1. Chapter 10 Nonlinear Models Undergraduated Econometrics Page 1

2. 10.1 Polynomial and Interaction Variables
Chapter Contents
10.1 Polynomial and Interaction Variables
10.2 A Simple Nonlinear-in-the-Parameters Model
10.3 A Logistic Growth Curve
10.4 Poisson Regression
Undergraduated Econometrics
Page 2
Chapter 10: Nonlinear Models

3. Polynomial and Interaction Variables
10.1
Polynomial and Interaction Variables
Undergraduated Econometrics
Page 3
Chapter 10: Nonlinear Models

4. Nonlinear models can be classifies into two categories.
10.1
Nonlinear Models
Nonlinear models can be classifies into two categories.
In the first category are models that are nonlinear in the variables, but still linear in terms of unknown parameters.
The second category of nonlinear models contains models that are nonlinear in the parameters and cannot be made linear in the parameters after a transformation.
Undergraduated Econometrics
Page 4
Chapter 10: Nonlinear Models

5. Consider the average cost equation
10.1
Nonlinear Models
10.1.1
Polynomial Terms In A Regression Model
Consider the average cost equation
And the total cost function Eq Eq
Undergraduated Econometrics
Page 5
Chapter 10: Nonlinear Models

6. FIGURE 10.1 (a) Total cost curve and (b) total product curve
Nonlinear Models
FIGURE 10.1 (a) Total cost curve and (b) total product curve
10.1.1
Polynomial Terms In A Regression Model
Undergraduated Econometrics
Page 6
Chapter 10: Nonlinear Models

7. 10.1
Nonlinear Models
FIGURE 10.2 Average and marginal (a) cost curves and (b) product curves
10.1.1
Polynomial Terms In A Regression Model
Undergraduated Econometrics
Page 7
Chapter 10: Nonlinear Models

8. The slope of the average cost curve Eq. 10.1.1 is:
Nonlinear Models
10.1.1
Polynomial Terms In A Regression Model
The slope of the average cost curve Eq is:
For this U-shaped curve, we expect β2 < 0 and β3 > 0 Eq
Undergraduated Econometrics
Page 8
Chapter 10: Nonlinear Models

9. The slope of the average cost curve Eq. 10.1.2 is:
Nonlinear Models
10.1.1
Polynomial Terms In A Regression Model
The slope of the average cost curve Eq is:
For a U-shaped marginal cost curve, we expect the parameter signs to be α2 > 0, α3 < 0, and α4 > 0 Eq
Undergraduated Econometrics
Page 9
Chapter 10: Nonlinear Models

10. An initial model would be:
10.1
Nonlinear Models
10.1.2
Interaction Between Two Continuous Variables
Suppose that we wish to study the effect of income and age on an individual’s expenditure on pizza
An initial model would be: Eq
Undergraduated Econometrics
Page 10
Chapter 10: Nonlinear Models

11. Implications of this model are:
10.1
Nonlinear Models
10.1.2
Interaction Between Two Continuous Variables
Implications of this model are:
: For a given level of income, the expected expenditure on pizza changes by the amount β2 with an additional year of age
: For individuals of a given age, an increase in income of $1,000 increases expected expenditures on pizza by β3
Undergraduated Econometrics
Page 11
Chapter 10: Nonlinear Models

12. Table 10.1 Pizza Expenditure Data
Nonlinear Models
Table 10.1 Pizza Expenditure Data
10.1.2
Interaction Between Two Continuous Variables
Undergraduated Econometrics
Page 12
Chapter 10: Nonlinear Models

13. That is, the effect of one variable is modified by another
10.1
Nonlinear Models
10.1.2
Interaction Between Two Continuous Variables
It is not reasonable to expect that, regardless of the age of the individual, an increase in income by $1 should lead to an increase in pizza expenditure by β3 dollars?
It would seem more reasonable to assume that as a person grows older, his or her marginal propensity to spend on pizza declines
That is, as a person ages, less of each extra dollar is expected to be spent on pizza
This is a case in which the effect of income depends on the age of the individual.
That is, the effect of one variable is modified by another
One way of accounting for such interactions is to include an interaction variable that is the product of the two variables involved
Undergraduated Econometrics
Page 13
Chapter 10: Nonlinear Models

14. 10.1
Nonlinear Models
10.1.2
Interaction Between Two Continuous Variables
We will add the interaction variable (AGE x INCOME) to the regression model
The new model is: Eq
Undergraduated Econometrics
Page 14
Chapter 10: Nonlinear Models

15. Implications of this revised model are: 1. 2.
10.1
Nonlinear Models
10.1.2
Interaction Between Two Continuous Variables
Implications of this revised model are: 1. 2.
Undergraduated Econometrics
Page 15
Chapter 10: Nonlinear Models

16. The estimated model is:
10.1
Nonlinear Models
10.1.2
Interaction Between Two Continuous Variables
The estimated model is:
Undergraduated Econometrics
Page 16
Chapter 10: Nonlinear Models

17. 10.1
Nonlinear Models
10.1.2
Interaction Between Two Continuous Variables
The estimated marginal effect of age upon pizza expenditure for two individuals—one with $25,000 income and one with $90,000 income is:
We expect that an individual with $25,000 income will reduce pizza expenditures by $6.06 per year, whereas the individual with $90,000 income will reduce pizza expenditures by $14.07 per year
Undergraduated Econometrics
Page 17
Chapter 10: Nonlinear Models

18. A Simple Nonlinear-in-the- Parameters Model
10.2
A Simple Nonlinear-in-the- Parameters Model
Undergraduated Econometrics
Page 18
Chapter 10: Nonlinear Models

19. Begin with the following artificial example
10.2
A Simple Nonlinear-in-the-Parameters Model
We turn to models that are nonlinear in the parameters and which need to be estimated by a technique called nonlinear least squares.
Begin with the following artificial example Eq
Undergraduated Econometrics
Page 19
Chapter 10: Nonlinear Models

20. 10.2
A Simple Nonlinear-in-the-Parameters Model
We set up a sum of squared errors function that, in the context of (10.2.1) Eq
Undergraduated Econometrics
Page 20
Chapter 10: Nonlinear Models

21. Figure 10.3 Sum of squares function for single-parameter example
10.2
A Simple Nonlinear-in-the-Parameters Model
Figure 10.3 Sum of squares function for single-parameter example
Undergraduated Econometrics
Page 21
Chapter 10: Nonlinear Models

22. 10.2
A Simple Nonlinear-in-the-Parameters Model
Using nonlinear least squares software, we find that the nonlinear least squares estimate and its standard error are
Undergraduated Econometrics
Page 22
Chapter 10: Nonlinear Models

23. A Logistic Growth Curve
10.3
A Logistic Growth Curve
Undergraduated Econometrics
Page 23
Chapter 10: Nonlinear Models

24. 10.3
A Logistic Growth Curve
A model that is popular for modeling the diffusion of technological change is the logistic growth curve
yt is the adoption proportion of a new technology. There is only one explanatory variable on the right-hand side, namely, time, t=1, 2…., T. Eq
Undergraduated Econometrics
Page 24
Chapter 10: Nonlinear Models

25. Figure 10.4 Logistic growth curve
10.3
A Logistic Growth Curve
Figure 10.4 Logistic growth curve
Undergraduated Econometrics
Page 25
Chapter 10: Nonlinear Models

26. 10.3
A Logistic Growth Curve
Using nonlinear least squares to estimate the logistic growth curve yields the results in Table We find that the estimated saturation share of the EAF technology is = The point of inflection, where the rate of adoption changes form increasing to decreasing, is estimated as
Undergraduated Econometrics
Page 26
Chapter 10: Nonlinear Models

27. Table10.4 Estimated Growth Curve for EAF Shre of Steel
10.3
A Logistic Growth Curve
Table10.4 Estimated Growth Curve for EAF Shre of Steel
Undergraduated Econometrics
Page 27
Chapter 10: Nonlinear Models

28. 10.3
A Logistic Growth Curve
Suppose that you wanted to test the hypothesis that the point of inflection actually occurred in The corresponding null and alternative hypotheses can be written as
From the very small p-values associated with both the F and the X²-statistics, we reject H0 and conclude that the point of inflection does not occur at 1980.
Undergraduated Econometrics
Page 28
Chapter 10: Nonlinear Models

29. 10.4 Poisson Regression Undergraduated Econometrics Page 29
Chapter 10: Nonlinear Models

30. 10.4
Poisson Regression
A distribution more suitable for count data is the Poisson distribution. Its probability density function is given by
Undergraduated Econometrics
Page 30
Chapter 10: Nonlinear Models

31. 10.4
Poisson Regression
A distribution more suitable for count data is the Poisson distribution. Its probability density function is given by
y= the number of times a household visits Lake Keepit per year
μ= the average number of visits per year for all households
y!= y×(y-1)×(y-2)×…×2×1. Eq
Undergraduated Econometrics
Page 31
Chapter 10: Nonlinear Models

32. 10.4
Poisson Regression
In Poisson regression, we improve on ( ) by recognizing that the mean μ is likely to depend on various household characteristics. Eq
Undergraduated Econometrics
Page 32
Chapter 10: Nonlinear Models

33. Recall that, in the simple linear regression model, we can write
10.4
Poisson Regression
Recall that, in the simple linear regression model, we can write
Equation (10.4.4) is linear in parameters which can be estimated via nonlinear least squares. Eq Eq
Undergraduated Econometrics
Page 33
Chapter 10: Nonlinear Models

34. Table 10.6 Estimated Model for Visits to Lake Keepit
10.4
Poisson Regression
Table 10.6 Estimated Model for Visits to Lake Keepit
Undergraduated Econometrics
Page 34
Chapter 10: Nonlinear Models

35. First we compute an estimate of the mean for this household
10.4
Poisson Regression
First we compute an estimate of the mean for this household
Then using the Poisson distribution, we have
Undergraduated Econometrics
Page 35
Chapter 10: Nonlinear Models