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ECMT Econometric Applications

Mid-Semester Practice Test

Question 1. [8 marks]

(i) Consider the following model used to explain the salary of chief executive officers:

salary = β0 + β1 sales + β2 mktval + β3 sales × mrkval + β4 sales2 + u

where sales is the company’s total sales (measured in $millions) and mrkval is the company’s market value (also measured in $millions). In terms of the parameters of the population model, what is the effect on expected salary of an extra $1million of sales? [2 marks]

(ii) What does attenuation bias’refer to, and what is the cause of this form of bias for OLS? [2 marks]

(iii) Describe what it means for an estimator to be consistent. Under what assumptions is the OLS estimator consistent? [2 marks]

(iv) Discuss the meaning of statistical significance and economic significance. [2 marks]

 

Question 2. [8 marks]

Let read denote the percentage of students at Queensland high schools who get a passing grade on a reading comprehension test. We are interested in the effect of school spending on the reading ability of students. A simple model is:

read = β0 + β1 expend + β2 enroll + β3 faminc + u                                (1)

where expend denotes school expenditure (per student) in $000s, enroll is the number of students enrolled at the school and faminc is the average family income of students at the school in $000s. The following table contains OLS estimates for this model:

Table 1: OLS Results

Variable

Estimate

S.E.

intercept

-23.14

24.99

expend

7.75

3.04

enroll

-1.26

0.58

faminc

0.324

0.36

n = 512

R2  = 0.1983

(i) What is the interpretation of the coefficient β2 ? [1 mark]

(ii) Test the statistical significance of β2 , against the alternative β2  < 0, using a 1% significance level. [2 marks]

(iii) Test whether this model has any explanatory power at the 1% significance level. [2 marks] Note: the F-test statistic is given by the formula:

F =         (Ru(2)r  Rr(2))/q       

(1 − Ru(2)r) / (n − 1)

where q is the number ofrestrictions, and ur and r stand for unrestricted and restricted models, respectively.

(iv) Suppose you want to test the null hypothesis H0  : β 1  = β3 against the alternative H1  : β 1  

β3 . Write down a transformed version of model (1) which you can estimate and do a t-test on a single parameter of the transformed model. Which parameter of the transformed model do you test? [2 marks]

(v) For this model R2  = 0.1983; does the size of the R2 statistic indicate that this is a good or bad econometric model? Explain your reasoning. [1 mark]

 

Question 3. [8 marks]

The following equation can be used to model the salary of graduates from different MBA programs in the US:

log (salary) = β0 + β1 testsc + β2 WAM + β3  log (libsize) + β4 rank + u             (2)

where salary is median annual salary for the graduating class, testsc is the median test score on a professional exam by members of the graduating class, WAM is the median weighted average

(course) mark for members of the graduating class, libsize is the number of volumes in the uni- versity library, and rank is world-wide rank of the MBA program (where rank = 1 is the best).

Based on a sample of 269 observations the following estimates were obtained:

log —(salary) = 11.000 + 0.050 testsc + 0.048 WAM + 0.040 log (libsize) − 0.0009 rank

(0.143)  (0.017)            (0.023)              (0.009)                         (0.0001)

n = 269,     R2  = 0.7927,     SSR = 0.40058

(i) From the information provided for the sample regression function, calculate the total sum of squares (SST) for the model. [1 mark]

(ii) Construct the 95% confidence interval for β2 . Is 0 within the confidence interval? [2 marks]

(iii) It is possible that model (2) is misspecified because of ignored non-linearities.  Outline the

steps required to carry out the RESET test. Clearly state the null and alternative hypotheses fo the test. [3 marks]

(iv) Are there any advantages from estimating model (2) using the Least Absolute Deviation (LAD)

estimator instead of OLS? Explain your reasoning. [1 mark]

(v) What benefit would we obtain from estimating model (2) using the quantile regression estima- tor for a set of quantiles: τ = {0.05, 0.1, 0.25, 0.5, 0.75, 0.9, 0.95} instead of just the median τ = 0.5? Explain your reasoning. [1 mark]

 

Question 4. [8 marks]

Consider the following model for wages in Australia:

log (wage) = β0 + β1 educ + β2 exper + β3 tenure + u                              (3)

where wage is the person’s wage (in dollars), educ is the level of education attainment (in years), exper is labour market experience (in years), and tenure is the time with the current employer (in years). Assume that E [u | educ, exper, tenure] = 0.

(i) You worry that model (3) might suffer from heteroskedasticity in the error variance. Outline the steps you would take to conduct the Breusch-Pagan test for heteroskedasticity. Be sure to state the null and alternative hypotheses and the conclusion you would make from a rejection

of the null hypothesis. [2 marks]

(ii) Assuming your result from (i) provided statistical evidence of heteroskedasticity in model (3). What are the implications for the coefficient estimates if you use OLS to estimate model (3)? [1 mark]

(iii) Now, suppose you know that the heteroskedasticity takes the following form:

Var (u | educ, exper, tenure) = σ 2  × exper2

Write a transformation of the model that has a homoskedastic error term. How would you estimate this model? [3 marks]

(iv) Instead, assume you do not know the functional form of the heteroskedasticity in model (3).

Explain the approach you would take to estimate this model. Are there any issues with this method? [2 marks]

 

Question 5. [8 marks]

The dynamic version of Okun’s Law relates changes in the unemployment rate (u) to changes in output (y) and has the following form:

∆ut  = α0 + α1 ∆yt + α2 ∆yt 1 + α3 ∆yt 2 + ηt                                                   (4)

(i) What is the interpretation of α 1  in model (4)?  What is the long-run propensity (LRP) for model (4)? [2 marks]

 

(ii) Using model (4) derive the transformed model you would need to estimate to obtain a direct

estimate of the LRP and its standard error.  Which coefficient in the transformed equation represents the LRP? [3 marks]

(iii) What does it mean for a variable to be non-stationary? What are the potential problems of

applying OLS to a dependent variable when it is non-stationary? [1 mark]

(iv) In order to test for the presence of a unit root in the process ut  the following model was estimated:

t  = 0.155 − 0.033 ut 1  − 0.010 time

(0.141)  (0.027)          (0.007)

T = 247,     R2  = 0.016

Based on these estimates, test whether ut has a unit root using a 5% significance level. What do you conclude, and how should your regression analysis of ut proceed? [2 marks]

Table 2: Critical Values of the Dickey-Fuller Coefficient Test

Quantiles     0.01   0.025     0.05     0.10     0.90     0.95   0.975     0.99

-3.42    -3.12   -2.86   -2.57   -0.44   -0.08     0.23     0.60

-3.96    -3.67   -3.41   -3.13   -1.25   -0.94    -0.66   -0.32

Note: The first row tabulates the critical values of the DF t-test of the null being a random walk with intercept; the second row tabulates the critical values of the DF t-test of the null being a random walk with both intercept and linear trend.