Hetroscedasticity

 

The variance of the disturbance term is assumed to be fixed (constant) for all explanatory variable values in the classical linear regression model.

That is

Technically this assumption is known as HOMOSCEDASTICITY.

Diagrammatically the phenomenon of Homosedasticity is given below:

In a scatter plot, the residual values cluster around the regression line without taking any particular shape due to homoscedasticity.

if the assumption of constant variance for explanatory variable values is violated and the variance of the disturbance term is not constant for all explanatory variable values, then the variance of the error term is hetroscedastic and the phenomenon is known as hetroscedasticity.

That is

The subscript “i” denotes the relationship between the variance of the disturbance term is and predictor. Then the variance of the disturbance term is called HETROSCEDASTIC.

Diagrammatically the phenomenon of Hetrosecdasticity is given below:

In homoscedasticity, we assume that response variable vary as predictor vary but the variance of   (which is equivalent to the variance of the disturbance term) is constant for all predictors. In heteroscedasticity, response variable vary as predictors vary but the variance of  the disturbance term is also vary.

This leads to the conclusion that Xi is a function of the variance of the disturbance term.

In heteroscedasticity, the residual values display a cone like structure.

Example: Consider regressing luxury spending (Y) on family income (X). 

Case I: For families with modest incomes

The residual will be quite small because almost all low-income families don’t spend much on luxuries. In this instance, the disturbance term's variance is homoscedastic.

Case II: For families with high incomes

The size of the residuals varies greatly among high income families. because some families spend significantly more on luxury goods than others, who tend to spend less. Because the extent of the error varies depending on the values of the independent variable, this situation displays heteroscedasticity.  

Different kinds of Heteroscedastic Structures

Three different heteroscedastic structures are present. As follows:

1.      If the disturbance term's variance rises as Xi increases.

      It can be expressed as:

     Graphic representations include the following:

2.      If the disturbance term's variance increases as Xi falls.

It can be written as;

     Graphic representations include the following:

3.      It can be stated as follows if the variance of the disturbance term oscillates with .

Then it can be written as:

Graphically it can be expressed as:

Heteroscedasticity's causes

The variance of the disturbance term may vary for a number of reasons. Below are a few of them:

1.      The Nature of the Phenomenon under Investigation

The nature of the phenomenon under study might have an increasing or decreasing tendency.

e.g., as income rises, the variety in food consumption patterns grows. In this case variance of the disturbance term is expected to increase.

As people learn their error of behaviour become smaller over time. In this case, variance of the disturbance tern is expected to decrease.

e.g., The number of typing mistakes decreases as the number of hours of typing practice increases.

2.      Streamline Data Collection & Processing Methods

The variance of error term is expected to decrease as the data collection and processing methods advance. Therefore, the institutions that use sophisticated techniques for data collection and processing have lower error rates.

1.      The Model & Incorrect data Transformation

Heteroscedasticity is also caused by faulty data transformation and functional form in the model. For instance, it is simpler to gather information about family-wide clothing expenses than it is about a single family member. Consider a straightforward linear regression model.

Data collection for each family member might be challenging, but data collection for the entire family is simpler. Consequently, Yij is is known as a whole.

The information on average spending for each family member is then found as follows, rather than per person expenditure:

This shows that the variance of the resulting disturbance term is depend on the family size “mij” and not constant. Heteroscedasticity therefore manifests in the data. The variance of the disturbance term will be constant, if all families have identical size. 

4.      Inaccurate model stipulation

Heteroscedasticity results from the model leaving out a significant variable for a variety of reasons.

5.      Existence of Outliers

The existence of outliers can also lead to heterosexuality.  Such observations might significantly change the outcomes of regression analysis, especially when the sample size is small.

• Read More: Detection of Hetroscedasticity
• Read More: Consequences of Hetroscedasticity