Statistics - Adjusted R-Squared


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R-squared measures the proportion of the variation in your dependent variable (Y) explained by your independent variables (X) for a linear regression model. Adjusted R-squared adjusts the statistic based on the number of independent variables in the model.${R^2}$ shows how well terms (data points) fit a curve or line. Adjusted ${R^2}$ also indicates how well terms fit a curve or line, but adjusts for the number of terms in a model. If you add more and more useless variables to a model, adjusted r-squared will decrease. If you add more useful variables, adjusted r-squared will increase.

Adjusted ${R_{adj}^2}$ will always be less than or equal to ${R^2}$. You only need ${R^2}$ when working with samples. In other words, ${R^2}$ isn't necessary when you have data from an entire population.

Formula

${R_{adj}^2 = 1 - [\frac{(1-R^2)(n-1)}{n-k-1}]}$

Where −

  • ${n}$ = the number of points in your data sample.

  • ${k}$ = the number of independent regressors, i.e. the number of variables in your model, excluding the constant.

Example

Problem Statement:

A fund has a sample R-squared value close to 0.5 and it is doubtlessly offering higher risk adjusted returns with the sample size of 50 for 5 predictors. Find Adjusted R square value.

Solution:

Sample size = 50 Number of predictor = 5 Sample R - square = 0.5.Substitute the qualities in the equation,

$ {R_{adj}^2 = 1 - [\frac{(1-0.5^2)(50-1)}{50-5-1}] \\[7pt] \, = 1 - (0.75) \times \frac{49}{44} , \\[7pt] \, = 1 - 0.8352 , \\[7pt] \, = 0.1648 }$


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