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Econometrics

Chapter 6: Multiple Regression Model

Population regression function

Assumptions of the classical regression

model

• Ideal conditions that guarantee that estimated parameters are unbiased, consistent, and attain the lowest variance among linear unbiased estimators.

Assumption 1: linearity

• Linear relationship between the dependent variable and k independent variables:

Assumption 3: Homoskedasticity

• The error terms have a constant variance (for all possible combinations of X (^) 1i , X (^) 2i , …, X (^) ki ):

Heteroskedastic vs. homoskedastic

error processes

Assumption 5: Nonstochastic X

• The X (^) i are nonstochastic (not random)
• Common violations:
  • measurement error
  • endogenous variables
• This assumption guarantees that the covariance between the independent variable and the error term will be zero.

Assumption 6 – no perfect

multicollinearity

• None of the independent variables can be written as an exact linear combination of the other independent variables

OLS estimation

• Ordinary least squares estimation:

OLS estimators

• Derivation requires the use of matrix algebra
• Estimators and standard errors are calculated by all statistical and econometric software packages

Properties of OLS estimators

• Under the conditions of the classical regression model, OLS estimators are: - consistent - linear - unbiased - best linear unbiased estimators (BLUE) (Gauss- Markov theorem)

Coefficient of Determination – R 2

With a bit of algebraic manipulation:

R^2 and the intercept

• R 2 may be appropriately computed as RSS/TSS only if an intercept term is included in the regression model

R^2 and degrees of freedom

• As additional variables are added to a regression, R 2 tends to increase even if there is no causal relation between the added variables and the dependent variable.
• R 2 = 1 when the # of estimated intercept and slope parameters = number of observations