Question

When is the more variability in the independent variable, the wider the confidence interval for the Least Square estimators ?

What do the values of the R² and the Adjusted R² measure? How are they related?

Answer 1

Part A

Confidence interval and variability

The amount of variation that is feasible in the independent variable(or any variable under study) is quantified with the help of confidence interval. When a confidence interval has larger length, it means that depending on the significance level that we specified(say 5%) , the given variable can vary in that given range of the confidence interval. Larger the length of the confidence interval, larger is the range of the permissible values. Thus the variability in the independent variable can be more of the confidence interval for the least square estimator is wider.

R squared and Adjusted Rsquared

The R squared value and adjusted Rsquared value both helps in finding out how good the model is. They are both the measures of how much variability in the dependent variable is captured by all the independent variables together.

Adjusted Rsquared over Rsquared

Neverthless, Adjusted R-squared has an advantage or benefit over R-squared. If we add an additional independent variable added to the model, the R-squared value will increase irrespective of the independent variable's contribution to explaining the dependent variable. While Adjusted R squared penalizes the value and thus increases in magnitude only if the newly added variable has a contribution in explaining the dependent variable better.