Econometric Theory/Assumptions of Classical Linear Regression Model

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The estimators that we create through linear regression give us a relationship between the variables. However, performing a regression does not automatically give us a reliable relationship between the variables. In order to create reliable relationships, we must know the properties of the estimators ${\hat {\alpha }},{\hat {\beta }}$ and show that some basic assumptions about the data are true. One must understand that having a good dataset is of enormous importance for applied economic research.

Unbiasedness

Under the following four assumptions, OLS is unbiased. This means that:
$E({\hat \alpha })=\alpha$
$E({\hat \beta })=\beta$

Linearity

The model must be linear in the parameters.
The parameters are the coefficients on the independent variables, like $\alpha$ and $\beta$ . These should be linear, so having $\beta ^{2}$ or $e^{\beta }$ would violate this assumption.

Sample Variation

The $x_{i}$ s cannot all have the same value. This is perfect multicollinearity, it is not allowed.

Random Sampling

The $x_{i}$ values must be randomly selected. In other words, there is no correlation between two different x values: $Cov(x_{i},x_{j})=0$ for $i\neq j$ .

Zero Conditional Mean

The mean of the error terms, given a specific value of the independent variable $x_{i}$ , is zero. $E(\epsilon _{i}|X_{i})=0$ .

Efficiency of OLS (Ordinary Least Squares)

Given the following two assumptions, OLS is the Best Linear Unbiased Estimator (BLUE). This means that out of all possible linear unbiased estimators, OLS gives the most precise estimates of $\alpha$ and $\beta$ .

With the third assumption, OLS is the Best Unbiased Estimator (BUE), so it even beats non-linear estimators. Also given this assumption, ${\hat \alpha }$ is distributed according to the Student's t-distribution about $\alpha$ , and ${\hat \beta }$ is distributed in such a way about $\beta$ .

No Heteroskedasticity

The variance of the Error terms are constant. $Var(u_{i}|x_{i})=\sigma ^{2}$ . This means that the variance of the error term $u_{i}$ does not depend on the value of $x_{i}$ . If this is the case, the error terms are called homoskedastic. This is not always the case in economic data, for example the variation in a person's wage will vary with their level of education -- someone who is a high-school dropout will not have much variation in their wage, where people with Ph.D.s may see very different wages.

No Serial Correlation

The error terms are independently distributed so that their covariance is 0. $Cov(u_{i},u_{j}|x_{i},x_{j})=0\forall i\neq j$ .

Normally Distributed Errors

The error terms are normally distributed. $u_{~}N(0,\sigma ^{2})$

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