# Sectoral Differences in the Choice of the Time Horizon during Estimation of the Unconditional Stock Beta

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Data and Methodology

^{th}observation $\forall \text{}i\text{}=\text{}1,\text{}\dots \text{},\text{}252$8. Returns on stocks and the market are estimated in continuous time using logarithmic differences of daily returns.

_{B}< T with T: the number of used observations and T

_{B}: the point in time where the structural break occurs.

_{max}= 14.

_{B}< T − 1. The date for which the estimated value of the t-statistic ${T}_{\widehat{a}}=\left(\widehat{a}-1\right)/s.e.(\widehat{a})$ is minimized, and for which the probability of rejecting the unit root null is maximized, is considered to be the endogenously determined break date of the examined series. In the context of the performed unit root test, the statistical significance of the unit root null without a break in series is tested against the alternative of a break-stationary process.

## 4. Results

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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^{1}Numerous articles have dealt with issues of heteroscedasticity, autocorrelation and the time variation of betas through the estimation of ARCH and GARCH models, GARCH conditional betas, stochastic volatility conditional betas, Kalman Filter approaches, Flexible Least Squares, Markov switching approaches [1,2,3,4,5]; see Hollstein and Prokopczuk [6] for a recent and comprehensive comparison of market beta estimation techniques. Another set of literature has concentrated on the estimation of realized betas to improve beta forecasts [7,8]. We refer to them in a separate footnote in the literature section of this paper.^{2}Some sites such as Bloomberg allows users to specify the period of estimation while other sites such as Compustat and Dow Jones do not.^{4}The reason for the appearance of these models was that while betas were recognized to be time-varying in nature (time invariance of beta is a basic assumption of the CAPM), there were no models or forecasting techniques that could outperform the constant beta model [28]. This advancement was a response to the availability of higher frequency financial data that allowed for the development of new estimators and evaluation criteria. Barndorff-Nielsen and Shephard [8] and Andersen et al. [26] provided the foundation for the computation of realized betas by assuming that security prices follow a multivariate continuous time stochastic volatility diffusion. While the estimation of realized betas does not fit the scope of this paper, we reflect on them in the conclusions as they present the next step of this research.^{5}Moreover, they found that the 12-month beta estimates from the constant beta model resulted in reduction of mean squared error forecasts in excess of 30% as compared to autoregressive models commonly reported in the literature.^{6}Past research has concentrated on the differences in beta amongst sectors. Rosenberg [32] noted that companies active in the Agriculture and Utilities industry show lower levels of betas while companies in the Electronics, Air Transport and Securities show higher. Liu [33] found that Real Estate shows high values of time varying betas.^{7}Using the realized volatility index $R{V}_{t}=\sqrt{\raisebox{1ex}{$252\times {{\displaystyle \sum}}_{i=0}^{21}{\left(\mathrm{ln}(\frac{SP{X}_{t-i}}{SP{X}_{t-i-1}})\right)}^{2}$}\!\left/ \!\raisebox{-1ex}{$21$}\right.\times 100}$, our first sample showed an index of 10.84 and our second sample 18.56. These were statistically different at a = 1%.^{8}252 daily observations were downloaded minus the 30 most recent observations allow for the estimation of 30-day betas all the way to 252 day betas.^{9}Average t-day betas were estimated for the whole sample of 2641 stocks in the NYSE and were found to be very close to 1.^{10}Only 2324 out of 2641 stocks in the NYSE were included in the analysis as some lacked an ICB classification match and some exhibited negative betas when few observations were included in the regression.^{11}The large standard deviation suggests that there are numerous stocks in each category both on the upper and the lower side of the average. This, however, represents the main reason that we took all of the stocks in NYSE so as to allow company specific events to average out, which allows us to glimpse at the sectoral averages.^{12}We would therefore need to look at ICB2-3-4 categories for more detailed information. Moreover, about 40% of the stocks in each category are non-stationary at the 1% level, 50% at the 5% level and 60% at the 10% level of significance.^{13}Fourth level ICB categories are available upon request only, as the great number of categories and the limited number of observations in many of these categories prevents us from either effectively discussing the results within the limits of an article or reaching useful conclusions.^{14}The analysis was repeated employing both market capitalization and ICB1 level analysis. Those results are available upon request.

Category Number | ICB1 Category | 2011–2012 | 2006–2007 | ||||||
---|---|---|---|---|---|---|---|---|---|

No. of Stocks | Av. Break | Av. Std | Perc. < 60 | No. of Stocks | Av. Break | Av. Std | Perc. < 60 | ||

1 | Basic Materials ^{2,6,10} | 143 | 101 | 54.2 | 14.0% | 117 | 113 | 44.5 | 19.7% |

2 | Consumer Goods ^{3,5,7,9,10} | 203 | 115 | 61.8 | 19.2% | 151 | 115 | 41.3 | 15.9% |

3 | Consumer Services ^{2,6,10} | 224 | 102 | 54.0 | 21.4% | 184 | 113 | 40.9 | 13.0% |

4 | Financials ^{6,10} | 887 | 106 | 55.8 | 20.6% | 829 | 112 | 39.9 | 14.8% |

5 | Health Care ^{2,6,10} | 105 | 100 | 51.1 | 27.6% | 95 | 107 | 40.5 | 15.8% |

6 | Industrials ^{1,3,4,5,7,9,10} | 356 | 113 | 57.4 | 18.8% | 322 | 107 | 44.6 | 22.7% |

7 | Oil & Gas ^{2,4,6,10} | 168 | 97 | 54.4 | 28.6% | 143 | 100 | 43.6 | 14.7% |

8 | Technology ^{10} | 88 | 110 | 60.2 | 26.1% | 76 | 109 | 42.2 | 19.7% |

9 | Telecommunications ^{6,10} | 50 | 96 | 49.4 | 24.0% | 46 | 111 | 43.6 | 21.7% |

10 | Utilities ^{1 through 9} | 100 | 131 | 61.5 | 20.0% | 90 | 104 | 40.0 | 8.9% |

Category Number | ICB1 | ICB2 | 2011–2012 | 2006–2007 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

No. of Stocks | Av. Break | Av. Std | Perc. < 60 | No. of Stocks | Av. Break | Av. Std | Perc. < 60 | |||

1a | Basic Materials | Basic Resources | 79 | 96 | 47.5 | 20% | 65 | 113 | 43.3 | 23% |

1b | Chemicals | 64 | 109 | 61.1 | 6% | 52 | 112 | 46.2 | 15% | |

2a | Consumer Goods | Automobiles & Parts | 34 | 112 | 62.0 | 9% | 23 | 114 | 39.6 | 13% |

2b | Food & Beverage | 51 | 99 | 54.5 | 22% | 39 | 116 | 40.4 | 15% | |

2c | Personal & Household Goods | 118 | 122 | 63.9 | 21% | 89 | 114 | 42.5 | 17% | |

3a | Consumer Services | Media | 64 | 97 | 49.6 | 28% | 41 | 115 | 39.6 | 7% |

3b | Retail | 96 | 103 | 55.0 | 18% | 82 | 115 | 39.9 | 10% | |

3c | Travel & Leisure | 64 | 104 | 57.3 | 20% | 61 | 107 | 43.3 | 21% | |

4a | Financials | Banks | 124 | 110 | 55.9 | 19% | 103 | 121 | 39.4 | 14% |

4b | Financial Services | 672 | 104 | 54.9 | 20% | 647 | 109 | 39.8 | 15% | |

4c | Insurance | 91 | 115 | 60.9 | 24% | 79 | 118 | 39.4 | 13% | |

5 | Health Care | Health Care | 105 | 100 | 51.1 | 28% | 95 | 107 | 40.5 | 16% |

6a | Industrials | Construction & Materials | 52 | 108 | 58.4 | 15% | 51 | 100 | 41.5 | 29% |

6b | Industrial Goods & Services | 304 | 114 | 57.3 | 19% | 271 | 109 | 45.0 | 21% | |

7 | Oil and Gas | Oil & Gas | 168 | 97 | 54.4 | 29% | 143 | 100 | 43.6 | 15% |

8 | Technology | Technology | 88 | 110 | 60.2 | 26% | 76 | 109 | 42.2 | 20% |

9 | Telecommunications | Telecommunications | 50 | 96 | 49.4 | 24% | 46 | 111 | 43.6 | 22% |

10 | Utilities | Utilities | 100 | 131 | 61.5 | 20% | 90 | 104 | 40.0 | 9% |

Category Number | ICB2 | ICB3 | 2011–2012 | 2006–2007 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

No. of Stocks | Av. Break | Av. Std | Perc. < 60 | No. of Stocks | Av. Break | Av. Std | Perc. < 60 | |||

1a | Basic Resources | Forestry & Paper | 14 | 109 | 57.1 | 21% | 11 | 81 | 42.2 | 64% |

1b | Industrial Metals | 36 | 100 | 54.3 | 28% | 28 | 113 | 40.0 | 18% | |

1c | Mining | 29 | 84 | 28.9 | 10% | 26 | 127 | 41.3 | 12% | |

2 | Chemicals | Chemicals | 64 | 109 | 61.1 | 6% | 52 | 112 | 46.2 | 15% |

3 | Automobiles & Parts | Automobiles & Parts | 34 | 112 | 62.0 | 9% | 23 | 114 | 39.6 | 13% |

4a | Food and Beverages | Beverages | 7 | 120 | 69.3 | 14% | 7 | 140 | 7.2 | 0% |

4b | Food Producers | 44 | 96 | 52.0 | 23% | 32 | 110 | 42.8 | 19% | |

5a | Personal and Household Goods | Household Goods | 57 | 110 | 58.6 | 23% | 44 | 116 | 42.7 | 18% |

5b | Leisure Goods | 22 | 136 | 67.8 | 27% | 15 | 116 | 40.0 | 7% | |

5c | Personal Goods | 31 | 141 | 66.1 | 10% | 24 | 111 | 45.4 | 17% | |

5d | Tobacco | 8 | 103 | 66.2 | 38% | 6 | 112 | 45.5 | 33% | |

6 | Media | Media | 64 | 97 | 49.6 | 28% | 41 | 115 | 39.6 | 7% |

7a | Retail | Food & Drug Retailers | 10 | 112 | 55.4 | 10% | 8 | 130 | 17.7 | 0% |

7b | General Retailers | 86 | 102 | 55.1 | 19% | 74 | 114 | 41.3 | 11% | |

8 | Travel & Leisure | Travel & Leisure | 64 | 104 | 57.3 | 20% | 61 | 107 | 43.3 | 21% |

9 | Banks | Banks | 124 | 110 | 55.9 | 19% | 103 | 121 | 39.4 | 14% |

10a | Financial Services | Equity Investment Instruments | 379 | 103 | 51.5 | 21% | 405 | 112 | 39.5 | 13% |

10b | General Financial | 126 | 109 | 58.1 | 23% | 116 | 108 | 41.7 | 18% | |

10c | Nonequity Investment Instrumen | 6 | 142 | 67.2 | 17% | 3 | 122 | 31.7 | 0% | |

10d | Real Estate | 161 | 102 | 59.5 | 16% | 123 | 103 | 38.9 | 22% | |

11a | Insurance | Life Insurance | 35 | 109 | 59.0 | 20% | 28 | 106 | 38.3 | 18% |

11b | Nonlife Insurance | 56 | 119 | 62.2 | 27% | 51 | 124 | 38.7 | 10% | |

12a | Health Care | Health Care Equipment & Servic | 78 | 99 | 48.7 | 26% | 68 | 107 | 38.1 | 15% |

12b | Pharmaceuticals & Biotechnolog | 27 | 103 | 58.6 | 33% | 27 | 105 | 46.8 | 19% | |

13 | Construction and Materials | Construction & Materials | 52 | 108 | 58.4 | 15% | 51 | 100 | 41.5 | 29% |

14a | Industrial Goods and Services | Aerospace & Defense | 26 | 111 | 51.7 | 12% | 28 | 116 | 42.1 | 14% |

14b | Electronic & Electrical Equipm | 51 | 114 | 63.8 | 25% | 44 | 110 | 48.0 | 20% | |

14c | General Industrials | 36 | 115 | 55.6 | 25% | 35 | 106 | 39.9 | 23% | |

14d | Industrial Engineering | 59 | 119 | 60.0 | 15% | 59 | 110 | 43.2 | 19% | |

14e | Industrial Transportation | 55 | 120 | 57.3 | 20% | 41 | 95 | 42.0 | 29% | |

14f | Support Services | 77 | 107 | 54.0 | 18% | 64 | 114 | 49.8 | 22% | |

15a | Oil and Gas | Alternative Energy | 1 | 64 | 0% | 1 | 128 | 0% | ||

15b | Oil & Gas Producers | 116 | 91 | 52.2 | 33% | 96 | 101 | 45.3 | 16% | |

15c | Oil Equipment, Services & Dist | 51 | 111 | 57.5 | 20% | 46 | 98 | 40.4 | 13% | |

16a | Technology | Software & Computer Services | 42 | 113 | 60.6 | 19% | 35 | 99 | 37.1 | 20% |

16b | Technology Hardware & Equipmen | 46 | 107 | 60.3 | 33% | 41 | 117 | 44.9 | 20% | |

17a | Telecommunications | Fixed Line Telecommunications | 28 | 110 | 57.0 | 21% | 25 | 111 | 40.6 | 20% |

17b | Mobile Telecommunications | 22 | 79 | 30.7 | 27% | 21 | 111 | 47.9 | 24% | |

18a | Utilities | Electricity | 66 | 132 | 60.0 | 15% | 54 | 99 | 41.0 | 11% |

18b | Gas, Water & Multiutilities | 34 | 128 | 65.1 | 29% | 36 | 110 | 38.1 | 6% |

Category Number | Market Capitalization | 2011–2012 | 2006–2007 | Sig | ||||
---|---|---|---|---|---|---|---|---|

Obs. | Av. Break | Av. Std | Obs. | Av. Break | Av. Std | |||

1 | Up to $50 million | 766 | 109 | 55.7 | 619 | 110 | 42.7 | 0.668 |

2 | $50 million–$300 million | 262 | 104 | 53.6 | 214 | 111 | 40.7 | 0.081 |

3 | $300 million–$2 billion | 527 | 107 | 55.8 | 438 | 114 | 41.9 | 0.033 |

4 | $2 billion–$10 billion | 487 | 110 | 60.3 | 436 | 107 | 40.7 | 0.445 |

5 | $10 billion–$200 billion | 379 | 105 | 57.8 | 360 | 108 | 42.4 | 0.389 |

6 | Above $200 billion | 10 | 84 | 43.7 | 10 | 110 | 49.2 | 0.218 |

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**MDPI and ACS Style**

Dadakas, D.; Karpetis, C.; Fassas, A.; Varelas, E. Sectoral Differences in the Choice of the Time Horizon during Estimation of the Unconditional Stock Beta. *Int. J. Financial Stud.* **2016**, *4*, 25.
https://doi.org/10.3390/ijfs4040025

**AMA Style**

Dadakas D, Karpetis C, Fassas A, Varelas E. Sectoral Differences in the Choice of the Time Horizon during Estimation of the Unconditional Stock Beta. *International Journal of Financial Studies*. 2016; 4(4):25.
https://doi.org/10.3390/ijfs4040025

**Chicago/Turabian Style**

Dadakas, Dimitrios, Christos Karpetis, Athanasios Fassas, and Erotokritos Varelas. 2016. "Sectoral Differences in the Choice of the Time Horizon during Estimation of the Unconditional Stock Beta" *International Journal of Financial Studies* 4, no. 4: 25.
https://doi.org/10.3390/ijfs4040025