Least Squares Criterion: What it is, How it Works

What Is the Least Squares Criterion?

The least squares criterion is a formula used to measure the accuracy of a straight line in depicting the data that was used to generate it. That is, the formula determines the line of best fit.

This mathematical formula is used to predict the behavior of the dependent variables. The approach is also called the least squares regression line.

Understanding the Least Squares Criterion

The least squares criterion is determined by minimizing the sum of squares created by a mathematical function. A square is determined by squaring the distance between a data point and the regression line or mean value of the data set.

A least squares analysis begins with a set of data points plotted on a graph. Independent variables are plotted on the horizontal x-axis while dependent variables are plotted on the vertical y-axis. The analyst uses the least squares formula to determine the most accurate straight line that will explain the relationship between an independent variable and a dependent variable.

Common Uses of Least Squares

Advances in computing power in addition to new financial engineering techniques have increased the use of least square methods and extended its basic principles.

Least squares and related statistical methods have become commonplace throughout finance, economics, and investing, even if its beneficiaries aren't always aware of their use.

For example, the robo-advisors now used by many investing platforms employ Monte Carlo simulation techniques to manage portfolios, though this is accomplished behind the scenes and out of the sight of the account holders who use them.

Other applications include time-series analysis of return distributions, economic forecasting and policy strategy, and advanced option modeling.

What Does Least Squares Tell You?

Instead of trying to solve an equation exactly, mathematicians use the least squares method to arrive at a close approximation. This is referred to as a maximum-likelihood estimate.

The least squares approach limits the distance between a function and the data points that the function explains. It is used in regression analysis, often in nonlinear regression modeling in which a curve is fit into a set of data.

The least squares approach is a popular method for determining regression equations, and it tells you about the relationship between response variables and predictor variables.

Modeling methods that are often used when fitting a function to a curve include the straight-line method, the polynomial method, the logarithmic method, and the Gaussian method.

Linear or ordinary least squares is the simplest and most commonly used linear regression estimator for analyzing observational and experimental data. It finds a straight line of best fit through a set of given data points.