Why we need an error term in regression model? What is its statistical distribution?

Why We Need an Error Term in Regression Models

The error term in a regression model is crucial for several reasons:

Model Accuracy and Fit

The error term helps in assessing the fit of the model. A smaller average error term indicates that the model explains a large portion of the variation in the dependent variable, suggesting a better fit145.

Inference and Statistical Tests

The error term is essential for performing statistical tests on the estimated parameters of the model. Its properties, such as its distribution, are critical in determining the efficiency and unbiasedness of parameter estimators14.

Model Specification

A systematic pattern in the error terms can indicate model misspecification, such as omitted variables, incorrect functional form, or heteroscedasticity. This helps in identifying and improving the model124.

Accounting for Unobserved Factors

The error term captures all the factors influencing the dependent variable that are not accounted for by the independent variables in the model. This includes unobservable, omitted, or unmeasurable factors145.

Statistical Distribution of the Error Term

In the context of regression analysis, particularly in the Classical Normal Linear Regression Model (CNLRM), the error term is typically assumed to follow certain statistical properties:

Normal Distribution

The error terms ((\epsilon)) are often assumed to be normally distributed, which is a key assumption for many statistical tests and inference procedures in regression analysis4.

Zero Mean

The error terms are assumed to have a mean of zero, indicating that, on average, the errors do not systematically overestimate or underestimate the true values124.

Constant Variance (Homoscedasticity)

The error terms are assumed to have constant variance across all levels of the independent variables. This assumption is crucial for the validity of many statistical tests12.

Independence

The error terms are assumed to be independent of each other, meaning that the error for one observation does not influence the error for another observation124.

By assuming these properties, the error term allows for robust statistical analysis and inference in regression models. However, it is important to note that the actual error term itself is unobservable and can only be inferred through the residuals, which are the observable estimates of the error term124.