Application of Chow, Cusum and rolling window in testing stability of systematic risk of companies listed in WIG-ESG in 2019-2022

Mikołajek-Gocejna, Magdalena Article Application of Chow, Cusum and rolling window in testing stability of systematic risk of companies listed in WIG-ESG in 2019-2022 Journal of Banking and Financial Economics (JBFE) Provided in Cooperation with: Faculty of Management, University of Warsaw Suggested Citation: Mikołajek-Gocejna, Magdalena (2023) : Application of Chow, Cusum and rolling window in testing stability of systematic risk of companies listed in WIG-ESG in 2019-2022, Journal of Banking and Financial Economics (JBFE), ISSN 2353-6845, University of Warsaw, Faculty of Management, Warsaw, Iss. 20, pp. 1-29, https://doi.org/10.7172/2353-6845.jbfe.2023.2.1 This Version is available at: https://hdl.handle.net/10419/313474 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Banking and Financial Economics Journal of Banking and Financial Economics Volume 2023 Number 20 Article 1 March 2024 Application of Chow, Cusum and Rolling Window in Testing Application of Chow, Cusum and Rolling Window in Testing Stability of Systematic Risk of Companies Listed in WIG-ESG in Stability of Systematic Risk of Companies Listed in WIG-ESG in 2019–2022 2019–2022 Magdalena Mikołajek-Gocejna SGH Warsaw School of Economics, Institute of Value Management, Poland , [email protected].pl Follow this and additional works at: https://press.wz.uw.edu.pl/jbfe Part of the Finance and Financial Management Commons Recommended Citation Recommended Citation Mikołajek-Gocejna, Magdalena (2024) "Application of Chow, Cusum and Rolling Window in Testing Stability of Systematic Risk of Companies Listed in WIG-ESG in 2019–2022," Journal of Banking and Financial Economics : Vol. 2023: No. 20, Article 1. DOI: 10.7172/2353-6845.jbfe.2023.2.1 Available at: https://press.wz.uw.edu.pl/jbfe/vol2023/iss20/1 This Scholarly Research Article is brought to you for free and open access by Sekcja Wydawnicza Wydziału Zarządzania Uniwersytetu Warszawskiego/University of Warsaw Faculty of Management Press. It has been accepted for inclusion in Journal of Banking and Financial Economics by an authorized editor of Sekcja Wydawnicza Wydziału Zarządzania Uniwersytetu Warszawskiego/University of Warsaw Faculty of Management Press. 1 1 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) Journal of Banking and Financial Economics 2(20)2023, 1–29 DOI: 10.7172/2353-6845.jbfe.2023.2.1 Application of Chow, Cusum and Rolling Window in Testing Stability of Systematic Risk of Companies Listed in WIG-ESG in 2019–2022 Magdalena Mikołajek-Gocejna SGH Warsaw School of Economics, Institute of Value Management, Poland [email protected]aw.pl htt ps://orcid.org/0000-0002-8979-2491 Received: 26 June 2023 / Revised: 25 November 2023 / Accepted: 4 December 2023 / Published online: 28 December 2023 ABSTRACT The aim of the article is to analyze the stability of beta coeffi cients of companies listed in WIG-ESG. There are many studies on the stability of companies’ systematic risk, but the literature and research lack an analysis of the stability of the beta coeffi cient for ESG companies. We examined beta coeffi cients for 57 companies listed in WIG-ESG, established for sets of daily rates of return between September 3, 2019, to June 6, 2022 (period including COVID-19 crisis and asset price infl ation, Russian invasion of Ukraine). We estimate the beta coeffi cient for the whole as a result of which we obtain the average value of the beta coeffi cient over the entire analyzed period, and subperiods with fi xed length rolling window, resulting in a time series of beta coeffi cients. To assess beta stability, we used the Chow test with the F statistic, the Cusum test based on generalized fl uctuations test framework, and the Wald-Wolfowitz runs test of randomness around the mean for the time series beta coeffi cients obtained in the rolling window. The considered tests argue for the instability of the time series of beta coeffi cients in most of the companies tested: 93% short-term instability cases confi rmed by the Chow test, 100% short-term instability cases confi rmed by the Wald-Wolfowitz runs test. The paper is an initial attempt to bridge the gap that presently exists between the theoretical and empirical literature on the stability of ESG companies’ systematic risk. It cannot be ruled out (hypothesis) that the beta coeffi cient for companies listed in the WIG-ESG index is/will be stable over longer periods of time. JEL Classifi cation: G11, G12, G13 Keywords: Capital Asset Pricing Model (CAPM), beta coeffi cient, systematic risk, ESG, environment, social and governance criteria, Cusum Test, Chow Test, rolling window. 1. INTRODUCTION In recent years, sustainable fi nance has become one of the most important trends, especially in developed capital markets. Investors, market supervisory authorities and companies, by considering ESG (environmental, social and corporate governance) factors, respond to global challenges that we all face and will face in the coming decades. ESG factors, although they present 2 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 current data, refer primarily to the future, because they show how to effectively manage longterm risk and create value not only for shareholders, but for all stakeholders of the company. The companies that meet the social, environmental, and corporate governance criteria are more aware of the changes taking place in the world, thanks to which they better forecast their future situation, and their operations are more stable and sustainable. In the fi fth edition of the GPW survey on the impact of ESG factors on investment decisions, 81% of professional stock market investors in Poland assessed that companies that have implemented the ESG strategy are perceived as entities with lower risk. (GPW, 2019). Moreover, companies with a strong ESG profi le are less vulnerable to systematic market shocks and therefore show lower systematic risk (Mikołajek-Gocejna, 2022, pp. 597–615). Identifying and measuring risk have been of constant interest to both fi nancial theoreticians and practitioners. Various theories have been propounded for pricing of assets considering the risk element. The most common and widely accepted method has been the capital asset pricing model (CAPM) model, which takes into consideration the systematic risk of the asset, measured as the beta coeffi cient. The beta coeffi cient is defi ned as the ratio of the covariance of the rate of return of the examined fi nancial instrument Ri and the rate of return of the market portfolio Rm to the variance of the rate of return of the market portfolio (Tofallis, 2008, p. 1359): () () (, ) () () ,cov var R cor R R var R var R RR im im im m i #b== , (1) where: Ri – measures the rate of return of the fi nancial instrument, Rm – measures the rate of return of the market portfolio, cov (Ri, Rm) is the covariance between the rates of return. In general, the calculation of the beta coeffi cient is based on comparing volatility of the rate of return from shares of a specifi c company in the adopted unit of time with volatility of the rate of return from the stock exchange portfolio (index) adopted for comparison (Dharmaratne, Harris, 2006, pp. 68–61). Since volatility – in this case, of the rate of return – refl ects the risk of their realization, the measurement of the beta coeffi cient means the measurement and comparison of risks related to the investment in the shares of a given entity and the average, previously defi ned market portfolio, respectively (this measurement should concern the expected rate of return, practice shows however, that beta is calculated on the basis of historical, i.e. realized rate of return). The beta coeffi cient is also an estimator of the parameter of simple linear regression equation proposed by Sharpe (1963). Therefore, the rate of return on shares of the i-th company in the t-th period can be written as (Elton, Gruber, 1998, p. 154; Jajuga, Jajuga, 1998, p. 63): Rit = αi + βi Sharp Rmt + εit , (2) where: Rit – rate of return of shares of the i-th company, Rmt – rate of return on an index of the market, α i – the free expression of the model, which is a component of the return on shares of the company and independent of the market situation, β i Sharp – the direction coeffi cient constant over time which measures the expected change in R i depending on the change in Rm, εit – is Gaussian noise N (0, σi) with zero as expected value and standard deviation σi, t – number of observations of the time series. 3 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 In the Capital Asset Pricing Model, there is an additional variable: risk-free rate of return RF: (Treynor (1961), Sharpe (1964), Lintner (1965a, 1965b), Mossin (1966)) Ri = RF + βi CAMP (Rm – RF) + εi . (3) In the equation, the risk-free return RF can be a deterministic constant or a random variable. CAPM is the most frequently and most willingly model of estimation of the cost of capital used in practice, due to its easy implication and interpretation. The beta coeffi cient is also called stock aggressiveness. Malkiel and Xu (2006) identifi ed this type of risk as the systematic risk, which is undiversifi able. Possibilities of using beta in the practice of investment processes are closely related not only to the correctness of its estimation, but also its stability over time (Wright, Mason, and Miles 2003). The Sharpe model and Capital Asset Pricing Model assume that beta is stable and predictable over time. (Treynor, 1965, pp. 63–75). Thus, the main hypothesis of the article is that beta coeffi cients of ESG companies listed on the Polish capital market are not stable in short time. Despite the problem of beta stability is quite well described in the literature, results of the stability tests carried out over the years by various researchers are ambiguous, inconclusive, and contradictory. Moreover, literature and research lack an analysis of the stability of the beta coeffi cient for ESG companies. This paper is an initial attempt to bridge the gap that presently exists between the theoretical and empirical literature on the stability of ESG companies’ systematic risk. 2. STABILITY OF ESG COMPANIES BETA – LITERATURE REVIEW An important issue from the point of view of forecasting and the possibility of making investment decisions on the basis is the analysis of beta stability over time and the study of the sensitivity of its assessments to changes in the method of estimating the model and measurement of variables. Beta instability causes low predictive effi ciency of the model, as makes it impossible to use the dependencies described by the model in the future. Moreover, inference based on a model with unstable parameters may result in large errors. 2.1. Systematic risk of ESG companies Literature and research lack an analysis of the beta coeffi cient stability for ESG companies. Thus, two groups of publications were analyzed. The fi rst covered research on the risk of ESG companies, the second, stability of beta coeffi cients. It was necessary to combine the two issues and carry out studies on the stability of the systemic risk for ESG companies. In the literature, there are not many cases of studies analyzing systematic risk of ESG companies or the relationship between ESG factors and company-specifi c risk (Sassen, Hinze, and Hardeck, 2016; Mikołajek-Gocejna, 2022). Most studies show, that involvement in social and environmental activities leads to improvement in an organization’s image, and its credit ratings, as well as lowering the cost of capital (Gangi et al., 2020; Xue et al., 2020), caused largely by a decrease in risks measured appropriately, e.g., by the standard deviation of rates of return or the beta coeffi cient. Boutin-Dufresne and Savaria research (2004) showed that corporate social responsibility activities can help diminish the overall business risk of a company, and even improve its longterm risk-adjusted performance. 4 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Negative correlation between systematic risk and CSR was also confi rmed by Jo and Na (2012). Orlitzky and Benjamin (2001) reviewed 18 American cases of studies on the relationship between corporate social performance (CSP) and risk, indicating that integration of ethical factors in corporate management leads to their lower exposure to fi nancial risk. Similar results were obtained by Boutin-Dufresne and Savaria for Canadian fi rms (2004). Albuquerque et al. (2019) examined the relationship between CSR and fi rms’ systematic risk using a sample of 28578 annual observations of the United States companies over the period 2003–2015 and found that the level of systematic risk is lower for companies with better CSR performance. Similar results were obtained by Shakil (2021), Rehman et al. (2020) and Zhou et al. (2020). Analysis conducted by Hassan et al. (2021) showed that companies that follow stricter ESG principles are more resilient to systematic market shocks regardless of their country of origin. The authors analyzed 4624 non-fi nancial fi rms from Africa, Asia, Europe, Latin America, North America, and Oceania over the period 2002–2018. Moreover, Dunna et al. (2018), concluded that high-scoring ESG stocks have lower volatility and betas than lower scoring ESG stocks. Research conducted by Bouslah, Kryzanowski, and Mzali (2011) showed that not all ESG aspects affect the systematic risk of companies. Employee relations, environment, human rights and corporate governance negatively affect fi rm risk, but other dimensions (community, diversity and product) do not signifi cantly impact fi rm risk. Thus, next to the studies that used aggregated ESG measures, there are studies based on individual ESG measures as explanatory variables. For example, Sharfman and Fernando (2008) confi rmed the negative correlation between the cost of equity (beta coeffi cient) and the quality of environmental management in American companies. Zaman et al. (2021) found that eco-innovative companies are less risky. Xue et al. (2020) claimed that involvement in environmental activities can consequently reduce fi nancial risk. Similar results were obtained by Salama et al. (2011). Moreover, Zaman et al. (2021) found a negative relationship between eco-innovation and stock price crash risk. In turn, research conducted by Chen et al. (2020) showed that there is a negative correlation between the dominant role of institutional investors in the shareholding structure of a company and its risk. 2.2. Stability of systematic risk The problem of beta stability is quite well described in the literature, however, the results of stability tests carried out over the years by various researchers are ambiguous, inconclusive, and contradictory. Most of the analyses were conducted in developed markets, but there are also studies on the stability of systematic risk for companies listed on developing markets. They include both studies on individual stocks as well as portfolios. Results of empirical work on beta instability can be divided into three groups: those that confi rm that beta is stable over time, those that confi rm its instability and those that give ambiguous indications (Table 1). The existence of stability of beta over different phases of the market was confi rmed by analyses conducted by Shamsher et al. (1994), Fabozzi and Francis (1977), Fisher and Kamin (1985) Faff (2001), Das (2008), George and Bainy (2012), Harish and Mallikarjunappa (2019). Several studies documented that beta is time varying because of the infl uence of microeconomic and macro-economic factors. The time varying nature of beta at the New York Stock Exchange was fi rst discovered by Blume (1971). Instable betas were also confi rmed by researches conducted by Sunder (1980), Bos and Newbold (1984) Russel, Impson and Imre (1994), Braun et al. (1995), Brooks et al. (1998), Faff, Hillier, Hillier (2003), Shah, Moonis, (2003), Irala (2007), Sarma and Sarmah (2008), Attya and Eatz (2011), Simon et al. (2012), Mazowina (2013), Celik (2013), Wijethunga and Dayaratne (2015), Ye (2017), Gupta (2020) 5 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Contradictory results in beta stability were obtained by: Baesel (1974), Levy (1971), Witkowska (2008), Singh (2008), Ray (2010), Deb and Mistra (2011), Terceño et al. (2011), Dubey (2014), Dębski et al. (2011), Ye (2017), Mikołajek-Gocejna (2021). One of the most widely used methods to estimate beta as a time series process is the Kalman Filter (Kalman, 1960). It has been applied for the estimation of betas and tests for beta constancy in several studies (e. g. Bos, Newbold, 1984; Fisher, Kamin, 1985; Shah, Moonis, 2003). Kalman fi lters for beta estimation also presented diffi culties, due to their failure to deal with the problem of heteroskedasticity (Fisher, Kamin, 1985). 3. METHODOLOGY AND DATA 3.1. Systematic risk estimation and data In the study, we will estimate the beta coeffi cient as an estimator of the parameter of simple linear regression equation proposed by Sharpe (1963). Rit = αi + βi Sharp Rmt + εit , (4) where t is the index of the moments of time from the period T from which samples of the analyzed rate of returns for the i-th company are derived. We examined beta coeffi cients for 57 companies listed in WIG-ESG, established for the sets of daily rates of return between September 3, 2019, to June 6, 2022 (period including COVID-19 crisis and asset price infl ation, Russian invasion of Ukraine). To obtain an up-to-date beta rating, the model should be estimated over a relatively short period of time, while maintaining the estimation sample size requirements. Therefore, our studies prefer daily quotations, however we are aware of the limitations of the approach.1 According to the theoretical assumptions of the Sharp/CAPM model, the market index should cover the broadest spectrum of investment instruments available to investor. Thus, we choose the rate of return from the WIG Index (market index) as the variable explaining the rates of return of individual ESG companies. We estimate the beta coeffi cient for: 1) the whole, as a result of which we obtain the average value of the beta coeffi cient over the entire analysis period, 2) and subperiods with fi xed length rolling windows, resulting in a time series of beta coeffi cients. In the study covering the whole period, we used the beta coeffi cient estimation by the OLS regression of the Sharp equation (4), which ensures that estimators are unbiased (or at least asymptotic, unbiased and consistent when the variable Rm is random): 1,RR RR i iSharp mm mi a b=-ll = 6 G @ (5) where: Ri – (n x 1) vector of daily return on assets i, R m – (n x 2) matrix of daily return on a market portfolio proxy with 1 in the fi rst column (for intercept). 1 The use of daily returns avoids the dilemma of how to estimate them that accompanies longer intervals. In addition, aggregating daily returns to e.g., monthly returns causes a loss of important information. An important argument for the use of high-frequency data is also the possibility of obtaining a relatively long sample for a short period of time (i.e., many observations, which gives relatively low standard errors) 6 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 This method is the simplest computationally, although it is numerically less effi cient than the one used of defi nition (1) and effi cient recursive algorithms for calculating moments. Due to the purpose of the research, we prioritize the ease of calculations over their effi ciency. In the rolling regression (Zivot and Wang, 2006, pp. 342–349), period T is divided into sub-periods: 1) containing the same number of 20 observations, 2) which we shift in the time domain by one observation (rolling window) from the beginning to the end of the period T, 3) beta coeffi cient βi Sharp(t) estimated for the data from a given subperiod (window) is assigned to the end of t of the subperiod: 1 () () () () () (), t tRtRt RtRt i iSharp mm mi a b=-ll > 6 H @ (6) where: Ri(t) is an (20 x 1) vector of daily return on assets i in which the fi rst element is Ri t – 19 and the last is Ri t, Rm(t) is an (20 x 1) matrix of daily return on a market portfolio proxy in which the fi rst row is the vector (1, Rm t – 19) and the last is (1, Rm t). As a result of the procedure, we obtain a time series of estimated beta coeffi cients. The 20-day length of the time window is dictated by the length of the series (686 days), the daily data frequency that corresponds to the average length of month, and by the desire to obtain a given degree of data smoothing, and the number of regressions required (667 for each of the 57 companies). Assigning the result of the beta parameter estimation to the end of the interval, results in no beta assigned to the initial 19 days period. In the estimation, we assume that the random regression component εi is normally distributed. 3.2. Stability testing The issue of beta stability can be treated as a problem of invariance of their estimates, and it applies both to its stability over time, as well as to no sensitivity to changes in the method and frequency of measurement of variables and methods of model estimation (Tarczyński et al., 2013, p. 71). To assess beta stability, we used: 1) Chow test (Chow, 1960), with the F statistic, 2) Cusum test (Ploberger and Kramer, 1992), based on the generalized fl uctuations test framework, 3) Wald-Wolfowitz runs test of randomness around the mean for the time series beta coeffi cients obtained in rolling windows. In the Chow test period T with daily data is divided into two parts T1 and T2 with a shifting time of division from the 20th day from the beginning to the 20th day before the end of period T. Thus, the division point covers all possible dates for dividing the series into two disjoint parts with a minimum number of 20 observations in each part. The test compares OLS residuals estimated (just like (5)) from models estimated separately in T 1 and T 2 subsamples with OLS residuals estimated for the whole series2: 2 Here and further designations in the equation adapted to the designations of the variables in the article. 7 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 () , () () ( ) () () Ft e u te t n k uetet 2 i ii ii i i T TT =- - tt tt t t (7) where: t = 20 … (n – 19) is the division point of period T, ûi – OLS residuals from the model, where parameters are fi tted for all observations, êi(t) – OLS residuals from the full model, where coeffi cients in subsamples 1 ≤ T1 ≤ t and t + 1 ≤ T2 ≤ n are estimated separately, n – number of observations, k – number of regression coeffi cients (k = 2), F i (t) – has an asymptotic χ 2 distribution with k degrees of freedom (F i (t)/k has a F distribution with k and n – 2k degrees of freedom). The Cusum test (Ploberger and Kramer, 1992) is based on the recursive residuals. We estimate a simple OLS model (just like (5)) for sub-periods from the fi rst observation to the end of the sub-period. The end of the sub-period varies from the third observation (k + 1, where k = 2) to the one before last one (n – 1). Based on the obtained estimators for data up to the moment t – 1, we predict the value of the rate of return Rit of the fi nancial instrument for the moment t with an error: ûit = Rit – [αi(t – 1) + βi Sharp (t – 1)Rmt], (8) where: αi(t – 1), βi Sharp (t – 1) – OLS estimate (6) based on all observations up to t – 1 of assets i, t = (1 + k) … n, (for k = 2, t = 3 … n) The variance of predictor is σ2 1 ()() ,RRt Rt R 11 1 1 1 mt m m mt +-- - l ^ e h o 6 = @ G where: Rm(t – 1) is a (t – 1 x 2) matrix of monthly return on a market portfolio proxy in which the fi rst row is the vector (1, Rm1) and the last is (1, Rm (t – 1)), and σ2 is the variance of disturbance. After scaling the ûit errors, we get recursive residuals: 1 11 1 1 ,w RRt Rt R u 1 it mt m m mt it = +-- - l t ^^ _ hh i 6 = @ G (9) with the zero expected value and constant variance σ2 (homoscedasticity)3. We cumulatively sum the standardized recursive residuals obtaining: ,wW 1 iit t3 2 hv =x x= + u/ (10) where: η = n – 2 is the number of recursive residuals, τ = 1 … η is the index of cumulative sums of recursive residuals, nww 2 1 it j n2 3 v=--k = u ^ h / is the variance estimate of wit . 3 OLS residuals are not homoscedastic, even if the variance of the disturbance is constant. 14 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Table 1 Beta stability in academic research Sta bility of beta Author Market Period of analysis Number of stocks Method of testing Stable beta Fabozzi and Francis (1977) NYSE January 1966 to December 1971 700 Dummy as variable Fisher and Kamin (1985) NYSE 1926–1979 All listed cross-sectional regression, variance analysis, Kalman Filter Faff (2001) Australian Stock Exchange 1974 to 1995 (monthly return) 24 industry portfolio Dummy variable, Regression analysis Sromon, Das (2008) Indian Stock Exchange NSE Nifty February 1999 to September 2007 39 stocks Time as variable Dummy as variable George and Bainy (2012) Indian Stock Exchange BSE100 Index 1996–2009 169 stocks Time as variable Dummy as variable Harish and Mallikarjunappa (2019) Indian Stock Exchange S&P BSE Sensex companies 2000–2014 30 stocks Chow test, multiple breaking point test, CUSUM test Instable beta Blume (1971) NYSE January 1926 to June 1968 All listed Regression analysis Analysis of beta correlations Sunder (1980) NYSE 1926–1975 127 stocks Variance analysis Bos, Newbold (1984) Kalman Filter Russell, Impson and Imre (1994) NYSE 500, 250, 200, 125, and 100 trading days 2,497 stocks Variance analysis Braun et al. (1995) NYSE July 1926–December 1990 CRSP NYSE Rolling windows, GARCH Brooks et al. (1998) Singapore Stock Exchange 1986 to 1993 247 stocks OLS regression estimates Faff, Hillier, Hillier (2003) UK 1st January, 1969 to 30th April, 1998 32 industry sector portfolios GARCH model, rolling window Shah, Moonis (2003) India’s Bombay Stock Exchange 1st May 1996 to 30th March 2000 50 stocks GARCH, Kalman Filter Irala (2007) Indian Stock Market BSE April 1994 to March2006 660 stocks statistical signifi cance of the CAPM model (t-Student test) Sarma and Sarmah (2008) Indian Stock Market BSE December 2001 to November2006 5 stocks Chow test Razvan et al. (2009) Bukarest Stock Exchange 20th January to 20th July 2009 10 stocks statistical signifi cance of the CAPM model(t-Student test) Instable beta Javid & Ahmad (2011) Pakistan Stock Market Karachi Stock Exchange 1993–2007 50 stocks Dummy as variable Mazowina (2013) Zimbabwean Stock Market February 2009 to 31 December 2012 66 stocks Chow test 15 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Sta bility of beta Author Market Period of analysis Number of stocks Method of testing Instable beta Celik (2013) Turkish Stock Market 03.01.2005–31.12.2009 Istanbul Stock Exchange (ISE) sector indices and the ISE-30 All Share Index Rolling windows Simon et al. (2012) Brazilian stock exchange (BM&FBOVESPA) 2002 and 2011 All stocks Analysis of beta correlations Wijethunga A.W.G. Dayaratne D.A. (2015) Colombo stock exchange 2005–2013 26 stocks Rolling windows Ye (2017) China’s Stock Exchange Shanghai Stock Exchange and the Shenzhen Stock Exchange Before 2008 208 stocks Dummy as a variable Gupta (2020) Bombay Stock Exchange January 2006 to January 2018 11 sectors Chow Test Dummy Variable Levy 1971 NYSE 1962–1970 500 stocks Correlation analysis Baesel (1974) NYSE January 1950 to 1967 160 stocks Chow-test Witkowska (2008) Warsaw Stock Exchange 2000–2006 8 stocks t-Student test Contractionary results Singh (2008) Bombay Stock Exchange BSE 1991–2002 158 stocks Regression analysis Ray (2010) Bombay Stock Exchange BSE100 January 1996 to December 2009 100 stocks Time as variable Dummy as variable Chow test Deb and Mistra (2011) Indian Stock Market BSE 1996 to 2010 158 stocks Dummy as variable Terceño et al. (2011) Hong Kong Stock Exchange 01.01.2005 and 06.31.2009 All stocks OLS regression estimates Dubey (2014) National Stock Exchange of India June 15, 2001 to March 31, 2010 t 25 stocks OLS regression estimates, wavelet fi lters Dębski et al. (2016) Warsaw Stock Exchange 2005–2013 134 stocks Chow test Dębski et al. (2017) Warsaw, Frankfurt and Paris Stock Exchange 2005–2015 37 stocks 28 stocks 36 stocks t-Student test, Chow test Ye (2017) Shenzhen Stock Exchange which January 2008 to December 2013 208 stocks t-Student test dummy variable Mikołajek-Gocejna (2021) emerging markets 2005–2021 25 emerging markets indexes Chow-test Cusumtest Rolling-window Source: Own collaboration. Table 1 – continued 16 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Table 2 Chow and Cusum Test Walor Chow.date Chow.sup.F Chow.sup.F.pvalue Cusum.S Cusum.S.pvalue IIB 382 15.08185528 0.019973285 0.647821148 0.327845666 DNP 286 30.52889011 1.08E-05 0.491109827 0.646765335 LTS 368 49.32294455 5.43E-10 0.791890783 0.145420383 MIL 378 113.9468673 00.734618166 0.204967266 ING 283 77.83149039 1.11E-16 0.534963667 0.548012718 OPL 359 23.97115944 0.00029592 0.409997695 0.824710486 MBK 402 73.54352641 9.90E-16 0.596196782 0.420722389 PGE 365 46.80376457 2.10E-09 0.861671606 0.092397623 CCC 363 88.51940363 00.835298751 0.110174712 ABS 129 9.077899211 0.233223383 0.320920321 0.9488163 KRU 129 28.0469548 3.84E-05 0.763552274 0.172902915 ALR 387 155.275043 00.9380663 0.053787773 EAT 287 23.49410408 0.000374523 0.708004418 0.238267184 PLW 360 46.23915774 2.84E-09 0.582102012 0.448527549 KTY 438 53.41596002 5.99E-11 0.378141994 0.882660864 BHW 379 114.6822427 00.450062631 0.739790413 JSW 381 48.63765734 7.85E-10 0.654193277 0.317402503 GTC 276 9.238391044 0.220097067 0.472763041 0.688630113 CAR 379 39.67107573 9.29E-08 0.534859982 0.548240886 ATT 378 49.23252259 5.70E-10 0.733330974 0.206492187 EUR 377 40.61439883 5.64E-08 0.461204888 0.714829501 BDX 367 22.50915189 0.000607508 0.467051358 0.701606453 ENG 425 19.91837428 0.002127018 0.36816124 0.898235543 KER 608 128.8762717 00.232403379 0.965952905 ENA 377 31.76535268 5.74E-06 0.628207876 0.361415019 TPE 359 31.42702312 6.84E-06 0.602723896 0.408187535 FMF 381 39.34008645 1.11E-07 0.511410835 0.600601617 CMR 544 41.60377911 3.34E-08 0.334056038 0.939119414 LCC 377 50.27374761 3.26E-10 0.420610376 0.803214462 WPL 378 55.69859382 1.74E-11 0.454188632 0.730590552 ECH 357 49.21092465 5.77E-10 0.344339182 0.929026721 GPW 316 36.13063626 5.96E-07 0.357433279 0.913294154 17 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Walor Chow.date Chow.sup.F Chow.sup.F.pvalue Cusum.S Cusum.S.pvalue PKP 373 62.34990402 4.68E-13 0.68056555 0.276606314 VRG 130 16.25462029 0.011777334 0.626268015 0.364851298 CIE 386 32.05733983 4.94E-06 0.48603112 0.658363247 BFT 292 43.38643443 1.30E-08 0.457931241 0.72219986 MAB 367 12.8237853 0.053278843 0.514295356 0.59409052 AMC 371 51.24419302 1.93E-10 0.797223443 0.140659091 FTE 378 11.14458497 0.106267508 0.551025104 0.51311024 LVC 462 27.41579518 5.29E-05 0.500232976 0.625962749 LWB 463 38.86331504 1.42E-07 0.756345023 0.180500265 BRS 352 17.89436681 0.005531515 0.54891527 0.517641872 STP 289 33.46658137 2.38E-06 0.65642274 0.313802547 PXM 358 44.69984055 6.45E-09 0.584578514 0.443570835 GNB 391 5.76518121 2.89E-11 0.430552857 0.782337045 CIG 373 16.75431427 0.009373255 0.599674445 0.414016321 TRK 368 25.21049507 0.000159893 0.566419866 0.480587001 GTN 396 27.7096287 4.56E-05 1.268167143 0.003028214 PKO 368 193.8597071 00.648530667 0.32667157 PZU 373 115.5760748 00.673814359 0.286678652 PKN 368 82.93981804 0 1.075136042 0.018119942 CDR 349 33.33283174 2.56E-06 0.362188842 0.90684615 LPP 379 91.49276578 00.647212353 0.328855338 SPL 373 103.3113881 00.641553246 0.338340222 KGH 368 83.82726214 00.659523834 0.308841615 CPS 347 35.68615614 7.52E-07 0.44308895 0.755198423 PGN 377 32.00106162 5.09E-06 0.85915987 0.093981604 Source: own estimation. Table 2 – continued 18 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Table 3 Linear trend regression Company a a.se a.t a.pvalue b b.se b.t b.pvalue betats.se betats.R2 betats.df betats.F DW pv_of_DW IIB –0.10934 0.039239 –2.78645 0.0055 0.000773 0.000102 7.591129 1.08E-13 0.506131 0.079744 665 57.62525 0.135391 3.45E-129 DNP –0.22538 0.038838 –5.80316 1.01E-08 0.00126 0.000101 12.50971 2.14E-32 0.50096 0.190498 665 156.4929 0.142529 3.21E-128 LTS –0.53456 0.039991 –13.3672 2.88E-36 0.002805 0.000104 27.04442 3.18E-109 0.515826 0.523776 665 731.4004 0.110361 1.30E-132 MIL –0.61177 0.063329 –9.66015 9.51E-21 0.003704 0.000164 22.54674 4.72E-84 0.816862 0.433249 665 508.3553 0.080081 8.15E-137 ING –0.5202 0.036921 –14.0898 1.22E-39 0.002388 9.58E-05 24.93442 2.15E-97 0.476226 0.483184 665 621.7251 0.085091 4.08E-136 OPL –0.36444 0.024154 –15.0882 1.87E-44 0.001569 6.27E-05 25.03753 5.67E-98 0.311554 0.485245 665 626.8779 0.210043 3.02E-119 MBK –0.43055 0.059367 –7.25233 1.14E-12 0.002994 0.000154 19.44058 5.34E-67 0.765749 0.362377 665 377.936 0.083031 2.11E-136 PGE –0.65123 0.053985 –12.0632 1.92E-30 0.003359 0.00014 23.99133 4.11E-92 0.696327 0.463962 665 575.584 0.128772 4.33E-130 CCC –1.11246 0.064932 –17.1328 8.90E-55 0.004509 0.000168 26.77214 1.07E-107 0.837532 0.518725 665 716.7475 0.069055 2.31E-138 ABS –0.23277 0.028858 –8.06603 3.39E-15 0.000832 7.49E-05 11.11845 1.86E-26 0.372232 0.156755 665 123.62 0.180573 4.01E-123 KRU –0.56067 0.047961 –11.6901 7.64E-29 0.002366 0.000124 19.02112 9.65E-65 0.618633 0.352359 665 361.8029 0.179437 2.84E-123 ALR –0.81117 0.057181 –14.1862 4.24E-40 0.004168 0.000148 28.09997 3.96E-115 0.737552 0.542832 665 789.6083 0.078982 5.72E-137 EAT –0.44618 0.039715 –11.2346 6.18E-27 0.002164 0.000103 21.00701 1.55E-75 0.512267 0.398894 665 441.2944 0.141319 2.20E-128 PLW –0.23091 0.043952 –5.25359 2.01E-07 0.001608 0.000114 14.1014 1.07E-39 0.566922 0.23019 665 198.8495 0.140928 1.95E-128 KTY –0.37574 0.036678 –10.2442 5.75E-23 0.001764 9.51E-05 18.54246 3.49E-62 0.473099 0.340816 665 343.8229 0.124189 1.03E-130 BHW –0.439 0.038992 –11.2588 4.91E-27 0.00227 0.000101 22.44798 1.67E-83 0.502938 0.431095 665 503.9119 0.081999 1.51E-136 JSW –0.85591 0.098975 –8.64777 3.92E-17 0.004553 0.000257 17.73435 6.51E-58 1.276644 0.321087 665 314.5073 0.085206 4.24E-136 GTC –0.11969 0.036165 –3.30943 0.000985 0.000739 9.38E-05 7.879995 1.34E-14 0.466479 0.085401 665 62.09431 0.07637 2.46E-137 CAR –0.35157 0.041364 –8.49942 1.25E-16 0.001759 0.000107 16.39514 5.47E-51 0.533544 0.287857 665 268.8006 0.127661 3.06E-130 ATT –0.49452 0.056158 –8.80581 1.12E-17 0.002243 0.000146 15.40004 5.43E-46 0.724366 0.262881 665 237.1612 0.096747 1.71E-134 EUR –0.12467 0.041921 –2.97386 0.003047 0.001648 0.000109 15.15995 8.30E-45 0.540721 0.256837 665 229.8241 0.117675 1.32E-131 BDX –0.50655 0.039252 –12.9052 3.66E-34 0.001714 0.000102 16.82985 3.26E-53 0.506293 0.298704 665 283.244 0.120805 3.53E-131 ENG –0.14249 0.018359 –7.76142 3.18E-14 0.000711 4.76E-05 14.93518 1.05E-43 0.236805 0.251176 665 223.0597 0.173244 4.26E-124 KER –0.55356 0.047807 –11.579 2.25E-28 0.002423 0.000124 19.53706 1.61E-67 0.616645 0.364668 665 381.6967 0.126997 2.48E-130 ENA –0.42754 0.048882 –8.74644 1.80E-17 0.002521 0.000127 19.88542 2.08E-69 0.630513 0.372896 665 395.4299 0.109046 8.56E-133 TPE –0.41522 0.068442 –6.06673 2.19E-09 0.002383 0.000178 13.42528 1.55E-36 0.882804 0.21324 665 180.2382 0.093357 5.78E-135 FMF –0.29596 0.057144 –5.17916 2.96E-07 0.00258 0.000148 17.40339 3.50E-56 0.737075 0.31293 665 302.8781 0.108318 6.80E-133 CMR –0.44025 0.038053 –11.5692 2.48E-28 0.001823 9.87E-05 18.47179 8.29E-62 0.490838 0.339102 665 341.2069 0.12469 1.20E-130 LCC –0.23598 0.046138 –5.11457 4.12E-07 0.001223 0.00012 10.22093 7.08E-23 0.595123 0.135766 665 104.4673 0.081938 1.48E-136 19 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Company a a.se a.t a.pvalue b b.se b.t b.pvalue betats.se betats.R2 betats.df betats.F DW pv_of_DW WPL –0.57783 0.043125 –13.399 2.05E-36 0.00245 0.000112 21.89962 1.84E-80 0.556254 0.419008 665 479.5935 0.121088 3.86E-131 ECH –0.23806 0.034685 –6.86346 1.54E-11 0.001299 9.00E-05 14.4402 2.60E-41 0.447389 0.238712 665 208.5193 0.104903 2.30E-133 GPW –0.35757 0.024844 –14.3925 4.40E-41 0.001215 6.44E-05 18.84996 7.97E-64 0.320456 0.348244 665 355.3211 0.149209 2.56E-127 PKP –0.38135 0.046959 –8.12093 2.25E-15 0.002668 0.000122 21.90694 1.68E-80 0.60571 0.41917 665 479.9141 0.101601 8.03E-134 VRG –0.25261 0.032935 –7.67011 6.13E-14 0.001151 8.54E-05 13.46807 9.86E-37 0.424815 0.214309 665 181.3888 0.148726 2.21E-127 CIE –0.52295 0.046032 –11.3605 1.86E-27 0.002109 0.000119 17.65967 1.60E-57 0.593754 0.31925 665 311.8638 0.115442 6.50E-132 BFT –0.24151 0.038776 –6.22829 8.36E-10 0.001276 0.000101 12.68881 3.42E-33 0.500155 0.194921 665 161.006 0.121956 5.08E-131 MAB –1.06978 0.096004 –11.1431 1.47E-26 0.004043 0.000249 16.23379 3.61E-50 1.238328 0.283819 665 263.5359 0.16929 1.27E-124 AMC –0.47292 0.036745 –12.8702 5.26E-34 0.001827 9.53E-05 19.17004 1.53E-65 0.473962 0.355926 665 367.4904 0.13 6.37E-130 FTE –0.21564 0.057772 –3.73259 0.000206 0.001419 0.00015 9.466455 4.93E-20 0.745177 0.118754 665 89.61378 0.105587 2.86E-133 LVC –0.48436 0.042387 –11.427 9.79E-28 0.001795 0.00011 16.32939 1.18E-50 0.546735 0.286212 665 266.649 0.141167 2.10E-128 LWB –0.61397 0.072098 –8.51577 1.10E-16 0.003455 0.000187 18.47396 8.07E-62 0.929971 0.339155 665 341.2873 0.062795 3.03E-139 BRS –0.23199 0.026468 –8.76469 1.55E-17 0.00121 6.87E-05 17.62468 2.45E-57 0.341406 0.318389 665 310.6292 0.247096 1.84E-114 STP –0.40903 0.053849 –7.59578 1.04E-13 0.00272 0.00014 19.47433 3.51E-67 0.694585 0.363179 665 379.2497 0.120484 3.19E-131 PXM –0.39289 0.063313 –6.20561 9.58E-10 0.002432 0.000164 14.80953 4.28E-43 0.816648 0.248012 665 219.3222 0.090427 2.26E-135 GNB –1.05458 0.082906 –12.7201 2.48E-33 0.004577 0.000215 21.28337 4.67E-77 1.069378 0.405178 665 452.9817 0.141582 2.39E-128 CIG –0.27678 0.05756 –4.80856 1.88E-06 0.0014 0.000149 9.376604 1.05E-19 0.742444 0.116773 665 87.92071 0.159514 6.24E-126 TRK –0.50191 0.051612 –9.72467 5.47E-21 0.00203 0.000134 15.16011 8.29E-45 0.665722 0.256841 665 229.829 0.159826 6.87E-126 GTN –0.85955 0.057786 –14.8747 2.06E-43 0.003045 0.00015 20.3124 9.88E-72 0.745361 0.382884 665 412.5934 0.153256 8.99E-127 PKO –0.59322 0.044165 –13.4321 1.44E-36 0.003524 0.000115 30.76182 6.43E-130 0.569665 0.587287 665 946.2896 0.060767 1.57E-139 PZU –0.64264 0.02658 –24.1776 3.73E-93 0.002849 6.89E-05 41.32759 7.33E-186 0.342844 0.71976 665 1707.97 0.096994 1.85E-134 PKN –0.54446 0.039955 –13.6268 1.81E-37 0.002964 0.000104 28.5984 6.54E-118 0.515367 0.551545 665 817.8687 0.094282 7.77E-135 CDR –0.2245 0.051361 –4.37095 1.44E-05 0.001865 0.000133 13.99659 3.35E-39 0.662494 0.227557 665 195.9045 0.123868 9.27E-131 LPP –0.56185 0.055261 –10.1672 1.14E-22 0.003434 0.000143 23.95832 6.29E-92 0.712791 0.463277 665 574.0013 0.088847 1.36E-135 SPL –0.60266 0.04691 –12.8471 6.69E-34 0.003295 0.000122 27.08268 1.94E-109 0.605076 0.524481 665 733.4715 0.072525 7.11E-138 KGH –0.83565 0.05662 –14.7588 7.54E-43 0.003914 0.000147 26.64717 5.35E-107 0.730327 0.516389 665 710.0716 0.065246 6.72E-139 CPS –0.41877 0.033395 –12.5398 1.57E-32 0.001524 8.66E-05 17.59572 3.47E-57 0.430751 0.317675 665 309.6093 0.101721 8.34E-134 PGN –0.34212 0.030527 –11.2072 8.01E-27 0.001952 7.92E-05 24.64875 8.57E-96 0.393756 0.477432 665 607.561 0.165295 3.71E-125 Source: Own estimation. Table 3 – continued 20 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Table 4 Run tests Company R n1 n2 E(R) VAR(R) z-value p-value IIB 49 348 319 333.8695652 165.8697866 –22.1188 2.08E-108 DNP 46 296 371 330.2833583 162.3096797 –22.3141 2.70E-110 LTS 27 310 357 332.844078 164.8478198 –23.8209 2.03E-125 MIL 12 312 355 333.113943 165.1164523 –24.9899 7.88E-138 ING 22 322 345 334.1034483 166.103309 –24.2164 1.50E-129 OPL 52 317 350 333.6836582 165.6842834 –21.8837 3.71E-106 MBK 32 301 366 331.3328336 163.3475197 –23.4206 2.64E-121 PGE 30 361 306 332.2323838 164.2397293 –23.5832 5.74E-123 CCC 14 326 341 334.3313343 166.3310017 –24.8378 3.50E-136 ABS 48 368 299 330.9310345 162.9497845 –22.1643 7.60E-109 KRU 54 302 365 331.5247376 163.5376538 –21.7017 1.98E-104 ALR 22 302 365 331.5247376 163.5376538 –24.204 2.02E-129 EAT 40 336 331 334.4812594 166.4808845 –22.8231 2.70E-115 PLW 56 334 333 334.4992504 166.498875 –21.5833 2.58E-103 KTY 23 325 342 334.2833583 166.2830534 –24.1397 9.58E-129 BHW 27 289 378 328.5622189 160.6147823 –23.7949 3.77E-125 JSW 35 331 336 334.4812594 166.4808845 –23.2107 3.55E-119 GTC 56 331 336 334.4812594 166.4808845 –21.5831 2.59E-103 CAR 40 306 361 332.2323838 164.2397293 –22.8029 4.30E-115 ATT 28 309 358 332.7001499 164.7046386 –23.7421 1.32E-124 EUR 48 296 371 330.2833583 162.3096797 –22.1571 8.91E-109 BDX 41 326 341 334.3313343 166.3310017 –22.7443 1.64E-114 ENG 46 359 308 332.5502249 164.5555577 –22.338 1.58E-110 KER 38 231 436 302.9970015 136.4867746 –22.6827 6.63E-114 ENA 25 322 345 334.1034483 166.103309 –23.9836 4.12E-127 TPE 46 341 326 334.3313343 166.3310017 –22.3566 1.04E-110 FMF 36 326 341 334.3313343 166.3310017 –23.132 2.21E-118 CMR 48 290 377 328.826087 160.8740499 –22.1409 1.28E-108 LCC 48 306 361 332.2323838 164.2397293 –22.1786 5.53E-109 WPL 34 322 345 334.1034483 166.103309 –23.2853 6.25E-120 ECH 39 308 359 332.5502249 164.5555577 –22.8837 6.75E-116 GPW 48 375 292 329.3358321 161.3754997 –22.1466 1.13E-108 PKP 34 314 353 333.3598201 165.3613967 –23.2796 7.13E-120 21 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Company R n1 n2 E(R) VAR(R) z-value p-value VRG 45 341 326 334.3313343 166.3310017 –22.4341 1.83E-111 CIE 37 297 370 330.5052474 162.528833 –23.0224 2.78E-117 BFT 45 358 309 332.7001499 164.7046386 –22.4175 2.66E-111 MAB 68 272 395 323.1589205 155.3516684 –20.4716 3.85E-93 AMC 45 317 350 333.6836582 165.6842834 –22.4275 2.12E-111 FTE 56 366 301 331.3328336 163.3475197 –21.5428 6.19E-103 LVC 51 309 358 332.7001499 164.7046386 –21.95 8.66E-107 LWB 33 254 413 315.5487256 148.0876157 –23.2185 2.96E-119 BRS 38 279 388 325.5937031 157.7124301 –22.9006 4.59E-116 STP 47 300 367 331.1349325 163.1515597 –22.2448 1.27E-109 PXM 15 306 361 332.2323838 164.2397293 –24.7536 2.84E-135 GNB 37 262 405 319.1709145 151.5233633 –22.9231 2.74E-116 CIG 51 350 317 333.6836582 165.6842834 –21.9614 6.74E-107 TRK 34 353 314 333.3598201 165.3613967 –23.2796 7.13E-120 GTN 34 326 341 334.3313343 166.3310017 –23.287 6.00E-120 PKO 8 319 348 333.8695652 165.8697866 –25.3023 3.01E-141 PZU 10 305 362 332.0644678 164.072999 –25.1434 1.67E-139 PKN 14 315 352 333.4737631 165.4749691 –24.8353 3.73E-136 CDR 36 324 343 334.2293853 166.2291199 –23.1311 2.25E-118 LPP 10 297 370 330.5052474 162.528833 –25.1403 1.81E-139 SPL 16 330 337 334.4632684 166.462895 –24.6832 1.62E-134 KGH 8 301 366 331.3328336 163.3475197 –25.2984 3.32E-141 CPS 40 329 338 334.4392804 166.4389105 –22.8228 2.72E-115 PGN 34 312 355 333.113943 165.1164523 –23.2778 7.45E-120 Mean 36.07018 316.9123 350.0877 331.3998527 163.4544162 –23.1002 6.76E-95 Min 8 231 292 302.9970015 136.4867746 –25.3023 3E-141 Max 68 375 436 334.4992504 166.498875 –20.4716 3.85E-93 Source: Own estimation Table 4 – continued 22 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Figure 1 Chow test  23 © 2023 Authors. This is an open access article distributed under the Creative Commons BY 4.0 license (https://creativecommons.org/licenses/by/4.0/) DOI: 10.7172/2353-6845.jbfe.2023.2.1 Magdalena Mikołajek-Gocejna • Journal of Banking and Financial Economics 2(20)2023, 1–29 Figure 2 Cusum test