Exam 8: Nonlinear Regression Functions

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: $\widehat { U r _ { i } ^ { 95 } } = 3.19 + 0.27 \times U r _ { i } ^ { 85 } , R ^ { 2 } = 0.21 , S E R = 1.031$ $\quad$$\quad$$\quad$$(0.56) \quad(0.07)$ (a) Sketch the regression line and add a $45 ^ { \circ }$ line to the graph. Interpret the regression results. What would the interpretation be if the fitted line coincided with the $45 ^ { \circ }$ line?

The textbook shows that $\ln ( x + \Delta x ) - \ln ( x ) \cong \frac { \Delta x } { x }$ Show that this is equivalent to the following approximation $\ln ( 1 + y ) \cong y$ if y is small. You use this idea to estimate a demand for money function, which is of the form $m = \beta _ { 0 } \times G D P ^ { \beta _ { 1 } } \times ( 1 + R ) ^ { \beta _ { 2 } } \times e ^ { u } ,$ where m is the quantity of (real) money, G D P 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 G D P , and on (1+R) . Can you for see a problem in using this transformation?

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

The following are properties of the logarithm function with the exception of a. $\ln ( 1 / x ) = - \ln ( x )$ . b. $\ln ( a + x ) = \ln ( a ) + \ln ( x )$ . c. $\ln ( a x ) = \ln ( a ) + \ln ( x )$ . d. $\ln \left( x ^ { a } \right) a \ln ( x )$ .

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) $Y _ { i } = \beta _ { 0 } X _ { 1 i } ^ { \beta _ { 1 } } X _ { 2 i } ^ { \beta _ { 2 } }$

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 $\left( X _ { 1 , T } \text { and } X _ { 1,0 } \right)$ investment in human capital or education $\left( X _ { 2 , T } \text { and } X _ { 2,0 } \right)$ and per capita income in the initial period $\left( Y _ { 0 } \right)$ You run the following regression: $\ln \left( Y _ { T } \right) = \beta _ { 0 } + \beta _ { 1 } X _ { 1 , T } + \beta _ { 2 } X _ { 1,0 } + \beta _ { 3 } X _ { 2 , T } + \beta _ { 4 } X _ { 1,0 } + \beta _ { 5 } \ln \left( Y _ { 0 } \right) + u _ { T }$ 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?

You have estimated the following equation: $\widehat { \text { TestScore } } = 607.3 + 3.85 \text { Income } - 0.0423 \text { Income } { } ^ { 2 } \text {, }$ where TestScore is the average of the reading and math scores on the Stanford 9 standardized test administered to $5 ^ { \text {th } }$ 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 a. suggests a positive relationship between test scores and income for most of the sample. b. is positive until a value of Income of $610.81$ . c. does not make much sense since the square of income is entered. d. suggests a positive relationship between test scores and income for all of the sample.

An extension of the Solow growth model that includes human capital in addition to physical capital, suggests that investment in human capital (education)will increase the wealth of a nation (per capita income).To test this hypothesis, you collect data for 104 countries and perform the following regression: =0.046-5.869\times gpop +0.738\times+0.055\times Educ, =0.775, SER =0.1377 (0.079) (2.238) (0.294) (0.010) where RelPersInc is GDP per worker relative to the United States, gpop is the average population growth rate, 1980 to1990, sK is the average investment share of GDP from 1960 to1990, and Educ is the average educational attainment in years for 1985.Numbers in parentheses are for heteroskedasticity-robust standard errors. (a)Interpret the results and indicate whether or not the coefficients are significantly different from zero.Do the coefficients have the expected sign?

An example of a quadratic regression model is a. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X + \beta _ { 2 } Y ^ { 2 } + u _ { i }$ . b. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } \ln ( X ) + u _ { i }$ . c. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X + \beta _ { 2 } X ^ { 2 } + u _ { i }$ . d. $Y _ { i } ^ { 2 } = \beta _ { 0 } + \beta _ { 1 } X + u _ { i }$ .

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\timesEduc,=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"?

An example of the interaction term between two independent, continuous variables is a. $\quad Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } D _ { i } + \beta _ { 3 } \left( X _ { i } \times D _ { i } \right) + u _ { i }$ . b. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$ . c. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } D _ { 1 i } + \beta _ { 2 } D _ { 2 i } + \beta _ { 3 } \left( D _ { 1 i } \times D _ { 2 i } \right) + u _ { i }$ . d. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + \beta _ { 3 } \left( X _ { 1 i } \times X _ { 2 i } \right) + u _ { i }$ .

The following interactions between binary and continuous variables are possible, with the exception of a. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } D _ { i } + \beta _ { 3 } \left( X _ { i } \times D _ { i } \right) + u _ { i }$ . b. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } \left( X _ { i } \times D _ { i } \right) + u _ { i }$ . c. $Y _ { i } = \left( \beta _ { 0 } + D _ { i } \right) + \beta _ { 1 } X _ { i } + u _ { i }$ . d. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } D _ { i } + u _ { i }$ .

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?