ECO2151 Introduction to Econometrics - Winter 2023 Homework 2
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Homework 2
ECO2151 Introduction to Econometrics - Winter 2023
The assignment is due on Thursday, March 2nd, by 11:59 pm. You are expected to submit a total of 2 iles: your written answers to the questions in a .pdf ile + the codes/results from your statistical software printed in a .pdf ile (look for the “Print Results” button on the top left of the STATA window and select the option to print to .pdf). You can collect the two iles together if you prefer. The ile(s) with your answers and STATA codes/results are to be submitted through the Brightspace.
Note: it is your responsibility to present the results in an organized way and such that it is clear where the answer to each question is.
1. (12 points) In this question, we are going to perform some empirical analysis using the data in the “data hmw2”iles posted on the Brightspace. The .dta ile can be opened directly in STATA; alternatively you can copy and paste the data from the Excel ile into the STATA Data Editor. The “Data Description” ile contains a description of the variables (please note that our data is a subsample of the original dataset, so some of the variables have been dropped).
Type in STATA: sum to get some summary statistics about your variables.
Note: in all the subpoints below, you can refer to the results that are printed in the ile with your STATA codes/results in your answers - you do not need to re-write the parameters that appear in the tables or re-compute the values of t-statistics that are provided.
a. (3 points) You are interested in studying the relationship between the age of a person and their earnings. Run the following regression using heteroskedasticity robust standard errors:
earningsi = β0 + β1 agei + ui
Report the OLS coe
cients and the standard errors. Test the H0 that age has no impact on a person’s earnings. Use a 5% signiicance level and a two-sided alternative hypothesis. Can you reject the H0? Explain your answer.
b. (3 points) You are concerned that the model estimated in part a. may be afected by omitted variable bias. In particular, you think that the variable educ could have an impact on earnings and be correlated with the age of a person at the same time. To check the correlation between educ and age, type: corr educ age (recall that the correlation between the two variables is the number that appears in the low left corner of the table). If your regression in a. is afected by omitted variable bias, what would the sign of this bias be? As a consequence, would your estimated 
1 be overestimating or underestimating the true parameter β 1 ? Explain your answers.
c. (3 points) To address your concern of a possible omitted variable bias, you decide to run the regression:
earningsi = β0 + β1 agei + β2 educi + ui
Report the OLS coe
cients and their heteroskedasticity-robust standard errors. Does the new value
of 
1 relect what you were expecting based on your discussion about the sign of the omitted variable
bias? Explain your answer.
d. (3 points) Finally, you think that work experience could have an impact on earnings as well and be another omitted variable. You want to include work experience in the regression but, unfortunately, you don’t have data about this variable. Since people usually start going to school when they are 6 years old, you decide to compute work experience as: expi = agei − educi − 6. Generate the variable exp in STATA using this formula. Then run the following regression with heteroskedasticity-robust standard errors:
earningsi = β0 + β1 agei + β2 educi + β3 expi + ui
What happens in STATA? Are you able to obtain estimates for all the parameters in the model? If not, can you explain why?
2. (13 points) Consider the estimated OLS regression:
Yi = 
0 + 
1Xi + 
i (1)
where (Yi,Xi ), i = 1, ...,n, are the observations in your sample, 
0 and 
1 are the OLS estimators of the true parameters β0 and β 1 , and 
i are the residuals of the regression.
a. (6 points)
n
i. Theory. Show that, by construction: 对 
i = 0
(Hint: use the deinition of 
i and then the formula of the OLS estimator 
0 .)
ii. Practice. For this question, type in STATA: set type double. This command increases the precision of the results.
Using the same data that you employed for Question 1, estimate again the linear regression model: earningsi = β0 + β1 agei + ui
In the notation of (1), earningsi = Yi and agei = Xi .
Now type in STATA: predict uhat, residual. This command creates a variable, called “uhat”,
which includes the residuals 
i .
Finally, type in STATA: sum uhat or sum uhat, detail. Does the theoretical result from part a.i. of the question hold? Explain your answer.
b. (7 points) Consider the alternative speciication of (1):
Yi = 
0 + 
1Zi + 
i (2)
where 
0 and 
1 are the OLS estimators of the true parameters V0 and V1 and 
i are the residuals of the regression. The data for the variable Yi are the same as in (1), while the data for the variable Zi are computed from Xi as: Zi = aXi, where a 
0 is a parameter. In other words, the data for Zi are just a re-scaling of our original data for Xi .
i. Theory. Use the formulas of the OLS estimators to show that, by construction: 
0 = 
0 and
1 = a
1 .
ii. Practice. We are going to use again the same data employed for Question 1. In STATA, generate the variable agem = 12 ∗ age. The variable agem measures the age of each observation in months rather than years. Now estimate the linear regression model
earningsi = V0 + V1agemi + vi
Compare your estimated 
0 and 
1 with the values of 
0 and 
1 obtained in part a. Does the theoretical result from part b.i. of the question hold? Explain your answer.
2023-03-02