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; from here we derive β1:
; from here we derive β2:
According to the variance definition,
Assumptions depicts that ; and ; and
The variance of can be found using the same way:
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:
The disturbances ui and uj are uncorrelated, i.e., no serial correlation. This means that, given xi, the deviations of any two Y values from their mean value do not exhibit patterns.
Assumption 6: Zero covariance between ui and Xi
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.
The followings are the assumptions:
; ; ; ;
; from this equation we found out βhat:
; ; ;
By using assumption we prove that the coefficient of OLS is unbiased
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