Basic Statistics and Data Analysis

Statistics Lecture Notes, Online MCQs

Tagged: Heteroscedasticity

OLS Assumptions

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Homoscedasticity: Assumption of constant variance of a random variable

The assumption about the random variable μ (error term) is that its probability distribution remains the same for all observations of X and in particular that the variance of each μ is the same for all values of the explanatory variables, i.e the variance of errors is the same across all levels of the independent variables. Symbolically it can be represented as

$Var(\mu) = E\{\mu_i – E(\mu)\}^2 = E(\mu_i)^2 = \sigma_\mu^2 = \mbox(Constant)$

This assumption is known as the assumption of homoscedasticity or the assumption of constant variance of the error term μ‘s. It means that the variation of each μi around its zero means does not depend on the values of X (independent) because the error term expresses the influence on the dependent variables due to

  • Errors in measurement
    The errors of measurement tend to be cumulative over time. It is also difficult to collect the data and check its consistency and reliability. So the variance of μi increases with increasing the values of X.
  • Omitted variables
    Omitted variables from the function (regression model) tend to change in the same direction as X, causing an increase in the variance of the observation from the regression line.

The variance of each μi remains the same irrespective of small or large values of the explanatory variable i.e. $\sigma_\mu^2$ is not a function of Xi i.e $\sigma_{\mu_i^2} \ne f(X_i)$.

Homoscedasticity or Constant Variance

Consequences if Homoscedasticity is not meet

If the assumption of homoscedastic disturbance (Constant Variance) is not fulfilled, the following are the consequence

  1. We cannot apply the formula of the variance of the coefficient to conduct tests of significance and construct confidence intervals. The tests are inapplicable $Var(\hat{\beta}_0)=\sigma_\mu^2 \{\frac{\sum X^2}{n \sum X^2}\}$ and $Var(\hat{\beta}_1) = \sigma_\mu^2 \{\frac{1}{\sum X^2}\}$
  2. If μ (error term) is heteroscedastic the OLS (Ordinary Least Square) estimates do not have minimum variance property in the class of Unbiased Estimators i.e they are inefficient in small samples. Furthermore, they are inefficient in large samples (that is, asymptotically inefficient).
  3. The coefficient estimates would still be statistically unbiased even if the μ‘s are heteroscedastic. The $\hat{\beta}$’s will have no statistical bias i.e $E(\beta_i)=\beta_i$ (coefficient’s expected values will be equal to the true parameter value).
  4. The prediction would be inefficient because the variance of prediction includes the variance of μ and of the parameter estimates which are not minimal due to the incidence of heteroscedasticity i.e. The prediction of Y for a given value of X based on the estimates $\hat{\beta}$’s from the original data, would have a high variance.

Tests for Homoscedasticity

Some tests commonly used for testing the assumption of homoscedasticity are:

Reference:
A. Koutsoyiannis (1972). “Theory of Econometrics”. 2nd Ed.

Heteroscedasticity / OLS Assumptions / Test of Heteroscedasticity

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Breusch Pagan Test for Heteroscedasticity

Breusch Pagan test (named after Trevor Breusch and Adrian Pagan) is used to check for the presence of heteroscedasticity in a linear regression model.

Assume our regression model is $Y_i = \beta_1 + \beta_2 X_{2i} + \mu_i$ i.e we have simple linear regression model, and $E(\mu_i^2)=\sigma_i^2$, where $\sigma_i^2=f(\alpha_1 + \alpha_2 Z_{2i})$,

That is $\sigma_i^2$ is some function of the non-stochastic variable Z‘s. The f() allows for both the linear and non-linear forms of the model. The variable Z is the independent variable X or it could represent a group of independent variables other than X.

Step to Perform Breusch Pagan test

  1. Estimate the model by OLS and obtain the residuals $\hat{\mu}_1, \hat{\mu}_2+\cdots$
  2. Estimate the variance of the residuals i.e. $\hat{\sigma}^2=\frac{\sum e_i^2}{(n-2)}$
  3. Run the regression $\frac{e_i^2}{\hat{\sigma^2}}=\beta_1+\beta_2 Z_i + \mu_i$ and compute explained sum of squares (ESS) from this regression
  4. Test the statistical significance of ESS/2 by $\chi^2$-test with 1 df at appropriate level of significance (α).
  5. Reject the hypothesis of homoscedasticity in favour of heteroscedasticity if $\frac{ESS}{2} > \chi^2_{(1)}$ at the appropriate level of α.

Note that the

  • Breusch Pagan test is valid only if μi‘s are normally distributed.
  • For k independent variables, ESS/2 have ($\chi^2$) Chi-square distribution with k degree of freedom.
  • If the μi‘s (error term) are not normally distributed, the White test is used.

References:

  • Breusch, T.S.; Pagan, A.R. (1979). “Simple test for heteroscedasticity and random coefficient variation”. Econometrica (The Econometric Society) 47 (5): 1287–1294.

See the Numerical Example of the Breusch-Pagan Test for the Detection of Heteroscedasticity

Heteroscedasticity / Test of Heteroscedasticity

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Goldfeld Quandt Test: Comparison of the Variances of Error Terms

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 μi.

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 Procedure:

The procedure of 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. (n – c)/2, one sub-group includes small values of X and other sub-group include the large values of X, a data set is arranged according to the magnitude of X.
  3. Now Fit the separate regression to each of the sub-group, and obtain the sum of squared residuals form each of them.
    So $\sum c_1^2$ show sum of squares of Residuals from sub-sample of low values of X with $(n – c)/2 – K$ df, where K is the total number of parameters.$\sum c_2^2$ show sum of squares of Residuals from 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 $u_i$.

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.

Correlation and Regression Analysis / Heteroscedasticity / OLS Assumptions

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Introduction, Reasons and Consequences of Heteroscedasticity

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.

Homoscedasticity

Heteroscedasticity

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.

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