Overview

Fitting a line through the origin to a set of data points is a specific type of linear regression where the y-intercept is fixed at zero. This type of model is particularly useful in situations where the dependent variable theoretically passes through the origin when the independent variable is zero. This constraint simplifies the model and can enhance interpretability, but it also restricts its flexibility.

Mathematical Formulation

Ordinary Linear Regression

In typical linear regression, we aim to find a line defined by the equation:

$$y = mx + b$$

where:

• $y$ is the dependent variable,
• $x$ is the independent variable,
• $m$ is the slope,
• $b$ is the y-intercept.

Line Through the Origin

For a line that passes through the origin, the equation simplifies to:

$$y = mx$$

because $b = 0$. The goal here is to estimate the value of $m$ that minimizes the sum of squared differences between observed values and the predicted values based on the line.

Calculating the Slope

The slope, $m$, for a line through the origin can be determined using the following formula:

$$m = \frac{\sum (x_i \cdot y_i)}{\sum (x_i^2)}$$

where $\sum$ indicates summation over all data points, $x_i$ are the observed values of the independent variable, and $y_i$ are the observed values of the dependent variable.

Implementation

This technique can be implemented using statistical software or programming languages like Python. Below is an example using Python:

• Assumptions: Ensure that it is appropriate for the line to pass through the origin given the context of your data.
• Bias and Variance: By forcing the line through the origin, you might add bias if the true model includes an intercept, affecting the model's accuracy.
• Data Transformation: Sometimes transforming data (e.g., standardizing) can affect the suitability of modeling through the origin.
• Physics: Where relationships often inherently pass through the origin (e.g., Ohm’s Law, where the voltage drop across a conductor is proportional to the current through it).
• Finance: In cases of returns with no investment (initial input) leading to zero returns.
• Economics: Situations where supply/demand start from the origin in models without initial bias.