3.6 Assumptions

Like the simple linear regression, the Multiple linear regression requires:

• Assumptions 1 and 2 to ensure unbiasedness.
• Assumptions 3 and 4 to enable the computation of the estimators’ variance matrix.
  
• The assumption 5 is needed when the regressors are all random.
  
• The assumption 6 to justify the linear form of the model.
  
• The assumption 7 of the constancy of the parameters to ensure that no break exists in the DGP model.
  
• The assumption 8 of normality to obtain tests for various hypotheses.

Definition 3.5 (Assumptions)

• A1: Regressors are non random.
  
• A2: The vector mean of errors is equal to zero \(E(\varepsilon)=0\).
  
• A3: The variance of errors is constant \(E(\varepsilon^t)=\sigma^2\). This assumption is called Homoskedasticity.
  
• A4: No correlation between errors \(E(\varepsilon_t\varepsilon_s)=0\).
  
• These two last assumptions (3,4) can be rewritten together in matrix form: \(\Omega_{\varepsilon}=E(\varepsilon\varepsilon^t)=\sigma^2I_n\).
  
• A5: No correlation between the errors and the regressors \(E(X^t\varepsilon)=0\), only required when the regressors are random.
  
• A6: Linearity in terms of parameters.
  
• A7: The model parameters are all constant.
• A8: The errors are normally and identically distributed \(\varepsilon\sim NID(0, \sigma^2I)\).

In addition to the above assumptions required for both simple and multiple linear regression, the following assumptions are needed only for the latter.

Definition 3.6 (Assumptions)

• A9: No strong correlations between regressors to enable the invertibility of the matrix \(X^tX\), called no multicollinearity.
  
• A10: The number of paramters should be less than the sample size.