Heteroscedasticity

Lecture notes about Heteroscedasticity, its consequences, detection, and remedy. Hetero or Heteroscedasticity is a violation of the OLS assumption of consistent variance.

Goldfeld Quandt Test: Comparison of Variances of Error Terms

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The Goldfeld Quandt test is one of two tests proposed in a 1965 paper by Stephen Goldfeld and Richard Quandt. Both parametric and nonparametric tests are described in the paper, but the term “Goldfeld–Quandt test” is usually associated only with the parametric test.
Goldfeld-Quandt test is frequently used as it is easy to apply when one of the regressors (or another r.v.) is considered the proportionality factor of heteroscedasticity. Goldfeld-Quandt test is applicable for large samples. The observations must be at least twice as many as the parameters to be estimated. The test assumes normality and serially independent error terms $u_i$.

The Goldfeld Quandt test compares the variance of error terms across discrete subgroups. So data is divided into h subgroups. Usually, the data set is divided into two parts or groups, and hence the test is sometimes called a two-group test.

Goldfeld Quandt Test: Comparison of Variances of Error Terms

Before starting how to perform the Goldfeld Quand Test, you may read more about the term Heteroscedasticity, the remedial measures of heteroscedasticity, Tests of Heteroscedasticity, and Generalized Least Square Methods.

Goldfeld Quandt Test Procedure:

The procedure for conducting the Goldfeld-Quandt Test is;

  1. Order the observations according to the magnitude of $X$ (the independent variable which is the proportionality factor).
  2. Select arbitrarily a certain number (c) of central observations which we omit from the analysis. (for $n=30$, 8 central observations are omitted i.e. 1/3 of the observations are removed). The remaining $n-c$ observations are divided into two sub-groups of equal size i.e. $\frac{(n-2)}{2}$, one sub-group includes small values of $X$ and the other sub-group includes the large values of $X$, and a data set is arranged according to the magnitude of $X$.
  3. Now Fit the separate regression to each of the sub-groups, and obtain the sum of squared residuals from each of them.
    So $\sum c_1^2$ shows the sum of squares of Residuals from a sub-sample of low values of $X$ with $(n – c)/2 – K$ df, where K is the total number of parameters.$\sum c_2^2$ shows the sum of squares of Residuals from a sub-sample of large values of $X$ with $(n – c)/2 – K$ df, where K is the total number of parameters.
  4. Compute the Relation $F^* = \frac{RSS_2/df}{RSS_2/df}=\frac{\sum c_2^2/ ((n-c)/2-k)}{\sum c_1^2/((n-c)/2-k) }$

If variances differ, F* will have a large value. The higher the observed value of the F*-ratio the stronger the heteroscedasticity of the $u_i$.

Goldfeld Quandt Test of

References

  • Goldfeld, Stephen M.; Quandt, R. E. (June 1965). “Some Tests for Homoscedasticity”. Journal of the American Statistical Association 60 (310): 539–547
  • Kennedy, Peter (2008). A Guide to Econometrics (6th ed.). Blackwell. p. 116

Numerical Example of the Goldfeld-Quandt Test.

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Heteroscedasticity Definition, Reasons, Consequences (2012)

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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).

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:

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

Reasons for Heteroscedasticity

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

  1. 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.
  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 and incorrect functional form (linear or log-linear model) are also the sources of heteroscedasticity
Heteroscedasticity Definition

Consequences of Heteroscedasticity

  1. The OLS estimators and regression predictions based on them remain 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 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.

FAQS about Heteroscedasticity

  1. Define heteroscedasticity.
  2. What are the major consequences that may occur if heteroscedasticity occurs?
  3. What does mean by the constant variance of the error term in linear regression models?
  4. What are the possible reasons that make error term variance a variable?
  5. In what kind of data are problems of heteroscedasticity is likely to exist?
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