10 Simple Linear Regression

• The estimated slope coefficient in simple linear regression represents the change in the dependent variable for a one-unit change in the independent variable.
• Formula for the estimated slope coefficient is Cov(X,Y) / σX^2, where Y is the dependent variable and X is the independent variable.
• To predict the value of the dependent variable (Y) given an independent variable (X) use the formula Y = 2.83 + 1.5X.
• When a linear relationship exists between an independent variable and the relative change in the dependent variable, a lin-log model Ina regression of the form In Y = bo + b₁X is appropriate.
• R² is calculated as the sum of squares regression divided by the sum of squares total.
• R² = sum of squares regression / sum of squares total = 556 / 1,235 = 0.45.

Coefficient of Determination

• The coefficient of determination for a linear regression describes the percentage of variation in the dependent variable explained by the variation of the independent variable.

Heteroskedasticity

• Heteroskedasticity occurs when the variance of the residual term in a regression is not constant across all observations.

Confidence Interval

• To determine a confidence interval around a predicted value from a simple linear regression, the degrees of freedom are n - 2.

Linear Regression Assumptions

• A crucial assumption in linear regression is that error terms are independently distributed, implying correlations between them should be zero.
• Constant variance of error terms and no correlation between the independent variable and the error term are typical assumptions.

Simple vs. Multiple Linear Regression

• Simple linear regression models the relationship between one dependent and one independent variable.
• Multiple regression models the relationship between one dependent variable and more than one independent variable

ANOVA table

• F-statistic = sum of squares regression / mean squared error = 550 / 19.737 = 27.867
• The F-statistic tests the fit of the model.

Logarithmic Variables

• Functional forms of simple linear regressions are available to account for logarithmic variables if either or both the dependent and independent variables are logarithmic.
• A log-lin model is appropriate if the dependent variable demonstrates logarithmic behavior, while the independent variable is linear.
  • A lin-log model is appropriate if the independent variable shows logarithmic behavior, while the dependent variable is linear.
  • A log-log model is appropriate if both the independent and dependent variables are logarithmic.

Correlation Coefficient

• To measure the strength of the relationship, as measured by the correlation coefficient (r):
• r = √0.599 = 0.774

Least Squares Criterion

• The least squares criterion in a simple regression model minimizes the sum of squared differences between the predicted and actual values of the dependent variable.
• Serves as the squared vertical distances.