This test has two parts totaling 100 points. The test is OPEN BOOK,OPEN NOTES. Useful formulas and tables of distributions can be found on the course formula sheet and critical values documents posted to D2L. You MAY NOT share the test questions with any other individuals. You MUST work on the exam by yourself. The test is open book/open notes. The internet is NOT to be used. Calculators are permitted. Either print the answer sheet posted on D2L or use your own blank paper to write up your answers. You may handwrite your answers, type them, or both. Show all of your work as partial credit will be given. Please scan your answer sheets and submit using the D2L dropbox using pdf format if possible.
In WORDS, briefly describe and discuss the following. Try to limit your answer to no more than three or four sentences. You may use formulas if they are helpful, but a formula without an explanation will receive no credit!
a) What are the implications of heteroskedasticity for OLS (assuming MLR.1, MLR.2, MLR.3 and MLR.4 hold)?
b) What are the implications for generalized least squares in a transformed regression that uses the WRONG model for hi (assuming MLR.1, MLR.2, MLR.3 and MLR.4 hold). For example, the true model has hi = educ2 but the transformed regression is based on hi = educi where educi is years of education.
c) Can the Gauss-Markov Theorem be used to rank the variances of the OLS estimator and the IV estimator when MLR.4 holds (assuming MLR.1, MLR.2, MLR.3 and MLR.5 also hold)? Explain why or why not.
d) Can the Gauss-Markov Theorem be used to rank the variances of the OLS estimator and the IV estimator when MLR.4 fails? Explain why or why not.
e) Are there conditions under which a variable, z1, can serve as both a valid instrument for an included regressor, x1, AND a valid proxy for an omitted variable, x2, that is correlated with the included regressor? In math notation, in the regression yi = βO + β1xi1 + ui where ui contains an omitted variable xi2 that is correlated with xi1, are there conditions under which zi1 can be a valid instrument for xi1 AND a valid proxy for the omitted variable, xi2? Explain why or why not.
This part of the test asks a series of questions about the relationship between years of education, educ, and the natural log of hourly wages, log(rage). Additional variables available to the empirical researcher are years of experience on the job, exper, and years of education of the person's mother, motheduc, and years of education of the person's father, ƒ atheduc.
A random sample of 428 individuals was used to estimate three regression models of increasing complexity:
log(ragei) = βO + β1educi + ui (1)
log(ragei) = βO + β1educi + β2experi + ui (2)
log(ragei) = βO + β1educi + β2experi + β3exper 2 + ui (3)
OLS estimates of the three regressions are as follows. Numbers in parentheses ( ) are OLS standard errors. Numbers in brackets [ ] are heteroskedasticity robust standard errors.
log(ragei) = —.185 +.1086educi R2 = .118 (.185) (.014) SSR = 197.0
[.171] [.013]
log(ragei) = —.400 +.1095educi +.0157exper i R2 = .148 (.190) (.014) (.004) SSR = 190.2
[.183] [.013] [.004]
l^og(ragei) = —.522 +.1075educi +.0416exper i —.0008exper 2 R2 = .157
(.199) (.014) (.013) (.0004) SSR = 188.3
[.202] [.013] [.015] [.0004]
a) Notice that as the model becomes less complex (going from (3) to (2) to (1)), the SSR increases and the R2 decreases. Provide a theoretical explanation for these patterns. (4 points)
b) Using the estimated model (3), what is the impact, on average, of one additional year of education on wages for a person with 6 years of education (educ = 6)? What is the impact for a person with 12 years of education? What is the impact for a person with 16 years of education? In answering these questions, clearly state what in being held constant (you may assume MLR.4 holds). (3 points)
c) Using the estimated regression model (3), what is the impact, on average, of one additional year of experience on wages for a person with 1 year of experience (exper = 1)? [Hint: modify the log-level result on page 3 of the formula sheet for the case where an x2 term is in the model: log(y) = βO + β1x + β2x2.] What is the impact for a person with 10 years of experience (exper = 10)? What is the impact for a person with 20 years of experience (exper = 20)? What is happening to the impact of an additional year of experience on wages as experience increases? Does this make sense economically? Why or why not? (5 points)
d) Assuming MLR.5 holds, test the hypothesis that an additional year of education is associated with a 5% increase in wages. Use regression model (3). Clearly state your null and alternative hypotheses in terms of β1. Clearly state your rejection rule. Carry out your test at the 5% significance level. (5 points)
e) In a meeting to determine tuition increases, a college university president says that education is the 'only' factor that matters for wages and claims that years of experience on the job has no effect on wages. Test this hypothesis using regression model (3). Clearly state your null and alternative hypotheses in terms of the slope parameters. Clearly state your rejection rule. Carry out your test at the 5% significance level. Continue to assume MLR.5 holds. (5 points)
To assess whether regression model (3) satisfies MLR.5, both the Breusch-Pagan and White tests for heteroskedasticity were computed for regression (3). The F -statistic version of the Breusch-Pagan test yielded 3.98 and the F -statistic version of the White test yielded 1.79.
f) Let ui be the OLS residuals from regression model (3). Write down the regression model that was used to compute the Breusch-Pagan test F -statistic. Carry out the Breusch-Pagan test at the 5% significance level. (4 points)
g) Write down the regression model that was used to compute the White test F -statistic. Carry out the White test at the 5% significance level. (4 points)
h) Based on the heteroskedasticity tests, is the use of OLS standard errors in parts (d) and (e) appropriate? Why or why not? Does your answer to part (d) change in any substantial way if you use the robust standard error? Explain. (4 points)
Suppose we think that in regression model (3) the error has heteroskedasticity of the form
var(ui|educi, experi, exper 2) = σ2hi,
where
hi = .885 — .055exper i + .0012exper 2.
i) Under the assumption this model of hi is correct, what is the transformed version of regression model (3) that removes the heteroskedasticity and can be used to obtain the generalized least squares (GLS) estimators of the β parameters? What would you need to check about all the values of hi before trying to estimate this transformed regression? (4 points)
Using hi from part (i), OLS estimation of the transformed model (GLS) resulted in the fitted model:
l^og(ragei) = —.533 +.112educi +.0357exper i —.0006exper 2 R2 = .156
(.199) (.014) (.013) (.0004) SSR = 188.49
[.194] [.013] [.014] [.0004]
The Breusch-Pagan statistic for testing the null hypothesis of no heteroskedasticity in the transformed model resulted in a p-value of 0.9505.
j) Is there evidence that the assumed form of hi is correct and that the GLS transformation has removed heteroskedasticity from regression model (3)? Explain. (3 points)
k) If you were doing this empirical analysis as part of your job, which estimated parameters would you report to your supervisor, OLS, GLS or both? Why? Carefully outline the logic and justifi- cation behind your decision. (4 points)
One concern with regression model (3) is the potential for MLR.4 to fail because of omitted variables such as ability. Suppose we plan to use one or both of the parental education variables, motheduc and ƒ atheduc to solve the MLR.4 problem. The remaining questions explore how these variables could be used to solve the MLR.4 problem.
l) To be specific, suppose that the error in regression model (3) depends on ability and ability is correlated with educ. What would this imply about the statistical properties of the OLS estimators of regression model (3)? Why? (3 points)
m) An empirical researcher suggests that motheduc and ƒ atheduc be used as proxies for ability. For motheduc and ƒ atheduc to be good proxies for ability, how must they be related to educ and ability? (4 points)
The researcher estimated the following regression. Only robust standard errors are reported.
l^og(ragei) = —.470 +.120educi +.0412exper i —.0008exper 2 —.016motheduci —.005ƒ atheduci (4)
[.194] [.013] [.014] [.0004] [.012] [.011]
R2 = .163 SSR = 186.90
n) Is there any statistical evidence that wages of a person are related to motheduc and ƒ atheduc given we have controlled for educ, exper, and exper 2? Be precise in your answer. Is the estimated slope on educ substantially different from what OLS estimation of regression model (3) gave? Are you surprised? Why or why not? (5 points)
o) Another empirical researcher suggests that motheduc and ƒ atheduc be used as instruments for educ. For motheduc and ƒ atheduc to be valid instruments, how must they be related to educ and to ability? (4 points)
This researcher estimates regression model (3) using 2SLS. In the first stage, the researcher uses exper,
exper 2, motheduc and ƒ atheduc to construct a proxy for educ:
eˆduci = 8.367 +.0853exper i —.0019exper 2 +.186motheduci +.185ƒ atheduci R2 = .262
[.280] [.026] [.0009] [.026] [.024]
In the second stage, the researcher uses eˆduci as a proxy for educi and obtains the IV estimated model
l^og(ragei) = .0481 +.061eˆduci +.0442exper i —.0009exper 2
R2 = .146
(5)
[.427] [.033] [.015] [.0004]
Only robust standard errors are reported in both cases.
p) Notice that the IV estimated slope of β1 (coefficient on educ) is substantially smaller than what OLS estimation of regression model (3) gave. Provide an explanation for this. (4 points)
q) Is there any indication that motheduc and ƒ atheduc are weak instruments? Be very precise in your answer and back up your conclusions with empirical evidence. (6 points)
r) If you were doing this empirical analysis as part of your job, which estimated parameters would you report to your supervisor, regression model (4), the IV estimated model (5), or both? Justify your answer with evidence and logical arguments. (4 points)
CS 340 Milestone One Guidelines and Rubric Overview: For this assignment, you will implement the fundamental operations of create, read, update,
Retail Transaction Programming Project Project Requirements: Develop a program to emulate a purchase transaction at a retail store. This
7COM1028 Secure Systems Programming Referral Coursework: Secure
Create a GUI program that:Accepts the following from a user:Item NameItem QuantityItem PriceAllows the user to create a file to store the sales receip
CS 340 Final Project Guidelines and Rubric Overview The final project will encompass developing a web service using a software stack and impleme
