## The Meaning Of The Term Linear

Since this text is concerned primarily with linear models like (2.2.2), it is essential to know what the term linear really means, for it can be interpreted in two different ways.

The first and perhaps more "natural" meaning of linearity is that the conditional expectation of Y is a linear function of Xi, such as, for example, (2.2.2).6 Geometrically, the regression curve in this case is a straight line. In this interpretation, a regression function such as E(Y | Xi) = fa1 + faXf is not a linear function because the variable X appears with a power or index of 2.

The second interpretation of linearity is that the conditional expectation of Y, E(Y | Xi), is a linear function of the parameters, the fa's; it may or may not be linear in the variable X.7 In this interpretation E(Y | Xi) = fa1 + fa2X2 is a linear (in the parameter) regression model. To see this, let us suppose X takes the value 3. Therefore, E(Y | X = 3) = fai + 9fa2, which is obviously linear in fa1 and fa2. All the models shown in Figure 2.3 are thus linear regression models, that is, models linear in the parameters.

Now consider the model E(Y | Xi) = fa1 + fa2 Xi. Now suppose X = 3; then we obtain E(Y | Xi) = fa1 + 3fa2, which is nonlinear in the parameter fa2. The preceding model is an example of a nonlinear (in the parameter) regression model. We will discuss such models in Chapter 14.

Of the two interpretations of linearity, linearity in the parameters is relevant for the development of the regression theory to be presented shortly. Therefore, from now on the term "linear" regression will always mean a regression that is linear in the parameters; the fa's (that is, the parameters are raised to the first power only). It may or may not be linear in the explanatory variables, the X's. Schematically, we have Table 2.3. Thus, E(Y | Xi) = fa1 + fa2Xi, which is linear both in the parameters and variable, is a LRM, and so is E(Y | Xi) = fa1 + fa2 X2, which is linear in the parameters but nonlinear in variable X.

6A function Y = f (X) is said to be linear in X if X appears with a power or index of 1 only (that is, terms such as so on, are excluded) and is not multiplied or divided by any other variable (for example, X ■ Z or X/Z, where Z is another variable). If Ydepends on X alone, another way to state that Y is linearly related to X is that the rate of change of Y with respect to X (i.e., the slope, or derivative, of Y with respect to X, dY/dX) is independent of the value of X. Thus, if Y = 4X, dY/dX = 4, which is independent of the value of X. But if Y = 4X2, dY/dX = 8X, which is not independent of the value taken by X. Hence this function is not linear in X.

7A function is said to be linear in the parameter, say, fa1, if fa1 appears with a power of 1 only and is not multiplied or divided by any other parameter (for example, fa1fa2, P2/P1, and so on).

Linearity in the Variables

Linearity in the Parameters

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