What are Our Assumptions? Simple Regression (SR) Assumptions
Set Up: We want to minimize this – the mean squared distance between the true dependent variable and our (as-yet) hypothetical regression line. Note that assumption SR3 states that Solving for Going forward, our goal is to get the above equation so that Note the "equals zero" on the Right Hand Side, so the 2 and the negative one can get divided out without issue. And we can multiply it all by 1/n without a problem, since it's just zero on the other side. Also, let's break the summation into three separate terms. Note that Note that Note that Now, solving for We know how to find the mean of y and the mean of x, but we'll need some more work to find beta-hat-one. Solving for As a reminder, going forward, our goal is to get the above equation so that Plus we'll do some work to get the equation equal to something simple and easy to deal with. All the little tricks and oddities to come help us achieve that goal. We can get rid of the 2 and negative one since it's all equal to zero. Note that we know that Let's open up one of the parentheses in the summation, distributing the negative sign. Let's break the summation into two different terms. Note that the To the term above, we'll subtract, " This term is equal to zero, so it's chill. On This shows that all we're doing is adding zero. The same works for Let's put those two "zero" terms in: Now let's distribute them Recall our goal is to solve this equation in terms of Moving the second term above to the other side of the equation, we get: Now, distribute that negative sign in front of Notice that " Now solve in terms of Note that And that To Recap, our goal was to solve that minimization equation in terms of Beta-hat-sub-nought, Beta-hat-sub-one, And you can use your answer here to find a number for So, if you want to estimate a line that best a sample of |