Regression Analysis Dr. Dmitri M. Medvedovski

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Classical Normal Linear Regression Model

The Normality Assumption for Ui

Mean: E(Ui )=0 Variance: E[Ui E(Ui )]2 = E(Ui2 ) = G2 Cov: (Ui ,Ui ) : E[(Ui E / Ui )][U j E(U j)] = E(UiUj ) = 0 Ui ~ N(O,G2 ) if i j

Why Normality Assumption?

1) Ui represents the combined influence on the dependent variable of a large number of independent variables that are not explicitly introduced in the regression model. We hope that the influence of these omitted or neglected variables is small and at best random.

By the Central Limit theorem (CLT) of statistics, it can be shown that if there are a large number of independent and identically distributed random variables, then, with the few exceptions, the distribution of their sum tends to a normal distribution as a member of such variables increases indefinitely. It is the CLT that provides a theoretical justification for the assumption of normality of Ui .

2) Even if the number of variables is not very large or if these variables are not strictly independent, their sum may still be normally distributed.

3) The probability distributions of OLS estimators can be easily derived because any linear function of normally distributed variables is itself normally distributed. OLS estimators 1 and 2 are linear functions of Ui . Therefore, if Ui are normally distributed, so are 1 and 2 , which makes hypothesis testing very straightforward.

4) The normal distribution is comparatively simple distribution involving only two parameters (mean and variance). It is very well known and its properties are extensively studied in mathematical statistics. Many phenomena seem to follow the normal distribution.

5) Finally, if we are dealing with a small, or finite, sample size, say data of less than 100 observations, the normality assumption not only helps us to derive the exact probability distributions of OLS estimators but also enables us to use t , F statistical tests for regression models. If sample size is reasonably large, we may be able to relax the normality assumption.