Consider the problem described at the end of Section 2-6,...

Consider the problem described at the end of Section $2-6$, running a regression and only estimating an intercept. i. Given a sample $\left\{y_{i}: i=1,2, \ldots, n\right\},$ let $\tilde{\beta}_{0}$ be the solution to $$\min _{b_{0}} \sum_{i=1}^{n}\left(y_{i}-b_{0}\right)^{2}$$ Show that $\tilde{\beta}_{0}=\bar{y},$ that is, the sample average minimizes the sum of squared residuals. (Hint: You may use one-variable calculus or you can show the result directly by adding and subtracting $\bar{y}$ inside the squared residual and then doing a little algebra.) ii. Define residuals $\tilde{u}_{i}=y_{i}-\bar{y} .$ Argue that these residuals always sum to zero.

Consider the problem described at the end of Section $2-6$, running a regression and only estimating
an intercept.
i. Given a sample $\left\{y_{i}: i=1,2, \ldots, n\right\},$ let $\tilde{\beta}_{0}$ be the solution to
$$\min _{b_{0}} \sum_{i=1}^{n}\left(y_{i}-b_{0}\right)^{2}$$
Show that $\tilde{\beta}_{0}=\bar{y},$ that is, the sample average minimizes the sum of squared residuals. (Hint:
You may use one-variable calculus or you can show the result directly by adding and subtracting $\bar{y}$ inside the squared residual and then doing a little algebra.)
ii. Define residuals $\tilde{u}_{i}=y_{i}-\bar{y} .$ Argue that these residuals always sum to zero.

Key Concepts

Properties of Residuals

Residuals are defined as the differences between the observed data and the fitted values from the model. In the intercept-only regression, the fitted value for every observation is the sample mean. A notable property of these residuals is that they sum to zero, which arises from the fact that the sample mean minimizes the total squared deviation and the average of the deviations is zero. This property holds in any regression model that includes a constant term.

Least Squares Estimation

This concept involves finding the parameter values that minimize the sum of squared differences between the observed data and the values predicted by a model. In the context of regression, the objective function (often called the residual sum of squares) is minimized to obtain the best fit parameters. The minimization can be carried out using calculus or algebraic manipulations, leading to estimates that have desirable statistical properties.

Intercept-Only Regression

An intercept-only regression is a special case of the regression model where no independent variables are included; the model simply estimates a constant level for the dependent variable. In this case, the regression function reduces to estimating only the intercept term. The task then becomes finding the constant that minimizes the sum of squared deviations from the observed values.

Sample Average as Minimizer

Within an intercept-only model, minimization of the sum of squared differences between observed values and a constant leads directly to the sample average. By taking the derivative of the sum of squared residuals with respect to the constant parameter and setting it to zero, one obtains the result that the sample mean minimizes the sum of squared deviations, making it the best estimate of the intercept in the least squares sense.

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