# What is the chronology of estimating OLS coefficients?

In a simple linear regression, Deriving OLS estimators by maximum likelihood principle (shown in this link Derivation of OLS Estimator) takes the form

Does this mean that coefficient estimates are dependant on other coefficients; therefore chronology matters?

• If no, how do I solve for b0 and b1 from the above equations?
• If yes, which coefficient gets estimated first? (Is it the intercept first followed by predictor estimates? or other way round? Also within the 'predictor coefficients' does order matter?,If yes: which one gets estimated first?)

(I believe the matrix form i.e. normal equation used in OLS suggests that any coefficient estimate is dependant on 'other predictors' but not their 'beta values'. So chronology shouldn't matter)

\begin{align} \hat{\beta}_0 &= \bar{y} - \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2} \cdot \bar{x}, \\[10pt] \hat{\beta}_1 &= \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2}. \\[6pt] \end{align}
Coefficient estimates depend on which independent variables (i.e. the $$x_i$$'s) you include in your regression (not the coefficients explicitly). Which independent variables you include impacts how the 'terrain' of the cost function (i.e. the error sum of squares) looks, and hence where the minimum of the cost function occurs.