# Heteroscedasticity

An important assumption of OLS is that the disturbances μi appearing in the population regression function are homoscedastic (Error term have the same variance).
i.e. The variance of each disturbance term μi, conditional on the chosen values of explanatory variables is some constant number equal to $\sigma^2$. $E(\mu_{i}^{2})=\sigma^2$; where $i=1,2,\cdots, n$.
Homo means equal and scedasticity means spread.

Consider the general linear regression model
$y_i=\beta_1+\beta_2 x_{2i}+ \beta_3 x_{3i} +\cdots + \beta_k x_{ki} + \varepsilon$

If $E(\varepsilon_{i}^{2})=\sigma^2$ for all $i=1,2,\cdots, n$ then the assumption of constant variance of the error term or homoscedasticity is satisfied.

If $E(\varepsilon_{i}^{2})\ne\sigma^2$ then assumption of homoscedasticity is violated and heteroscedasticity is said to be present. In the case of heteroscedasticity, the OLS estimators are unbiased but inefficient.

Examples:

1. The range in family income between the poorest and richest family in town is the classical example of heteroscedasticity.
2. The range in annual sales between a corner drug store and general store.  ## Reasons for Heteroscedasticity

There are several reasons when the variances of error term μi may be variable, some of which are:

1. Following the error learning models, as people learn their error of behaviors becomes smaller over time. In this case $\sigma_{i}^{2}$ is expected to decrease. For example the number of typing errors made in a given time period on a test to the hours put in typing practice.
2. As income grows, people have more discretionary income and hence $\sigma_{i}^{2}$ is likely to increase with income.
3. As data collecting techniques improve, $\sigma_{i}^{2}$ is likely to decrease.
4. Heteroscedasticity can also arise as a result of the presence of outliers. The inclusion or exclusion of such observations, especially when the sample size is small, can substantially alter the results of regression analysis.
5. Heteroscedasticity arises from violating the assumption of CLRM (classical linear regression model), that the regression model is not correctly specified.
6. Skewness in the distribution of one or more regressors included in the model is another source of heteroscedasticity.
7. Incorrect data transformation, incorrect functional form (linear or log-linear model) is also the source of heteroscedasticity

# Consequences of Heteroscedasticity

1. The OLS estimators and regression predictions based on them remains unbiased and consistent.
2. The OLS estimators are no longer the BLUE (Best Linear Unbiased Estimators) because they are no longer efficient, so the regression predictions will be inefficient too.
3. Because of the inconsistency of the covariance matrix of the estimated regression coefficients, the tests of hypotheses, (t-test, F-test) are no longer valid.

Note: Problems of heteroscedasticity is likely to be more common in cross-sectional than in time series data.

Reference
Greene, W.H. (1993). Econometric Analysis, Prentice–Hall, ISBN 0-13-013297-7.
Verbeek, Marno (2004.) A Guide to Modern Econometrics, 2. ed., Chichester: John Wiley & Sons.
Gujarati, D. N. & Porter, D. C. (2008). Basic Econometrics, 5. ed., McGraw Hill/Irwin. Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

### 20 Responses

1. zunaira says:

aslam o alikum
sir Detecting Autocorrelation ka topic chaye.

• Muhammad Imdad Ullah says:

Walaikum us Salam
Hope, soon topics related to autocorrelation will be updated.

2. Devasish hazarika says:

These r found in Gujarati’s book. But there is no reference.

• Muhammad Imdad Ullah says:

Thanks. Missing reference is now added.

3. Neil says:

I found a good example of this recently. My son conducted an experiment in his school chemistry class into the rate of decomposition of hydrogen peroxide in the presence of a catalyst. The concentration of H2O2 against time follows a half-life rule. This means that the log of H202 concentration vs time should be linear. However, it wasn’t, it was curved so the residuals were greater at the extreme ends of the curve than at the middle. This suggests that there was some other variable affecting the rate of decomposition that wasn’t accounted for by the simple model. I’ve got some R code to illustrate it for anyone who’s interested.

4. sarmad chandio says:

sir how can we get SPSS to practice on..

• Muhammad Imdadullah says:

Try to perform analysis with options available in each dialog box of different analysis.

5. Dipak raj joshi says:

who we solve the problem of numerical example in google

• Muhammad Imdadullah says:

I am trying to write a post for numerical computation in the google search.

6. Alabi says:

pls kindly give me forms of heteroscedacity that are commomly in use by researchers

7. Isaiah AMEH says:

what is the nature of heteroscedasticity?

8. Khalil Abdulkadir Usman says:

Thank you beyond measure.

9. subhash davar says:

excellent write up.

10. Adama keita says:

hi please am final year undergraduate student working on my project topic
investigating the nature of distribution of heteroscedacity date.
please if you help me with data set on heteroscedasticity data.

• Muhammad Imdadullah says:

lot of Heteroscedasticity data is available in different text book. There are online data banks where from you can search it. Google it such as heteroscedasticd data, econometric data, etc.

11. watum bright willy says:

what are the causes of heteroscedasticity? any the notes are good

• Muhammad Imdadullah says:

Thanks for liking. Hope soon will update. If you have some, you can share, with your name as co-author.

• HAMID says:

HOW TO DETECT HETEROSCEDISTICITY?

• ANN says:

Heteroscedasticity is caused by different variability of data e.g. When one gain more experience the error become less, Also as income for richer increases you expect the gap between the poor and the richer to widen.

12. hailu says:

heteroscedasticity

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