Question

Is there any relationship between intercept and the r square, beta?

Answer 1

First we need to understand what are these figures mean!

Consider a simple linear regression model:

here, is intercept which has no meaning until x=0 and   is slope of the regression line which is the change in y per unit change in x. We can determine the regression model by determining the betas.

now,

R square (or r square) is called COEFFICIENT OF DETERMINATION . It indicates proportion of variation in dependent variable explained by regressors ( independent variable). The higher the value of R square, the greater the goodness of regression model. The problem with R^2 is that it will either stay same or increases as we introduce new variable to the model even if the newly introduced variables do not have any relationship with the response variable thus giving the false impression of the goodness of the model.( hence the concept of adjusted R^2 comes).

While dealing with simple linear regression model, where an intercept is included, we use r^2 as square of correlation coefficient between dependent and independent variable . When we introduced more independent var. to the model,we use R^2 as square of coefficient of multiple correlation.

Note that R ^2 and beta are two different measures used in regression model but they are somewhat related. R^2 measures how closely the response and the independent var. are correlated with each other whereas beta measures how large the changes in response due to independent var. are.