implemented to predict the quantum system's behavior or quantum mechanics, especially for the prediction of the momentum of energetic electrons (Schrödinger 1926). This quantum mechanics based on the idea of Louis de Broglie's wave–particle duality in 1924 was played an important role that has significantly influenced the development of Schrödinger's wave

function (Recherches sur la théorie des quanta Researches on the quantum theory). Basically, energy can be expressed by a simple equation, as shown below:

$$E = KE + PE\tag{1}$$

where *E* = Energy, *KE* = Kinetic Energy, *PE* = Potential Energy, *KE* = <sup>1</sup> <sup>2</sup> *MV*<sup>2</sup> , *M* = Mass (kg), *V* = Velocity (m/s). Consequently, we obtain

$$E = \frac{1}{2}MV^2 + \mathcal{U}, PE = \mathcal{U} \tag{2}$$

The momentum of *P* = *MV*, the object is empowered to move by *P*<sup>2</sup> = *M*2*V*<sup>2</sup> then *V*<sup>2</sup> = *<sup>P</sup>*<sup>2</sup> *<sup>M</sup>*<sup>2</sup> . However, Equation (2) can be rewritten by substitution of *<sup>V</sup>*<sup>2</sup> and this is presented in Equations (3) and (4) as follows:

$$E = \frac{1}{2}M\frac{P^2}{M^2} + \mathcal{U}\_\prime \tag{3}$$

$$E = \frac{P^2}{2M} + \mathcal{U}.\tag{4}$$

Once again, the wave function from Schrödinger's equation's original idea used *ψ* to represent the energy of the particle-wave duality movement. This can be demonstrated as follows:

$$
\psi = e^{i(kx - \omega t)} \tag{5}
$$

$$\frac{d\psi}{d\chi} = ik e^{i(kx - \omega t)} = ik\psi \tag{6}$$

$$\frac{d^2\Psi}{d\chi^2} = (ik)^2 \epsilon^{i(kx-\omega t)}\tag{7}$$

$$\frac{d^2\psi}{dx^2} = i^2k^2e^{i(kx-\omega t)}\tag{8}$$

$$\frac{d^2\psi}{d\mathbf{x}^2} = i^2k^2e^{-i(\omega t - k\mathbf{x})} \tag{9}$$

$$\frac{d^2\psi}{dx^2} = -k^2 e^{-i(\omega t - kx)}\tag{10}$$

and Equation (10) is defined that: *k* = *<sup>P</sup>* -, (de Broglie relation) then

$$\frac{d^2\psi}{d\mathfrak{x}^2} = - (\frac{P^2}{\hbar^2})\psi , \psi = e^{-i(\omega t - kx)} . \tag{11}$$

Equation (11) is multiplied by -<sup>2</sup> on both sides, we obtain

$$-\hbar^2 \frac{d^2 \Psi}{d\mathbf{x}^2} = P^2 \Psi \tag{12}$$

From Equation (4), it can be modified, and it can be rewritten by the equation as displayed below:

$$E = \frac{P^2}{2M} + \mathcal{U}\_\prime \tag{13}$$

$$E\psi = \frac{P^2\psi}{2M} + lI\psi,\tag{14}$$

$$E\psi = \frac{-\hbar^2}{2M} \frac{d^2 \psi}{dx^2} + lI\psi\_\prime - \hbar^2 \frac{d^2 \psi}{dx^2} = P^2 \psi\_\prime \tag{15}$$

Equation (15) was mentioned by Schrödinger to implement the quantum mechanics prediction for electrons moved by energy relied on the case of time-independence. For making a sensible computation, the time-dependent case can be done by starting from the Planck–Einstein relation (Griffiths 1995) as presented that *E* = *ω* = *h f* , *E* = *hv* = *h f* , *<sup>h</sup>* = Planck constant (6.626 × <sup>10</sup><sup>−</sup>34). The Planck–Einstein relation suggests that whenever energy is empowered, frequencies (*v*) are parallel increments, the Planck constant *h* is stable. The proof of the following equations can express electrons' energic movements:

$$\frac{d\psi}{dt} = -i\omega e^{i(kx - \omega t)}\tag{16}$$

$$\frac{d\psi}{dt} = -i\omega\psi, \psi = e^{i(kx - \omega t)}\tag{17}$$

From the Planck–Einstein relation, we obtain

$$E = \hbar \omega,\tag{18}$$

$$E\psi = \hbar\omega\psi\tag{19}$$

Multiplied <sup>−</sup>*<sup>i</sup>* into Equation (19) on both sides,

$$\frac{-i}{\hbar}E\psi = -i\omega\psi\_{\prime}(\frac{d\psi}{dt} = -i\omega\psi),\tag{20}$$

$$\frac{-i}{\hbar}E\psi = \frac{d\psi}{dt} \tag{21}$$

$$E\psi = \frac{\hbar}{-i}\frac{d\psi}{dt}.\tag{22}$$

The time-independence should be transformed into the Schrödinger equation for the time-dependent by replacing Equation (22) into (15). The finalized result of these steps is Equation (23):

$$\frac{\hbar}{-i}\frac{d\psi}{dt} = \frac{-\hbar^2}{2M}\frac{d^2\psi}{dx^2} + \mathcal{U}\psi\tag{23}$$

and

$$i\hbar\frac{d\psi}{dt} = \frac{-\hbar^2}{2M}\frac{d^2\psi}{dx^2} + \mathcal{U}\psi\tag{24}$$

The finalized equation is the fundamental of the Schrödinger equation for predicting the momentum of wave-particle dualities in different cases. However, the right-hand and left-hand sides of those equations can be substituted by *H*ˆ (Hamiltonian OPERATOR, *i d dt* - - *ψ*(*t*)- = *H*ˆ - - *ψ*(*t*)- (Time-dependent), *H*ˆ - *ψ*- = *E* - *ψ*- (Time-independent)) for forecasting the total systematic energy. In particular, the behavior for the quantum mechanism. In terms of the Schrödinger wave function's interpretation, this is the highlight for this article to apply the periodic function for measuring the momentum of returns of stock markets in ASEAN countries. Mathematically, we start with

$$
\psi = A \sin(\frac{2\pi}{\lambda} x),
\tag{25}
$$

where *ψ* represents the prediction value of total energy for the momentum of return movements during observable periods. *A* is the amplitude of Equation (25) and *λ* is the wavelength included in the equation simultaneously.

Figure 6 implies the concept of quantum mechanics applying for stock return predictions. In other words, this cognition is applied from the concept of sound amplification mathematically explained in Equation (25). The upper regime (high energy zone) is a quadrant of positive positions, which explains that returns are still moved. In this case, the high energy zone stands for the explanation of Bull market momentums (Ahn et al.

2018; Ataullah et al. 2008). Conversely, the fall of returns (Bear market momentum) is the negative quadrant—compared with the case of low energy with no evidence or information to push up. Interestingly, this is a huge challenge from quantum mechanics' performance to figure out the better frontier for understanding stock return fluctuations, especially when extremes data are intensively mentioned, but distribution is elusive to find.

**Figure 6.** The wave function is based on the Schrödinger equation.

#### *4.2. Extreme Value Analysis*

Many distributions have been mentioned to model share returns as a whole observation (normal tail distributions). However, the weakness of the whole distribution is the missing of extreme tail losses. The Generalized Pareto Distribution (GDP) introduced by Pickands (1975) intentionally focuses on the threshold of the extreme losses by taking the negative of the log-returns and then choosing a positive threshold. The model assumes observations under the threshold, *μ*, is from a certain distribution with parameters *η*. *H*(·|*η*) is from a GPD. Thus, the distribution function *f* of any sample *x* can be expressed following Behrens et al. (2004) as

$$f(\mathbf{x}|\eta, \mathbf{\tilde{y}}, \sigma, \boldsymbol{\mu}) = \begin{cases} \begin{array}{c} H(\mathbf{x}|\eta), \\ H(\mathbf{x}|\eta) + (1 - H(\boldsymbol{\mu}|\eta)) G(\mathbf{x}|\boldsymbol{\xi}, \sigma, \boldsymbol{\mu}), \quad \mathbf{x} \ge \boldsymbol{\mu}. \end{array} \tag{26}$$

For an observation of size *η*, *x* = (*x*1, ...., *xn*) from *f* , parameter vector *θ* = (*η*, *σ*, *ξ*, *μ*), *N* = [*i* : *xi* < *μ*], and *P* = [*i* : *xi* ≥ *μ*], the likelihood equation is

$$L(\theta; \mathbf{x}) = \prod\_{N} H(\mathbf{x}|\eta) \prod\_{P} \left(1 - H(\mathbf{x}|\eta)\right) \left[\frac{1}{\sigma} \left(1 + \frac{\xi(\mathbf{x}\_{i} - \mu)}{\sigma}\right)\_{+}^{-(1+\xi)/\xi}\right],\tag{27}$$

for *ξ* = 0, and for *ξ* = 0,

$$\overset{\circ}{L}(\theta; \dot{\mathfrak{x}}) = \underset{\circ}{\prod} \overset{\circ}{H}(\mathfrak{x}|\eta) \overset{\circ}{\prod}\_{\mathcal{P}} (1 - H(\mathfrak{x}|\eta)) [(1/\sigma) \exp\{(\mathfrak{x}\_i - \mu)/\sigma\}].$$

The threshold *μ* is the point where the density has a disruption. Depending on the parameters, the density jump can fluctuate positively or negatively, and in each case, the choice of which observations will be defined as exceedances that can be more obvious or obscure.

#### *4.3. Bayesian Inference and Simulations for Value at Risk (VaR) and Expected Shortfall (ES)*

Recall that the parameters in the extreme value model are *θ* = (*η*, *σ*, *ξ*, *μ*). The prior and posterior distributions are respectively described as follows

#### 4.3.1. The Origin of the Prior Information

Since expressing prior beliefs directly in terms of GPD parameters is a difficult task. The idea to deal with this problem is information within a parameterization on which

experts are familiar. Equation (1) can be re-written as an inversion; thus, we obtain the 1-*p* quantile of the distribution as follows

$$q = \mu + \frac{\sigma}{\tilde{\xi}} \left( P^{-\tilde{\xi}} - 1 \right), \tag{28}$$

where *q* is defined as the level of returns associated with a return period of 1/*p* time units. The elicitation of the prior information is expressed in terms of (*q*1, *q*2, *q*3), referring to as the location-scale parameterization of GPD, for specific values of *p*<sup>1</sup> > *p*<sup>2</sup> > *p*3. Consequently, parameters are ordered and *q*<sup>1</sup> < *q*<sup>2</sup> < *q*3. Therefore, the prior information is suggested by setting the median and 90% quantile estimations for specific values of *p*, for example.

Next, the elicited parameters are transformed to gain the equivalent gamma parameters, *di* ∼ *Ga*(*ρi*, *γi*) where *i* = 1, 2, 3 and the physical lower bound of the factor is *e*<sup>1</sup> = *q*0. *e*<sup>1</sup> = 0 is preferable. The following gamma distributions with hyper parameters: *d*<sup>1</sup> = *q*<sup>1</sup> ∼ *Ga*(*ρ*1, *γ*1) and *d*<sup>2</sup> = *q*<sup>2</sup> − *q*<sup>1</sup> ∼ *Ga*(*ρ*2, *γ*2), knowing as the marginal prior distribution for *σ* and *ξ*, which is expressed as follows

$$\begin{array}{l} \pi(\sigma,\xi)a\left[\mu+\frac{\sigma}{\xi}\left(p\_{1}^{-\frac{\sigma}{\xi}}-1\right)\right]^{p\_{1}-1}\exp\left[-\gamma\_{1}\left\{\mu+\frac{\sigma}{\xi}\left(p\_{1}^{-\frac{\sigma}{\xi}}-1\right)\right\}\right] \\ \quad \times \left[\frac{\sigma}{\xi}\left(p\_{2}^{-\frac{\sigma}{\xi}}-p\_{1}^{-\frac{\sigma}{\xi}}\right)\right]^{p\_{2}-1}\exp\left[-\gamma\_{2}\left\{\frac{\sigma}{\xi}\left(p\_{2}^{-\frac{\sigma}{\xi}}-p\_{1}^{-\frac{\sigma}{\xi}}\right)\right\}\right] \\ \quad \times \left[-\frac{\sigma}{\xi^{2}}\left\{(P\_{1}P\_{2})^{-\frac{\sigma}{\xi}}(\log P\_{2}-\log P\_{2})-P\_{2}^{-\frac{\sigma}{\xi}}\log P\_{2}+P\_{1}^{-\frac{\sigma}{\xi}}\log P\_{1}\right\}\right], \end{array} \tag{29}$$

where *ρ*1, *ρ*2, *γ*1, and *γ*<sup>2</sup> are hyper parameters obtained from the prior information. In the form of the median and some percentiles, the correspondences are the return periods of <sup>1</sup> *p*1 and <sup>1</sup> *p*2 . The prior for *q*<sup>1</sup> is in the principle depended on *μ*. This dependence is substituted by the dependence on the prior mean of *μ*. Interestingly, in some cases, the situation where *ξ* = 0 is considered. For example, a positive probability is set, and the prior distribution evaluates a probability *q* if *ξ* = 0 and 1 − *q* if *ξ* = 0.

#### 4.3.2. The Prior Density for Parameters at the Threshold

Apart from the information above the threshold, *u* is assigned to follow a truncated normal distribution with parameters *uμ*, *σ*<sup>2</sup> *μ* , curtailed from below at *e*<sup>1</sup> with density as Equation (30)

$$\pi \left( \mu \left| u\_{\mu}, \sigma\_{\mu}^{2}, e\_{1} \right> \right) = \frac{1}{\sqrt{2 \pi \sigma\_{\mu}^{2}}} \times \left\{ \frac{\exp \left[ -0.5 \left( \mu - u\_{\mu} \right)^{2} / \sigma\_{\mu}^{2} \right]}{\Omega \left[ - \left( \varepsilon\_{1} - u\_{\mu} \right) / \sigma\_{\mu} \right]} \right\},\tag{30}$$

with *u<sup>μ</sup>* is included in some high percentile and *σ*<sup>2</sup> *<sup>μ</sup>* is sufficient to present a reasonably noninformative prior. In other words, de Zea Bermudez et al. (2001) suggested that the higher level to set the prior distribution for *μ*, and this requires setting a prior distribution for the hyper thresholds.

#### 4.3.3. Posterior Density Estimations

From the expression of the likelihood in Equation (27) and the priors, the posterior distribution is given from using Bayes theorem, which is combined with simulations (the MCMC methods via Metropolis–Hastings algorithms, (Metropolis et al. 1953)). To get hold

of a gamma distribution for data below the threshold, the functional form on the logarithm scale is derived as follows

log *<sup>p</sup>*(*θ*|*x*) <sup>=</sup> *<sup>K</sup>* <sup>+</sup> *<sup>n</sup>* ∑ *i*=1 *I*(*xi* < *μ*)[*α* log *β* − log *τ*(*α*) + (*α* − 1)log *xi* − *βxi*] + *n* ∑ *i*=1 *<sup>I</sup>*(*xi* <sup>≥</sup> *<sup>μ</sup>*)log 1 − *μ* 0 *βα <sup>τ</sup>*(*α*)*t <sup>α</sup>*−1*e*−*β<sup>t</sup> dt* − *n* ∑ *i*=1 *I*(*xi* ≥ *μ*)log *σ* −1+*<sup>ξ</sup> ξ n* ∑ *i*=1 *<sup>I</sup>*(*xi* <sup>≥</sup> *<sup>μ</sup>*)log 1 + *<sup>ξ</sup>*(*xi*−*μ*) *σ* +(*<sup>a</sup>* <sup>−</sup> <sup>1</sup>)log *<sup>α</sup>* <sup>−</sup> *<sup>b</sup><sup>α</sup>* <sup>+</sup> (*<sup>c</sup>* <sup>−</sup> <sup>1</sup>)log *<sup>α</sup> <sup>β</sup>* − *d α β* + log *<sup>α</sup> β*2 −1 2 *μ*−*u<sup>μ</sup> σμ* − *b*<sup>1</sup> *μ* + *<sup>σ</sup> ξ p*−*<sup>ξ</sup>* <sup>1</sup> − 1 +(*a*<sup>2</sup> <sup>−</sup> <sup>1</sup>)log *μ* + *<sup>σ</sup> ξ p*−*<sup>ξ</sup>* <sup>2</sup> <sup>−</sup> *<sup>p</sup>*−*<sup>ξ</sup>* 1 <sup>−</sup> *<sup>b</sup>*<sup>2</sup> *u* + *<sup>σ</sup> ξ p*−*<sup>ξ</sup>* <sup>2</sup> <sup>−</sup> *<sup>p</sup>*−*<sup>ξ</sup>* 1 + log −*σ ξ* (*p*<sup>1</sup> *p*2) −*ξ* (log *<sup>p</sup>*<sup>2</sup> <sup>−</sup> log *<sup>p</sup>*1) <sup>−</sup> *<sup>p</sup>*−*<sup>ξ</sup>* <sup>2</sup> log *<sup>p</sup>*<sup>2</sup> <sup>+</sup> *<sup>p</sup>*−*<sup>ξ</sup>* <sup>1</sup> log *p*<sup>1</sup> , (31)

where *k* is the normalizing constant. For the computation, making analytical posterior distributions depends on the convergence rate in each MCMC simulations case.

#### 4.3.4. Risk Measurement

As the goal of the article is the risk analysis for a financial context. The famous Value at Risk (VaR) can summarize the worst loss over a target horizon with a given level of confidence and outline the overall market risk faced by an institution (Assaf 2009).

For extreme data analyses, the GPD continues to boundlessness. This kind of extreme distribution is not known with certainty in practice, but the Bayesian framework allows us to quantify this uncertainty. Expressly, the posterior predictive distribution follows

$$p\left(\mathbf{x}^f \middle| \mathbf{x}\right) = \int\_{\theta} p\left(\mathbf{x}^f \middle| \theta\right) p(\theta | \mathbf{x}) d\theta. \tag{32}$$

If uncertainty regarding an unknown parameter is captured in a posterior distribution, a predictive distribution for any quantity *μ* that depends on the unknown parameter, through a sampling distribution, can be achieved by the Equation (33). In this case, *p x f* - - *x* mentions to an updated GDP observation obtained a set of parameters. The following transformation gives the updated information:

$$\mathcal{U}\mathcal{U} \sim \mathcal{U}\\
1\\
form(0,1) \ge \log p(\theta|\mathbf{x}) = \left[ \left( \mathcal{U}^{-\varepsilon} - 1 \right) + \frac{\sigma}{\varepsilon} + \mu \right] \sim \mathcal{G}\\
PD(\mu, \sigma, \varepsilon) \tag{33}$$

By the MCMC methods, a large number of large updating samples can be stimulated. In terms of the GDP, the Value at Risk (VaR) and Expected Shortfall (ES) can be the expression as follows:

$$VaR(1 - \mathfrak{a}) = \left(\mathfrak{a}^{-\varepsilon} - 1\right)\frac{\sigma}{\mathfrak{e}} + \mathfrak{p},\tag{34a}$$

$$ES(1 - \alpha) = VaR(1 - \alpha) + \frac{\sigma \alpha^{-\varepsilon}}{1 - \varepsilon}.\tag{34b}$$

These measures are ordered to obtain quantiles to create intervals. Note that since negative log share returns are included, which are the GPD above a suitable threshold, it is crucial to rescale *α* by multiplying by the divide between the number of observations and number of exceedances.

#### **5. Computational and Comparative Results**

#### *5.1. Data Visualization*

In this section, the historical data tries to explain the type of non-normal distributions. Table 1 details the log-return transformation, which assures the set of observed data

are stationary in long-term periods, and the Phillips-Perron (PP) unit-root test confirms this. Additionally, the expression to define the data set is not normally distributed is represented by the significant level of the Jarque–Bera test. All financial indexes reject the null hypothesis, which refers to the normal distribution.


**Table 1.** Descriptive statistics (original data).

Source: authors.

#### *5.2. The Distribution Outlook Comparison*

In this crucial section, adapting from the contribution conducted by Gençay and Selçuk (2004), the threshold is set as 6%, which refers to the approximately understandable return of the stock exchanges. This is the prior information at the threshold line level that explicitly separates exceedance samples and risk-free samples. Shape and scale parameters are estimated from two comparative sources—an original observed distribution (Gaussian random walk) and quantum-wave distribution from each selected financial index. To compute the VaR model at 99% confidence and the corresponding expected shortfall, Table 2 represents the comparative outcome that indicates the modified distribution by applied quantum computing for Bayesian extreme value forecasting can capture missing information more efficiently than the traditional econometrics (Gaussian random walk process) because every DIC values indicate that the quantum distribution of five stock markets in five ASEAN countries is appropriate with the model of Bayesian extreme value prediction compared with data distribution based on the Gaussian random walk process.

**Table 2.** The model validation by Deviance Information Criterion (DIC).


Noted: \* stands for the minimum value of DIC calculations. Source: authors.

#### *5.3. Risk Measures by the Quantum Distribution*

In Table 3, it seems to be clear that the risk projections estimated from data sourced by quantum-wave distributions; risk measurements calculated by the VaR model and corresponding expected shortfalls are reported by this table. First, the strong advantage of Bayesian posterior densities is the ability to provide random parametric intervals, which are more suitable for quantile settings in the GPD and random distributions in financial sectors. The 2.5% interval can be applied to stand for the case of risk aversions, the mean (50%) indicates the risk-neutral case is mentioned, and the 97.5% interval pinpoints the risk lovers. In terms of the investors who need to maximize profits from the markets and protect the minimum risk as much as possible. For the predominant case, investing in Malaysia seems safer than the other four countries, the chance of failures is 16.78% in the case of risk taking, and the opportunity to loss equals 11.37% in risk-avoiding. This is supported by (Pero and Apandi 2018) to introduce Malaysia's leadership role in ASEAN. Malaysia can be deemed as a leader within ASEAN, championing several important policies in the international arena. On the other hand, Thailand's stock exchange is indicated to have the highest rate of losses in both the risk lover case and risk-avoiding case in ASEAN financial markets. The forecasting results are between 16.56% and 30%. Since the financial market partially depends on the situation of business confidence and the political atmosphere. The Thai stock exchange seems to absorb risks more than other ASEAN countries. For Singapore, the Philippines, and Indonesia, risk and return predictions are ranked in the third, fourth, and fifth, respectively. In the scenario of maximizing profit, 24.72% to 26.22% are approximately the taking losses when focusing on the investment in these three markets. Conversely, 14.32% to 15.20% are the chance of losses for the case of risk aversion.

**Table 3.** The extreme value prediction of Value at Risk (VaR) and Expected Shortfall (ES) is based on quantum mechanics.


Source: authors.

#### **6. Conclusions**

Most economic collapses have appeared in many computational predictions relied on raw observed distributions, which are still common sense for traditional econometrics research. The concept of the Gaussian random walk process continues to be suspicious, especially econometrics for stock predictions. In other words, the assumption of distributional normality is sensibly understandable, but it is sensitive to face suspicious predicted outcomes. At the center of the research gap to determine the origin of real data distributions, this article contributes to quantum mechanics applied for matching the wave function, which is relevant to the fundamental processes of thoughts. For this article, risk management in financial analyses is one of the top issues people have struggled to eliminate unquestionably.

Every level of complexities in data science potentially empowers the ability to capture missing information of the quantum-wave distribution. Expressly, the distribution generated by the quantum mechanics done in the wave equation is compatible with Bayesian inference for measuring risks and expected shortfall predictions, especially when exploring for preciseness in ASEAN financial markets. The comparison of DIC strongly secures this statement. With the quantum distribution, it is sensible to state that a realistic parameter is found, a harmonic inference regarding humans' decision making can be computed, and a meticulous estimation for dealing with extreme tails information in raw data can be demonstrated numerically. In conclusion, the quantum distribution can potentially fix the

gap of missing information in data analyses, especially modern econometrics in financial research.

For upcoming research, applying quantum computations in social science is more challenging. The clue that the quantum distribution can give more real observations is the research changer in the age of big-data analyses. The future plan for installing the novel distribution into behavioral economic research and financial econometrics is the major issue that critically confronts traditional aspects.

**Author Contributions:** This contribution is originated conceptually by C.C. In other words, C.C. has done major parts of the paper, including investigation, methodology, software, supervision, and project administration. He is also the corresponding author. For the sections regarding data curation, writing–original draft preparation, review and editing, and visualization, S.W. takes charge of these tasks. He is the co-author. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Data in finance, business and economics.

**Acknowledgments:** This research was supported by a research grant from Chiang Mai University.

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

#### **References**


Hooker, Cliff. 2011. *Philosophy of Complex Systems*, 1st ed. Edited by Dov M. Gabbay, Paul Thagard and John Woods. Handbook of the Philosophy of Science. vol. 10. ISBN 9780080931227. Available online: https://www.sciencedirect.com/book/9780444520760/ philosophy-of-complex-systems (accessed on 2 February 2021).


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