16.04.2024 Multiple Regression#

Why multiple Regression?

  • control for other factors

  • explain more of the variation of y

  • flexible functional form

Example:

\[ wage = \beta_0 + \beta_1 educ+\beta_2 experience + \epsilon \]

Interpretation \(\beta_1\) = change in y w.r.t \(x_1\), holding all other factors fixed

Assumption here: \(E(u | x_1,x_2) = 0\) :

  • u same for all combinations of \(x_1, x_2\)

OLS Expected Value#

Assumptions:

  1. parameters (\(\beta_i\)​) are linear (variables can be nonlinear)

  2. random sample of population

  3. no perfect collinearity

    • variables are not constant

    • no perfect linearity of variables

    • nonlinear functions of same variable are allowed, e.g \(y = \beta_0+\beta_1 x + \beta_2 x^2+u\)

  4. zero conditional mean: \(E(u | x_1,x_2,...) = 0\)

=> estimate = truth: \(E(\hat{ \beta }) = \beta\)

Misspecification#

Irrelevant Variables

  • no effect on unbiasedness

  • effect on variance of OLS-estimator

Omitted Variable Bias

  • true model: \(y = \beta_0+\beta_1 x_1 + \beta_2 x_2\)

  • our model: \(\hat{ y } = \hat{ \beta_1 } + \hat{ \beta_2 } \hat{ \delta }\) (delta = slope between \(x_1\) and \(x_2\))

  • => \(Bias(\beta_1) = E(\beta_1)-\beta_1\)

OLS Variance#

5th assumption: Homoskedasticity

  • error u = same variance for all given values

  • \(Var(u | x_1,...) = \sigma^2\)

Variance formula

\[ Var (\hat{ \beta_1 }) = \frac{ \sigma^2 }{SST_j(1-R_j^2)} \]

  • SST = total sample variation

  • R = R-sqaured