[This exercise requires the material in the chapter...

[This exercise requires the material in the chapter appendix.] Suppose that the regression model
$$
y=\beta_0+\beta_1 x_1+\beta_2 x_2+\varepsilon
$$
is estimated by least squares. Show that the residuals, $e_{i,}$ from the fitted model sum to 0 .

Key Concepts

Least Squares Estimation

Least squares estimation is a method used in regression analysis to find the line (or hyperplane) that minimizes the sum of squared differences between the observed dependent variable values and those predicted by a linear model. This method results in parameter estimates that optimally fit the given data under the assumption of normally distributed errors and has many attractive mathematical properties including unbiasedness and minimum variance among linear unbiased estimators under certain conditions.

Residuals

Residuals are the differences between the observed values and the fitted values produced by a regression model. They capture the part of the data that is not explained by the model and are fundamental for model diagnostics. A noteworthy property in least squares regression, particularly when an intercept is included in the model, is that the residuals sum to zero, ensuring that the model does not systematically over- or under-predict the dependent variable.

Normal Equations

The normal equations arise from setting the partial derivatives of the sum of squared residuals with respect to each parameter to zero in order to find the optimal parameter estimates. They provide a system of equations that the estimated parameters must satisfy. One of the results of these equations, when an intercept is included in the regression model, is that the sum of the residuals equals zero, reflecting the orthogonality between the residuals and the constant term of the model.

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