Econometric Theory/Assumptions of Classical Linear Regression Model

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.

Assumption 1

The relationship between the dependent variable Y and the independent variable(s) X are linear. This does not mean that the independent variable cannot be a non-linear relationship (like log(X), X²) but the parameters must be linear.

Assumption 2

Sample_Var(X) > 0. Without multiple X values in the sample we have no way to show that there is a relationship between the Y and the X variables.

There is no way to know the Y intercept

Assumption 3

The mean of the error terms, given the independent varialbe X, is zero. $E({ϵ}_i|X_i)=0$. Here we are assuming that on average, our X variables are on our regression line.

Assumption 4

Our X variables are given and non-random. Although we may take a random sample, the X variables are due to something. It is not random that a class of 2nd graders are 7 and 8 years old.

Assumption 5

The variance of the Error terms are constant. $Var({ϵ}_i|X_i)={\sigma }^2$ The variance does not change with different samples. This is known as homoscedastic errors, and means the X terms are equally scattered above and below the regression line through out the line.

Assumption 6

The error terms are independently distributed so that their Covariance = 0. $Cov({ϵ}_i,{ϵ}_j|X_i)=0\mathrm{\forall }i\ne j$. This is known as serial independence.

BLUE

The First 6 assumptions are the requirements for a Best Linear Unbiased Esimator. This says that the estimators are the most efficient ones amongst the set of unbiased linear estimators.

Assumption 7

The number of observations is greater than the number of estimators. This is so the regression has enough degrees of freedom to be performed.

Assumption 8

The error terms are Normally distributed amongst the estimated X variable. ${ϵ}_{i}N(0,{\sigma }^2)\Rightarrow {ϵ}_{i}N(\alpha +\beta X_i,{\sigma }^2)$