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The least squares estimator is a fundamental concept in linear regression, used to identify the best-fitting line through a set of data points. It minimizes differences between the data points and the line predicted by the model. In simpler terms, it finds the slope of the line that most accurately represents the data.
To derive the least squares estimator for the slope, denoted as \(\beta_1\), we minimize the **sum of squared deviations** between observed values and predicted values, \(y_i - b_1 x_i\). This minimizes the error that the line makes when predicting data.
The least squares estimator in our specific scenario is the trial value \(b_1\) which makes this error the smallest possible. We derive it using calculus by setting the derivative of the **sum of squared errors (SSE)** to zero. This gives us an optimal formula:

• \(b_1 = \frac{\sum_{i=1}^n x_i y_i}{\sum_{i=1}^n x_i^2}\)

This result ensures that the line fits the data through the origin \((0,0)\) with minimal error.

Understanding the sum of squared errors (SSE) is essential when performing linear regression. The SSE measures how well the regression line fits the data. It is the sum of the squares of the differences between observed values \(y_i\) and predicted values \(b_1 x_i\), where \(b_1\) is the assumed slope.
The formula for SSE is as follows:

• \( SSE(b_1) = \sum_{i=1}^n (y_i - b_1 x_i)^2 \)

Here, the goal is to find the value for \(b_1\) that makes the SSE as small as possible. This is crucial because a smaller SSE indicates a better fit to the data, meaning predictions of sales revenue are closer to actual values.
In practice, making this sum as small as possible means that the predicted line closely follows the trend of the data points, ensuring accuracy in our regression model.

In regression through the origin, the line of best fit is constrained to pass through the point \((0,0)\). This approach is often used when it is logical for the relationship to naturally go through the origin, for instance, when both variables should be zero if the input variable has no effect.
For this type of regression, the equation simplifies to \(Y=\beta_1 x+\epsilon\), omitting the intercept. The focus then solely lies on determining the correct slope \(\beta_1\), which tells us how much \(Y\) changes per unit change in \(x\).
By constraining this line through the origin, it often simplifies the calculations and provides straightforward interpretations.

• This method assumes no other influencing factors at play when \(x\) equals zero.
• The fit calculated through this form of regression helps in understanding pure relationships, often resulting in simpler models with confined parameters.

It is critical to consider whether a regression through the origin is appropriate for your data, as it removes the flexibility of adjusting for non-zero baselines.