Ordinary Least Square Method of Regression

Ordinary Least Squares (OLS) method is a fundamental technique in statistical modelling and linear regression analysis. It aims to find the best-fitting line or hyperplane that minimizes the sum of squared differences between observed and predicted values. This technique is widely used due to its simplicity, interpretability, and mathematical properties.

Concept of OLS:

The primary goal of OLS is to determine the linear relationship between a dependent variable YYY and one or more independent variables XXX. In simple linear regression with one independent variable, the relationship is modeled as:

Y = β0+ β1X + ϵ

where:

• Y is the dependent variable.
• X is the independent variable.
• β0​ is the intercept of the regression line.
• β1​ is the slope of the regression line.
• ϵ is the error term, representing the difference between observed and predicted values.

Properties of OLS:

• Unbiasedness:

OLS estimators are unbiased, meaning that, on average, they correctly estimate the true parameters.

• Efficiency:

Among all linear unbiased estimators, OLS estimators have the smallest variance, making them the Best Linear Unbiased Estimator (BLUE) under the Gauss-Markov theorem.

• Consistency:

As the sample size increases, OLS estimators converge to the true parameter values.

• Normality:

If the errors ϵ are normally distributed, the OLS estimators follow a normal distribution, allowing for hypothesis testing and confidence interval estimation.

Assumptions of OLS:

The reliability of OLS estimators depends on several assumptions:

1. Linearity: The relationship between dependent and independent variables is linear.
2. Independence: The residuals are independent of each other.
3. Homoscedasticity: The residuals have constant variance.
4. Normality: The residuals are normally distributed.

Violation of these assumptions can lead to inefficient or biased estimates, necessitating diagnostic tests and potentially alternative methods.

Applications of OLS:

OLS regression is widely used in various fields such as economics, finance, social sciences, and engineering. It helps in understanding relationships between variables, forecasting, and making data-driven decisions.