Homoskedastic

What is Homoskedastic?

Homoskedastic is the situation in a regression model in which the residual term for each observation is constant for all observations. It essentially means that as the value of the dependent variable changes, the error term does not vary much for each observation.

However, when the residual term’s size differs across an independent variable’s values, it means that homoskedasticity has been violated. The condition is referred to as heteroskedastic, implying that each observation variance is different and may lead to inaccurate inferential statements.

A regression model lacking homoskedasticity may need to add a predictor variable to explain the observations’ dispersion. Homoskedasticity can also be expressed differently in general linear models that all diagonals of a variance-covariance matrix ϵ must bear the same number.

How Homoskedastic Works

Homoskedasticity is one of the critical assumptions under which the Ordinary Least Squares (OLS) gives an unbiased estimator, and the Gauss–Markov Theorem applies. Linear regression modeling typically tries to explain the occurrences with a single equation.

For example, OLS assumes that variance is constant and that the regression does not necessarily pass through the origin. In such a case, the OLS seeks to minimize residuals and eventually produces the smallest possible residual terms. By definition, OLS gives equal weight to all observations, except in the case of heteroskedasticity.

Similarly, the Gauss–Markov Theorem gives the best linear unbiased estimator of a standard linear regression model using independent and homoskedastic residual terms. If attention is restricted to the linear estimators of the independent variable’ values, the theorem holds true. Thus, homoskedasticity is required for the efficiency of OLS and the Gauss–Markov Theorem and their standard errors to be consistent and unbiased to make accurate statistical inferences.

Reliability of the Homoskedastic Assumption

Basically, the homoskedastic assumption is required in linear regression models to ensure asymptotic covariance and standard error accuracy. With homoskedasticity, although the residual terms remain constant (unbiased and consistent), the resulting covariance matrix among the estimated parameters is bound to be incorrect. It can result in inflated Type I error rates or low statistical power.

The residuals are needed in order to detect the violation of homoskedasticity. When the residual terms’ distributions are approximately constant across all observations, the homoskedastic assumption is said to be tenable. Conversely, when the spread of the error terms is no longer approximately constant, heteroskedasticity is said to occur.

Special Considerations

Formally, a simple regression model for N observations and ρ predictors consists of four terms. It can be compactly expressed in a matrix form as:

Y = Xβ + ϵ

Where:

• Y is the independent variable, which represents the phenomenon under study
• X is the constant
• Β is the predictor variable, and
• ϵ is the residual term or error term, which shows the level of dispersion that the predictor variable does not explain

Testing for a Homoskedastic Assumption

There are various methods of testing fitted simple linear regression models for homoskedasticity. One method is the traditional graphic residual analysis. However, because of the complexity associated with such an approach, other relatively simple and methodological approaches are available.

They include the Neter-Wasserman / Goldfeld-Quandt Test (NWGQ), the Neter-Wasserman / Ramsey / Spearman Rho T-Test (NWRS), the White-Test (W), The Breusch-Pagan / Cook-Weisberg Scores Test (BPCW), and the Glejser / Mendenhall-Sincich test (GMS). It is important to supplement the graphical method with an appropriate confirmatory approach to enhance model development.

Example of Homoskedasticity

Suppose a researcher wants to explain the market performance of several companies using the number of marketing approaches adopted by each. In such a case, the dependent variable would be market performances, and the predictor variable would be the number of marketing methods. The error term would give the value of variance regarding market performance.

If the variance is homoskedastic, it would mean the model may be a suitable explanation for market performance, explaining it with regard to the number of marketing methods. However, the homoskedastic assumption may be violated by the variance.

A graphical representation of the residual term may show a large number of marketing strategies corresponding with high market performance. One predictor variable would not be able to explain the variance of the scores, which is explained by a single predictor variable – the number of marketing approaches.

Improving Regression Models to Exhibit Homoskedasticity

There are some underlying factors in homoskedasticity, and the regression model may be modified to make it possible to identify the factors. Further investigation may reveal that some established companies have the upper hand, since they have previously tested the marketing strategies, and they already know which strategies work and those that have the least impact. It places start-up companies on the receiving end due to their lack of past exposure with the marketing strategies.

The additional explanatory variable would be added to improve the regression model, leading to two explanatory variables – the number of market strategies and whether a company had previous experience with a certain method. With the two variables, the market performance variance would be explained with homoskedasticity defining the residual term variance.

Additional Resources

CFI is the official provider of the global Commercial Banking & Credit Analyst (CBCA)™ certification program, designed to help anyone become a world-class financial analyst. To keep advancing your career, the additional resources below will be useful:

• Durbin Watson Statistic
• Heteroskedasticity
• Random Variable
• Variance Formula
• See all data science resources