Exam 8: Nonlinear Regression Functions

The best way to interpret polynomial regressions is to

In the regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } D _ { i } + \beta _ { 3 } \left( X _ { i } \times D _ { i } \right) + u _ { i }$ where X is a continuous variable and D is a binary variable, $\beta _ { 3 }$

One of the most frequently estimated equations in the macroeconomics growth literature are so-called convergence regressions.In essence the average per capita income growth rate is regressed on the beginning-of-period per capita income level to see if countries that were further behind initially, grew faster.Some macroeconomic models make this prediction, once other variables are controlled for.To investigate this matter, you collect data from 104 countries for the sample period 1960-1990 and estimate the following relationship (numbers in parentheses are for heteroskedasticity-robust standard errors): =0.020-0.360\times gpop +0.004\times Educ -0.053\times,=0.332, SER =0.013 (0.009)(0.241)(0.001)(0.009) where g6090 is the growth rate of GDP per worker for the 1960-1990 sample period, RelProd60 is the initial starting level of GDP per worker relative to the United States in 1960, gpop is the average population growth rate of the country, and Educ is educational attainment in years for 1985. (a)What is the effect of an increase of 5 years in educational attainment? What would happen if a country could implement policies to cut population growth by one percent? Are all coefficients significant at the 5% level? If one of the coefficients is not significant, should you automatically eliminate its variable from the list of explanatory variables?

Show that for the following regression model $Y _ { t } = e ^ { \beta _ { 0 } + \beta _ { 1 } x d + u }$ where $t$ is a time trend, which takes on the values $1,2 , \ldots , T , \beta _ { 1 }$ represents the instantaneous ("continuous compounding") growth rate. Show how this rate is related to the proportionate rate of growth, which is calculated from the relationship $Y _ { t } = Y _ { 0 } \times ( 1 + g ) ^ { t }$ when time is measured in discrete intervals.

Give at least three examples from economics where you expect some nonlinearity in the relationship between variables.Interpret the slope in each case.

In the case of perfect multicollinearity, OLS is unable to estimate the slope coefficients of the variables involved. Assume that you have included both $X _ { 1 } \text { and } X _ { 2 }$ as explanatory variables, and that $X _ { 2 } = X _ { 1 } ^ { 2 }$ , so that there is an exact relationship between two explanatory variables. Does this pose a problem for estimation?

For the polynomial regression model, a. you need new estimation techniques since the OLS assumptions do not apply any longer. b. the techniques for estimation and inference developed for multiple regression can be applied. c. you can still use OLS estimation techniques, but the $t$ -statistics do not have an asymptotic normal distribution. d. the critical values from the normal distribution have to be changed to $1.96 ^ { 2 } , 1.96 ^ { 3 }$ , etc.

You have been asked by your younger sister to help her with a science fair project. During the previous years she already studied why objects float and there also was the inevitable volcano project. Having learned regression techniques recently, you suggest that she investigate the weight-height relationship of $4 ^ { \text {th } }$ to $6 ^ { \text {th } }$ graders. Her presentation topic will be to explain how people at carnivals predict weight. You collect data for roughly 100 boys and girls between the ages of nine and twelve and estimate for her the following relationship: $\widehat { \text { Weight } } = 45.59 + 4.32 \times$ Height $4 , \quad R ^ { 2 } = 0.55$ , SER $= 15.69$ $\quad$$\quad$$\quad$$\quad$$\quad$$( 3.81 )\quad ( 0.46 )$ where Weight is in pounds, and Height 4 is inches above 4 feet. (a)Interpret the results.

Being a competitive female swimmer, you wonder if women will ever be able to beat the time of the male gold medal winner.To investigate this question, you collect data for the Olympic Games since 1910.At first you consider including various distances, a binary variable for Mark Spitz, and another binary variable for the arrival and presence of East German female swimmers, but in the end decide on a simple linear regression.Your dependent variable is the ratio of the fastest women's time to the fastest men's time in the 100 m backstroke, and the explanatory variable is the year of the Olympics.The regression result is as follows, $\widehat { \text { TFover } M } = 4.42 - 0.0017 \times \text { Olympics, }$ where TFoverM is the relative time of the gold medal winner, and Olympics is the year of the Olympic Games.What is your prediction when females will catch up to men in this discipline? Does this sound plausible? What other functional form might you want to consider?

After analyzing the age-earnings profile for 1,744 workers as shown in the figure, it becomes clear to you that the relationship cannot be approximately linear. You estimate the following polynomial regression model, controlling for the effect of gender by using a binary variable that takes on the value of one for females and is zero otherwise: = -795.90+82.93\times Age -1.69\timesAg+0.015\timesAg-0.0005\times (283.11)(29.29)(1.06)(0.016) -163.19 Female, =0.225, SER =259.78 (12.45) (a) Test for the significance of the $A _ { g e } e ^ { 4 }$ coefficient. Describe the general strategy to determine the appropriate degree of the polynomial.

(Requires Calculus) In the equation $\widehat { \text { TestScore } } = 607.3 + 3.85 \text { Income } - 0.0423 \text { Income } { } ^ { 2 }$ the following income level results in the maximum test score

Labor economists have extensively researched the determinants of earnings.Investment in human capital, measured in years of education, and on the job training are some of the most important explanatory variables in this research.You decide to apply earnings functions to the field of sports economics by finding the determinants for baseball pitcher salaries.You collect data on 455 pitchers for the 1998 baseball season and estimate the following equation using OLS and heteroskedasticity-robust standard errors: =12.45+0.052\times Years +0.00089\times Innings +0.0032\times Saves (0.08)(0.026)(0.00020) (0.0018) -0.0085\timesERA,=0.45, SER =0.874 (0.0168) where Earn is annual salary in dollars, Years is number of years in the major leagues, Innings is number of innings pitched during the career before the 1998 season, Saves is number of saves during the career before the 1998 season, and ERA is the earned run average before the 1998 season. (a)What happens to earnings when the pitcher stays in the league for one additional year? Compare the salaries of two relievers, one with 10 more saves than the other.What effect does pitching 100 more innings have on the salary of the pitcher? What effect does reducing his ERA by 1.5? Do the signs correspond to your expectations? Explain.

The interpretation of the slope coefficient in the model $\ln \left( Y _ { i } \right) = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$ is as follows:

In the regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } D _ { i } + \beta _ { 3 } \left( X _ { i } \times D _ { i } \right) + u _ { i }$ where X is a continuous variable and D is a binary variable, to test that the two regressions are identical, you must use the

In the regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } D _ { i } + \beta _ { 3 } \left( X _ { i } \times D _ { i } \right) + u _ { i }$ where X is a continuous variable and D is a binary variable, $\beta _ { 2 }$

Sketch for the log-log model what the relationship between Y and X looks like for various parameter values of the slope, i.e., $\beta _ { 1 } > 1 ; 0 < \beta _ { 1 } < 1 ; \beta _ { 1 } = ( - 1 )$

You have been told that the money demand function in the United States has been unstable since the late 1970 . To investigate this problem, you collect data on the real money supply ( m=M / P ; where M is $M _ { 1 }$ and P is the GDP deflator), (real) gross domestic product (G D P) and the nominal interest rate (R) . Next you consider estimating the demand for money using the following alternative functional forms: (i) $\quad m = \beta _ { 0 } + \beta _ { 1 } \times G D P + \beta _ { 2 } \times R + u$ (ii) $\quad m = \beta _ { 0 } \times G D P ^ { \beta _ { 1 } } \times R ^ { \beta _ { 2 } } \times e ^ { u }$ (iii) $\quad m = \beta _ { 0 } \times G D P ^ { \beta _ { 1 } } \times ( 1 + R ) ^ { \beta _ { 2 } } \times e ^ { u }$ Give an interpretation for $\beta _ { 1 } \text { and } \beta _ { 2 }$ in each case. How would you calculate the income elasticity in case (i)?

A nonlinear function

In estimating the original relationship between money wage growth and the unemployment rate, Phillips used United Kingdom data from 1861 to 1913 to fit a curve of the following functional form $\left( \frac { \dot { W } } { W } + \beta _ { 0 } \right) = \beta _ { 1 } \times u r ^ { \beta _ { 2 } } \times e ^ { u } ,$ where $\frac { \dot { W } } { W }$ is the percentage change in money wages and u r is the unemployment rate. Sketch the function. What role does $\beta _ { 0 }$ play? Can you find a linear transformation that allows you to estimate the above function using OLS? If, after taking logarithms on both sides of the equation, you tried to estimate $\beta _ { 1 } \text { and } \beta _ { 2 }$ using OLS by choosing different values for $\beta _ { 0 }$ by "trial and error procedure" (Phillips's words), what sort of problem might you run into with the left-hand side variable for some of the observations?

Earnings functions attempt to predict the log of earnings from a set of explanatory variables, both binary and continuous.You have allowed for an interaction between two continuous variables: education and tenure with the current employer.Your estimated equation is of the following type: $\widehat { \ln ( \text { Earn } ) } = \widehat { \beta _ { 0 } } + \widehat { \beta _ { 1 } } \times \text { Femme } + \widehat { \beta _ { 2 } } \times \text { Educ } + \widehat { \beta _ { 3 } } \times \text { Tenure } + \widehat { \beta _ { 4 } } \times ( \text { Educ } \times \text { Tenure } ) + \bullet$ where Femme is a binary variable taking on the value of one for females and is zero otherwise, Educ is the number of years of education, and tenure is continuous years of work with the current employer.What is the effect of an additional year of education on earnings ("returns to education")for men? For women? If you allowed for the returns to education to differ for males and females, how would you respecify the above equation? What is the effect of an additional year of tenure with a current employer on earnings?