Introduction

In linear regression, Ordinary Least Squares (OLS) is a widely used method for estimating the parameters of the model. To ensure the reliability and validity of OLS estimates, certain assumptions need to be met. These assumptions provide the foundation for the underlying statistical theory and guide the interpretation of the regression results. Let’s delve into the key assumptions of OLS parameter estimation.

Assumptions of Linear Regerssion

Assumption 1: Linearity
The relationship between the independent variables and the dependent variable is assumed to be linear. This means that any change in the independent variables results in a proportional change in the dependent variable. For instance, in a simple linear regression equation, $y=\beta_0+\beta_1 x+\epsilon$, the relationship between $x$ and $y$ should be linear.

Assumption 2: Independence of Errors
The errors $(\epsilon)$ should be independent of each other. This assumption ensures that the errors from one observation do not influence the errors of another observation. Violation of this assumption could lead to biased and inefficient coefficient estimates.

Assumption 3: Homoscedasticity
Homoscedasticity refers to the constant variance of the errors across all levels of the independent variables. In other words, the spread of the residuals should remain consistent throughout the range of the predictor variables. Heteroscedasticity, where the spread of residuals changes, can impact the accuracy of coefficient estimates.

Assumption 4: No Perfect Multicollinearity
Multicollinearity occurs when two or more independent variables are highly correlated, leading to redundancy in the model. Perfect multicollinearity, where one independent variable is a perfect linear combination of others, can result in numerical instability and make interpretation difficult.

Assumption 5: Zero Mean of Residuals
The sum of the residuals should be close to zero. This ensures that the model is not systematically overestimating or underestimating the dependent variable across observations.

Assumption 6: Normality of Errors
The errors ($\epsilon$) are assumed to follow a normal distribution. This assumption is crucial for hypothesis testing and confidence interval estimation. Deviations from normality can affect the validity of statistical tests.