Does Linear Regression Assume Normality?

Introduction to Linear Regression and OLS

When discussing linear regression, it's crucial to distinguish between the algorithm itself (Ordinary Least Squares, or OLS) and the statistical assumptions underlying a linear regression model. OLS is an algorithm that aims to find the best-fitting line through a cloud of points in an n-dimensional space, where the line minimizes the sum of squared Euclidean distances between the points and the line. This is achieved without any assumptions about the distribution of the data.

Statistical Assumptions of Linear Regression

Let's dive into the statistical assumptions of linear regression. If we assume that some dimensions of the data represent independent, identically distributed (i.i.d.) random variables, and the last dimension represents a random variable determined by the function y b varepsilon, where varepsilon is a normally distributed random variable, we're delving into a more nuanced version of the model.

1. Unbiasedness: Under these assumptions, the predictions made by the regression (denoted as hat{y}) are an unbiased estimator of the true value y. This means that the expected value of the predicted values equals the true value, or E[hat{y}] y.

2. Gauss-Markov Theorem: If we further assume that the errors are homoscedastic and uncorrelated, we arrive at the Gauss-Markov theorem. This theorem states that among all unbiased estimators, the least squares estimators have the smallest variance. This means that the linear regression model provides the most precise estimates under these conditions.

3. Normality of Errors: When the error term is normally distributed, the linear regression model can be interpreted as a maximum likelihood estimator. This leads to important statistical properties, such as the fact that the regression coefficients follow a Student's t-distribution. This allows us to estimate standard errors, construct confidence intervals, and perform hypothesis tests on the coefficients.

Key Assumptions of Linear Regression

Linear regression models make several assumptions about the data:

Data Treatment: The predictor variables (X) are treated as fixed values rather than random variables. This means that predictor variables are assumed free of measurement errors. Linearity: The response variable is a linear combination of parameters and predictor variables. Homoscedasticity: The errors have constant variance across all values of the response variables. No Autocorrelation: The errors are uncorrelated with each other. Unique Solution: There exists a unique solution for the parameters/coefficients of the model.

While Ysim N(mu, sigma^2) at any given values x_{i1}, x_{i2}, …, x_{ip} is a common assumption, it is not strictly required for OLS. However, normality of errors is often desirable for making inferences and constructing confidence intervals.

Conclusion

In summary, while the linear regression algorithm itself does not assume normality, the statistical properties and inferences derived from linear regression do benefit from certain assumptions about the data, including normality of errors. Understanding these assumptions is crucial for the appropriate application and interpretation of linear regression models.