Question

Discuss all of the assumptions
Briefly what problem when they are not met
How to solve it

The seven classical assumptions are:
I. The regression model is linear, is correctly specified, and has an
additive error term
II. The error term has a zero population mean
III. All explanatory variables are uncorrelated with the error term
IV. Observations of the error term are uncorrelated with each other
(no serial correlation)
V. The error term has a constant variance (no heteroskedasticity)
VI. No explanatory variable is a perfect linear function of any other
explanatory variable(s) (no perfect multicollinearity)
VII. The error term is normally distributed (this assumption is optional
but usually is invoked)

Answer 1

1. If the regression model is not linear or error is not additive, then the model is misspecified i.e. there is a specification error. The problem can be solved by assuming the appropriate statistical relationship.

2. If the error term does not have zero population mean, then the estimator will be biased and inefficient. To solve this problem, one has to use a method that gives an unbiased estimator with minimum variance.

3. If an explanatory variable is correlated with the error term, then the estimator will give biased results. In such cases, one can use a method such as the instrumental variable method to estimate the coefficients.

4. The presence of serial correlation leads to the autocorrelation problem. It affects the minimum variance property. First Differences Procedure and Cochrane-Orcutt Procedure can be used to solve the problem.

5. If the error term does not have constant variance, then there is heteroskedasticity problem. This problem makes the estimates inefficient as the variance of coefficients is no longer minimum. The problem can be solved using Weighted Least square method or Generalized Least square method.

6. If the explanatory variables are perfectly correlated i.e. there is perfect multicollinearity problem, then the coefficients become undetermined. The problem can be solved by omitting one of the variables. For example, if we take as many dummy variables as there are number of categories, we have perfect multicollinearity problem (also called dummy trap). By taking n-1 dummy variables for n categories can solve this problem.

7. If the error term is not normally distributed, then the significance tests of regression coefficients are not valid. The problem may be solved by transforming the data or trying another statistical relation (eg. try non-linear regression). If the problem is not solved, one may assume the distribution that the error term closely resembles and use maximum likelihood or another method to estimate the equation.