### Heteroscedasticity Definition

An important assumption of OLS is that the disturbances $u_i$ appearing in the population regression function are homoscedastic (Error terms have the same variance).

i.e. The variance of each disturbance term $u_i$, conditional on the chosen values of explanatory variables is some constant number equal to $\sigma^2$. $E(u_{i}^{2})=\sigma^2$; 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 the 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:**

- The range in family income between the poorest and richest families in town is the classical example of heteroscedasticity.
- The range in annual sales between a corner drug store and a general store.

### Reasons for Heteroscedasticity

There are several reasons why the variances of error term $u_i$ may be variable, some of which are:

- Following the error learning models, as people learn their errors of behavior become smaller over time. In this case $\sigma_{i}^{2}$ is expected to decrease. For example the number of typing errors made in a given period on a test to the hours put in typing practice.
- As income grows, people have more discretionary income, and hence $\sigma_{i}^{2}$ is likely to increase with income.
- As data-collecting techniques improve, $\sigma_{i}^{2}$ is likely to decrease.
- 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.
- Heteroscedasticity arises from violating the assumption of CLRM (classical linear regression model), that the regression model is not correctly specified.
- Skewness in the distribution of one or more regressors included in the model is another source of heteroscedasticity.
- Incorrect data transformation and incorrect functional form (linear or log-linear model) are also the sources of heteroscedasticity

### Consequences of Heteroscedasticity

- The OLS estimators and regression predictions based on them remain unbiased and consistent.
- 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.
- 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 are 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.

Learn R Programming Language

zunairaaslam o alikum

sir Detecting Autocorrelation ka topic chaye.

Muhammad Imdad UllahWalaikum us Salam

Hope, soon topics related to autocorrelation will be updated.

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

Muhammad Imdad UllahThanks. Missing reference is now added.

NeilI 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.

sarmad chandiosir how can we get SPSS to practice on..

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

Dipak raj joshiwho we solve the problem of numerical example in google

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

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

Isaiah AMEHwhat is the nature of heteroscedasticity?

Khalil Abdulkadir UsmanThank you beyond measure.

subhash davarexcellent write up.

Adama keitahi 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 Imdadullahlot 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.

watum bright willywhat are the causes of heteroscedasticity? any the notes are good

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

HAMIDHOW TO DETECT HETEROSCEDISTICITY?

ANNHeteroscedasticity 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.

hailuheteroscedasticity