Step 1: Understanding the Question:
The question asks for the main objective of the Ordinary Least Squares (OLS) method, which is the most common technique for fitting a linear regression model.
Step 2: Key Formula or Approach:
In linear regression, we try to find a line (or hyperplane) that best fits the data. The model is of the form \( \hat{y} = \beta_0 + \beta_1 x \).
The difference between the actual value \(y_i\) and the predicted value \( \hat{y}_i \) is called the residual, \( e_i = y_i - \hat{y}_i \).
The OLS method aims to find the coefficients (\( \beta_0, \beta_1 \)) that minimize the Sum of Squared Residuals (SSR), also known as the Residual Sum of Squares (RSS).
\[ \text{SSR} = \sum_{i=1}^{n} e_i^2 = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2 \]
Step 3: Detailed Explanation:
The goal of OLS is to find the line that is "closest" to all the data points simultaneously.
The "distance" from each point to the line is measured by the residual. Some residuals will be positive (point is above the line) and some negative (point is below the line).
To prevent positive and negative errors from canceling each other out, we square them. Squaring also has the desirable property of penalizing larger errors more heavily.
By minimizing the sum of these squared residuals, OLS finds the unique best-fitting line for the given data.
Step 4: Final Answer:
The primary goal of the OLS method is to minimize the sum of squared residuals.