# Basic Statistics and Data Analysis

## Breusch-Pagan Test for Heteroscedasticity

Breusch–Pagan test (named after Trevor Breusch and Adrian Pagan) is used to test for 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. 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 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, 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.

# GoldFeld-Quandt Test of Heteroscedasticity

The Goldfeld-Quandt test is one of two tests proposed in a 1965 paper by Stephen Goldfeld and Richard Quandt. Both a parametric and nonparametric test 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 in h subgroups. Usually data set is divided into two parts or groups, and hence the test is sometimes called a two-group test.

The procedure of conducting 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 includes the large values of X, as 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)/2K df, where Kis 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)/2K df, where K is 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 differs, F* will have a large value. The higher the observed value of F* ratio the stronger the hetro of the μi‘s.

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
• Cook, R. Dennis; Weisberg, S. (April 1983). “Diagnostics for heteroscedasticitiy in regression”. Biometrika 70 (1): 1–10.

# Heteroscedasticity

An important assumption of OLS is that the disturbances μi appearing in the population regression function are homoscedastic (Error term have 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 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 of 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 grow, people have more discretionary income and hence $\sigma_{i}^{2}$ is likely to increase with income.
3. As data collecting techniques improves, $\sigma_{i}^{2}$ is likely to decrease.
4. Heteroscedasticity can also arises 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