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What is the purpose of the homoscedasticity assumption in regression analysis?

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If you have ever used linear regression to model the relationship between a dependent variable and one or more independent variables, you may have encountered the term homoscedasticity. But what does it mean and why is it important for regression analysis? In this article, you will learn about the concept of homoscedasticity, its role in the validity and efficiency of regression estimates, and how to check and correct for it.

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1 What is homoscedasticity?

Homoscedasticity is a statistical assumption that means that the variance of the error term in a regression model is constant across all levels of the independent variables. In other words, the variability of the dependent variable around the regression line is the same for all values of the predictors. This implies that the errors are independent and identically distributed, which is one of the conditions for applying the ordinary least squares (OLS) method to estimate the regression coefficients.

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Eren Unlu

Co-Founder
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In context of regular regression, particularly OLS, comes from the Gauss-Markov theorem, to achieve the best unbiased regressor. For general case, for gaussian assumptions where everything is modeled with two parameters : mean and variance, the linear modeling on such input data cannot generate different variances for different parts.

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Tom Bzik

ASTM E11 Quality and Statistics Leadership Roles
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Regression modeling is built upon assumed models about the nature of the modeled data. Homoscedasticity or homogeneity of variance or the variability of data about the regression model having constant variance that does not depend on its location is one such model. In the pre-computer and early computer eras it greatly simplified the computations involved in model estimation and model summary statistics. As with all assumptions, if true, it works. If false (heterogeneity of variance) it fails to a degree which depends on the degree of heterogeneity of variance. Heterogeneity of variance requires either specific data substructure to estimate the variance at each point or a model that describes it. More effort, but more reward when needed.

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Natalie Rodrigue

Skilled Biostatistician & Epidemiologist - Mentor, Experienced, Knowledgeable & Passionate Instructor - DoE - Expert in the Analysis of Genomic Data & Statistical Modelling. + 30 yrs experience.
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In short, homoscedasticity means that every observation is subject to the same amount of experimental error. There are situations in which this expected and others where it clearly is not expected.

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2 Why is homoscedasticity important?

Homoscedasticity is important because it affects the validity and efficiency of the regression estimates. If the homoscedasticity assumption is met, then the OLS method produces unbiased, consistent, and efficient estimates of the regression coefficients and their standard errors. This means that the estimates are close to the true population values, they converge to the true values as the sample size increases, and they have the smallest possible variance among all unbiased estimators. On the other hand, if the homoscedasticity assumption is violated, then the OLS method still produces unbiased and consistent estimates, but they are no longer efficient and their standard errors are biased. This means that the estimates have a larger variance than necessary, and the confidence intervals and hypothesis tests based on them are unreliable.

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3 How to check for homoscedasticity?

There are several ways to check for homoscedasticity in a regression model, both graphically and numerically. One of the simplest graphical methods is to plot the residuals (the difference between the observed and predicted values of the dependent variable) against the fitted values (the predicted values of the dependent variable) or against each independent variable. If the plot shows a random scatter of points around zero, then the homoscedasticity assumption is likely to be met. If the plot shows a pattern, such as a funnel shape, a curve, or a cluster, then the homoscedasticity assumption is likely to be violated. One of the most common numerical methods is to perform a Breusch-Pagan test, which tests whether the variance of the residuals is related to the fitted values or any of the independent variables. If the test rejects the null hypothesis of homoscedasticity, then there is evidence of heteroscedasticity (the opposite of homoscedasticity).

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4 How to correct for heteroscedasticity?

If there is evidence of heteroscedasticity in a regression model, there are several ways to correct for it and obtain valid and efficient estimates. One of the most popular methods is to use robust standard errors, which adjust the standard errors of the regression coefficients to account for the heteroscedasticity. Robust standard errors do not change the estimates of the coefficients themselves, but they make the confidence intervals and hypothesis tests more reliable. Another method is to use weighted least squares (WLS), which assigns different weights to the observations based on their variance, so that the observations with higher variance have less influence on the estimates. WLS does change the estimates of the coefficients, but it also improves their efficiency.

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Layton Franko

adjunct professor at Queens college
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She I deal with consumer products This problem becomes enormous. S as y hare forecasting sales for a consumer product; u ha e population. Disposable income, and a few other variables all of which are related thus a heteroscedastivity problem. So what do u do( if u don’t do something u are looking at a bunch of meaningless coefficients. )So what do u do? I would difference all the data and then do the regression

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5 What are the limitations of homoscedasticity?

Homoscedasticity is a useful assumption for regression analysis, but it is not always realistic or necessary. In some cases, the variance of the error term may depend on the level of the independent variables, especially when the dependent variable is bounded, skewed, or measured on different scales. In such cases, homoscedasticity may not be appropriate or meaningful, and other methods, such as transforming the dependent variable or using generalized linear models, may be more suitable. Moreover, homoscedasticity is not a sufficient condition for the validity and efficiency of the regression estimates, as there are other assumptions that need to be met, such as linearity, normality, and independence. Therefore, homoscedasticity should be checked and corrected for, but not taken for granted or overstated.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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What is the purpose of the homoscedasticity assumption in regression analysis?

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3

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6

1 What is homoscedasticity?

2 Why is homoscedasticity important?

3 How to check for homoscedasticity?

4 How to correct for heteroscedasticity?

5 What are the limitations of homoscedasticity?

6 Here’s what else to consider

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What is the purpose of the homoscedasticity assumption in regression analysis?

1

2

3

4

5

6

1 What is homoscedasticity?

2 Why is homoscedasticity important?

3 How to check for homoscedasticity?

4 How to correct for heteroscedasticity?

5 What are the limitations of homoscedasticity?

6 Here’s what else to consider

Statistics

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