The Multiple Regression Model

General form of the model

$$Y={\beta }_0+{\beta }_1{X}_1+\cdots +{\beta }_k{X}_k+ϵ;ϵ\sim NIID(0,{\sigma }^2)$$

Where:

• $Y:$ Response variable
• $X_j:$ $j$-th predictor variable
• ${\beta }_0:$ Intercept
• ${\beta }_j:$ $j$-th regression coefficient
• $ϵ:$ Error component
• $k$ : Amount of predictor variables

Model assumptions

• $y_i$ mutually indepentent between each other
• $\mathrm{Var}(y_i)={\sigma }^2:$ The model has constant variance
• $Y,ϵ:$ Probabilistic part of the model
• $E(Y|X)={\beta }_0+\cdots +{\beta }_k{X}_k:$ Deterministic part of the model

The assumptions for the error component ${ϵ}_i$ is the same as the one on simple linear regression model: Assumptions for the Error Component.

Interpretation of model components

• Slope: ${\beta }_0$ is the mean value of $Y$ at $X_1=X_2=\cdots =X_p=0$
• Regression coefficients: For an increase in $X_i$ by 1 unit, then mean of $y$ increases by ${\beta }_i$ (assuming other predictors are constant)

Elasticity

Elasticity measures the relative change in the dependent variable $Y$ due to a relative change in $X_k$.

Semi-elasticity measures the relative change in the dependent variable $Y$ due to an (absolute) one-unit-change in $X_k$.

For a linear regression, the elasticity of $Y$ with respect to $X_k$ is

$$\begin{array}{rl}\frac{\partial E[y_i|x_i]/E[y_i|x_i]}{\partial x_{ik}/x_{ik}}=& \frac{\partial E[y_i|x_i]}{\partial x_{ik}}\cdot \frac{x_{ik}}{E[y_i|x_i]}\\ =& \frac{x_{ik}}{x_i^{\prime }\beta }{\beta }_k\end{array}$$

Linear regression for $\log (y_i)=x_i^{\prime }\beta +{\epsilon }_i$, the elasticity of $Y$ with respect to $X_k$ is

$$\frac{\partial E[y_i|x_i]/E[y_i|x_i]}{\partial x_{ik}/x_{ik}}={\beta }_k{x}_{ik}$$

${\beta }_k$ measures the relative change in $Y$ due to a change in $X_k$ by one unit. Here, $b_k$ is called the semi-elasticity of $Y$ with respect to $X_k$.