OLS simple and multiple regression properties, assumptions, and estimator formulas

S.R. Mechanical Property #1

The sum of the OLS residuals is zero

S.R. Mechanical Property #2

The OLS residuals are uncorrelated with the explanatory values.

S.R. Mechanical Property #3

The OLS simple regression line cuts through the variable sample means

S.R. Mechanical Property #4

The mean of the OLS fitted values is equal to the mean of the dependent variable of interest

S.R. Mechanical Property #5

The OLS fitted values are uncorrelated with the OLS residuals

SST

SST = sum (yi - y_bar)^2

SSE

SSE = sum (y_hat - y_bar)^2

SSR

SSR = sum e_hat^2

R^2

R^2 = SSE / SST

1 - SSR / SST

Statistical Property #1

B0_hat and B1_hat are unbiased estimators of true parameters B0 and B1

Statistical Property #2

var(B1_hat) = s.e.^2 / sum (xi - x_bar)^2

var(B0_hat) = s.e.^2 n^-1 sum ( xi^2) / sum (xi - x_bar)^2

Statistical Property #3

s.e._hat^2 is an unbiased estimator of s.e.^2

Assumption #1

The population relation is linear

Assumption #2

The sample is random

Assumption #3

There is variation in the explanatory variable, x

Assumption #4

The unconditional error mean is zero

Assumption #5

The conditional error mean is zero

Assumption #6

Homoskedasticity

The variance of the error term does not change depending on the value of explanatory variable x