Econ 570 - Spring 2012

We studied the EasyMetrics 2-variable regression simulation to see how the errors affect the estimated slope. The following algebra lays out a more mathematical explanation. The estimated coefficient is not equal to the true coefficient because of the sample covariance of the errors and the regressor.

The errors are part of the true model. They are unobservable because we do not know the true coefficients. We can use the estimated coefficients to calculate the residuals, which you can think of as "estimated errors."

The sample covariance of the residual and the regressor is zero by construction.

By the definition of the residual, which can be thought of a forecast error, the observed dependent variable is the sum of the predicted value from the estimated regression and the residual. We have just seen that the covariance of the regressor (and, hence, the prediction) and the error is zero by construction. This allows us to decompose the variance of the dependent variable into "explained" and "unexplained" variances. The R-squared is the explained variance as a percentage of the total variance.