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2. Discrete Choice Application: Suppose we have the following data fiom a single question about vehicle choice with the following attributes: p price (in thousands)...

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2. Discrete Choice Application: Suppose we have the following data fiom a single question about vehicle choice with the following attributes: p price (in thousands) 1 fuel economy (in mpg) . andx2 = 0-60 acceleration time (in seconds). where n; is the number of people who chose altera Itive j and 0 is the no-choice option (the outside good).Al 15 25p (S1OOOs) (mpg) T2 (sec)15 3520 2525 35502024Assume a logit model with utility linear in the parameters, so that "j = BoPs + Bxij + B,1zj + Bs

2. Discrete Choice Application: Suppose we have the following data fiom a single question about vehicle choice with the following attributes: p price (in thousands) 1 fuel economy (in mpg) . andx2 = 0-60 acceleration time (in seconds). where n; is the number of people who chose altera Itive j and 0 is the no-choice option (the outside good). Al 15 25 p (S1OOOs) (mpg) T2 (sec) 15 35 20 25 25 35 50 20 24 Assume a logit model with utility linear in the parameters, so that "j = BoPs + Bxij + B,1zj + BsS Vj e{AB.CD.O} %G =] for the outside good and 0 otherwise, and B; is the utility of the outside good. a) Write the maximum (log) likelihood formulation as a standard optimization problem b) Use Excel Solver to determine the betas that maximize likelihood Report the resulting betas with their standard ertors Does the solution satisfy first and second order optimality conditions? Justify YOUT answer: Do the signs of the beta values match what you would expect? Explain why o why not: For each of the attributes (price, Ipg; and acceleration). does the entire 95% credible interval for beta have the same sign? What could you do to further reduce uncertainty in the betas? f) What is the expected average willingness to pay for a 1 mpg increase ceteris paribus (you do not need to compute confidence intervals)? Suppose a new product N were added to this market with price S15.000. efficiency 30mpg; and acceleration seconds. Simulate expected share for N given a choice among the altematives {AB,C,DN,O} ? What is the expected value and the 95% credible interval of the expected share for N? h) What is the maximum price that could be charged for N while still capturing at least 10% expected share? 1) What is the maximum price for N at which the 95% credible interval of expected share lies above 10%2 j) What value of price maximizes expected revenue?



Answers

Use the data in APPLE for this exercise. These are telephone survey data attempting to elicit the
demand for a (fictional) "ecologically friendly" apple. Each family was (randomly) presented with a
set of prices for regular apples and the ecolabeled apples. They were asked how many pounds of each
kind of apple they would buy.
(i) Of the 660 families in the sample, how many report wanting none of the ecolabeled apples at
the set price?
(ii) Does the variable ecolbs seem to have a continution over strictly positive values?
What implications does your answer have for the suitability of a Tobit model for ecolbs?
(iii) Estimate a Tobit model for ecolbs with ecoprc, raminc, and hhsize as explanatory
variables. Which variables are significant at the 1$\%$ level?
(iv) Are faminc and hisize jointly significant?
(v) Are the signs of the coefficients on the price variables from part (iii) what you expect? Explain.
(vi) Let $\beta_{1}$ be the coefficient on ecoprc and let $\beta_{2}$ be the coefficient on regprc. Test the hypothesis $\quad \mathrm{H}_{0} :-\beta_{1}=\beta_{2}$ against the two-sided alternative. Report the $p$ -value of the test. to refer to Section $4-4$ if your regression package does not easily compute such tests.)
(vii) Obtain the estimates of E(ecolbs, x) for all observations in the sample. [See equation $(17.25) .1$ Call these $\overline{e c o l b s}_{i}$ . What are the smallest and largest fitted values?
(viii) Compute the squared correlation between $e c o l b s_{i}$ and $\widehat{e c o l b s}_{i}$
(ix) Now, estimate a linear model for ecolbs using the same explanatory variables from part (iii). Why are the OLS estimates so much smaller than the Tobit estimates? In terms of goodness-of
fit, is the Tobit model better than the linear model?
(x) Evaluate the following statement: "Because the $R$ -squared from the Tobit model is so small, the estimated price effects are probably inconsistent."

First one. There are 248 families do not want the apples at any price Or two. So distribution is not continuous, There is focal points and rounding for example. Many people report on powder and either 2/3 of a pound or one and a third pounds. This the fact that the distribution of quantity demanded is not continuous violates the underlying assumption of the topic model. Yeah. Which is the Layton error has normal distribution, but we will still explore the Tobin approach in this context. Yeah. It may work better than the linear model for estimating the expected demand function, do you? And Along with Part eight. The estimates from their topic and L. S.. Models are reported in the same table or the tablet model. The price variables ali them are Statistically significant at the one level. The sign over these prize coefficients are in accordance with the demand theory, the own price effect is negative and across price effect is positive. Cross price is the price of this substitute good, which is regular apples part Let's do part 6. 1st part six we will obtain their fitted values and we find that the ranged from 27 8 798 to you. 3.33 at five. The null hypothesis is later one plus beta two equals zero. This is something you can easily test regardless of your statistical package. Yeah. Yeah, you should get a small T statistic about minus point you and a P value of buying eight. So we are unable to reject the North part seven. The squirt correlation between it is E call B. S and it's fitted value is about .04 and that is there are square hard eight. Given the linear model estimates, even this result, we find that the old LS estimate are smaller than the top bit estimate. And in terms of our square oops, you compare the goodness of fit between the two models we look at. There are square mhm. There are squared of the topic. Model is still smaller, slightly smaller than the old LS model and serve. We can conclude that the topic is yeah, no better than the old LS. It doesnt suite the data better. The Last Part, Part nine. The statement is simply incorrect so you could run into a uh counter example. We have valid price effects, but we cannot explain much of the variation in the dependent variable. It's simply difficult to estimate the demand for a fictitious product.

Part one. To estimate the demand equation, we need at least one exhaustion is variable that appears in their supply equation part to you. For the two variables Wave to T. And wave three T. To be valid instruments for the lock of average price. We need to assumptions First, these two can be excluded from the demand equation. Yeah. This may not be entirely reasonable as wave heights are determined partly by weather. Yeah. And the demand at a local fish market could also depend on whether the second assumption at least One of wave two and wave three appears in the supply equation. There is evidence of this in part three, as the two variables are jointly significant in the reduced form for lock of average price, Part three, these are the L. L. S. estimates of the reduced form. The lock of irish price depends on weekday dummies, we have monday, Tuesday Wednesday thursday. The two explanatory variables are the main ones are wave two and weigh three. There are 97 observations and our square is .3. We test the not hypothesis. The coefficient of way to equals the coefficient of way three. All together equal zero With an f statistic of 19.1 and AP value roughly zero. We are able to retract the null hypothesis and we conclude that two variables wave t way to and weigh three are yeah, George, lee pictures very significant. Mhm. Uh huh. In part four we will estimate the demand function by two stage least square. This is what we get. Mhm. So total quantity in lug has a negative relationship with average price and with monday Tuesday Wednesday dummies, total quantity has a positive relationship with her stay dummy. Again we use 97 observations and our square is .193. The demand elasticity is the coefficient of a lot of average price. This one The 95 confidence interval for the demand elasticity is about -1.47 two minus point 17 at this point When is 1.47 to -17. The point estimate point -12 seems reasonable. Mm hmm. A 10% increase in price reduces quantity demanded. Yeah. Bye. 8.2 percent. Part five. We find a strong you find strong evidence of positive serial correlation because we estimate the coefficient of you IT -1 to be point two 94 with a centered barrel of .103. Yeah. So we could fix this problem by estimating a new we west centered errol for the two stage least square instead of the usual standard error. Hard 6th. To estimate the supply elasticity, we would have to assume that the weekday dummies do not appear in the supply equation. Yeah, yeah. But yeah, they do appear in the demand equation. Part three Shows. Is that, yeah, there are weekday effects in the demand function, but we cannot know about the supply function. Part seven. In the estimation of the reduced forum for log of average price In part three, the weekday dummies are jointly insignificant. The value of the f statistic is .53 very small and the p value is .71. This means some of these dummies could show up in the demand equation, but they canceled out in a way that they do not affect the equilibrium price Once we've two and wave three are in the equation, so we can't estimate the supply equation without more information.

Part one. In this part, we're going to define a new variable. ICO, by equal by takes a value of one. If Iko Lbs is greater than zero and takes a value of zero. If equal LBS is zero equal by indicates whether at the prices given a family would buy any ecologically friendly apple, the fraction of families claim they would buy equal labeled airports is you will take the number of cases that equal by equals one, which is 412 divided by the total number of observations, which is 660 and you wouldn't get 0.6 to 4. This is the old LS estimates of the linear probability model. We went interpret the prices variables Yeah, which are ICO prices and regular prices. The coefficient of ICO prices is minus 0.8 and the coefficient of regular prices is point 72 So it means if iko price increases by, say, 10 cent se $0.1 then the probability of buying ICO labeled apples falls by about 0.8 point eight for regular prices. If the regular prices increase bye, 10 cent, then the probability of buying eco labeled apples increases about 0.72 We will evaluate their joint significance off their non price variables, which are family income, household size, educational level and age. We will do an F test. The F statistic is 4.43 and the two degrees of freedom is four and 653. The P value of the test is 0.15 So we are able to reject the non hypotheses. In other words, these factors jointly significant at the 1% level. Among the four non price variables, education appears to have the most important effect. The coefficient of education is 0.25 with a very small standard. Errors, a difference of four years of education, increased the probability of buying ICO labeled apples by 0.1, so we can expect that more highly educated people are more open to buying produced that is environmental friendly. Another variable that is also important is household size. The coefficients of household size is 1024 with a small standard error, comparing a couple with two Children to one that has no Children. Other factors equal the couple with two Children has a point though, for a higher probability of buying equal label apples. We've been compared to models, the original model in Part two and a new model where you replace family income with its lock value. The first thing we could notice is the are square. The art square in the original model has increased from 0.110 2.112 So the model the new models fits the data likely better. The second thing you can notice is there coefficient of family income and lock family income. This coefficient for family income is 0.1 But for lock of family income is point 04 or five. We could interpret this coefficient as if the lock of family income increased by 10%. Okay, if lack of family income increases by 0.1, which means almost a 10% increase in family income, then the probability of buying an ICO labeled product is estimated to increase by about 0.45 a small effect. After we estimate the model, we can get the fitted probability which we denote ICO by hat, we win count. The number of cases we're ICO by is greater than one, and we will count the number of cases where ICO, by it's less than zero ICO by is the probability of buying equal label apples, so it should be between zero and one inclusive. But we just estimate a linear probability model, and one limitation of this kind of model is that it produced predicted values that fall outside the range of zero and one. Now, in this part, we're going to see if this model produced strange results, and you could find that there are two fitted probabilities above one, and there is no cases where the fitted probability is smaller than zero. The number of cases or the number of observations is 660. So given that we have to wrong values, is this not a problem? For past six, we will define a new variable. A binary variable. Geico by squiggle or tilda equal by chilled er gets a value of one. If the fitted value of equal by is greater than or equal 2.5 and ICO by Tilda, takes a value of zero. If the fitted value of ICO by ISS less than 0.5 okay in the next step within tabulate. Okay. The values of ICO body, Tilda and ICO by not the fitted value, the actual probability of purchasing ICO labeled apples. This is the tabulation result or we have a confusion matrix. We have two roles predicted not by and predicted by thes rose are corresponded to two values of equal by Tilda. And we have two columns, the actual not purchase and the actual purchase of ICO labeled products. These two columns relate to the two values of the binary variable. The original dependent variable equal by the first cell is 102. The next to it is 72 and then we have 146 then 340. So these two, the highlighted cells, are the cases where the model predicts, currently 248. At two cases where people did not purchase equal labeled apples, and 412 cases where people actually purchase the equal label apples. The model predicts correctly 41% of the cases where people did not purchase equal label products and 82.5% of cases where people purchased equal label products. Yeah, we can calculate the overall percent correctly predicted by taking the number of cases that the model predicts correctly divided by the number of observations. So we have 102 plus 3 40 all divided by 6 60 and we get 67%. The model does pretty well, and it does a much better job predicting the decision to buy equal label apples.


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