Simple regression is the two-variable regression with one dependent variable, Y_i, and one independent variable, X_i.

A multivariate regression has more than one independent variable.

*** the assumptions are the same as for the 2 variable linear regression model, but it also assumes that the independent variables are not collinear.

- a linear relationship exists between the dependent and independent variables

- no correlation with the error term for the independent variables

- Conditional expectation of the error term is 0

- Homoskedasticity (the variance of the error terms is constant for all observations)

- The error term for one observation is not correlated with that of another observation

- The model only includes the appropriate independent variables and does not omit variables.

- The independent variables are not collinear (the independent variables are not random, and there is no exact linear relation between any two or more independent variables).

it refers to the condition when two or more of the independent variables or linear combinations of the independent variables in a multiple regression are highly correlated with each other.

This condition distorts the standard error of estimate and the coefficient starndard errors. Results in a greater chance of type II errors.

The coefficient of multiple correlation is the square root of R-squared. R-square is defined below...

R² = ESS / TSS

= sum (Yhat - Y_mean)² / sum(Y_i - Y_mean)²

= 1 - (RSS / TSS)

F statistic is used for testing of the joint hypothesis that both slope coefficients equal 0.

F = (ESS / df) / (RSS /df)

df = n-number of slope coefficients - 1

R² will always go up if a new variable with any explanatory power is added to the regression. Thus R² may reflect the impact of a large set of independent variables rather than how well the set explains the dependent variable.

Adjusted R² uses df as follows:

adjusted R² = 1-(1-R²) * [(n-1)/(n-k-1)]