# Homoscedasticity

**homoscedastichomogeneity of variancehomoskedastichomoskedasticityHomogeneityHomoscedasticity and heteroscedasticityinhomogeneity**

In statistics, a sequence (or a vector) of random variables is homoscedastic if all its random variables have the same finite variance.wikipedia

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### Heteroscedasticity

**heteroscedasticheteroskedasticityheteroskedastic**

The complementary notion is called heteroscedasticity.

Thus heteroscedasticity is the absence of homoscedasticity.

### Goldfeld–Quandt test

Testing for groupwise heteroscedasticity requires the Goldfeld–Quandt test.

In statistics, the Goldfeld–Quandt test checks for homoscedasticity in regression analyses.

### Gauss–Markov theorem

**best linear unbiased estimatorGauss–Markovbest linear unbiased estimation**

As used in describing simple linear regression analysis, one assumption of the fitted model (to ensure that the least-squares estimators are each a best linear unbiased estimator of the respective population parameters, by the Gauss–Markov theorem) is that the standard deviations of the error terms are constant and do not depend on the x-value.

The errors do not need to be normal, nor do they need to be independent and identically distributed (only uncorrelated with mean zero and homoscedastic with finite variance).

### Simple linear regression

**simple regressioni.e. regression linelinear least squares regression with an intercept term and a single explanator**

As used in describing simple linear regression analysis, one assumption of the fitted model (to ensure that the least-squares estimators are each a best linear unbiased estimator of the respective population parameters, by the Gauss–Markov theorem) is that the standard deviations of the error terms are constant and do not depend on the x-value.

It is also possible to evaluate the properties under other assumptions, such as inhomogeneity, but this is discussed elsewhere.

### Variance

**sample variancepopulation variancevariability**

In statistics, a sequence (or a vector) of random variables is homoscedastic if all its random variables have the same finite variance.

### Linear discriminant analysis

**discriminant analysisDiscriminant function analysisFisher's linear discriminant**

One popular example is Fisher's linear discriminant analysis.

LDA instead makes the additional simplifying homoscedasticity assumption (i.e. that the class covariances are identical, so ) and that the covariances have full rank.

### Homogeneity (statistics)

**homogeneityhomogeneousheterogeneity**

For example, considerations of homoscedasticity examine how much the variability of data-values changes throughout a dataset.

### Bartlett's test

**Bartlett**

Equal variances across populations is called homoscedasticity or homogeneity of variances.

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, a sequence (or a vector) of random variables is homoscedastic if all its random variables have the same finite variance.

### Sequence

**sequencessequentialinfinite sequence**

In statistics, a sequence (or a vector) of random variables is homoscedastic if all its random variables have the same finite variance.

### Random variable

**random variablesrandom variationrandom**

### Goodness of fit

**goodness-of-fitfitgoodness-of-fit test**

Serious violations in homoscedasticity (assuming a distribution of data is homoscedastic when in reality it is heteroscedastic ) may result in overestimating the goodness of fit as measured by the Pearson coefficient.

### Pearson correlation coefficient

**correlation coefficientPearson product-moment correlation coefficientPearson correlation**

Serious violations in homoscedasticity (assuming a distribution of data is homoscedastic when in reality it is heteroscedastic ) may result in overestimating the goodness of fit as measured by the Pearson coefficient.

### Breusch–Pagan test

**Cook–Weisberg testBreusch–Pagan statisticCook-Weisberg test**

Residuals can be tested for homoscedasticity using the Breusch–Pagan test, which performs an auxiliary regression of the squared residuals on the independent variables.

### Normal distribution

**normally distributedGaussian distributionnormal**

Two or more normal distributions, are homoscedastic if they share a common covariance (or correlation) matrix,.

### Covariance matrix

**variance-covariance matrixcovariance matricescovariance**

Two or more normal distributions, are homoscedastic if they share a common covariance (or correlation) matrix,.

### Correlation and dependence

**correlationcorrelatedcorrelations**

Two or more normal distributions, are homoscedastic if they share a common covariance (or correlation) matrix,.

### Pattern recognition

**pattern analysispattern detectionpatterns**

Homoscedastic distributions are especially useful to derive statistical pattern recognition and machine learning algorithms.

### Machine learning

**machine-learninglearningstatistical learning**

Homoscedastic distributions are especially useful to derive statistical pattern recognition and machine learning algorithms.

### Homogeneity and heterogeneity

**heterogeneoushomogeneousheterogeneity**

### Ordinary least squares

**OLSleast squaresOrdinary least squares regression**

The OLS estimator is consistent when the regressors are exogenous, and optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated.

### Financial models with long-tailed distributions and volatility clustering

**Stable and tempered stable distributions with volatility clustering – financial applications**

These classical models of financial time series typically assume homoskedasticity and normality cannot explain stylized phenomena such as skewness, heavy tails, and volatility clustering of the empirical asset returns in finance.

### Levene's test

**Levene**

It tests the null hypothesis that the population variances are equal (called homogeneity of variance or homoscedasticity).

### Robust regression

**robust estimationRobustrobust linear model**

In the homoscedastic model, it is assumed that the variance of the error term is constant for all values of x.