• Instead of one regressor, we can have multiple ones.

• For example, how do horse power (hp) and weight (wt) relate to miles per gallon? \[ mpg_i = b_0 + b_1 hp_i + b_2 wt_i + e_i \]

• Well, how?

• Let’s have a look at the 3D figure in the book

• Multiple means there can be more than 2 regressors as well.
• In fact, there could be \(K\) of them. \[ \begin{aligned} \widehat{y}_i &= b_0 + b_1 x_{1i} + b_2 x_{2i} + \dots + b_K x_{Ki}\\ e_i &= y_i - \widehat{y}_i \end{aligned} \]
• The partial effect of regressor \(j\) is the partial derivative \[ \frac{\partial y}{\partial x_j} = b_j \]

• How does \(mpg\) change if we change \(hp\)?
• This question only makes sense if \(wt\) does not change.
• Holding the others constant: partial derivative. \[ \frac{\partial mpg_i}{\partial hp_i} = b_1 \]

• Do this!

  lm(formula = mpg ~ wt + hp, data = mtcars)

• Multiple regression requires that variables are linearly independent.
• Imagine we had wt_plus defined as wt + 1 above.
• wt_plus wouldn’t add any information - hence it’s redundant
• If we only have one x, this translates into \(Var(x)\neq0\).
• Gives rise to the rank condition, i.e. that \(N\geq K+1\)

• Let’s go back to the Caschool dataset.
• Could testscr also depend on average income in the school area? \[ \text{testscr}_i = b_0 + b_1 \text{str}_i + b_2 \text{avginc}_i + e_i \]

• We simply add avginc to our formula:

  library("Ecdat") # reload the data
  fit_multivariate <- lm(formula = "testscr ~ str + avginc", data = Caschool)
  summary(fit_multivariate)

• Let’s again look at the plot

• We can relax the ceteris paribus assumption.
• If we want to allow the effect of one variable to depend on the value of another one, we can do this.
• Is the effect of str dependent on avginc? Is the effect stronger in richer areas?
• Go back to the book

library(ScPoEconometrics)
runTutorial("lm-example")