Question 1: Spreadsheet Modelling Murdock realty manages real estate investment properties, such as, condominiums and commercial real estat. For each rental property, the company has two associated revenue streams: administrative fees assocaited with maintaining the investment properties and fees associated with managing each investment property. In addition to the revenue streams, Murdock realty has several cost outlays associated with managing each individual property: labor costs, leasing costs for the building Murdock realty operates out of, marketing costs associated with the Murdock realty website, and additional expenses. Use the above information and Figure 1 to answer the following questions. (A) Murdock realty is currently trying to expand into new markets. To do so, they need to raise new capital, which will require borrowing money from a bank. As part of their loan application the bank has asked Murdock realty to explain their business model, including all revenue and cost streams. Dram an influence diagram that accurately represents this information and how these different pieces relate to company profit. (4 pt)
(B) Consider the spreadsheet model in Figure 1. Currently, Murdock realty charges a flat management fee per managed property. The current fee is 250$. Determine, using Figure 1, how profit will change if the management fee is increased to 325$ (assuming the same number of clients is retained). Do you expect the same number of clients to be retained after Murdock realty increases the management fee? (5 pt) (2pt) Revenue will increase and costs will stay the same, therefore profit will increase. (2pt) πold = 500, π new = 1250 which yields a percentage increase of 150% (1pt) However, basic economics says that, ceteris parabus, as price increases quantity demanded decreases. Therefore, it is unlikely that the same number of clients will be retained with the new pricing scheme. (C) Murdock realty currently has labor costs and revenues given in Figure 1. Murdock realty would like to give its’ employees a raise. Using Figure 1, determine how much Murdock realty can increase their employees hourly wage before earning zero profit (holding everything else constant). Explain how this can be answered using native spreadsheet functions in excel. (6 pt)
The simplest way to solve this is to use guess and verify. I showed them how that if x ∈ [a, b] and f (·) is continuous on [a, b], with f (a) > 0, f (b) < 0, f (·) crosses zero at a value λ (Rolls’ theorem, aka, mean value theorem). If we start 1
off with 18$ we get a positive number, going to 19$ we get a negative. The answer must therefore lie somewhere in the middle. The actual answer is 18.57. So, Murdock realty can increase the wage of every employee by 3.57. If they get close to this, and state how they obtained the answer, the student should receive full credit. This can easily be answered in excel using the “goal seek” function. (3pt) They should recognize that this is similar to what I have done in the review lecture and so they should mention somthing like: if x ∈ [a, b] and f (·) is continuous on [a, b], with f (a) > 0, f (b) < 0, f (·) crosses zero at a value λ. Words here is fine. (2pt) Answer is 18.57 (1pt). So, Murdock realty can increase the wage of every employee by 3.57.(1pt) (1pt) They must mention goal seek as this is the answer to the last part of the question.
Figure 1: Awesome Image
Question 2: Regression Analysis Nearly forty years of economic research says that wages are positively related to an employees level of education and job experience. The goal of this question is to determine the exact relationship between wages, education, and experience. We record data related to wages, experience, education, and age for a representative survey of 753 individuals in Australia. The precise definition of the recorded variables in the data set are as follows: lwage, the logarithm of the individuals stated wage; Age , age of the individual at the time of the survey; Educ, the individuals level of education at the time of the survey; Exp , the individuals level of experience at the time of the survey. 2
(A) From the output given in Figure 2, state the regression model that is being estimated. (1 pt) Ans: ln(wage) = β 0 + β 1 ∗ age + β 2 ∗ exp + β 3 ∗ educ + they either get it or they don’t.
Figure 2: Awesome Image (B) What percentage of the variation in ln(wage) is explained by this model and state whether or not this model is a good fit? (2 pt)
About 15% of the variation can be explained by the three covariates. With only 428 observations the model fit is adequate but not spectacular. (1pt) for correct percentage: 15% of the variation can be explained by the three covariates (1pt) for correct interpretation of fit. Accept statements that say fit is adequate or not great. (C) Calculate the coefficient estimate of education. (3 pt)
The coefficient has not been given so it will need to be backed-out using t − val = ˆ stdβ . Using this formula we have that β educ = 7.69489 ∗ .014201 ≈ .10927. β/ (2pt) for those who get the right formula to back out the coefficient. (1pt) for correct answer.
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(D) Using the information in Figure 2 and your knowledge of the student t-distribution, and standard normal distribution, report an approximate 95% confidence interval for the coefficient of experience (3pt). Please state why you are only able to build an approximate 95% confidence interval (2 pt). (total = 5pt)
By design, the t table will not be given so the students will not be able to get the exact critical value. However,in this example we can use tLarge,.025 ≈ 1.96 for the critical value to obtain the approximate interval: (.11 − 1.96 ∗ .0142, .11 + 1.96 ∗ .0142) = (.082168, .137832) (1pt) for correct approximation for t-distribution of 1.96. (1pt) for correct formula for confidence interval using approximation 1.96. (1pt) for correct answer (.082168, .137832). (1pt) for knowing that the interval is approximate because they are using a normal when they should be using a student-t (1pt) for knowing that the correct degrees of freedom for the student-t is 428 − 3 − 1 = 424. (E) We know from class that 95% CIs can be used to conduct two-sided hypothesis tests. Is a two-sided hypothesis test for education appropriate in this example (2 sentence maximum please)? (3 pt)
A two-sided hypothesis test does not account for any prior information we have about the effect of the variable of interest on the outcome variable. If this were the case with education, we would be saying that an additional year of education could have a negative impact on wages. No, a two-sided test is not appropriate (1pt) for those who note the stem of the question gives prior information about the sign of education in this regression, and hence the appropriateness of the two-sided test. (1pt) for those who acknowledge that testing β educ = 0 implies the effect of this sign could be negative. (1pt) for those who just answer no. (F) Conduct a one-sided approximate hypothesis test for a positive educational affect at the 95% level (please state all testing steps) (4 pt). Why is the test approximate? (1 pt) (total = 5) (1pt) for those who answer something like: Since t428,.05 is not given we can use the corresponding z .05 = 1.645 value to conduct the test. This is why the test is approximate. (1pt) for each of the following four pieces of the test. These are either correct or not, no partial credit. (1) Hypothesis. Ans: H 0 : β educ = 0 vs. H a : β educ > 0 (2) T.S. Ans: teduc = 7.69489
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(3) C.V. Ans: The critical value is t428,.05 since we are not given this value we can use Use 1.645 (since N is large). (4) Decision rule: Reject H 0 if teduc > tC.V. . Ans: Reject t educ > 1.645. (G) In the observed sample there are many missing values for wages, as not all individuals in the sample were employed at the time the sample information was collected. (G.1) State the missing data method used to generate the regression results in Figure 2. (1 pt) (1pt) Discarding individuals from the sample that are unemployed. (G.2) Does the missing data method in [G.1] place any restrictions on how the coefficients in the regression output can be interpreted? (3 pt) Yes, similar to the College football revenue example in class, the regression estimates can only be interepreted accurately for individuals in the sample. Which means that these results are only valid for individuals who were employed at the time of the sample, which is a rather sever restriction in terms of policy impact. (2pt) for knowing that the coefficients can only be interpreted for individuals who were employed. (1pt) for noting that this is a severe restriction. (G.3) State two possible ways of dealing with the missing values for lwage. For one of the two methods, give a short description as to why this method may be appropriate. (2 pt) As in the previous regression results, we could simply throw out those individuals for which the data are missing. In addition, we could impute values for the missing observations using some model. If they give reasonable answers give them the points. (H) Figure ?? contains regression output associated with an alternative way of dealing with the missing values. (H.1) From the given output, obtain and interpret the coefficient estimate for education (3 pt) Unlike the previous regression output, it seems now we are given even less information. But we are now given the excel confidence intervals which tells us ˆ − 1.96 ∗ stdβ =⇒ β ˆ = C I l + 1.96stdβ ≈ .0615. CI l = β
The coefficient can be interpreted in the standard way since we have imputed the missing values (Note that N=753). (1pt) for noting that they have to use the normal approximation here. (1pt) for correct formula to obtain coefficient estimate. It must include the normal value since they won’t have the t-value memorized (1pt) for the correct approximate answer. Anything close to .0615 give them the points.
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(H.2) Using the regression output in Figure ??, test the null hypothesis that there is no effect of education on wages. (please include all testing steps) (3pt) (1) Hypothesis. Ans: H 0 : β exper = 0 vs. H a : β exper =0 (2) T.S. Ans: Must use the confidence interval since no exact t-stat is given (3) Decision rule: Reject H 0 if 0 ∈ (CI l , CI h). Ans: Reject H 0 (2pt) for listing all testing steps correctly. Their can be variations in the above since we are using the confidence interval approach to testing the null. (1pt) for correct conclusion (H.3) Compare and contrast the coefficient estimates from education and experience across the two methods for dealing with the missing data. What do these results say about the appropriateness of the two missing data methods? (4 pt) The first thing they will need to do is back out, using the same method as above, the coefficient from experience. Which is about β exper ≈ .0065 (1pt). Both coefficient estimates are smaller than their corresponding counterparts obtained when simply throwing out the observations of nonemployed individuals. However, they are not strikingly different (2pt). The second method yields coefficient estimates that are about half as large as those obtained by the first method, but both sets are significant and they have the same sign. The coefficient on age and the intercept change, but neither are significant. This says that both methods may be appropriate, depending on how you interpret “large” (1pt) (1pt) for backing out the additional coefficient correctly (2pt) for correctly comparing the coefficient estimates. (1pt) for correctly interpreting what this means in terms of the missing data methods.
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