Exam 8: Nonlinear Regression Functions

Females,it is said,make 70 cents to the dollar in the United States.To investigate this phenomenon,you collect data on weekly earnings from 1,744 individuals,850 females and 894 males.Next,you calculate their average weekly earnings and find that the females in your sample earned $346.98,while the males made $517.70. (a)Calculate the female earnings in percent of the male earnings.How would you test whether or not this difference is statistically significant? Give two approaches. (b)A peer suggests that this is consistent with the idea that there is discrimination against females in the labor market.What is your response? (c)You recall from your textbook that additional years of experience are supposed to result in higher earnings.You reason that this is because experience is related to "on the job training." One frequently used measure for (potential)experience is "Age-Education-6." Explain the underlying rationale.Assuming,heroically,that education is constant across the 1,744 individuals,you consider regressing earnings on age and a binary variable for gender.You estimate two specifications initially: = 323.70 + 5.15 × Age - 169.78 × Female,R2=0.13,SER=274.75 (21.18)(0.55)(13.06) = 5.44 + 0.015 × Age - 0.421 × Female,R2=0.17,SER=0.75 (0.08)(0.002)(0.036) where Earn are weekly earnings in dollars,Age is measured in years,and Female is a binary variable,which takes on the value of one if the individual is a female and is zero otherwise.Interpret each regression carefully.For a given age,how much less do females earn on average? Should you choose the second specification on grounds of the higher regression R2? (d)Your peer points out to you that age-earning profiles typically take on an inverted U-shape.To test this idea,you add the square of age to your log-linear regression. = 3.04 + 0.147 × Age - 0.421 × Female - 0.0016 Age2, (0.18)(0.009)(0.033)(0.0001) R2 = 0.28,SER = 0.68 Interpret the results again.Are there strong reasons to assume that this specification is superior to the previous one? Why is the increase of the Age coefficient so large relative to its value in (c)? (e)What other factors may play a role in earnings determination?

Including an interaction term between two independent variables,X1 and X2,allows for the following except:

In the model Yi = β0 + β1X1 + β2X2 + β3(X1 × X2)+ ui,the expected effect is

In the regression model Yi = β0 + β1Xi + β2Di + β3(Xi × Di)+ ui,where X is a continuous variable and D is a binary variable,β2

An example of the interaction term between two independent,continuous variables is

To decide whether Yi = β0 + β1X + ui or ln(Yi)= β0 + β1X + ui fits the data better,you cannot consult the regression R2 because

Sports economics typically looks at winning percentages of sports teams as one of various outputs,and estimates production functions by analyzing the relationship between the winning percentage and inputs.In Major League Baseball (MLB),the determinants of winning are quality pitching and batting.All 30 MLB teams for the 1999 season.Pitching quality is approximated by "Team Earned Run Average" (ERA),and hitting quality by "On Base Plus Slugging Percentage" (OPS). Summary of the Distribution of Winning Percentage,On Base Plus Slugging Percentage, and Team Earned Run Average for MLB in 1999 Your regression output is: = -0.19 - 0.099 × teamera + 1.490 × ops,R2=0.92,SER = 0.02. (0.08)(0.008)(0.126) (a)Interpret the regression.Are the results statistically significant and important? (b)There are two leagues in MLB,the American League (AL)and the National League (NL).One major difference is that the pitcher in the AL does not have to bat.Instead there is a "designated hitter" in the hitting line-up.You are concerned that,as a result,there is a different effect of pitching and hitting in the AL from the NL.To test this hypothesis,you allow the AL regression to have a different intercept and different slopes from the NL regression.You therefore create a binary variable for the American League (DAL)and estimate the following specification: = - 0.29 + 0.10 × DAL - 0.100 × teamera + 0.008 × (DAL× teamera) (0.12)(0.24)(0.008)(0.018) + 1.622*ops - 0.187 *(DAL× ops),R2=0.92,SER = 0.02. (0.163)(0.160) What is the regression for winning percentage in the AL and NL? Next,calculate the t-statistics and say something about the statistical significance of the AL variables.Since you have allowed all slopes and the intercept to vary between the two leagues,what would the results imply if all coefficients involving DAL were statistically significant? (c)You remember that sequentially testing the significance of slope coefficients is not the same as testing for their significance simultaneously.Hence you ask your regression package to calculate the F-statistic that all three coefficients involving the binary variable for the AL are zero.Your regression package gives a value of 0.35.Looking at the critical value from you F-table,can you reject the null hypothesis at the 1% level? Should you worry about the small sample size?

There has been much debate about the impact of minimum wages on employment and unemployment.While most of the focus has been on the employment-to-population ratio of teenagers,you decide to check if aggregate state unemployment rates have been affected.Your idea is to see if state unemployment rates for the 48 contiguous U.S.states in 1985 can predict the unemployment rate for the same states in 1995,and if this prediction can be improved upon by entering a binary variable for "high impact" minimum wage states.One labor economist labeled states as high impact if a large fraction of teenagers was affected by the 1990 and 1991 federal minimum wage increases.Your first regression results in the following output: = 3.19 + 0.27 × ,R2 = 0.21,SER = 1.031 (0.56)(0.07) (a)Sketch the regression line and add a 450 line to the graph.Interpret the regression results.What would the interpretation be if the fitted line coincided with the 450 line? (b)Adding the binary variable DhiImpact by allowing the slope and intercept to differ,results in the following fitted line: = 4.02 + 0.16 × - 3.25 × DhiImpact + 0.38 × (DhiImpact× ), (0.66)(0.09)(0.89)(0.11) R2 = 0.31,SER=0.987 The F-statistic for the null hypothesis that both parameters involving the high impact minimum wage variable are zero,is 42.16.Can you reject the null hypothesis that both coefficients are zero? Sketch the two regression lines together with the 450 line and interpret the results again. (c)To check the robustness of these results,you repeat the exercise using a new binary variable for the so-called mining state (Dmining),i.e. ,the eleven states that have at least three percent of their total state earnings derived from oil,gas extraction,and coal mining,in the 1980s.This results in the following output: = 4.04 + 0.15× - 2.92 × Dmining + 0.37 × (Dmining × ), (0.65)(0.09)(0.90)(0.10) R2 = 0.31,SER=0.997 How confident are you that the previously found effect is due to minimum wages?

The binary variable interaction regression

The figure shows is a plot and a fitted linear regression line of the age-earnings profile of 1,744 individuals,taken from the Current Population Survey. (a)Describe the problems in predicting earnings using the fitted line.What would the pattern of the residuals look like for the age category under 40? (b)What alternative functional form might fit the data better? (c)What other variables might you want to consider in specifying the determinants of earnings?

Table 8.1 on page 284 of your textbook displays the following estimated earnings function in column (4): = 1.503 + 0.1032 × educ - 0.451 × DFemme + 0.0143 × (DFemme×educ) (0.023)(0.0012)(0.024)(0.0017) + 0.0232 × exper - 0.000368 × exper2 - 0.058 × Midwest - 0.0098 × South - 0.030 × West (0.0012)(0.000023)(0.006)(0.006)(0.007) n = 52.790, 2 = 0.267 Given that the potential experience variable (exper)is defined as (Age-Education-6)find the age at which individuals with a high school degree (12 years of education)and with a college degree (16 years of education)have maximum earnings,holding all other factors constant.

The textbook shows that ln(x + Δx)- ln(x)≅ .Show that this is equivalent to the following approximation ln(1 + y)≅ y if y is small.You use this idea to estimate a demand for money function,which is of the form m = β0 × ×, × eu where m is the quantity of (real)money,GDP is the value of (real)Gross Domestic Product,and R is the nominal interest rate.You collect the quarterly data from the Federal Reserve Bank of St.Louis data bank ("FRED"),which lists the money supply and GDP in billions of dollars,prices as an index,and nominal interest rates in percentage points per year You generate the variables in your regression program as follows: m = (money supply)/price index;GDP = (Gross Domestic Product/Price Index),and R = nominal interest rate in percentage points per annum.Next you perform the log-transformations on the real money supply,real GDP,and on (1+R).Can you for see a problem in using this transformation?

An example of a quadratic regression model is

Indicate whether or not you can linearize the regression functions below so that OLS estimation methods can be applied: (a)Yi = (b)Yi = + ui

Show that for the following regression model Yt = where t is a time trend,which takes on the values 1,2,…,T,β1 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 Yt = Y0 × (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.

You have collected data for a cross-section of countries in two time periods,1960 and 1997,say.Your task is to find the determinants for the Wealth of a Nation (per capita income)and you believe that there are three major determinants: investment in physical capital in both time periods (X1,T and X1,0),investment in human capital or education (X2,T and X2,0),and per capita income in the initial period (Y0).You run the following regression: ln(YT)= β0 + β1X1,T + β2X1,0 + β3X2,T + β4X1,0 + ln(Y0)+ uT One of your peers suggests that instead,you should run the growth rate in per capita income over the two periods on the change in physical and human capital.For those results to be a parsimonious presentation of your initial regression,what three restrictions would have to hold? How would you test for these? The same person also points out to you that the intercept vanishes in equations where the data is differenced.Is that correct?

In the case of regression with interactions,the coefficient of a binary variable should be interpreted as follows:

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 ( + β0)= β1 × × eu, where is the percentage change in money wages and ur is the unemployment rate.Sketch the function.What role does β0 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 β1 and β2 using OLS by choosing different values for β0 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 find the determinants of earnings,using both continuous and binary variables.One of the central questions analyzed in this relationship is the returns to education. (a)Collecting data from 253 individuals,you estimate the following relationship = 0.54 + 0.083 × Educ,R2 = 0.20,SER = 0.445 (0.14)(0.011) where Earn is average hourly earnings and Educ is years of education. What is the effect of an additional year of schooling? If you had a strong belief that years of high school education were different from college education,how would you modify the equation? What if your theory suggested that there was a "diploma effect"? (b)You read in the literature that there should also be returns to on-the-job training.To approximate on-the-job training,researchers often use the so called Mincer or potential experience variable,which is defined as Exper = Age - Educ - 6.Explain the reasoning behind this approximation.Is it likely to resemble years of employment for various sub-groups of the labor force? (c)You incorporate the experience variable into your original regression = -0.01 + 0.101 × Educ + 0.033 × Exper - 0.0005 × Exper2, (0.16)(0.012)(0.006)(0.0001) R2 = 0.34,SER = 0.405 What is the effect of an additional year of experience for a person who is 40 years old and had 12 years of education? What about for a person who is 60 years old with the same education background? (d)Test for the significance of each of the coefficients of the added variables.Why has the coefficient on education changed so little? Sketch the age-(log)earnings profile for workers with 8 years of education and 16 years of education. (e)You want to find the effect of introducing two variables,gender and marital status.Accordingly you specify a binary variable that takes on the value of one for females and is zero otherwise (Female),and another binary variable that is one if the worker is married but is zero otherwise (Married).Adding these variables to the regressors results in: = 0.21 + 0.093 × Educ + 0.032 × Exper - 0.0005 × Exper2 (0.16)(0.012)(0.006)(0.0001) - 0.289 × Female + 0.062 Married, (0.049)(0.056) R2 = 0.43,SER = 0.378 Are the coefficients of the two added binary variables individually statistically significant? Are they economically important? In percentage terms,how much less do females earn per hour,controlling for education and experience? How much more do married people make? What is the percentage difference in earnings between a single male and a married female? What is the marriage differential between males and females? (f)In your final specification,you allow for the binary variables to interact.The results are as follows: = 0.14 + 0.093 × Educ + 0.032 × Exper - 0.0005 × Exper2 (0.16)(0.011)(0.006)(0.001) - 0.158 × Female + 0.173 × Married - 0.218 × (Female × Married), (0.075)(0.080)(0.097) R2 = 0.44,SER = 0.375 Repeat the exercise in (e)of calculating the various percentage differences between gender and marital status.