Exam 8: Nonlinear Regression Functions

Consider the polynomial regression model of degree Y_i = ?₀ + ?₁X_i + ?₂ $x _ { i } ^ { 2 }$ + ...+ ?_r $x _ { i } ^ { r }$ + u_i. According to the null hypothesis that the regression is linear and the alternative that is a polynomial of degree r corresponds to

In the log-log model, the slope coefficient indicates

Suggest a transformation in the variables that will linearize the deterministic part of the population regression functions below. Write the resulting regression function in a form that can be estimated by using OLS. (a)Y_i = β₀ $X _ { 1 i } ^ { \beta _ { 1 } }$ $x _ { 2 i } ^ { \beta _ { 2 } }$ (b)Y_i = $\frac { X _ { i } } { \beta _ { 0 } + \beta _ { 1 } X _ { i } }$ (c)Y_i = $\frac { e ^ { \beta _ { 0 } + \beta _ { 1 } X _ { 1 } } } { 1 + e ^ { \beta _ { 0 } + \beta _ { 1 } X _ { 1 } } }$ (d)Y_i = β₀ $X _ { 1 i } ^ { \beta _ { 1 } }$ ${ e } ^ { \beta _ { 2 } \beta _ { 2 } X _ { 2 i } }$

Choose at least three different nonlinear functional forms of a single independent variable and sketch the relationship between the dependent and independent variable.

In the regression model Y_i = β₀ + β₁X_i + β₂D_i + β₃(X_i × D_i)+ u_i, where X is a continuous variable and D is a binary variable, β₃

Assume that you had estimated the following quadratic regression model $\widehat{\text { TestScore }}$ = 607.3 + 3.85 Income - 0.0423 Income². If income increased from 10 to 11 ($10,000 to $11,000), then the predicted effect on testscores would be

Your task is to estimate the ice cream sales for a certain chain in New England. The company makes available to you quarterly ice cream sales (Y)and informs you that the price per gallon has approximately remained constant over the sample period. You gather information on average daily temperatures (X)during these quarters and regress Y on X, adding seasonal binary variables for spring, summer, and fall. These variables are constructed as follows: DSpring takes on a value of 1 during the spring and is zero otherwise, DSummer takes on a value of 1 during the summer, etc. Specify three regression functions where the following conditions hold: the relationship between Y and X is (i)forced to be the same for each quarter; (ii)allowed to have different intercepts each season; (iii)allowed to have varying slopes and intercepts each season. Sketch the difference between (i)and (ii). How would you test which model fits the data the best?

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 M₁ and P is the GDP deflator), (real)gross domestic product (GDP)and the nominal interest rate (R). Next you consider estimating the demand for money using the following alternative functional forms: (i)m = β₀ + β₁ × GDP + β₂ x R+ u (ii)m = β₀ × $G D P ^ { \beta 1 }$ x ${ R } ^ { \beta 2 }$ × e^u (iii)m = β₀ × $G D P ^ { \beta 1 }$ x $1 + R ^ { \beta _ { 2 } }$ × e^u Give an interpretation for β₁ and β₂ in each case. How would you calculate the income elasticity in case (i)?

For the polynomial regression model,

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\times ERA, R2=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. (b)Are the individual coefficients statistically significant? Indicate the level of significance you used and the type of alternative hypothesis you considered. (c)Although you are quite impressed with the fit of the regression, someone suggests that you should include the square of years and innings as additional explanatory variables. Your results change as follows: =12.15+0.160\times Years +0.00268\times Innings +0.0063\times Saves (0.05) (0.039)(0.00030)(0.0010) -0.0584\timesERA-0.0165\times Years 2-0.00000045\times (0.0165)(0.0026)(0.00000012) =0.69,SER=0.666 What is her reasoning? Are the coefficients of the quadratic terms statistically significant? Are they meaningful? (d)Calculate the effect of moving from two to three years, as opposed to from 12 to 13 years. (e)You also decide to test the specification for stability across leagues (National League and American League)by including a dummy variable for the National League and allowing the intercept and all slopes to differ. The resulting F-statistic for restricting all coefficients that involve the National League dummy variable to zero, is 0.40. Compare this to the relevant critical value from the table and decide whether or not these additional variables should be included.

In the regression model Y_i = β₀ + β₁X_i + β₂D_i + β₃(X_i × D_i)+ 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 model ln(Y_i)= ?₀ + ?₁X_i + u_i, the elasticity of E(Y|X)with respect to X is

The interpretation of the slope coefficient in the model ln(Y_i)= β₀ + β₁ ln(X_i)+ u_i is as follows:

The interpretation of the slope coefficient in the model ln(Y_i)= β₀ + β₁X_i + u_i is as follows:

You have estimated an earnings function, where you regressed the log of earnings on a set of continuous explanatory variables (in levels)and two binary variables, one for gender and the other for marital status. One of the explanatory variables is education. (a)Interpret the education coefficient. (b)Next, specify the binary variables and an equation, where the default is a single male, without allowing for interaction between marital status and gender. Indicate the coefficients that measure the effect of a single male, single female, married male, and married female. (c)Finally allow for an interaction between the gender and marital status binary variables. Repeat the exercise of writing down the various effects based on the female/male and single/married status. Why is the latter approach more general than the former?

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\timesU,=0.21,=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\timesU-3.25\times Dhilmpact +0.38\times DhiImpact \timesU , (0.66) (0.09) (0.89)(0.11) =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: $U r_{i}^{95}=4.04+0.15 \times U r_{i}^{85}-2.92 \times \text { Dmining }+0.37 \times\left(\text { Dmining } \times U r_{i}^{85}\right) \text {, }$ $\quad$$\quad$$\quad$$\quad$ (0.65) (0.09) $\quad$$\quad$$\quad$$\quad$$\quad$ (0.90) $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$ (0.10) $R ^ { 2 } = 0.31 , S E R = 0.997$ How confident are you that the previously found effect is due to minimum wages?

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, $\widehat{\text { TFoverM }}$ = 4.42 - 0.0017 × 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?

You have estimated the following equation: $\widehat{\text { TestScore }}$ = 607.3 + 3.85 Income - 0.0423 Income², where TestScore is the average of the reading and math scores on the Stanford 9 standardized test administered to 5^th grade students in 420 California school districts in 1998 and 1999. Income is the average annual per capita income in the school district, measured in thousands of 1998 dollars. The equation

The textbook shows that ln(x + Δx)- ln(x)≅ $\frac { \Delta x } { x }$ 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 × $G D P ^ { \beta 1 }$ ×, $( 1 + R ) ^ { \beta _ { 1 } }$ × e^u 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 the interaction term between two independent, continuous variables is