Chapter 10 Nonlinear Models Undergraduated Econometrics Page 1

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

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

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

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. 10.1.1 Eq. 10.1.2 Undergraduated Econometrics Page 5 Chapter 10: Nonlinear Models

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

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

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. 10.1.1 is: For this U-shaped curve, we expect β2 < 0 and β3 > 0 Eq. 10.1.3 Undergraduated Econometrics Page 8 Chapter 10: Nonlinear Models

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. 10.1.2 is: For a U-shaped marginal cost curve, we expect the parameter signs to be α2 > 0, α3 < 0, and α4 > 0 Eq. 10.1.4 Undergraduated Econometrics Page 9 Chapter 10: Nonlinear Models

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. 10.1.5 Undergraduated Econometrics Page 10 Chapter 10: Nonlinear Models

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

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

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

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. 10.1.6 Undergraduated Econometrics Page 14 Chapter 10: Nonlinear Models

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

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

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

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

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. 10.2.1 Undergraduated Econometrics Page 19 Chapter 10: Nonlinear Models

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. 10.2.2 Undergraduated Econometrics Page 20 Chapter 10: Nonlinear Models

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

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

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

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. 10.3.1 Undergraduated Econometrics Page 24 Chapter 10: Nonlinear Models

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

10.3 A Logistic Growth Curve Using nonlinear least squares to estimate the logistic growth curve yields the results in Table 10.4. We find that the estimated saturation share of the EAF technology is = 0.46. 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

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

10.3 A Logistic Growth Curve Suppose that you wanted to test the hypothesis that the point of inflection actually occurred in 1980. 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

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

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

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. 10.4.1 Undergraduated Econometrics Page 31 Chapter 10: Nonlinear Models

10.4 Poisson Regression In Poisson regression, we improve on (10..4.1) by recognizing that the mean μ is likely to depend on various household characteristics. Eq. 10.4.2 Undergraduated Econometrics Page 32 Chapter 10: Nonlinear Models

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. 10.4.3 Eq. 10.4.4 Undergraduated Econometrics Page 33 Chapter 10: Nonlinear Models

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

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