ECO671, Spring 2006 , Second homework assignment



ECO671, Spring 2015, Third homework assignment.Prof. Bill EvenThe assignment is due Monday 5/5 by 5 p.m. Submit your assignment via email to evenwe@mimaioh.edu by the deadline. Name the file hw4_xx_yy – where xx and yy are the uniqueids for the two team members. Late assignments will be penalized at the rate of 20 percentage points for every day (or part thereof) that the assignment is overdue. All team members will receive the same grade unless someone convinces me that I should do otherwise. Provide a type-written response to all the questions. Paste the relevant portion of the stata log (both the stata commands and the output) beneath the relevant part of each question in this word document and then provide a type-written explanation (or leave adequate space for handwritten explanations beneath relevant stata code and results). Be sure to include enough Stata code that I can determine exactly how you generated your data, variables, and results. If I am unable to determine what you did, I will assume that it is wrong.I. Multinomial logit and Probit. (50 points) In this problem, you are asked to examine data on travel mode choice between Sydney and Melbourne. This data set is made available by Stata to illustrate the use of alternative specific multinomial logit and probit models. The variables in the data are described below:Dependent variables:mode travel mode alternatives (1=air, 2=train, 3=bus, 4=car)choice travel mode chosen (1 if mode chosen, 0 otherwise)Alternative specific variables:termtime time spent in terminal time (0 for car)invehiclecost in-vehicle costtraveltime travel timetravelcost generalized cost of travel (foregone wages plus explicit travel expense)Case specific variablesincome household incomepartysize party size travellingid: person identifier.The data for the first two people in the data are listed below:idchoicetermtimeinvehicle costtravel time travel costincomepartysize mode 106959 10070351air 103431 37271351train 103525 41770351bus 11010 18030351car 2064586868302air 20443135484302train 20532539985302bus 2101125550302carNotice that for each person, there are 4 lines of data corresponding to the 4 possible choices. The “case specific” (alternative invariant) variables for a given person do not vary across alternatives (e.g. a person’s income is constant across the 4 travel modes); the alternative specific variables (e.g. travel time) vary across travel modes for each person. The reference manuals for asclogit and asmprobit provide examples of how to estimate multinomial logit/probit models with the type of data The Stata reference manuals for the asmprobit and asclogit routines are available on my website here .1. Using the asclogit model, estimate the relationship between the travel mode choice all of the control variables listed (alternative specific and case specific). Provide estimates with air as the base outcome, and for a model with bus as the base outcome. Output these coefficients using the outreg2 command and paste the results below. Describe how the estimates change when you switch the base group and explain why this happens. 2. Using the estimated model, compute the probability that a person with the average characteristics uses each mode of transportation. [See predict option in post-estimation commands.] Note that this creates a single variable containing the predicted probabilities for each of the four modes. How do the mean of the predicted probabilities compare with the fraction of workers that choose each mode in a single table. 3. The margins command is not compatible with asclogit or asmprobit. To compute marginal effects after estimating either model, you must use estat mfx. The default option with estat mfx is to calculate the marginal effect of each control variables on the probability of choosing each of the four modes at the mean value of all the control variables. Use the estat mfx command to estimate marginal effects. Based on the results, how would a 10 percent increase in income affect the probability that a person with the average characteristics would choose each mode of travel? Describe how you used the results from mfx to derive these estimates.4. Define travel-cost as the “price” of a mode of travel. Use your asclogit estimates to to derive a. the own-price (travel-cost) elasticity of demand for airline travel (i.e. %ch in air demand / % ch in air price)) b. the cross-price elasticity of demand between airline and bus travel (i.e. % ch in air demand / % ch in bus price)5. Suppose that there is a proposal to increase the price (travel-cost) of airline travel by 10% . Estimate the effect of such a change on the fraction of people that would choose each of the 4 modes of transportation. Summarize your results in a clearly labeled table.6. It is conceivable that the elasticity of demand for a travel mode varies with income. Estimate an asclogit model that allows the effect of price to vary for people according to income level. Based on your results, is the demand for a trave mode more or less elastic among high income households? Explain and provide a test of the null hypothesis that the own-price elasticity does not vary with income. 7. Stratify your sample into a high and low income group according to whether the person’s income is above or below median income. Use Stata’s lrtest to generate a test of the null hypothesis that the coefficients from the asclogit model are equal for high and low income workers. Discuss your conclusion. 8. Suppose that the train blows up and is no longer a travel option. Compute the predicted fraction of workers that would use each of the remaining three travel modes and compare the distribution of travel choices before and after the train disappears in a clearly labeled table. Provide a brief description of how you derived your new estimates. [Hint: IIA.]9. Unlike the multinomial logit, the multinomial probit (asmprobit in Stata) allows for error terms to be correlated across choices and can thus relax the IIA assumption. If corr(independent) is the option chosen for estimating the model, IIA is imposed. If corr(unstructured) is the option chosen, IIA is not imposed. Estimate the same model as in (1) with independent and unstructured correlation. Use a likelihood ratio test to test whether the IIA assumption is valid in the asmprobit. Interpret your results. 10. Use the asmprobit model to repeat the exercise in (2). Copy the table from (2) and add a column showing how the estimates compare in the logit versus probit model. II. Heckman sample selection model (50 points). For this problem, use the following stata data set constructed from the SCF. The data set is g:\eco\evenwe\eco671\data\scf671_2.dta and is a combination of data from the 7 cross sectional surveys conducted between 1989 and 2007 (SCF is collected every 3 years). The variable you will be examining is the mortgage rates paid by various households. The variables contained in the data set are given below: mortgage1 Has first mortage dummymortrate1 Interest rate on first mortgage x 100year year of surveyhhage age of head of householdmarried currently marrieddivorce currently divorced or separated dummyrisk attitude toward financial risk: 1=willing to take high risk; 4=unwilling to take any risknw household networth in $1000skids # of children in familymort30atloan 30 year mortgage rate on conventional loan when loan was originatedlvratio loan-to-house value at time loan was originatedrincome real household incomeblackdummy for black raceHispanicdummy for Hispanic ethnicityFor people without a mortgage (mortgage1=0), no mortgage rate is reported. 11. Estimate a linear regression model of mortgage rates controlling for all of the control variables above with the exception of the risk and number of children. Include year dummies in your regression.12. Use the Heckit model (see heckman in Stata) with the two-step option to re-estimate the mortgage rate equation. In the sample selection equation, use all the controls that were in the mortgage rate equation except loan-to-value ratio and mort30atloan (these don’t exist for people without loans). Also, add dummy variables for attitudes toward risk to the sample selection equation and the number of children. 13. Do the Heckit results suggest positive or negative sample selection? Is sample selection statistically significant? Provide the basis for your answer.14. Compare the estimated effect of the household head’s age and income in the OLS versus Heckman model. a. What does this tell you about the direction of the sample-selection bias in the OLS estimates? b. Given the estimated parameters in the Heckit model, is this what you would have predicted? Why?15. Estimate the predicted mortgage rate for everyone in the sample using:a. the OLS modelb. the sample selection model without conditioning on whether the person actually takes a mortgage (the unconditional prediction)iii. the sample selection model conditional on the person taking a mortgage (the conditional prediction). 16. For the group that has a mortgage, a. how do the OLS predictions and the unconditional predictions of the mortgage rate from the sample selection model compare? b. how do the conditional and unconditional predictions of the mortgage rate compare? c. In light of the above comparisons, how might you interpret OLS estimates when there is sample selection? Explain.17. To investigate the importance of how the sample selection problem differs across households, based on the Heckman model results, compute the probability that each person in the sample has a mortgage (see Heckman postestimation to find the appropriate prediction term). Divide the sample into two groups: those with a probability of receiving mortgage<.25 and those with probability>.75. The gap between the conditional and unconditional mortgage rate can be thought of as the “sample selection effect” on mortgages. Where is the “sample selection effect” largest? Why should you expect this? Explain. ................
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