# Statistical Inference for the Beta Coefficient

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## Abstract

**:**

## 1. Introduction

## 2. Estimated Beta Coefficient and Its Distributional Properties

#### 2.1. Sample Distribution of the Estimated Beta Coefficient

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1:**

#### 2.2. Interval Estimation and Test Theory

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2:**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**3:**

## 3. Empirical Illustration

## 4. Robustness to the Violation of the Normality Assumption

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Power function for $n\in \{60,\phantom{\rule{0.166667em}{0ex}}120,\phantom{\rule{0.166667em}{0ex}}250,\phantom{\rule{0.166667em}{0ex}}500\}$ and $\gamma =0.05$.

**Figure 2.**Sample estimators and confidence intervals for the beta coefficient of the equally-weighted portfolio constructed for the first $k=5$

**(top left**), $k=10$ (

**top right**), $k=15$ (

**middle left**), $k=20$ (

**middle right**), $k=25$ (

**bottom left**), and $k=30$ (

**bottom right**) assets included into the DAX index in alphabetical order.

**Figure 3.**Kernel density estimators for $n\in \{60,\phantom{\rule{0.166667em}{0ex}}120,\phantom{\rule{0.166667em}{0ex}}250,\phantom{\rule{0.166667em}{0ex}}500,\phantom{\rule{0.166667em}{0ex}}1000,\phantom{\rule{0.166667em}{0ex}}2000\}$ and the asymptotic density of the standardized estimator for the beta coefficient in the case of the $k=5$ (

**top left**), $k=10$ (

**top right**), $k=15$ (

**middle left**), $k=20$ (

**middle right**), $k=25$ (

**bottom left**), and $k=30$ (

**bottom right**) dimensional equally-weighted portfolio. The asset returns are drawn from the multivariate t-distribution with 5 degrees of freedom.

**Figure 4.**Kernel density estimators for $n\in \{60,\phantom{\rule{0.166667em}{0ex}}120\}$ and the asymptotic density of the standardized estimator for the beta coefficient in the case of the $k=5$ (

**top left**), $k=10$ (

**top right**), $k=15$ (

**middle left**), $k=20$ (

**middle right**), $k=25$ (

**bottom left**), abd $k=30$ (

**bottom right**) dimensional equally-weighted portfolio. The asset returns are drawn from the multivariate t-distribution with 10 degrees of freedom.

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## Share and Cite

**MDPI and ACS Style**

Bodnar, T.; Gupta, A.K.; Vitlinskyi, V.; Zabolotskyy, T.
Statistical Inference for the Beta Coefficient. *Risks* **2019**, *7*, 56.
https://doi.org/10.3390/risks7020056

**AMA Style**

Bodnar T, Gupta AK, Vitlinskyi V, Zabolotskyy T.
Statistical Inference for the Beta Coefficient. *Risks*. 2019; 7(2):56.
https://doi.org/10.3390/risks7020056

**Chicago/Turabian Style**

Bodnar, Taras, Arjun K. Gupta, Valdemar Vitlinskyi, and Taras Zabolotskyy.
2019. "Statistical Inference for the Beta Coefficient" *Risks* 7, no. 2: 56.
https://doi.org/10.3390/risks7020056