Handbook of Regression Analysis With Applications in R. Samprit Chatterjee

alt="images"/> corresponds to the simple regression model, and is consistent with the representation in Figure 1.1. The solid line is the true regression line, the expected value of

given the value of

. The dotted lines are the random errors

that account for the lack of a perfect association between the predictor and the target variables.

      1.2.2 ESTIMATION USING LEAST SQUARES

      The true regression function represents the expected relationship between the target and the predictor variables, which is unknown. A primary goal of a regression analysis is to estimate this relationship, or equivalently, to estimate the unknown parameters

. This requires a data‐based rule, or criterion, that will give a reasonable estimate. The standard approach is least squares regression, where the estimates are chosen to minimize

(1.2)

Figure 1.2 gives a graphical representation of least squares that is based on Figure 1.1. Now the true regression line is represented by the gray line, and the solid black line is the estimated regression line, designed to estimate the (unknown) gray line as closely as possible. For any choice of estimated parameters

, the estimated expected response value given the observed predictor values equals

FIGURE 1.2: Least squares estimation for the simple linear regression model, using the same data as in Figure 1.1. The gray line corresponds to the true regression line, the solid black line corresponds to the fitted least squares line (designed to estimate the gray line), and the lengths of the dotted lines correspond to the residuals. The sum of squared values of the lengths of the dotted lines is minimized by the solid black line.

      and is called the fitted value. The difference between the observed value

and the fitted value

is called the residual, the set of which is represented by the signed lengths of the dotted lines in Figure 1.2. The least squares regression line minimizes the sum of squares of the lengths of the dotted lines; that is, the ordinary least squares (OLS) estimates minimize the sum of squares of the residuals.

In higher dimensions (

), the true and estimated regression relationships correspond to planes (

) or hyperplanes (

), but otherwise the principles are the same. Figure 1.3 illustrates the case with two predictors. The length of each vertical line corresponds to a residual (solid lines refer to positive residuals, while dashed lines refer to negative residuals), and the (least squares) plane that goes through the observations is chosen to minimize the sum of squares of the residuals.

FIGURE 1.3: Least squares estimation for the multiple linear regression model with two predictors. The plane corresponds to the fitted least squares relationship, and the lengths of the vertical lines correspond to the residuals. The sum of squared values of the lengths of the vertical lines is minimized by the plane.

The linear regression model can be written compactly using matrix notation. Define the following matrix and vectors as follows:

      The regression model (1.1) is then

      (1.3)

The normal equations [which determine the minimizer of 1.2] can be shown (using multivariate calculus) to be

      which implies that the least squares estimates satisfy

      (1.4)

      The fitted values are then

      (1.5)

where

is the so‐called “hat” matrix (since it takes

to

). The residuals

thus satisfy

      (1.6)

      or

      

      1.2.3 ASSUMPTIONS

      The least squares criterion will not necessarily yield sensible results unless certain assumptions hold. One is given in (1.1) — the linear model should be appropriate. In addition, the following assumptions are needed to justify using least squares regression.

      1 The expected value of the errors is zero ( for all ). That is, it cannot be true that for certain observations the model is systematically too low, while for others it is systematically too high. A violation of this assumption will lead to difficulties in estimating . More importantly, this reflects that the model does not include a necessary systematic component, which has instead been absorbed into the error terms.

      2 The