Assumptions of Multiple Regression Model

Multiple regression models are built on the following assumptions.

• The relationship between the dependent variable, \(Y\), and the independent variables, \(X_{1}, X_{2},…, X_{k}\) is linear.
• The independent variables \((X_{1}, X_{2},…, X_{k})\) are not random.
• There exists no definite linear relationship between two or more of the independent variables. A high correlation between two or more independent variables is known as multicollinearity.
• The expected value of the error term, conditional on the independent variables, is equal to 0.

$$E(\epsilon|X_{1}, X_{2},…, X_{k})=0$$

• The variance of the error term is equal for all observations.

$$E(\epsilon_{i}^{2})=\sigma_{\epsilon}^{2}, i=1,2,…,n$$

           (This is known as the homoskedasticity assumption.)

• The error term is uncorrelated across all observations.

$$E(\epsilon_{i}\epsilon_{j})=0 {}∀ i≠j$$

Question

Which of the following is least likely an assumption of the multiple linear regression model?

    A. The independent variables are not random.

    B. The error term is correlated across all observations.

    C. The expected value of the error term, conditional on the independent variables, is equal to zero.

Solution

The correct answer is B.

The error term is uncorrelated across all observations.

$$E(\epsilon_{i}\epsilon_{j})=0 ∀i≠j$$

Other assumptions of the classical normal multiple linear regression model include:

    i. The independent variables are not random. Additionally, there is no exact linear relationship between two or more of the independent variables.

   ii. The error term is normally distributed.

  iii. The expected value of the error term, conditional on the independent variables, is equal to 0.

  iv. The variance of the error term is the same for all observations.

  v. A linear relation exists between the dependent variable and the independent variables.

Reading 2: Multiple Regression

LOS 2 (f) Explain the assumptions of a multiple regression model.