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Beta regression is a form of regression which is used when the response variable, , takes values within (,) and can be assumed to follow a beta distribution. [1] It is generalisable to variables which takes values in the arbitrary open interval ( a , b ) {\displaystyle (a,b)} through transformations. [ 1 ]
In statistics, standardized (regression) coefficients, also called beta coefficients or beta weights, are the estimates resulting from a regression analysis where the underlying data have been standardized so that the variances of dependent and independent variables are equal to 1. [1]
The normal equations can be derived directly from a matrix representation of the problem as follows. The objective is to minimize = ‖ ‖ = () = +.Here () = has the dimension 1x1 (the number of columns of ), so it is a scalar and equal to its own transpose, hence = and the quantity to minimize becomes
The capital asset pricing model uses linear regression as well as the concept of beta for analyzing and quantifying the systematic risk of an investment. This comes directly from the beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or (0, 1) in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
The beta family includes the beta of the first and second kind [7] (B1 and B2, where the B2 is also referred to as the Beta prime), which correspond to c = 0 and c = 1, respectively. Setting c = 0 {\displaystyle c=0} , b = 1 {\displaystyle b=1} yields the standard two-parameter beta distribution .
The beta-binomial distribution is the binomial distribution in which the probability of success at each of n trials is not fixed but randomly drawn from a beta distribution. It is frequently used in Bayesian statistics , empirical Bayes methods and classical statistics to capture overdispersion in binomial type distributed data.
In this case, models are not specified and the estimates for the beta weights suffer from omitted variable bias. That is, the beta weights may change from one sample to the next, a situation sometimes called the problem of the bouncing betas. It is this problem with bouncing betas that makes unit-weighted regression a useful method.