1. Byron Gangnes Econ 427 lecture 3 slides

2. Byron Gangnes A scatterplot

3. Byron Gangnes A regression line

4. Byron Gangnes A regression line

5. Byron Gangnes Linear Regression We assume that y is linearly related to x, with an independently and identically distributed (iid) disturbance term with a zero mean and constant variance: t = 1,…, T

6. Byron Gangnes Linear Regression The regression function gives an estimate of y, given x, which is just the conditional expectation of y given x = x*,

7. Byron Gangnes Linear Regression Since we don’t know the true (population) relationship, we estimate it from the data by calculating the parameters that minimize squared errors:

8. Byron Gangnes Linear Regression Then we used the estimated parameters to get a fitted value (also called an “in-sample forecast” of y, given x: where the “hats” indicate estimated values. The in-sample forecast errors are just:

9. Byron Gangnes sum of squared residuals SSR is the sum of squared residuals of the regression (the minimized value that OLS searches for)

10. Byron Gangnes R-squared A standard measure of overall goodness of fit is R2 (R-squared), technically the percentage of the variance of y explained by the variables in the model:

11. Byron Gangnes Adjusted R-squared the problem with R2 is that it always goes up when you add more variables. To avoid “overfitting” (any model will fit the data if there are enough RHS variables), we normally adjust for the degrees of freedom using the “adjusted R- squared”:

12. Byron Gangnes F-statistic F-statistic is a test of whether all model coefficients are jointly zero—an overall test of the significance of the regression:

13. Byron Gangnes Durbin-Watson statistic The Durbin-Watson statistic is a measure of serial correlation in the regression errors. (Why do we care about whether errors are serially correlated?)

14. Byron Gangnes Durbin-Watson statistic is a test of whether there is first-order autocorrelation of the model errors, i.e. Values for DW fall in the [0,4] interval and values significantly below 2 (below 1.5 say) are indicative of serial correlation.

15. Byron Gangnes Moments Mean Variance Skewness Kurtosis