Ordinary Least Squares (OLS) is a linear regression method used to estimate the relationship between one or more independent variables (predictors) and a dependent variable (response). The main objective of OLS is to minimize the sum of the squared differences between the observed values of the dependent variable and the values predicted by the linear model. In other words, OLS tries to find the best-fitting straight line through the data points that minimizes the squared residuals.
The OLS method is based on several assumptions:
- Linearity: The relationship between the independent variables and the dependent variable is assumed to be linear.
- Independence: The observations are assumed to be independent of each other.
- Homoscedasticity: The variance of the residuals (errors) is assumed to be constant across all levels of the independent variables.
- Normality: The residuals (errors) are assumed to be normally distributed.
Given a dataset with n observations and p independent variables, the OLS method estimates the coefficients (parameters) of the linear model by minimizing the residual sum of squares (RSS), which is the sum of the squared differences between the observed and predicted values of the dependent variable.
The OLS estimates can be computed using various techniques, such as solving the normal equations, matrix inversion, or using iterative algorithms like the gradient descent.
Once the OLS estimates are obtained, you can use the fitted linear model to make predictions, assess the goodness-of-fit, and perform hypothesis tests or build confidence intervals for the model parameters.
It is important to note that while OLS is a widely used and simple method for linear regression, it may not always be the most appropriate choice, especially when the assumptions mentioned above are not met or when dealing with multicollinearity, heteroscedasticity, or other issues in the data. In such cases, alternative methods like ridge regression, LASSO, or robust regression might be more suitable.