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Method of Least Squares: Minimizing Deviations

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The Formal Theorem

Let Y=Xβ+ϵ Y = X\beta + \epsilon be a linear model where Y Y is an n×1 n \times 1 vector of observations, X X is an n×p n \times p matrix of known constants with rank p p , β \beta is a p×1 p \times 1 vector of unknown parameters, and ϵ \epsilon is a vector of random errors. The Ordinary Least Squares (OLS) estimator β^ \hat{\beta} is the vector that minimizes the residual sum of squares S(β)=(Y−Xβ)T(Y−Xβ) S(\beta) = (Y - X\beta)^T(Y - X\beta) . The unique solution is given by the normal equations:
β^=(XTX)−1XTY \hat{\beta} = (X^T X)^{-1} X^T Y

Analytical Intuition.

Imagine the response vector Y Y as a beam of light in an n n -dimensional space, seeking its reflection in the reality we can measure. The columns of our design matrix X X span a lower-dimensional subspace—a hyper-plane of possibility. Most often, the observed truth Y Y does not lie on this plane; it is suspended in the void, displaced by the chaotic turbulence of stochastic noise ϵ \epsilon . The Method of Least Squares acts as a mathematical gravity, pulling Y Y down to its closest relative on the plane. This point of impact is the orthogonal projection, Y^ \hat{Y} . By minimizing the squared Euclidean distance (the sum of squared deviations), we ensure that the error vector—the distance between our model and reality—is perfectly perpendicular to our predictors. This geometric purity guarantees that we have extracted every ounce of linear signal, leaving behind only uncorrelated noise. It is the cinematic process of collapsing a complex, high-dimensional observation into its most efficient linear shadow.
CAUTION

Institutional Warning.

Students often confuse the unobservable population error ϵ \epsilon with the observable sample residual e e . While ϵ \epsilon represents the true deviation from the population mean, e e is merely the distance from the estimated projection, constrained by the geometry of the specific sample data.

Academic Inquiries.

01

Why do we minimize squared deviations instead of absolute deviations?

Squaring results in a continuously differentiable objective function with a closed-form analytical solution. Under the Gauss-Markov assumptions, it also yields the Best Linear Unbiased Estimator (BLUE) with minimum variance.

02

What happens if the matrix XTX X^T X is not invertible?

This occurs during perfect multicollinearity (rank deficiency). In such cases, the OLS estimator is not unique, and we must utilize a generalized inverse (Moore-Penrose) or regularization techniques like Ridge regression.

03

Is the OLS estimator biased if the errors are not normally distributed?

No. The OLS estimator β^ \hat{\beta} remains unbiased as long as the expected value of the errors is zero, regardless of the distribution's shape; normality is only required for exact inference (t-tests and F-tests).

Standardized References.

  • Definitive Institutional SourceCasella, G., & Berger, R. L., Statistical Inference.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Method of Least Squares: Minimizing Deviations: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/statistical-inference-i/method-of-least-squares--minimizing-deviations

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