Linear Models

Picture a mathematical framework that outlines the relationship between a dependent variable and independent variables. This is a linear model. Presupposing a linear function to describe the interplay between these variables, the model draws a straight line in two-dimensional space. Across the fields of statistical analysis, econometrics, and machine learning, these models hold a key role. They predict outcomes, unravel causal relationships, and measure the degree to which various factors affect the result.

The Basic Linear Model

Delving into the essence of a linear model, we encounter the simple linear regression. Here, we deal with a single dependent variable and one independent variable. This model takes shape in the equation:

y = β₀ + β₁x + ε

In the context of this equation:

y stands for the dependent variable (the outcome)
x acts as the independent variable (the predictor)
β₀ is the intercept (value of y when x equals 0)
β₁ indicates the slope (change in y per one-unit increment in x)
ε represents the error term (difference between the observed and the forecasted values)

Multiple Linear Regression

The narrative of linear models continues with the multiple linear regression. Building on the simple linear regression, it accommodates a greater number of independent variables. Its equation reads as:

y = β₀ + β₁x₁ + β₂x₂ +... + βₚxₚ + ε

Here:

x₁, x₂,..., xₚ denote the independent variables
β₁, β₂,..., βₚ are the coefficients that mark the effect of each independent variable on the dependent variable

Estimation and Assumptions

Estimating coefficients in a linear model commonly employs the ordinary least squares (OLS) method. This aims to minimize the sum of squared differences between actual and predicted values. The method rests on a few key assumptions:
Linearity: There is a linear relationship between dependent and independent variables.
Independence: Observations exist independently of one another.
Homoscedasticity: The variance of the error term stays constant across independent variable values.
Normality: The error term subscribes to a normal distribution.

Applications and Limitations

Linear models find wide-ranging application across various disciplines, spanning economics, finance, social sciences, and engineering. They lend themselves to predicting outcomes, discerning causal relationships, and assessing policies or interventions.

However, despite their widespread use, linear models also have their shortcomings:
Linearity assumption: The assumption of linearity inherent in these models may not always be valid in real-world scenarios.
Multicollinearity: The presence of highly correlated independent variables can give rise to unstable coefficient estimates, thus undermining the model's predictive capacity.