Question

2. () Explain what is meant by hetroscedasticity in a regression model Y Xp+e and causes a problem with inference in OLS (ii) In what type of data is heteroscedasticity most likely to be a problem: time series or cross- sectional data? How might you check for its presence? (5 marks) (ii) What is meant by Weighted Least Squares'? Explain how to use it in practice. (5 marks) (5 marks) (iv) Suppose you discover heteroscedasticity is a problem in your empirical work, and an estimation based on squared residuals suggests E(E/)-25 X', how could you correct for the problem? Explain your answer. (5 marks)
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Answer 1

1)Heteroscedasticity means unequal scatter. In regression analysis, we talk about heteroscedasticity in the context of the residuals or error term. Specifically, heteroscedasticity is a systematic change in the spread of the residuals over the range of measured values. Heteroscedasticity is a problem because ordinary least squares(OLS) regression assumes that all residuals are drawn from a population that has a constant variance (homoscedasticity).

Its causes are given below:

(a) It occurs more often in datasets that have a large range between the largest and smallest observed values. While there are numerous reasons why heteroscedasticity can exist, a common explanation is that the error variance changes proportionally with a factor. This factor might be a variable in the model.

2) In cross sectional data :

In some cases, the variance increases proportionally with this factor but remains constant as a percentage. For instance, a 10% change in a number such as 100 is much smaller than a 10% change in a large number such as 100,000. In this scenario, you expect to see larger residuals associated with higher values. That’s why you need to be careful when working with wide ranges of values!

A plot of the standardized residuals against the predictor variable points up the presence of heteroscedastic errors.

3) Linear regression models with heteroscedastic errors can also be fitted by method called the weighted least squares(WLS), when parameter estimates are obtained by minimizing a weighted sum of squares of residuals where the weights are inversely proportional to the variance errors. This is in contrast to OLS where the parameter estimates are obtained by minimizing equally weighted sum of squares esiduals.

4) One is to apply an appropriate transformation - derived, for example, from the family of Box-Cox transformations. This will help the 'transformed' data to have equal variance, and, as usually happens, will also make the transformed data to follow a normal distribution.