Exam 8: Nonlinear Regression Functions

Consider the polynomial regression model of degree Yi = β0 + β1Xi + β2 + ...+ βr + ui.According to the null hypothesis that the regression is linear and the alternative that is a polynomial of degree r corresponds to

The following are properties of the logarithm function with the exception of

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, = 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?

The best way to interpret polynomial regressions is to

In the log-log model,the slope coefficient indicates

In the regression model Yi = β0 + β1Xi + β2Di + β3(Xi × Di)+ ui,where X is a continuous variable and D is a binary variable,to test that the two regressions are identical,you must use the

A polynomial regression model is specified as:

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

A nonlinear function

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: = 0 + 1 × Femme + 2 × Educ + 3 × Tenure + 4 x (Educ × Tenure)+ ∙∙∙ 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?

The exponential function

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

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?

Sketch for the log-log model what the relationship between Y and X looks like for various parameter values of the slope,i.e. ,β1 > 1;0 < β1 < 1;β1 = (-1).

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 4th to 6th 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: = 45.59 + 4.32 × Height4,R2 = 0.55,SER = 15.69 (3.81)(0.46) where Weight is in pounds,and Height4 is inches above 4 feet. (a)Interpret the results. (b)You remember from the medical literature that females in the adult population are,on average,shorter than males and weigh less.You also seem to have heard that females,controlling for height,are supposed to weigh less than males.To see if this relationship holds for children,you add a binary variable (DFY)that takes on the value one for girls and is zero otherwise.You estimate the following regression function: = 36.27 + 17.33 × DFY + 5.32 × Height4 - 1.83 × (DFY × Height4), (5.99)(7.36)(0.80)(0.90) R2 = 0.58,SER = 15.41 Are the signs on the new coefficients as expected? Are the new coefficients individually statistically significant? Write down and sketch the regression function for boys and girls separately. (c)The medical literature provides you with the following information for median height and weight of nine- to twelve-year-olds: Median Height and Weight for Children,Age 9-12 Insert two height/weight measures each for boys and girls and see how accurate your predictions are. (d)The F-statistic for testing that the intercept and slope for boys and girls are identical is 2.92.Find the critical values at the 5% and 1% level,and make a decision.Allowing for a different intercept with an identical slope results in a t-statistic for DFY of (-0.35).Having identical intercepts but different slopes gives a t-statistic on (DFYHeight4)of (-0.35)also.Does this affect your previous conclusion? (e)Assume that you also wanted to test if the relationship changes by age.Briefly outline how you would specify the regression including the gender binary variable and an age binary variable (Older)that takes on a value of one for eleven to twelve year olds and is zero otherwise.Indicate in a table of two rows and two columns how the estimated relationship would vary between younger girls,older girls,younger boys,and older boys.

Assume that you had estimated the following quadratic regression model = 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

In nonlinear models,the expected change in the dependent variable for a change in one of the explanatory variables is given by

The interpretation of the slope coefficient in the model ln(Yi)= β0 + β1Xi + ui is as follows:

For the polynomial regression model,

The interpretation of the slope coefficient in the model ln(Yi)= β0 + β1 ln(Xi)+ ui is as follows: