5 Assumptions regarding the Error term in Linear Regression

Any simple linear regression model can be presented in the form of a simplified equation as shown below

Y_t =α + β*X_t + ε_t

Here the ε_t is the called the residual which is not explained by the model.More conventionally this is known as the error term.This basically captures the randomness of the omitted variables.Now this error term holds some assumptions as described below:

Zero Mean : E(ε_i) = 0 for all i. So,for any given X ; ε may have different values but on average it is zero.

Homoskedasticity : Var ( ε_i ) = σ² which is constant. So basically, residuals would not have higher variance for higher values of X.

Normality : ε_i is normally distributed. As mentioned earlier, ε_i captures the impact of omitted variables.If we have many such variables which are minor but are independently distributed random variables, the distribution of their sum tends to be normal as the number of such variables increases.

No autocorrelation : ε_i is independent such that different values of ε_i are not correlated.

Cov( ε_i , ε_j ) = E[ { ε_i -E( ε_i )} {ε_j – E(ε_j)}] = E( ε_i ε_j) = 0

Non-stochastic X : The values of X (i.e, the explanatory variables) are same in repeated samples.

The implication of violation of these assumptions will be discussed in a separate post.