Basics on Ekonometria.org — Econometrics Portal

Functions and their properties

Mon, 29 Jun 2026 00:00:00 +0000

The function is one of the most important concepts in all of mathematics and, at the same time, the foundation of econometrics, in which a model is a function linking explanatory variables to the explained variable. This chapter begins with the genesis of the concept, then builds it from the ground up — from a general relation to a function in the strict sense — and leads the reader through a complete catalogue of elementary functions, their properties, transformations, and the generalisation to functions of several variables and the space $\mathbb{R}^n$. We apply one rule consistently: every concept and every example is given its own figure.

Limits and continuity

Mon, 29 Jun 2026 00:00:00 +0000

The limit is the foundation of all of differential and integral calculus. It answers a single question: what value does a function approach as its argument approaches a given point? Without this concept one can define neither the derivative — the instantaneous rate of change — nor the integral — a sum of infinitely many infinitely small terms. This chapter builds the notion of the limit from its historical sources, through rigorous definitions, up to limits and continuity of functions of several variables. Every concept and every example is given its own figure, and the key theorems a proof.

Derivatives — the rate of change

Mon, 29 Jun 2026 00:00:00 +0000

The derivative measures the rate of change — the instantaneous speed with which the value of a function responds to a change in its argument. In economics it is marginal cost, marginal revenue, and elasticity; in econometrics it is the gradient of the objective function, whose vanishing determines the estimators. This chapter builds the differential calculus from the historical tangent problem, through the difference quotient and the rules of differentiation, up to Taylor’s formula and partial derivatives and the gradient in the space $\mathbb{R}^n$. Every concept and every example is given its own figure, and the key theorems a proof.

Integrals — summing infinitely many terms

Mon, 29 Jun 2026 00:00:00 +0000

If the derivative measures the rate of change, then the integral measures accumulation — a sum of infinitely many infinitely small increments. Geometrically it is the area under a graph; in economics it is the total cost summed from the marginal cost, consumer surplus, or the present value of a stream of payments. This chapter builds the notion of the integral from the historical problem of area, through the Riemann sum and the fundamental theorem of calculus, up to improper integrals and multiple integrals in the space $\mathbb{R}^n$. Every concept and every example is given its own figure.

Linear algebra — vectors and matrices

Mon, 29 Jun 2026 00:00:00 +0000

Linear algebra is the language in which all of econometrics is written. The regression model — regardless of the number of variables — reduces to a single matrix formula $\hat{\boldsymbol\beta}=(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top\mathbf{y}$, and understanding it requires the calculus of vectors and matrices. This chapter builds that calculus from the ground up: from the genesis of the concept, through vectors, matrices, and their geometric meaning, up to eigenvalues and orthogonal projection in the space $\mathbb{R}^n$. Every concept and every example is given its own figure.

Descriptive statistics

Mon, 29 Jun 2026 00:00:00 +0000

Descriptive statistics is the art of summarising data: replacing thousands of numbers with a few quantities that capture their location, spread, and shape. It is the starting point of every econometric analysis — before building a model, we must know the data. This chapter builds the apparatus of descriptive statistics from its genesis, through measures of location and variability, up to the description of multivariate data in the space $\mathbb{R}^n$. Every concept and every example is given its own figure.

Variance and standard deviation

Mon, 29 Jun 2026 00:00:00 +0000

The mean tells us where the “centre” of the data lies, but says nothing about how widely the values are spread around it. This gap is filled by variance and standard deviation — the most important measures of variability in all of statistics and econometrics. This chapter derives them from the ground up: from the question of why the mean is not enough, through the construction of the formula step by step, up to covariance and the generalisation to the space $\mathbb{R}^n$. Every concept and every example is given its own figure.

Correlation and the Pearson coefficient

Mon, 29 Jun 2026 00:00:00 +0000

Correlation answers the question of whether an increase in one quantity is accompanied by an increase (or decrease) in another — and how strong that relationship is. It is one of the most frequently used and most frequently misused concepts in statistics. This chapter builds the correlation coefficient from the ground up: from its genesis, through the construction of the formula and its geometric meaning in the space $\mathbb{R}^n$, to the interpretive pitfalls — Anscombe’s quartet and the confusion of correlation with causation. Every concept and every example is given its own figure.

Probability distributions

Mon, 29 Jun 2026 00:00:00 +0000

A probability distribution describes which values of a random variable are possible and how probable they are. Four distributions — the normal, Student’s $t$, chi-square, and $F$ — form a single system on which all of statistical inference rests: hypothesis tests and confidence intervals. This chapter builds them from the ground up: from the genesis of probability theory, through the density function and the cumulative distribution function, up to the mutual relationships of the distributions and their generalisation to the space $\mathbb{R}^n$. Every concept and every example is given its own figure.

The normal distribution

Mon, 29 Jun 2026 00:00:00 +0000

The normal distribution — also called Gaussian, or the bell curve — is the most important distribution in statistics and econometrics. Human heights, measurement errors, asset returns: a great many phenomena take the same symmetric shape, densest at the centre and thin at the extremes. This chapter builds it from the ground up: from its genesis and the origin of the bell shape, through the density formula and standardisation, up to the multivariate distribution in the space $\mathbb{R}^n$. Every concept and every example is given its own figure.

The central limit theorem

Mon, 29 Jun 2026 00:00:00 +0000

The central limit theorem (CLT) explains why the normal distribution appears almost everywhere — and why hypothesis tests and confidence intervals work at all. Its content is surprising: the sample mean has an approximately normal distribution regardless of the shape of the distribution we sample from. This chapter builds the theorem from the ground up: from its genesis and the law of large numbers, through the precise statement and its illustration, up to the multivariate generalisation and the significance for econometrics. Every concept and every example is given its own figure.

The Method of Least Squares: History, Theory and Proofs

Sat, 27 Jun 2026 00:00:00 +0000

The method of least squares was not born as an exercise in algebra. It grew out of the most demanding computational problems of the eighteenth and nineteenth centuries — determining the figure of the Earth and the orbits of celestial bodies from observations corrupted by error. This article traces that entire path: from the precursors, through the priority dispute between Legendre and Gauss, to Gauss’s two distinct justifications and the birth of modern regression — with dates, names, original-language texts and full proofs.