The numerator in the second term in this expression can have the constant quantity factored out from under the summation sign, because it does not vary over observations. The second term then becomes S(X_i - )/c. Now the stuff inside this summation is the sum of the deviations from the mean, which is zero. Thus the second term in the re-write of the numerator of line 2 disappears.

The Y_i values are the only random variables in this story. The X values are treated as deterministic (non-random) amounts. As "constants," they can be factored out of the expectation operator (review the rules of expectations).

Now we can write each individual value of Y_i in terms of the parameters of the population regression function, since we assume that underlying our sample is a population wherein Y_i = B₁+B₂X_i+u_i.

The expectation operation can be distributed across a sum.

The E[B₁] is just B₁, because this parameter is some true, if unknown constant value. Since the X's are deterministic, the expectation E[B₂X_i] is just B₂X_i. By assumption E[u_i] is zero (namely, the population regression function runs through the mean of the relevant conditional distributions of Y. Next, you can take the three terms in the "curly" brackets and distribute them, along with the S (X_i - ), into three separate terms. The B₁ can be factored out of its summation, and B₁S (X_i - ) will be equal to zero because the sum of the deviations of a variable from its mean is zero. The third term disappears because E[u] is zero. All that is left is the middle term, where the B₂ can be factored out of the summation, but not the X_i, because that varies with i.

To get to the next step, we need to reverse the result we used in getting from line 1 to line 3. If we use X_i instead of the Y_i appearing above, it will be likewise true that S (X_i - )X_i is equal to S (X_i - )(X_i - ). This gets the algebra into a convenient form where it is easy to see that stuff cancels out, and we are left with the tidy result that E[B₂]=B₂.

Above, we have substituted for the mean of Y an expression for it in terms of the parameters of the population regression function. This involves writing the individual values of Y_i as B₁+B₂X_i+u_i, summing over all i = 1,...,n, and then dividing by n. This yields B₁+B₂+ubar. The expectation operator can be distributed across the four terms in the square brackets.

The expected value of a true but unknown constant is just that constant, so E[B₁]=B₁. The X's being deterministic, the second term is also always equal to its expectation. The expected value (average) of the population regression function error term is zero. We have shown above that the slope estimator is an unbiased estimator for B₂, so we can use that result here. Stuff cancels, and we see that the OLS intercept estimator is also an unbiased estimator for the true but unknown population intercept parameter.

This uses the same version of the formula for b₂ as appears within equation 3 above. This is a nice simple place to start the derivation of the variance of the OLS slope estimator. For the next line, we write out all the terms in the summation.

This is were we draw upon the fact that the OLS estimator is a "linear" estimator. The formulas for the parameters can be expressed as a linear function of the data on Y. The expanded summation can be viewed as nothing more than a conventional linear combination of independent random variables (i.e. coefficient time variable, plus coefficient times variable, plus coefficient times variable...). If our sample is randomly drawn, then the observations on Y_i are statistically independent. For the variance of this linear combination of independently distributed random variables, we need to take the squares of the coefficients and multiply them times the variances of the individual random variables. This is exactly where the assumption of "no serial correlation in the errors" comes into the calculation of interesting quantities in regression analysis. If the the Y values (and hence the error terms) are serially correlated, then we would have to deal with all the covariance terms in the general formula for the variance of a linear combination of random variables. In the absence of serial correlation in the Y values (and therefore the errors), all of these covariance terms are zero.

14           = ((X₁ - )/c)² Var(Y₁) + ((X₂ - )/c)² Var(Y₂) + ... + ((X_n - )/c)² Var(Y_n)

Now it is convenient to put the drawn-out sum of terms back into summation notation, with one term for each value of i in the sample. But if we are assuming that the conditional variance of Y is the same, regardless of the value of X at which the distribution of Y is being considered (homoscedasticity assumption), then all the Var[Y_i] terms are equal to the same s².

15           = S [ (X_i - )²/c² ] Var(Y_i) = S [ (X_i - )²/c² ] s ²

This common s² value for all terms in the summation will factor out of the summation, and we can now reinstate the full definition of the abbreviation c. This is exactly where the assumption of homoscedastic errors comes into the calculation of interesting quantities in regression analysis. If the conditional error variances differed from observation to observation, we could not factor out the common s² value as we do here.

16           = s ² S [ (X_i - )²/c² ] = s ² S [ (X_i - )²/ (S (X_i - )²)²]

17           = s ² / S x_i²  .... s.e.(b₂) = s /Ö (S x_i² )

18 Var(b₁) = Var [ - b₂ ] = Var [ B₁ + B₂ + - b₂ ]

19           = Var(B₁) + Var(B₂) + Var() + Var(b₂)

20           = 0 + 0 + Var (S u_i/n) + ² Var(b₂)

21           = s ²/n + ² ( s ² / S x_i²  )

22           = s ² [ (1/n) + ²/S (X_i - )² ], or can be alternatively expressed as

23           = s ² [ S X_i²  / nS x_i²  ] .... s.e.(b₁) = s Ö (S X_i²  / nS x_i² )

For s ², use sample variance with modified degrees of freedom (2 estimated parameters required before e_i can be ascertained)

24      s² = S e_i² /(n-2) = (1/(n-2)) S (Y_i - b₁ - b₂X_i)²