1
$\begingroup$

I was reading about $\beta$ in finance, and I found the formula pictured below on this website. Is the n-1 a typo or do some people actually define $\beta$ with a Bessel correction? enter image description here

$\endgroup$

1 Answer 1

0
$\begingroup$

The formula is correct.

The general unbiased estimator for the variance-covariance matrix is with the Bessel correction.

When the data is known to be normally distributed, the MLE is with the $n$ at the denominator.

It seems that $R_m$ is assumed to be Normal, while $(R_e, R_m)$ clearly is not a bivariate Normal random variable (so the Bessel correction applies).

You can find a derivation here: https://en.wikipedia.org/wiki/Estimation_of_covariance_matrices#Maximum-likelihood_estimation_for_the_multivariate_normal_distribution

Beta can be defined in three equivalent ways:

  1. OLS estimate $\hat{\beta}_1$ for the slope of the linear regression $$ y_i=\beta_0 + \beta_1 x_i + \epsilon_i, \quad i \in \{1, 2, \ldots, N\} $$

  2. using the expression $\frac{\text{cov}[X,Y]}{\text{var}[X]}$. This is equivalent to (1) for OLS.
    The least squared estimate for $\beta_1$ is

$$ \hat{\beta}_1=\frac{\sum_i x_i y_i - \frac{1}{N}\sum_i x_i \sum_j y_j}{\sum_i x_i^2-\frac{1}{N} \left(\sum_i x_i\right)^2} = \frac{\text{cov}[X,Y]}{\text{var}[X]} $$

  1. Using the Pearson's correlation $\rho=\frac{\text{cov}[X,Y]}{\sqrt{\text{var}[X]\text{var}[Y]}}$

$$ \hat{\beta}_1=\frac{\text{cov}[X,Y]}{\text{var}[X]}=\rho\frac{\sqrt{\text{var}[Y]}}{\sqrt{\text{var}[X]}} $$

$\endgroup$
15
  • $\begingroup$ so the top is the unbiased estimator for the covariance and the bottom is an ML estimator for the variance of one of the variables. But why take the quotient? What does that represent? Also do you know for sure this (with n-1) is what finance people mean by $beta$? I thought beta was just supposed to be the result of a 1 dimensional regression which seems like it shouldn’t have the n-1 $\endgroup$ Aug 3, 2020 at 9:40
  • $\begingroup$ “an* unbiased” & “the* MLE” $\endgroup$ Aug 3, 2020 at 9:47
  • $\begingroup$ @Timkinsella yes, the beta formula is correct. There are 3 ways to define beta, one is this one, the other is the coefficient of the linear regression and the third one is based on the correlation wallstreetmojo.com/beta-formula $\endgroup$
    – piplustwo
    Aug 3, 2020 at 10:12
  • $\begingroup$ ah so three nonequivalent definitions? $\endgroup$ Aug 3, 2020 at 10:17
  • 1
    $\begingroup$ OLS=ordinary least-squares en.wikipedia.org/wiki/Ordinary_least_squares It's the method that minimizes the squares of the differences between the observed and predicted dependent variables en.wikipedia.org/wiki/Ordinary_least_squares#Matrix/… $\endgroup$
    – piplustwo
    Aug 3, 2020 at 11:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.