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Residuals and Least-Squares Property01:11

Residuals and Least-Squares Property

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The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y. If the observed data point lies below the line, the residual is negative, and the line overestimates the actual data value for y.
The process of fitting the best-fit...
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Variance01:15

Variance

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The deviations show how spread out the data are about the mean. A positive deviation occurs when the data value exceeds the mean, whereas a negative deviation occurs when the data value is less than the mean. If the deviations are added, the sum is always zero. So one cannot simply add the deviations to get the data spread. By squaring the deviations, the numbers are made positive; thus, their sum will also be positive.
The standard deviation measures the spread in the same units as the data....
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Variation01:19

Variation

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An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation, which is the square root of variance.
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Standard Deviation01:10

Standard Deviation

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The most commonly used measure of variation is the standard deviation. It is a numerical value measuring how far data values are from their mean. The standard deviation value is small when the data are concentrated close to the mean, exhibiting slight variation or spread. The standard deviation value is never negative, it is either positive or zero. The standard deviation is larger when the data values are more spread out from the mean, which means the data values are exhibiting more variation.
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Correlation and Regression00:53

Correlation and Regression

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In statistics, correlation describes the degree of association between two variables. In the subfield of linear regression, correlation is mathematically expressed by the correlation coefficient, which describes the strength and direction of the relationship between two variables. The coefficient is symbolically represented by 'r' and ranges from -1 to +1. A positive value indicates a positive correlation where the two variables move in the same direction. A negative value suggests a...
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Residual Plots01:07

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A residual plot is a statistical representation of data used to analyze correlation and regression results. It helps verify the requirements for drawing specific conclusions about correlation and regression. To obtain the residual plot, first, the residual for each data value is calculated, which is simply the vertical distance between the observed and the predicted value obtained from the regression equation.
When the residual values are plotted against the variable x, it is called a residual...
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Impact of consumer confidence on the expected returns of the Tokyo Stock Exchange: A comparative analysis of consumption and production-based asset pricing models.

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Related Experiment Video

Updated: Nov 27, 2025

Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
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Relative Entropy and Minimum-Variance Pricing Kernel in Asset Pricing Model Evaluation.

Javier Rojo-Suárez1, Ana Belén Alonso-Conde1

  • 1Department of Business Administration, Rey Juan Carlos University, 28032 Madrid, Spain.

Entropy (Basel, Switzerland)
|December 8, 2020
PubMed
Summary

This study introduces relative entropy as a robust method for testing asset pricing models, addressing issues with traditional procedures. The entropy-based approach offers a more accurate decomposition of model performance compared to generalized least squares R-squared statistics.

Keywords:
CAPMFama-French modelHansen-Jagannathan boundKullback-Leibler divergenceconsumption-CAPMfactor-mimicking portfoliopricing kernelrelative entropy

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Area of Science:

  • Quantitative Finance
  • Econometrics
  • Financial Modeling

Background:

  • Traditional asset pricing model tests often yield spurious rejections due to model misspecification or data characteristics.
  • Existing methods struggle to accurately assess model performance and identify sources of error.

Purpose of the Study:

  • To propose relative entropy as an alternative framework for testing asset pricing models.
  • To investigate the relationship between pricing kernel entropy and mean-variance efficiency of factor-mimicking portfolios.
  • To develop an entropy-based decomposition for assessing model explanatory power.

Main Methods:

  • Utilizing the relative entropy of pricing kernels, specifically Kullback-Leibler divergence.
  • Examining the connection between the generalized least squares (GLS) R-squared statistic and pricing kernel entropy.
  • Developing an entropy-based decomposition to analyze model performance.

Main Results:

  • Relative entropy provides a robust alternative to traditional testing procedures for asset pricing models.
  • The entropy-based decomposition explicitly separates model performance into pricing kernel entropy and its correlation with asset returns.
  • While correlated with GLS R-squared, relative entropy offers a more nuanced understanding of model fit.

Conclusions:

  • Relative entropy is a versatile tool for constructing reliable tests in asset pricing.
  • The proposed decomposition enhances the interpretability of asset pricing model performance.
  • This framework addresses limitations in existing methods for evaluating financial models.