Lecture notes about Heteroscedasticity, its consequences, detection, and remedy.

Heteroscedasticity in Regression

Heteroscedasticity in Regression

Heteroscedasticity in Regression: The term heteroscedasticity refers to the violation of the assumption of homoscedasticity in linear regression models (LRM). In the case of heteroscedasticity, the errors have unequal variances for different levels of the regressors, which leads to biased and inefficient estimators of the regression coefficients. The disturbances in the Classical Linear Regression Model (CLRM) appearing in the population regression function should be homoscedastic; that is they all have the same variance.

Mathematical Proof of $E(\hat{\sigma}^2)\ne \sigma^2$ when there is some presence of hetero in the data.

For the proof of $E(\hat{\sigma}^2)\ne \sigma^2$, consider the two-variable linear regression model in the presence of heteroscedasticity,

Y_i=\beta_1 + \beta_2 X+ u_i, \quad\quad (eq1)

where $Var(u_i)=\sigma_i^2$ (Case of heteroscedasticity)


\hat{\sigma^2} &= \frac{\sum \hat{u}_i^2 }{n-2}\\
&= \frac{\sum (Y_i – \hat{Y}_i)^2 }{n-2}\\
&=\frac{(\beta_1 + \beta_2 X_i + u_i – \hat{\beta}_1 -\hat{\beta}_2 X_i )^2}{n-2}\\
&=\frac{\sum \left( -(\hat{\beta}_1-\beta_1) – (\hat{\beta}_2 – \beta_2)X_i + u_i \right)^2 }{n-2}\quad\quad (eq2)

Noting that

\beta_1 + \beta_2 X + u_i\, – \,\hat{\beta}_1 – \hat{\beta}_2X &=0\\
-(\hat{\beta}_1 -\beta_1) – X(\hat{\beta}_2-\beta_2) – u_i & =0\\
(\hat{\beta}_1 -\beta_1) &= – X (\hat{\beta}_2-\beta_2) + u_i\\
\text{Applying summation on both side}&\\
\sum (\hat{\beta}_1-\beta_1) &= -(\hat{\beta}_2-\beta_2)\sum X + \sum u_i\\
(\hat{\beta}_1 – \beta_1) &= -(\hat{\beta}_2-\beta_2)\overline{X}+\overline{u}

Substituting it in (eq2) and taking expectation on both sides:

\hat{\sigma}^2 &= \frac{1}{n-2} \left[ -(-(\hat{\beta}_2 – \beta_2) \overline{X} + \overline{u} ) – (\hat{\beta}_2-\beta_2)X_i + u_i  \right]^2\\
&=\frac{1}{n-2}E\left[(\hat{\beta}_2-\beta_2)\overline{X} -\overline{u} – (\hat{\beta}_2-\beta_2)X_i-u_i \right]^2\\
&=\frac{1}{n-2} E\left[ -(\hat{\beta}_2 – \beta_2)(X_i-\overline{X}) + (u_i-\overline{u})\right]^2\\
&= \frac{1}{n-2}\left[-\sum x_i^2 Var(\hat{\beta}_2) + E[\sum(u_i-\overline{u}]^2 \right]\\
&=\frac{1}{n-2} \left[ -\frac{\sum x_i^2 \sigma_i^2}{(\sum x_i^2)} + \frac{(n-1)\sum \sigma_i^2}{n} \right]

If there is homoscedasticity, then $\sigma_i^2=\sigma^2$ for each $i$, $E(\hat{\sigma}_i^2)=\sigma^2$.

The expected value of the $\hat{\sigma}^2=\frac{\hat{u}_i^2}{n-2}$ will not be equal to the true $\sigma^2$ in the presence of heteroscedasticity.

Heteroscedasticity in Regression

Read more on the Remedy of Heteroscedasticity

More on heteroscedasticity on Wikipedia

MCQs General Knowledge

R Programming Language

Goldfeld-Quandt Test Example

Data is taken from the Economic Survey of Pakistan 1991-1992. The data file link is at the end of the post “Goldfeld-Quandt Test Example for the Detection of Heteroscedasticity”.

Goldfeld-Quandt Test Example

For an illustration of the Goldfeld-Quandt Test Example, the data given in the file should be divided into two sub-samples after dropping (removing/deleting) the middle five observations.

Sub-sample 1 consists of data from 1959-60 to 1970-71.

Sub-sample 2 consists of data from 1976-77 to 1987-1988.

The sub-sample 1 is highlighted in green colour, and sub-sample 2 is highlighted in blue color, while the middle observation that has to be deleted is highlighted in red.

The Step-by-Step Procedure to Conduct the Goldfeld Quandt Test

Step 1: Order or Rank the observations according to the value of $X_i$. (Note that observations are already ranked.)

Step 2: Omit $c$ central observations. We selected 1/6 observations to be removed from the middle of the observations. 

Step 3: Fit OLS regression on both samples separately and obtain the Residual Sum of Squares (RSS) for each sub-sample.

The Estimated regression for the two sub-samples are:

Sub-sample 1: $\hat{C}_1 = 1010.096 + 0.849 \text{Income}$

Sub-sample 2: $\hat{C}_2 = -244.003 + 0.88067 \text{Income}$

Now compute the Residual Sum of Squares for both sub-samples.

The residual Sum of Squares for Sub-Sample 1 is $RSS_1=2532224$

The residual Sum of Squares for Sub-Sample 2 is $RSS_2=10339356$

The F-Statistic is $ \lambda=\frac{RSS_2/n_2}{RSS_1/n_1}=\frac{10339356}{2532224}=4.083$

The critical value of $F(n_1=10, n_2=10$ at a 5% level of significance is 2.98.

Since the computed F value is greater than the critical value, heteroscedasticity exists in this case, that is, the variance of the error term is not consistent, rather it depends on the independent variable, GNP.

Your assignment is to perform this the Goldfeld-Quandt Test Example using any statistical software and confirm the results.

Download the data file by clicking the link gnp and consumption expenditure data

Learn about White’s Test of Heteroscedasticity

Goldfeld-Quandt Test Example

Learn R Programming

Test Preparation MCQS

Consequences of Heteroscedasticity

The following are consequences of heteroscedasticity when it exists in the data.

  • 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.
Consequences of Heteroscedasticity

Learn about Remedial Measures of Heteroscedasticity

R Programming Language

Test Preparation MCQs

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