Let x=(1)/(n)(x1+⋯+xn) and y=(1)/(n)(y1+⋯+yn) . Show that the least-squares line for the data (x

step 1 We can write a generic regression line problem in this way, where we have on the first column, t

Let x=(1)/(n)(x1+⋯+xn) and y=(1)/(n)(y1+⋯+yn) . Show that...

Let $\overline{x}=\frac{1}{n}\left(x_{1}+\cdots+x_{n}\right)$ and $\overline{y}=\frac{1}{n}\left(y_{1}+\cdots+y_{n}\right) .$ Show that the least-squares line for the data $\left(x_{1}, y_{1}\right), \ldots,\left(x_{n}, y_{n}\right)$ must pass through $(\overline{x}, \overline{y}) .$ That is, show that $\overline{x}$ and $\overline{y}$ satisfy the linear equation $\overline{y}=\hat{\beta}_{0}+\hat{\beta}_{1} \overline{x}$ .[Hint: Derive this equation from the vector equation $\mathbf{y}=X \hat{\boldsymbol{\beta}}+\boldsymbol{\epsilon} .$ Denote the first column of $X$ by $\mathbf{1}$ . Use the fact that the residual vector $\epsilon$ is orthogonal to the column space of $X$ and hence is orthogonal to $\mathbf{1} . ]$

Key Concepts

Least Squares Regression

Least squares regression is a statistical method used to determine the best-fitting line through a set of data points by minimizing the sum of the squared differences between the observed values and the values predicted by the model. This method provides parameter estimates that are optimal in the sense that they minimize error in the vertical (y) direction when considering how well the line approximates the data.

Residual Orthogonality

A fundamental property of least squares estimation is that the vector of residuals, which are the differences between the observed values and the fitted values, is orthogonal to the column space of the design matrix. This means that the residuals have zero inner product with any vector in this subspace, ensuring that they are uncorrelated with the predictors. This orthogonality condition is used to derive key relationships in the regression model, such as the fact that the regression line passes through the point defined by the means of the predictor and response variables.

Design Matrix

In linear regression, the design matrix is a structured matrix that contains the input data for all observations. Typically, its first column consists of ones to account for the intercept term, and the subsequent columns contain the values of predictor variables. The use of the design matrix allows the regression model to be represented in vector and matrix terms, facilitating the derivation and computation of the least squares estimates.

Normal Equations

The normal equations are derived from the condition that the residual vector is orthogonal to the column space of the design matrix. By setting the derivative of the sum of squared residuals with respect to the parameters to zero, a system of linear equations is obtained. These equations provide the necessary conditions for the parameter estimates to minimize the sum of squared errors, and they lead to important insights about the regression line, such as its passage through the mean of the data points.

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