E Views 12 Regression Analysis: Breaking Down Statistical Measures

R-squared is a key metric in regression analysis that measures how well our model fits the data.Let's start with some data points and fit a regression line.R-squared represents the proportion of variance in the dependent variable that's explained by our independent variables.These green lines show the residuals - the differences between our predicted values and actual values.And these yellow lines show the total variance in our data - the differences from the mean.R-squared is calculated by comparing these residuals to the total variance. A value closer to 1 indicates a better fit.However, R-squared has a limitation: it always increases when we add more variables, even if they don't actually improve our model.This is where Adjusted R-squared comes in. It penalizes the addition of variables that don't significantly improve the model.Let's look at an example. As we add more variables, R-squared increases, but Adjusted R-squared may actually decrease.Notice how Model 3 has the highest R-squared but the lowest Adjusted R-squared, suggesting potential overfitting.The standard error of regression measures how far observed values typically deviate from our regression line.These vertical lines show the distances between actual values and predicted values on our regression line.The standard error of a coefficient tells us how precise our estimate is. Smaller standard errors indicate more precise estimates.The t-statistic is calculated by dividing the coefficient by its standard error.The t-distribution helps us construct confidence intervals around our estimates.The 95 percent confidence interval represents the range where we expect the true parameter value to lie.When the absolute t-statistic exceeds 1.96, we reject the null hypothesis that the coefficient equals zero.