Question

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, educ age, 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 statistically significant 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 now statistically different from zero? Explain. (vii) Are the education and age variables significant using the more robust standard errors? How do you 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 cigarette smoking. (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 functional forms) as in part (iii). Does the linear model or exponential model provide a better fit? Is either R-squared very large?

Name: Problem 10, Chapter 17: Introductory Econometrics
Uploaded: 2021-05-08T22:01:40-08:00
Duration: 13 min
Channel: Heather Duong

   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, educ
age, 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 statistically
significant 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 now
statistically different from zero? Explain.
(vii) Are the education and age variables significant using the more robust standard errors? How do
you 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 cigarette
smoking.
(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 functional
forms) as in part (iii). Does the linear model or exponential model provide a better fit? Is either R-squared very large?

Introductory Econometrics

Jeffrey M.… 6th Edition

Chapter 17, Problem 10

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Solved by Expert Heather Duong

05/08/2021

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There are 101 people who claim to smoke 20 cigarettes a day. The reason for the pileup at 20 cigarettes could be because one pack of cigarettes typically contains 20 cigarettes. Show more…

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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, educ age, 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 statistically significant 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 now statistically different from zero? Explain. (vii) Are the education and age variables significant using the more robust standard errors? How do you 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 cigarette smoking. (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 functional forms) as in part (iii). Does the linear model or exponential model provide a better fit? Is either R-squared very large?

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Key Concepts

Poisson Regression

Poisson regression is a model used for count data where the expected count is modeled using a log-linear specification with explanatory variables. Its formulation implies that the dependent variable follows a Poisson distribution conditional on the covariates, making it suitable for modeling events that occur randomly over a fixed period or space.

Robust Standard Errors

Robust standard errors are adjusted standard error estimates that take into account potential violations of the model assumptions, such as heteroskedasticity. They provide more reliable hypothesis tests and confidence intervals when the variability of the error terms does not conform to the classical assumptions.

Model Comparison and Fit Metrics

This concept involves the evaluation of how well different statistical models, such as a nonlinear exponential model versus a linear OLS model, fit the observed data. Metrics like R-squared, fitted values, and correlation coefficients are used to compare model performance, especially in predicting outcomes at extreme values.

Maximum Likelihood Estimation (MLE)

MLE is a statistical method for estimating the parameters of a model by maximizing the likelihood function, which represents the probability of observing the given sample data. In the context of Poisson regression, MLE is used to obtain parameter estimates that best explain the observed counts.

Count Data Models

Count data models are used to analyze outcomes that are counts (non-negative integers). They consider the particular distributional properties of count data, such as the discreteness and the possible presence of overdispersion, where the variance exceeds the mean.

Conditional Poisson Distribution

This concept refers to the assumption that, given a set of explanatory variables, the distribution of the count variable follows a Poisson process. This conditional assumption underlies the use of Poisson regression and is critical in determining whether the model is appropriate for a given dataset.

Elasticity

Elasticity in this context measures the percentage change in the expected count of the dependent variable when an explanatory variable, such as price or income, changes by one percent. It provides a way to interpret the sensitivity of the outcome variable in a multiplicative model like the Poisson regression.

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