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Linear Models and Linear Regression 357

Confidence intervals for the regression parameters in this case can also be
established following similar procedures employed in the case of simple linear
regression. Concerning hypotheses testing, it was mentioned in Section 11.1.5
that testing of simultaneous hypotheses is more appropriate in multiple linear
regression, and that we will not pursue it here.

11.3 OTHER REGRESSION MODELS

In science and engineering, one often finds it necessary to consider regression
models that are nonlinear in the independent variables. Common examples of
this class of models include
2
Y ˆ 0 ‡ 1 x ‡ 2 x ‡ E; 11:53†
Y ˆ 0 exp 1 x ‡ E†; 11:54†
2
2
Y ˆ 0 ‡ 1 x 1 ‡ 2 x 2 ‡ 11 x ‡ 22 x ‡ 12 x 1 x 2 ‡ E; 11:55†
1
2
x
Y ˆ 0 ‡ E: 11:56†
1
Polynomial models such as Equation (11.53) or Equation (11.55) are still
linear regression models in that they are linear in the unknown parameters
0 , 2 ,..., [etc. Hence, they can be estimated by using multiple linear
1 ,
2
regression techniques. Indeed, let x 1 ˆ x, and x 2 ˆ x in Equation (11.53), it
takes the form of a multiple linear regression model with two independent
variables and can thus be analyzed as such. Similar equivalence can be estab-
lished between Equation (11.55) and a multiple linear regression model with
five independent variables.
Consider the exponential model given by Equation (11.54). Taking logar-
ithms of both sides, we have

ln Y ˆ ln 0 ‡ 1 x ‡ E: 11:57†

In terms of random variable ln Y , Equation (11.57) represents a linear regres-
sion equation with regression coefficients ln 0 and 1 . Linear regression tech-
niques again apply in this case. Equation (11.56), however, cannot be conveniently
put into a linear regression form.

Example 11.7. Problem: on average, the rate of population increase (Y) asso-

ciated with a given city varies with x, the number of years after 1970. Assuming that
2
EfYgˆ 0 ‡ 1 x ‡ 2 x ;
compute the least-square estimates for 0 , 1 ,and 2 based on the data pre-
sented in Table 11.5.

TLFeBOOK