Econometrics problem set II.
Submission deadline: 4th November 2019, 8:15am To be submitted:
– solutions in an edited and transparent format (on canvas or on paper [before the consultation])
– R scripts in an edited and transparent format (on canvas)
Please work on your own. Different students receive slightly different problems (based on their Neptun code) so simply copying the solutions will not work.
Let K be the first number (numerical value) in your Neptun code and L be the total count of non- numeric (letter) characters in your code. If there is no number in your code, let K = 0.
The wage2 dataset, which is included in the “Wooldridge” package in R, is needed for problems 3-5. You should use a subset of the observations in the dataset, by omitting observations in the range of [80 ∗ 𝐾 + 1; 80 ∗ 𝐾 + 200] (this means omitting 200 observations).
1. The following demand equation was estimated for a particular product on the basis of 30 measurements (standard errors are in parentheses):
̂2 log𝑄=6.4−0.85∗log𝑃+0.8∗log𝑌 𝑛=30; 𝑅 =0.16
(9.2) (0.40) (2.6)
where 𝑄 is the quantity, 𝑃 is the price and 𝑌 is income. Assume that the classical linear model assumptions hold, in particular, 𝑃 and 𝑄 are exogenous.
a. Test at the 5% level whether the price elasticity is zero. Formulate the null and the alternative
hypotheses. Give the p-value of the test statistic.
b. Test at the 5% level whether the income elasticity is one. Formulate the null and the alternative
hypotheses. Give the p-value of the test statistic.
c. Test at the 5% level whether neither the price nor the income has any effect on quantity. Formulate
the null and the alternative hypotheses. Give the p-value of the test statistic.
2. Consider the following two regressions, estimated on a sample of 50 observations. Assume that the classical linear model assumptions hold. Test at the 1% level whether 𝛽2 = 0. Give the p-value of the test statistic.
𝑦̂ = 𝛽̂0 + 𝛽̂1𝑥1 + 𝛽̂2𝑥2 𝑅2 =0.44 𝑦̂ = 𝛾̂0 + 𝛾̂1𝑥1 𝑅2 =0.37
3. The wage2 dataset contains individual wages and worker characteristics. Let us model the logarithm of the wage as a function of years of education (educ), years of parental education
(pareduc, defined as the average of meduc and feduc), work experience (exper) and tenure at the
current employer (tenure).
a. Estimate a model with the above explanatory variables and report the results. Interpret the
parameter estimates and evaluate their statistical significance. Give 95% confidence intervals for
the parameters.
b. State the null hypothesis that another year of general workforce experience has the same effect on
log(wage) as another year of tenure with the current employer. Test this null hypothesis against a
two-sided alternative hypothesis at the 5% level.
c. What is the expected increase of the wage if an employee spends one more year at her current
employer? Give a 90% confidence interval for the estimate.
d. How does the estimated parameter of education change if parental education is not included in the
model? Explain your answer.
4.
a. Starting from the model in problem 3, construct a new model where the effect of education
depends on parental education. Formulate a parametrization of the model in which you can easily
interpret the resulting parameter estimates. Compare the estimates to those of problem 3.
b. Transform the model in part a. so that you can directly measure the wage effect of one more year
of education if parental education is 𝐿 + 5, where 𝐿 is defined above based on your Neptun code.
c. Predict the logarithmic wage and the wage of a worker who has 𝐿 + 8 years of education, the average of her parents’ years of education is 𝐿 + 5, has spent 7 years at the current employer, and has 11 years of general workforce experience. (Note: We have not covered in class how to transform precisely a prediction of log(y) to a prediction of y. Use a naive (common sense)
approach.)
5.
a. Model the years of education (educ) as a function of the number of siblings (sibs) and parental
education (pareduc) in the wage2 dataset. First estimate a model with only linear terms. Interpret
the parameter estimates.
b. Add the square of pareduc to the model. Interpret the parameter estimates of the new model. Give
the partial effect of one more year of parental education on the years of education of the child
when parental education is 12 years.
c. Add the square of the number of siblings as well as the interaction of sibs and pareduc to the
model. Use the model selection tools discussed in class to choose between the various models (ranging from the simplest specification to the specification containing all squares and the interaction term as well).