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In this study, we analyzed the changes in Beta over time for the leading indexes of Borsa Istanbul (XU100, XUHIZ, XUMAL, XUSIN, XUTEK) across 5-10 year and 15-year intervals from 2008 to 2023. We utilized Rolling regression and Recursive regression methods to estimate the fluctuations in Beta over time and compared the performance of these estimation techniques. To evaluate the effect of the estimation window length on Beta, we incorporated daily and weekly estimation windows of various lengths: 252 days, 126 days, 52 weeks, and 26 weeks. Additionally, we examined how data frequency affects Beta estimation using daily and weekly datasets. Our analysis showed that the Rolling regression method consistently outperformed the recursive method. Moreover, we found that employing daily datasets, instead of monthly datasets, significantly enhanced Beta forecast performance. We also found that a 126-day window is the most effective length for the estimation window.
ABSTRACT
In this study, we analyzed the changes in Beta over time for the leading indexes of Borsa Istanbul (XU100, XUHIZ, XUMAL, XUSIN, XUTEK) across 5-10 year and 15-year intervals from 2008 to 2023. We utilized Rolling regression and Recursive regression methods to estimate the fluctuations in Beta over time and compared the performance of these estimation techniques. To evaluate the effect of the estimation window length on Beta, we incorporated daily and weekly estimation windows of various lengths: 252 days, 126 days, 52 weeks, and 26 weeks. Additionally, we examined how data frequency affects Beta estimation using daily and weekly datasets. Our analysis showed that the Rolling regression method consistently outperformed the recursive method. Moreover, we found that employing daily datasets, instead of monthly datasets, significantly enhanced Beta forecast performance. We also found that a 126-day window is the most effective length for the estimation window.
Keywords: Time-Varying Beta, Beta Forecast, Forecast Performance, Financial Markets, Istanbul Stock Exchange.
JEL Classification Codes: G12, G17
Referencing Style: APA 7
(ProQuest: ... denotes formulae omitted.)
INTRODUCTION
The transformation in modern finance initiated by Markowitz's (1952) pioneering approach has paved the way for the development of the Capital Asset Pricing Model (CAPM), with significant contributions from studies by (Sharpe, 1964), (Lintner, 1965) and (Mossin, 1966). In CAPM, the return of an asset is understood as a function of its relevant risk. Beta, which quantifies the asset's responsiveness to market risk factors within the model, finds application across various domains of modern finance, including asset pricing, portfolio performance evaluation, risk management, investment preferences, and capital allocation decisions.
Practically, CAPM operates under the assumption that Beta remains constant over time, leading to the presumption of a linear relationship between the asset's return and the market portfolio. This linearity simplifies the model's application and enhances analytical clarity. However, sometimes such assumptions may not consistently hold true in financial markets. When deviations occur, it can result in substantial challenges, including a reduction in the model's validity and reliability, complications in risk assessment, and potential impacts on investment strategies.
Beta, one of the most important parameters guiding practitioners in the field of finance, needs to be estimated as it is not directly observable. Therefore, the accuracy and reliability of Beta estimation are of paramount importance (Brenner & Smidt, 1977). The most frequently used method for Beta estimation is to estimate covariances and variances from historical return data. However, the main problem encountered in calculating the Beta coefficient is the variation it shows over time. At this point, it would not be wrong to express Beta as a type of stochastic parameter based on the feature it shows (Collins, Ledolter, & Rayburn, 1987). The variation seen in Beta over time also leads to the elimination of the linear relationship between the market portfolio and asset return in CAPM. The reason for the mentioned variation is due to the non-stationarity in returns and risks. Ignoring these variations can lead practitioners to make substantial errors in their estimates of systematic risk (Estrada, 2000).
Researchers have been discussing the variation in Beta and its implications for nearly half a century (Meyers, 1973; Blume, 1975; Fabozzi & Francis, 1978; Sunder, 1980; Theobald, 1981; Bos & Newbold, 1984; Bollerslev, Engle, & Wooldridge, 1988; Brooks, Faff, & Lee, 1992; Gong, Firth, & Cullinane, 2006; Abdymomunov & Morley, 2011; Ciner, 2015; Rizvi & Arshad, 2018; Kalnina, 2022). A common finding in these studies is the significance of accounting for the variation in Beta when selecting the estimation methods used. Because variability in Beta not only complicates its interpretation but also presents challenges in estimation for both practical applications andthe use of the CAPM (Рай; Lee &Fry, 1992; Groenewold & Fraser, 1999).
The challenges encountered in forecasting Beta necessitate a careful approach. We argue that this approach should take into account not only the selection of Beta estimation methods but also factors such as the characteristics of the dataset (e.g., frequency selection) and the estimation windows employed in the application of these methods. Because Damodaran (1999)' and Daves et al. (2000)? demonstrated that the return interval used for forecasting Beta significantly impacts the results, as evidenced by examination of returns at varying frequencies. However, it cannot be said that a significant amount of research has been conducted on the effects of both data frequency and estimation window selection on Beta forecast. At this point, our study aims to expand the existing literature. Our research addresses 2 more research problems while evaluating the performance of Beta estimation methods: The first is to investigate whether the length of the estimation window affects Beta estimation, and the second is to investigate whether the frequency of the data selected for Beta estimation influences its performance. Beta forecast will be implemented using Rolling and Recursive regression methods across the leading indexes of Borsa Istanbul (BIST), utilizing four estimation windows based on weekly and daily frequencies: 252 days, 126 days, 52 weeks, and 26 weeks. In order to make a more comprehensive performance evaluation rather than using a specific interval like many studies in the literature when evaluating the forecast performance of the methods, the data were examined in three sub-periods as 31.12.2008 - 29.12.2023, 31.12.2013 - 29.12.2023 and 31.12.2018 - 29.12.2023 at daily and weekly frequencies for 15, 10 and 5 years, respectively. The subsequent sections of the study are organized as follows: the second section reviews existing literature on Beta forecast, While the third section presents information regarding the dataset and methodology employed. The fourth section contains the findings derived from the analyses conducted. Finally, the study concludes with a section that offers final evaluations and reflections.
LITERATURE REVIEW
The risk and return of an investment, capital structure decision, investment decision and many other financial decision-making have always been of interest to financial manager and academics. The main factor here is the widespread use of CAPM Betas as a risk indicator. Moonis & Shah (2003) stated that the Betas of most stocks fluctuates over time. In this context, it is seen that studies on Beta reveal the multiple approaches in its estimation due to its inherent characteristics. So studies examining the forecasting performance of different methods are quite common in the Beta literature. Groenewold & Fraser (1999) conducted a study estimating Beta values for 23 industries in Australia over a monthly frequency from 1979 to 1994. They analyzed the fluctuations in Beta over time utilizing Rolling regression, Recursive regression, and Kalman filter methodologies. Their findings revealed significant instability in the Beta values across the examined industries. Furthermore, the study indicated that the variations in Beta over time were influenced by the estimation method employed. They discovered that nearly half of the industry Betas determined using the Kalman Filter were non-stationary, while almost all Betas estimated via the Rolling and Recursive regression methods displayed variability. Marti (2006) investigated the performance of various methods for estimating time-varying Beta in the US stock market from 1980 to 2005. The methods analyzed included Rolling regression, GARCH (Generalized Autoregressive Conditional Heteroskedasticity), the Kalman filter, and the Schwert and Seguin model. The evaluation, based on Mean Square Error (MSE), revealed that the Kalman filter exhibited a lower error rate compared to the other methods throughout the examined period. Mergner & Bulla (2008) investigated the time-varying behavior of systematic risk across 18 European sector portfolios by employing various methodologies, including GARCH, Kalman filter, Monte Carlo simulations, and Markov Switching techniques. Their analysis, conducted at a weekly frequency from 1987 to 2005, revealed that the Kalman Filter outperformed the other methods in accurately estimating the Beta of the sector portfolios. Domenech, Orbe Mandaluniz & Zarraga (2011) employed Rolling regression, Nonparametric Estimation, and GARCH methods to estimate time-varying Beta in their research. They analyzed data from six portfolios consisting of companies listed on the Mexican stock market between 2003 and 2009. In their evaluation of forecast performance using the Mean-Variance approach, the authors found that the GARCH-based models yielded superior results. Zhou (2013) analyzed the estimation performance of four different methods for Beta forecast of the Real Estate Investment Trust index traded on the London Stock Exchange from 1999 to 2011. The methods employed included Rolling regression, Dynamic Conditional Correlation, the Schwert and Seguin model, and the State Space model. The results of the study were assessed using Mean Absolute Error (MAE) and Mean Squared Error (MSE) as performance metrics. The findings indicated that the State Space model exhibited better performance throughout the examined period. Wijethunga & Dayaratne (2015) investigated the timevarying Beta forecasts of 26 stocks listed in the Finance, Banking, and Insurance indices of the Sri Lankan Stock Market. They employed Rolling regression and Recursive regression methods across three sub-periods from 2005 to 2013. Their analysis revealed that Beta fluctuated over time within the examined sectors of the Sri Lankan stock market, indicating that the choice of method influenced the changing Beta forecasts. However, they did not compare the performance of these methods. Messis & Zapranis (2016) analyzed the time-varying Beta values of 135 stocks listed in the S&P 500 index from 1993 to 2011. They compared the performance of various Beta estimation methods, including UGARCH, BGARCH, the Kalman Filter, and the Schwert and Seguin Model, by proposing a new approach that treats the Beta coefficient as a function of market return. The findings revealed significant differences in Beta forecast results among the methods. Specifically, the Kalman Filter demonstrated superior forecast performance in the short sample period, while UGARCH and BGARCH excelled over a five-year span, and the Schwert and Seguin Model performed best in the long sample period. Notably, the newly proposed method also exhibited enhanced forecast performance. Mantsios 8: Xanthopoulos (2016) examined the existence of time-varying Beta forecast and range effect in the Athens stock market between 2007 and 2012 during the Greek debt crisis. In their study, where they estimated Beta with daily, weekly and monthly data using the OLS method, they found that there was an increase in the standard deviation as the estimation window increased. Chakrabarti 8:Das (2021) examined the temporal changes in sector Betas using the multivariate Diagonal VECH GARCH method. They selected six industries from both the Indian and US markets, analyzing daily frequency data from 1999 to 2017. Their findings revealed that Betas did fluctuate over time, with variations in magnitude across different sectors and periods. They also noted that the proliferation of methodologies capable of capturing these changes in Beta over time could have beneficial implications for market risk assessment and hedging strategies. Lopez Herrera et al. (2022) forecasted Beta values for 23 stocks listed on the Mexican Stock Market, examining both pre-pandemic and pandemic periods from 2019 to 2020 using the Rolling regression method. The study's findings highlight the potential implications of assuming a constant Beta in return estimation and provide recommendations for practitioners to employ methods that account for fluctuations in Beta. Drobetz et al (2023) conducted an examination of all companies listed on US stock markets (NYSE, AMEX, and NASDAQ) utilizing Machine Learning-based methods for Beta forecast. Theirfindings revealed thattheMachine Learning approach outperformed other methods, including Linear Regression, Tree-Based Models, and Neural Networks, in terms of Beta forecast results. Blasques, Francq & Laurent (2024) conducted an examination of the time-varying Betas for 16 companies listed on the US Stock Market between 1999 and 2017, employing the Autoregressive Conditional Beta and Least squares methods. Their findings, derived from the Mean Squared Error (MSE) approach, indicated that their model, when applied to high-frequency data, outperformed both the Conditional Beta and Least Squares methods.
As can be seen from the studies given above, many methods have been used in the Beta literature to examine forecast performance. However, we think that not only the estimation method but also the frequency of the data set and the estimation window should be taken into consideration by the practitioners when forecasting Beta. As a matter of fact, studies emphasizing the importance of the mentioned parameters can be found in the literature. Baesel (1974) showed that Beta stationarity depends on the length of the period over which is estimated. When the studies examining the effects of choosing the estimation window length in Beta forecast are examined, Groenewold & Fraser (2000) examined 18 industrial sectors operating in Australia at a monthly frequency between 1973 and 1998. In their research, where they examined the forecast performance with the Rolling regression technique in a 5-year period, it was determined that Betas that changed over time showed significant instability. In addition, in the analysis they conducted to determine the optimal length of the estimation window, they determined that it was more appropriate to apply a 3-year period in the sample period. Hollstein, Prokopczuk, and Simen (2019) analyzed optimal forecast windows using daily data from NYSE, AMEX, and NASDAQ stocks between 1963 to 2015. They examined 1, 3, 6, 12, 24, 36, and 60-month windows for daily frequency data and 12, 36, and 60-month windows for monthly frequency data. They used Historical Beta, EWMA Beta, Shrinkage Beta, Dimson Beta, Scholes-Williams Beta, Correlation Separated Beta and Macro Beta methods in Beta forecast. The results show that a Historical Estimator based on daily return data with exponential weighting scheme and simple shrinkage adjustments in 12-month estimation window with daily frequency data gives the best estimates for Beta. Akyatan & Cetin (2020) forecasted Beta for 30 randomly generated portfolios comprised of 154 stocks from the XUTUM index, utilizing daily frequency data from 2003 to 2013. The authors employed the Rolling regression method combined with MGARCH DVECH, MGARCH DBEKK, MGARCH CCC, and MGARCH DCC techniques, focusing on 30-60-120240 and 360-day windows for estimation. To evaluate forecast performance, they utilized RMSE, MAE, and MAPE methods. Their analyses revealed that the Beta estimation using dynamic methods outperformed traditional approaches. Notably, they found that the MGARCH CCC method yielded the most successful results. Additionally, the research indicated that a 120-day window length provided superior outcomes among the various rolling window estimations utilized. Agrrawal, Gilbert, & Harkins (2022) examined the optimal length and return range in Beta forecast in the US stock market between 2000-2021 with the OLS method at daily, weekly and monthly frequencies. They found that the tracking error in Beta forecast decreased as the data frequency increased. In addition, in their study where they examined sub-periods from 1 to 5 years as the estimation period, they could not reach an optimal combination, but they mentioned the existence of a trend in choosing a long estimation period with daily and weekly data.
DATA AND METHODOLOGY
Data
Economic arguments suggest that Beta can change over time, leading to its modeling as a time-varying parameter (Moonis & Shah, 2003, p.6). As stated by Hawawini (1983), the Beta of a security can vary significantly depending on whether it is estimated using daily, weekly, or monthly returns. Since the Beta parameter forecast results are directly affected by the number of observations, the following question comes to mind when making a decision: should the number of observations be high or low?
While extending the length of the estimation period may reduce the observed change in Beta, it may also lead to a result that increases the probability of financial decisions and practices that have the potential to cause changes in Beta factors. At this point, the estimation period length that strikes the optimal balance becomes crucial (Theobald, 1981, p.747). Alexander & Chervany (1980) stated that the optimal estimation interval is 4-6 years in their study. Daves et al. (2000) proposed that an estimation period of 2-3 years yields more accurate results for daily frequence in Beta forecast. Choudhry & Wu (2008) determined the estimation window length as 1 and 2 years at daily frequency. In another study conducted by Choudhry & Wu (2009), the estimation window length determined as 1 and 2 years at weekly frequency. Akyatan & Cetin (2020) determined the estimation window length as 30-60-120-240 and 360 days with daily frequency data. It can be seen that recent studies in the literature tend to narrow the estimation window length. Because extending the time interval may introduce variability issues. A change in Beta during the estimation period inevitably introduces bias into the forecasted Beta values (Bartholdy 8: Peare, 2001, p.4). Consequently, selecting a longer estimation period raises concerns regarding the accuracy of the estimated Beta values. In this context, opting for a shorter estimation period can facilitate a morereliable estimation (Frankfurter, Leung, & Brockman, 1994). In various studies on Beta forecast across different estimation window length, such as those by Fama and Macbeth (1973), Corhay & Rad (1993), Frankfurter et al. (1994), Levy, Guttman, & Tkatch (2001) and Brzeszczynski et al. (2011) etc., it was found that Beta varies depending on the time interval used for estimation. Therefore the estimation windows are set at 126 days and 252 days for daily frequency data, and 26 weeks and 52 weeks for weekly frequency data. To reveal the change in Beta and determine whether the methods have an impact on the forecast performance, we examined three sub-periods: 31.12.2008 - 29.12.2023, 31.12.2013 - 29.12.2023 and 31.12.2018 - 29.12.2023 for 15, 10, 5 years, respectively. The purpose of choosing 3 sub-periods for Beta forecast is to evaluate whether the estimation success of the methods differs in different time periods. The objective is to assess the performance of the methods across specific time intervals and to investigate whether prediction performance varies across different windows and frequencies. Hence we have the motivation that this approach provides a more comprehensive perspective when revealing the effects of method success and data set frequency choice.
In our research, we utilized the total return data for the indices under examination, as recommended by Cayirli et al. (2022)·. This comprehensive dataset was sourced from the Borsa Istanbul Data Center, ensuring accuracy and reliability. For a detailed overview of the indices used in the study, please see Table 1, which presents essential information about each index.
Methodology
The findings from practitioners who calculate Beta using historical data indicate that Beta is a somewhat constrained method for estimating market risk due to the variations observed over time (Rossi, 2016). Consequently, exploring different approaches to Beta forecast has made it essential to consider methods that could enhance forecast performance. As stated by Chakrabarti & Das (2021), employing various methods to capture changes in Beta over time is crucial for managing and calculating risk effectively.
Although there are numerous approaches to Beta forecastin the literature, we focused on only two methods. Our aim was not only to evaluate the performance of Beta forecast methods but also to examine the effects of data frequency and estimation window length on the performance of forecast, which makes our study different from mostother studies. Accordingly, Beta was forecasted using both Rolling regression and Recursive regression methods. The analysis was conducted using Stata, E-views, and MS Excel software. Detailed information on the methods used in the research is provided below.
Market Model
The formulation of the market model used in the research is given in Equation 1.
... (1)
... Expected return for index i
... Risk-free interest rate
... Systematic risk (Beta) for index i
I'm: Market return
The return calculation was carried out with the formula in Equation 2.
... (2)
Ry in Equation 2 represents the logarithmic return of index 1 on day t, Er represents the index closing value on day t, and Et-1 represents the index closing value of the previous day. The return data for the XUTUM index has been used as the market return (m). The risk-free interest rate was calculated by considering Turkey's 2-year bond yield·. The steps followed in the daily and weekly risk-free interest rate conversion are given in Equation 3 and Equation 4.
... (3)
... (4)
rra: Daily risk-free interest rate
... Weekly risk-free interest rate
... Risk-free interest rate for day/week (t)
... Risk-free interest rate for day/week (t-1)
Rolling Regression
Rolling regression is a method used in empirical studies to characterize parameters that change over time (Cai & Juhl, 2023, p.1447). The Rolling regression method employs a variable regression model on time series data by dividing the dataset into specific estimation windows, training a regression model for each window. By shifting these windows, the method enables the model to adapt over time and facilitates the observation of trends within the dataset and the Beta value is updated with each new observation, allowing for real-time estimation of Beta. The calculation of Beta follows the same methodology as the Least Squares (LSS) method, as outlined in Equation 5 (Yeo, 2001);
... (5)
Rit : Return of index i on day t
Rmt : Return of market portfolio (XUTUM) on day t
To provide a more detailed understanding of the methodology, let's consider a scenario where Beta forecast is performed using a 30-day estimation window. Initially, the first 30 days of data undergo regression analysis, yielding the Beta forecast result for the 30th day. For calculating the forecast for the subsequent day, data from the 2nd to the 31st day is utilized. In summary, for each new day to be estimated within the specified window, the observation sample is advanced by one observation (t+1). In this research, four different estimation windows were employed for the Beta forecast using the relevant method, consisting of 252 and 126 days for daily frequency, and 52 and 26 weeks for weekly frequency.
Recursive Regression
The Recursive regression method operates on a foundational logic that enables the estimation of future values by utilizing both current and past data. This method features a continuous adaptation and update mechanism, allowing for the incorporation of new data into previous estimates. It is important to establish the estimation period at the outset of the analysis. In this method, the model is trained by excluding the first 30 days of observation for the selected period (e.g., 30 days), after which the Beta forecast begins. Each Beta forecast is retained within the system, and the estimated values generated throughout the estimation period are preserved in the observation set. While this method shares similarities with the EKK method, akin to the Rolling regression method in its application, it differs in terms of sample size. In contrast to the Rolling regression method, the sample size continues to grow throughout the estimation period. The methodology underlying this method is given in Equation 6 (Yeo, 2001);
... (6)
Rit : Return of index i on day t
Rmt : Return of market portfolio (XUTUM) on day t
...
In this research, we applied a method using various estimation windows for Beta calculations, which commence at 252 and 126 days, as well as 52 and 26 weeks prior to the observation value. This approach depends on the length of the estimation period relative to the starting date of the examined timeframe.
Forecast Performance Measurement
To evaluate the forecast performance of two Beta estimation methods, we used Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE). The MAE measures the average absolute differences between estimations and actual values, indicating error magnitude. RMSE, on the other hand, quantifies deviation by taking the square root of the average squared differences, giving more weight to larger errors. Using both metrics allows for a comprehensive assessment of model performance. Lower MAE and RMSE values indicate better accuracy, while higher values suggest larger deviations from true values. Methodologies for these metrics are detailed in Equations 7 and 8.
... (7)
Yit : Observed return value of index i on trading day/ week t
Pit : Estimated return value of index i on trading day/ weekt
... Error series
... Number of observations
... (8)
... Observed return value of index i on trading day/ weekt
... Estimated return value of index i on trading day/ week t
... Error series
Number of observations
The calculation of the observed return values in Equations 7 and 8 is crucial for evaluating the forecast performance of the methods used. We adopted a returnbased approach to estimate these values. The primary rationale for choosing return over Beta as the observed and estimated value is to minimize potential issues that may arise. Two notable challenges include:
1. The difficulties associated with calculating covariance and variance at daily or weekly frequencies when determining observed Beta values,
2. The return data utilized in the Market model can fluctuateat very smallrates, leading to the potential for extreme values resulting from proportional differences in the forecasted Beta values.
Toillustrate the second point above, considera scenario where a return of 0.01 is calculated for a sub-index on day t, while the market return is only 0.001. When the daily Risk-free rate (Which is expected to be quite small) is included in the equation, it results in an extremely high Beta value of around 10. Such conditions inflate Beta estimates and lead to biased outcomes both daily and weekly observations, particularly when return differences between market and index are minimal. The following steps were taken to eliminate the aforementioned problem; The observed return calculation was made with the logarithmic return in Equation 2, and then the CAPM method in Equation 1 was used in the estimated return calculation. The forecasted Beta values obtained from the Rolling and Recursive regression results were substituted into the model and the expected return obtained was compared with the observed return results. We share the idea that this approach is eliminate our potential concerns expressed above.
FINDINGS
The results obtained from the analyses are given below.
When the summary statistics values in Table 2 are examined, it can be said that the data do not show a normal distribution, as can be understood from the skewness and kurtosis values, and therefore these values are preliminary information for the variation seen in Beta. The findings of the average R? and average Beta values obtained from the analyses are given in Table 3 and Figure 1.
Theanalysis of the average R" and Beta values presented above reveals that there are differences among the subindicesin different periods. Notably, the Rolling regression and Recursive regression methods yield distinct average R? and Beta values, highlighting variability in findings based on estimation window selection. Moreover, the fluctuations in average Beta values indicate that changes in the alignment of index returns with market returns affect average R values and the model's explanatory power. When the average B? values for all periods covered in the study are examined, the highest R? value in the results obtained from both methods was found in the XU100 index in all windows. This is followed by the XUMAL, XUSIN, XUHIZ and XUTEK indices. Given the dual influence of systematic and idiosyncratic risks on asset pricing, an increase in idiosyncratic risk is likely to reduce the variances explanatory capacity. Regarding standard deviation values, there is no specific order like R?. The rankings of the standard deviations of the indices differ across the examined windows and time periods.
One of the aims of this study is to explore the effect of estimation window frequency on Beta forecast outcomes. Analysis of the preliminary standard deviation values in Figure 1 indicates that daily frequency Beta forecasts yield lower standard errors across most periods and sub-indices. Additionally, average Beta values from daily frequence vary by estimation window. The Rolling regression methodshowsa decreasein standard deviation values as the estimation window increases, while the Recursive regression method yields similar results, with some exceptions. Notably, differences in forecasted Beta means between the two methods suggest that the choice of estimation technique significantly influences the results, as corroborated by the descriptive statistics. To assess the predictive accuracy of the methodologies employed in this research for forecasting time-varying Beta, the results derived from the MAE and RMSE metrics are presented in Table 4.
When the MAE and RMSE forecast performance results in Table 4 are examined, the following points stand out in terms of comparing performance between the methods:
When the MAE and RMSE values for the period 20082023 and 2013-2023 are examined, the index ranking from lowest to highest is as follows: XU100, XUTEK, XUMAL, XUSIN and XUHIZ. But this situation has changed for the 2018-2023 period. The index ranking is; XUTEK, XU100, XUSIN, XUHIZ and XUMAL.
For the period 2008-2023, the Rolling regression method consistently exhibits lower MAE and RMSE values relative to the Recursive regression method across nearly all indices. When analyzing the RMSE values obtained within the 126-day window, it is observed that this finding cannot be attributed solely to the XUTEK index results. The RMSE value associated with the XUTEK index was recorded as 0.00315 using the Rolling regression method, whereas it was measured at 0.00305 for the Recursive regression method. These results suggest that the Rolling regression method generally demonstrates superior performance in Beta forecast during the 20082023 period, with the exception of specific case noted.
For the period 2013-2023, it is seen that the forecast results performed with the Rolling regression method in almost all indices have lower MAE and RMSE values compared to Recursive regression. When the MAE and RMSE values obtained in the 252 and 126-day windows are examined, it is observed that this finding cannot be attributed solely to the XUTEK index results. The RMSE value obtained in the 252-day window was determined as 0.00240 in the Rolling regression method and 0.00210 in the Recursive regression method. In the 126-day window, the MAE value obtained was measured as 0.00164 in the Rolling regression method, 0.00148 in the Recursive regression method, and RMSE was measured as 0.00274 in the Rolling regression method and 0.00217 in the Recursive regression method. Except for the mentioned exception, it can be stated that the Rolling regression method performed better in Beta forecast for the period 2013-2023.
For the period 2018-2023, the findings remain similar; the Rolling regression method demonstrates lower error values across all examined indices, with the exception of the XUTEK index. Moreover, an assessment of the RMSE values within the 252-day window reveals that the Rolling regression method yields an RMSE value of 0.00164, while the Recursive regression method is associated with an RMSE value of 0.00140. Excluding the aforementioned exception, the data demonstrates that the Rolling regression method generally outperforms the Recursive regression method in Beta forecast for the 2018-2023 period.
To enhance the robustness of the evaluation of prediction performance success for the Rolling and Recursive regression methods in Beta estimation, a paired t-test was used to determine the statistical difference and method success. The findings are presented in Table 5.
Table 5 compares the predictive performance of different methods using RMSE and MAE metrics. The analysis shows t-values of -6.6917 for RMSE and -8.4918 for MAE, with corresponding p-values of 0.000 for both metrics. These results indicate a significant negative difference between the first group (Rolling) and the second group (Recursive), suggesting that the mean error of the Rolling method is lower than that of the Recursive method. Therefore, this evidence supports the conclusion that the rolling regression method is more accurate than the recursive method in terms of predictive performance.
In line with our motivation to determine whether the data set frequency affects the Beta forecast success of the methods, when the results in Table 4 are examined, it is seen that the Beta values estimated using daily frequency data have lower error values compared to those estimated using weekly frequency data in both methods and in all periods examined. Findings indicate that daily frequency data significantly improves the accuracy of Beta forecast, leading to reduced error compared to weekly frequency data for the specific periods and indices under investigation. It is recommended to prioritize daily data over weekly data when selecting the frequency for Beta forecast, as this will enhance the precision of the results.
In the findings obtained in Table 4 regarding whether the length of the estimation window determined in Beta forecast, which is another motivation of the study, has an effect on forecast success;
For the period 2008-2023, in all weekly frequency Beta forecast performances obtained with the Rolling regression method, it is observed that there is a decrease in forecast errors as the estimation window narrows. In daily frequency estimations, it is determined that there is a decrease in forecast errors as the estimation window narrows in all indices except the XUTEK index. While the 252-day MAE value was measured as 0.00175 in the XUTEK index, the MAE value was determined as 0.00194 in the 126-day window. It can be stated that from the forecast errors obtained with the Recursive regression method in this period, the extension or shortening of the estimation window length does not lead to a difference that will establish superiority in frequency selection.
For the period 2013-2023, in all weekly frequency Beta forecast performances obtained with the Rolling regression method, it is observed that there is a decrease in forecast errors as the estimation window narrows. In the daily frequency forecast performance results, it was determined that this situation was also valid for other indices except for the XUTEK index. While the 252-day MAE value was measured as 0.00140 in the XUTEK index, the MAE value was measured as 0.00164 in the 126-day window, and the RMSE value was measured as 0.00240 in the 252-day window, while the RMSE value was determined as 0.00274 in the 126-day window. When the forecast performances obtained with the Recursive regression method were examined at daily frequency, it was determined that there was a decrease in forecast errors in all other indices except for the XUTEK index as the estimation window narrowed. In the XUTEK index, the MAE value was measured as 0.00146 in the 252day window, while the MAE value was measured as 0.00148 in the 126-day window, and the RMSE value was measured as 0.00210 in the 252-day window, while the RMSE value was determined as 0.00217 in the 126-day window. In the weekly frequency, it is not possible to mention the existence of a definite superiority between the estimation window lengths.
For the 2018-2023 period, similar to other periods, it was determined that there was a decrease in forecast errors as the estimation window narrowed in the daily frequency forecasts obtained with the Rolling regression method, except for the XUTEK index. In XUTEK index, MAE value was measured as 0.00085 in 252-day window, MAE value was measured as 0.00105 in 126-day window, RMSE value was measured as 0.00164 in 252-day window, while RMSE value was determined as 0.00185 in 126-day window. In weekly frequency, it was found that as the estimation window narrowed for other indices except for XUHIZ index, forecast errors decreased. In XUHIZ index, MAE value was measured as 0.01070 in 52-week window, MAE value was measured as 0.01077 in 26week window, RMSE value was measured as 0.01397 in 52-week window, while RMSE value was determined as 0.01398 in 26-week window. In the Recursive regression method, it cannot be said that the estimation window length has a definite superiority over each other for the prediction results obtained from neither weekly nor daily frequency data.
Another finding from the study indicates that the estimation window demonstrating the highest success in prediction across all periods is the 126-day estimation window, noted for its lowest MAE and RMSE values for nearly all indices.
Based on the results obtained from our research, the key outcomes can be summarized as follows:
+ Practitioners who utilize traditional methods for estimating Beta values, while assuming that Beta remains constant over time in financial decisionmaking, risk making incorrect calculations and, consequently, flawed decisions. Therefore, it can be said that the Rolling regression method is more effective than the Recursive regression method for Beta forecast within the specified period and data set examined in this study.
* The length of the estimation window is another crucial factor influencing the results of Beta forecast. It appears that the most effective estimation window interval for the Rolling regression method is the 126-day window.
Additionally, the impact of data frequency selection on Beta forecast and its forecast performance is a significant consideration. Opting for daily frequency data rather than weekly data when forecasting Beta is likely to enhance forecast accuracy.
CONCLUSION
The forecast of Beta, a critical parameter for practitioners, poses a challenge due to its non-observable nature. The conventional methodology for forecasting Beta involves computing covariances and variances derived from historical return data. However, in many studies existing literature such as Fama €: French (1997), Groenewold 8: Fraser (1999), Moonis 8: Shah (2003), Zhou (2013) and Lopez Herrera et al. (2022) highlights significant concerns regarding the inherent instability of Beta, revealing variability over time that undermines the linear relationship posited by the CAPM between market portfolio returns and asset returns. This fluctuation signifies that a fundamental assumption of the CAPM is often violated in practice, which can adversely impact practitioners. The findings from the studies by Faff, Lee & Fry (1992), Groenewold & Fraser (1999), and Estrada (2000) also support the existence of these negative effects, particularly leading to issues in Beta forecasting. Therefore, it is crucial to adopt methodologies that effectively capture these temporal variations to improve estimation accuracy.
Along with the selection of Beta estimation methods, determining the data frequency and estimation window length is another crucial issue. This is because there are significant studies in the literature that provide important findings regarding the impact of data frequency and the selection of estimation windows on Beta forecast results, such as those by Damodaran (1999), Daves et al. (2000), Choudhry & Wu (2008), and Patton & Verardo (2012). At this point our aim was not only to evaluate the performance of Beta estimation methods but also to examine the effects of data frequency and estimation window length on the performance of Beta forecast.
We utilized Rolling and Recursive regression methods to estimate fluctuations in Beta of BIST leading sector indices and compared their forecasting performance using MAE and RMSE metrics. We analyzed three subperiods: 31.12.2008 - 29.12.2023, 31.12.2013 - 29.12.2023, and 31.12.2018 - 29.12.2023 to determine the impact of these methods over different time frames. We also assessed the effect of estimation window lengths with daily and weekly data, using 252 days, 126 days, 52 weeks, and 26 weeks, and examined how data frequency impacts Beta estimation.
The findings obtained from this study stand out in terms of providing ideas to financial managers and academics in the following aspects;
Using Rolling regression and Recursive regression techniques over various trading day (126, 252) and trading week (26, 52) windows, our findings confirm previous studies by Groenewold & Fraser (1999), Moonis & Shah (2003) and Wijethunga & Dayaratne (2015), Cenesizoglu et al. (2017), indicating that Beta is indeed sensitive to temporal fluctuations and occurs variably across the methodologies used.
The analysis of method performances with respect to error forecast metrics indicates that estimates obtained from the Rolling regression approach demonstrate significantly lower error values. This finding suggests a superior estimation performance in comparison to the Recursive regression method. The average B? and Beta values derived from the analysis, consistent with Dasgupta et al. (2010), indicate that changes in the alignment of index returns with market returns affect average B? values and the model's explanatory power. Contrary to Tuncel (2009) we find both the frequency of the data and the specified estimation window influence the estimation of Beta. The results obtained are consistent with the studies of Damodaran (1999), Patton & Verardo (2012) and Mestre (2023), which highlight that data frequency has an impact on forecasting outcomes. Additionally, they are consistent with the research conducted by Frankfurter, Leung, & Brockman (1994), Odabaşı (2003) and Brzeszczynski, Gajdka, & Schabek (2011), which suggest that the selection of the estimation window influences the forecast results. Another finding from the study is that examination of error forecast results across different estimation windows indicates that performance is contingent upon the window selection. This result is consistent with the conclusions drawn by Cenesizoglu et al. (2016) and Messis & Zapranis (2016). Notably, in the context of Rolling regression, the most favorable forecast error outcomes are predominantly associated with the 126-day estimation window, while the Recursive regression method fails to demonstrate a clear optimal window. This result is similar to the findings of Akyatan and Cetin (2020), who found the optimal estimation window length as 120 days.
Our results highlight the significance of data frequency in Beta forecast outcomes and shows that utilizing daily frequency data yields lower forecast errors compared to weekly data across the examined time frames and indices. This finding aligns with the assertions made by Daves et al. (2000) Igbal 8: Brooks (2007) and Patton & Verardo (2012), who advocate that high-frequency data facilitates more effective detection of Beta variability.
The findings obtained from the 5, 10, and 15-year estimation periods reveal that Beta estimates exhibit variation over the course of the selected estimation period. Also as the length of the estimation window increases, there is a decrease in the standard deviation values for the majority of Beta forecasts. In accordance with the findings presented by Theobald (1981) and Daves et al. (2000), we believe that decision-makers' preference for longer estimation periods based on standard deviation values may carry certain risks. We argue that selecting a longer estimation window may increase the likelihood of financial and operational decisions that could lead to biased Beta forecasts during the chosen extended period.
In conclusion, our findings suggest that practitioners who operate under the presumption that Beta remains constant may inadvertently produce biased analyses, resulting in improper financial decisions. Similar to Chakrabarti & Das (2021) and Lopez Herrera et al. (2022), we share the idea that the adoption of dynamic methodologies capable of capturing Beta over time may have positive effects on market risk and hedging behaviors.
As suggested by Doan et al. (2022) Phin et al. (2022) and Zhou (2025), we believe that Beta estimation using high-frequency data could yield more reliable results. In future studies, using different methods in Beta forecast and comparing their performance with intraday data will make valuable contributions to reduce forecast error in the Beta literature.
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1 The author stated that the return interval in Beta estimation can be calculated at different frequencies, such as annual or quarterly. However, he cautioned that shorter intervals may present challenges due to the asset's non-continuous trading, which impacts its correlation with the market index and, consequently, the accuracy of Beta estimation. To address the non-trading issue, he recommended using lower frequency intervals or high-frequency data with necessary corrections. For more details, see Damodaran (1999).
2 The authors noted that there is a dilemma that financial managers face in Beta estimation: the choice of forecast period and the choice ofreturn interval. As the number of observations in the forecast period increases, the standard error of beta decreases, but this increases the calendar period of the forecast period. The longer the calendar period, the more likely it is that the firm will encounter decisions and practices that will change its financial structure and cause changes in Beta.
3 In their studies, the authors investigated the impact of using price returns instead of total returns on the resulting outputs while exploring topics such as asset pricing models and portfolio selection. They demonstrated that tests assessing the validity of models constructed using price returns could lead to inaccurate results. And also they revealed that using price return causes a bias in portfolio selection in favor of assets with low dividend yields and against those with high dividend yields and showed that the share of price return in total return also has an effect on the estimated betas.
4 Similar to the approaches of Kaya and Soybilgen (2019), Buberkoku (2021) and Tanyildizi 8: Yuksel Yigiter (2024), the 2-year bond yield Was utilized, as it is the most frequently observed rate, exhibits the highest trading volume, and functions as a benchmark for borrowing costs in the Turkish market.
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