From the linear model we find expected u

; from here we derive β₁:



; from here we derive β₂:

2. 
   According to the variance definition,
   

as





Assumptions depicts that ; and ; and

The variance of can be found using the same way:

3. 
   The assumptions of Ordinary Least Square (OLS) model:
   

Assumption 1: Linear regression model (in parameters) means that the model’s coefficient must be in the degree of one.

Assumption 2: X values are fixed in repeated sampling, which means that regressor must be nonstochastic.

Assumption 3: Zero mean value of : , the value of the random disturbance term (error term) is zero.

Assumption 4: Homoscedasticity or equal variance of :

Theoretically, the equation characterizes the assumption of homoscedasticity, or equal variance. In other words, it means that the Y populations corresponding to various X values have the same variance.

Assumption 5: No autocorrelation between the disturbances:

With

The disturbances u_­i and u_j are uncorrelated, i.e., no serial correlation. This means that, given x_i, the deviations of any two Y values from their mean value do not exhibit patterns.

Assumption 6: Zero covariance between u_i and X_i

The error term u and right-hand side variable X are uncorrelated.

Assumption 7: The number of observations n must be greater than the number of parameters to be estimated.

Assumption 8: Variability in X values. They must not all be the same. If all values of X is the same, then , which makes the denominator 0 and calculation of β₁ and β₂ will be impossible.

Assumption 9: The regression model is correctly specified.in other words, the model that is used in empirical analysis must not have specification errors.

Assumption 10: There is no perfect multicollinearity between Xs. That is to say, the relationship between independent variables must not be perfect.

4. 
   The followings are the assumptions:
   

; ; ; ;

; from this equation we found out βhat:

; ;

;

; ; ;

By using assumption we prove that the coefficient of OLS is unbiased