Week Lecture 3Slide #1 Minimizing e 2 : Deriving OLS Estimators The problem Deriving b 0 Deriving b 1 Interpreting b 0 and b 1

Week Lecture 3Slide #2 Measuring Error: Residuals 2 Objective: to minimize ∑e. For computational reasons, we minimize ∑e 2 Y X eiei ejej

Week Lecture 3Slide #3 OLS Derivation of b 0 Use partial derivation in this step:

Week Lecture 3Slide #4 Derivation of b 0, step 2

Week Lecture 3Slide #5 Derivation of b 1 Step 1: Multiply out e 2

Week Lecture 3Slide #6 Derivation of b 1 Step 2: Differentiate w.r.t. b 1

Week Lecture 3Slide #7 Derivation of b 1 Step 3: Substitute for b 0

Week Lecture 3Slide #8 Derivation of b 1 Step 4: Simplify and Isolate b 1

Week Lecture 3Slide #9 Calculating b 0 and b 1 The formula for b 1 and b 0 allow you (or preferably your computer) to calculate the error-minimizing slope and intercept for any data set representing a bi-variate, linear relationship. No other line, using the same data, will result in as small a squared-error (e 2 ). OLS gives best fit.

Week Lecture 3Slide #10 Interpreting b 1 and b 0 For each 1-unit increase in X, you get b 1 units change in Y When X is zero, Y will be equal to b 0. Note that regression with no independent variables is simply the mean.

Week Lecture 3Slide #11 Calculate b 0 and b 1 for: Check results against Stata.