# Are in the model such tests may lead to the exclusion

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are in the model. Such tests may lead to the exclusion of certain variables from the model. On the other hand, other variables such as horsepower x 4 may be important and should be included. Then the model needs to be extended so that its predictive capability is increased. The model in Eq. (1.9) is quite simple and should provide a useful starting point for our modeling. Of course, we do not know the values of the model coef fi cients, nor do we know whether the functional representation is appropriate. For that we need data. One must keep in mind that there are only 38 observations and that one cannot consider models that contain too many unknown parameters. A reasonable strategy starts with simple parsimonious models such as the one speci fi ed here and then checks whether this representation is capable of explaining the main features of the data. A parsimonious model is simple in its structure and economical in terms of the number of unknown parameters that need to be estimatedfromdata,yetcapableofrepresentingthekeyaspectsoftherelationship. We will say more on model building and model checking in subsequent chapters. The introduction in this chapter is only meant to raise these issues. 1.3 A GENERAL MODEL In all of our examples, we have looked at situations in which a single response variable y is modeled as y = μ + ε (1.10a) The deterministic component μ is written as μ = β 0 + β 1 x 1 + β 2 x 2 + · · · + β p x p (1.10b) where x 1 , x 2 ,. . . , x p are p explanatory variables. We assume that the explana- tory variables are “fi xed ”— that is, measured without error. The parameter
Abraham Abraham ˙ C01 November 8, 2004 0:33 16 Introduction to Regression Models β i ( i = 1 , 2 ,. . . , p ) is interpreted as the change in μ when changing x i by one unit while keeping all other explanatory variables the same. The random component ε is a random variable with zero mean, E ( ε ) = 0 , and variance V ( ε ) = σ 2 that is constant for all cases and that does not depend on the values of x 1 , x 2 ,. . . , x p . Furthermore, the errors for different cases, ε i and ε j , are assumed independent. Since the response y is the sum of a deterministic and a random component, we fi nd that E ( y ) = μ and V ( y ) = σ 2 . WerefertothemodelinEq.(1.10)as linearintheparameters .Toexplainthe idea of linearity more fully, consider the following four models with deterministic components: i. μ = β 0 + β 1 x ii. μ = β 0 + β 1 x 1 + β 2 x 2 (1.11) iii. μ = β 0 + β 1 x + β 2 x 2 iv. μ = β 0 + β 1 exp ( β 2 x ) Models (i) (iii) are linear in the parameters since the derivatives of μ with re- spect to the parameters β i , μ/ ∂β i , do not depend on the parameters. Model (iv) is nonlinear in the parameters since the derivatives μ/ ∂β 1 = exp ( β 2 x ) and μ/ ∂β 2 = β 1 x exp ( β 2 x ) depend on the parameters. The model in Eqs. (1.10a) and (1.10b) can be extended in many different ways. First, the functional relationship may be nonlinear, and we may consider a model such as that in Eq. (1.11iv) to describe the nonlinear pattern. Second, we may suppose that V ( y ) = σ 2 ( x ) is a function of the explanatory variables.