Introduction

Linear regression is a powerful statistical method used to model the relationship between a dependent variable and one or more independent variables. It’s crucial to understand which types of relationships can be effectively estimated using linear regression and how they differ from relationships that are not suitable for this method. This chapter will delve into the characteristics of models that can be estimated using linear regression and shed light on scenarios where linear regression may not be appropriate.

Models Suitable for Linear Regression

Linear regression is well-suited for models where the relationship between the dependent variable and the independent variable(s) can be represented by a linear equation. This means that the change in the dependent variable is proportional to the change in the independent variable(s). Mathematically, the model takes the form:

$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+\ldots+\beta_k x_k+\epsilon$$

Here, $y$ is the dependent variable, $x_1, x_2, \ldots, x_k$ are the independent variables, $\beta_0, \beta_1, \beta_2, \ldots, \beta_k$ are the coefficients to be estimated, and $\epsilon$ represents the error term.

Models Not Suitable for Linear Regression

Linear regression is not appropriate for models with non-linear relationships between the variables. For example, if the relationship between the variables forms a curve, exponential growth, or logarithmic pattern, linear regression may produce inaccurate results. Additionally, linear regression assumes that the errors (residuals) are normally distributed and have constant variance (homoscedasticity). If these assumptions are violated, the model’s estimates may be biased or inefficient.