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Use the data in SMOKE for this exercise.(i) The variable cigs is the number of cigarettes smoked per day. How many people in the sample do not smoke at all? What fraction of people claim to smoke 20 cigarettes a day? Why do you think there is a pileup of people at 20 cigarettes?(ii) Given your answers to part (i), does cigs seem a good candidate for having a conditional Poisson distribution?(iii) Estimate a Poisson regression model for cigs, including log(cigpric), log( (income), white, educage, and $a g e^{2}$ as explanatory variables. What are the estimated price and income elasticities?(iv) Using the maximum likelihood standard errors, are the price and income variables statisticallysignificant at the 5$\%$ level?(v) Obtain the estimate of $\sigma^{2}$ described after equation $(17.35) .$ What is $\hat{\sigma}$ ? How should you adjust the standard errors from part (iv)?vi) Using the adjusted standard errors from part (v), are the price and income elasticities nowstatistically different from zero? Explain.(vii) Are the education and age variables significant using the more robust standard errors? How doyou interpret the coefficient on educ?(viii) Obtain the fitted values, $\hat{y}_{i}$ , from the Poisson regression model. Find the minimum and maximum values and discuss how well the exponential model predicts heavy cigarettesmoking.(ix) Using the fitted values from part (viii), obtain the squared correlation coefficient between $\hat{\jmath}_{i}$ . and $y_{i}$(x) Estimate a linear model for cigs by OLS, using the explanatory variables (and same functionalforms) as in part (iii). Does the linear model or exponential model provide a better fit? Is either R-squared very large?

(i) 497 do not smoke, 101 smoke 20 cigs a day, bc 1 pack contains 20 cigs (ii) no but we can take advantage of the robustness property of Poisson regression (iii) see video (iv) yes (v) 4.54 (vi) price and income insignificant (vii) education and age significant (viii) from .515 to 18.84 (ix) R-squared = .043 (x) see video, R-squareds are small

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Chapter 17

Limited Dependent Variable Models and Sample Selection Corrections

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as one. Yeah. We find that in the sample There are 400 97 people who do not smoke at our, There are 100 and one people Said that they smoked 20 cigarettes a day. There is a bunching or a focal point formed at 20 cigarettes. Because one pack of cigarettes contain 20 cigarettes. Yeah, part two. The Poisson distribution does not allowed for the types of focal points shown in cigarettes variable. However, we can use the robust properties of the Poisson regression model or three. A result of the requestion passing regression is given in this table and along with the old estimates of the same model in part 10. The standard errors are the usual one. For both models. The estimated electricity for price and income are the estimated coefficients respectively price It is -155 and for income it is going zero 85 heart thor. When we use the maximum likelihood estimation, standard errors. The T statistic on the log of cigarette price Is -2.47. And for longer income It is 4.25. These are large T statistic values. So these variables are statistically significant. March five. The standard error of the the regression is reported in the table At the top. And when you come back to the table you may find sigma had to be four 54 because this is a large measure Much larger than one. It is evidence of over this person. It means that all of the standard errors for paul song regression Should be multiplied by 4.54. And the t. statistics should be divided by 4.54. So we addressed the centered Iran's from their personal regression And We Redo Part four. We find that for their lot of cigarette price variable. The T statistic now is smaller, It has to be divided by 4.54. So now it is minus point by four. And for the lock of income. Mhm. The T Statistic is .94. His statistic becomes way smaller for either variables meaning these through now very insignificant. We evaluate the significance of yeah two other variables. Education and age using the A justice standard errors. The T statistic before variable education is three point something in absolute value. It's actually smaller than minus three because the coefficient of education is estimated to be negative for each T statistic is over five. So both of these variables are highly significant coefficient on education implies that one more year of education reduces the expected number of cigarettes smoke per day By about six hard eight. We get the predicted value and we find that um predicted values range from 0.515 two 18 point 84 So we predict that everyone in the sample yeah was doing some smoking. And we did not predict anyone to smoke more than or even one pack of cigarettes per day. It shows that smoking, it's difficult to predict based on the explanatory variables we use here. About nine. There are square reported in the above table is point 043 which is actually the squared correlation between cigarettes and it's predicted value. Her tent. We re estimate the equation with L. S. When we look at the art square we find that the old S. Model has almost the same. Mhm. So if I report Oh it should be .045 R squared of LS. Model is slightly higher. Then they are square of the Warsaw model but both of them are quite small. Yeah. Again is uh it's not correct to compare the are square of the two models because they have different objectives. LS try to maximize our square but Pasang does not

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