University of Nottingham

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1 University of Nottingham BUSINESS SCHOOL A LEVEL 2 MODULE, SPRING SEMESTER INTRODUCTORY ECONOMETRICS Time allowed TWO hours Candidates must NOT start writing their answers until told to do so Answer ALL questions in Section A and TWO questions from Section B Section A accounts for one third of the total marks available for this examination. Each Section B question carries equal weight. Percentage figures following each part indicate the proportionate weighting of that part within the question Only silent, self-contained calculators with a Single-Line Display or Dual-Line Display are permitted in this examination. Dictionaries are not allowed with one exception. Those whose first language is not English may use a standard translation dictionary to translate between that language and English provided that neither language is the subject of this examination. Subject specific translation dictionaries are not permitted. No electronic devices capable of storing and retrieving text, including electronic dictionaries, may be used. DO NOT turn examination paper over until instructed to do so ADDITIONAL MATERIAL: 1 handout comprising statistical tables 1 formula sheet TURN OVER

2 2 SECTION A 1. Using quarterly data for Canada from 1947, quarter 1, to 1996, quarter 4, a researcher performs a regression using aggregate consumption as the dependent variable and aggregate income as the independent variable. Below is a plot of the regression residuals against time Regression residuals residual (a) [50%] Which of the assumptions of OLS does the figure suggest may be violated? Explain your answer. (b) [50%] Given your answer to part (a), what would be the consequences of employing OLS to estimate the relationship between consumption and income using this data? 2. A sample of n identically and independently distributed observations X 1, X 2,..., X n is obtained from a population which has a population mean μ and a population variance σ 2. The following statistic is used to estimate μ: n =1 ˆθ = X 2n (a) [35%] Demonstrate that ˆθ is a biased estimator for μ. (b) [25%] Find the variance of ˆθ. (c) [20%] What is meant by the term a consistent estimator? (d) [20%] Explain whether ˆθ is a consistent estimator. 3. An organiser of a social event is considering what price to charge for tickets. She is aware that the number of people who will attend (Q) depends partly on the price charged (P), and partly on random phenomena. In fact the quantity attending is given by the equation Q = Z P where Z is random, E(Z) = 120 and V(Z) = 30. The organiser is concerned to raise as much revenue as possible. price times the number attending, ie Revenue (R) is the R = P Q = P (Z P) (a) [20%] If a price of 10 is charged, how many people would attend, in expectation? (b) [30%] For an arbitrary price, find the expression for the expected revenue. (c) [30%] For an arbitrary price, find the expression for the variance of the revenue. (d) [20%] What price maximises expected revenue?

3 3 4. The incidence of cigarette smoking among the UK s population can be represented by a discrete random variable X which takes the value 1 if an individual is a smoker and zero otherwise. (a) [25%] If the probability is p that a randomly selected individual is a smoker, write down the probability function for X. (b) [45%] Write down the probability function of the sample mean, assuming that samples of size 2 are taken randomly from the population. (c) [30%] Find the variance of the sample mean, either directly, or from what you know about the general relationship between the variance of the population and the variance of the sample mean. SECTION B 5. A study by J.E. Biddle and D.S. Hamermesh (1990), Sleep and the Allocation of Time, Journal of Political Economy, 98, examined the determinants of sleep behaviour for a sample of 706 individuals. The following results show how sleep is related to hours worked: SLEEP = (38.9) TOTWORKED (1) (0.016) where SLEEP is the number of minutes of sleep per week, and TOTWORKED is the number of minutes spent at work per week The figures in parentheses below the estimated coefficients are standard errors. (a) [15%] According to these estimates, how many minutes per week would an unemployed person sleep? (b) [15%] If someone were to work an extra hour in a given week, how much sleep, in minutes, would they give up in that week? (c) [15%] Using a 5% level of significance, test whether the population co-efficient of TOTWORKED is less than zero. (d) [15%] Briefly define heteroscedasticity. (e) [10%] Give a reason why heteroscedasticity might be a feature of the model estimated above. (f) [30%] To test for heteroscedasticity, the researcher estimated the equation: e 2 = TOTWORKED R 2 = n = 706 TOTWORKED is defined as before, and the variable e 2 represents the squared residuals from equation (1). Explain the purpose of this exercise and the researcher s conclusions. TURN OVER

4 4 6. In a study to explain CEO salaries, data were collected for 177 large US companies for the year Each observation in the dataset relates to a specific company, and the variables recorded are: the CEO s salary in thousands of dollars (salary); the CEO s age in years (age); the square of the CEO s age in years (sq_age); the number of years of tenure as CEO at the company (ceoten); the company s profit in millions of dollars (profits); a dummy variable taking the value 1 if the CEO had attended college and zero otherwise (college), and a dummy variable taking the value 1 if the CEO had attended graduate school and zero otherwise (grad). In this sample, no CEO attended graduate school who had not previously attended college. Standard errors are given in parentheses in the following table. Dependent variable is salary Independent Variable const age sq_age ceoten profits college grad R 2 RSS MODEL A ( ) ( ) (0.3744) ( ) (0.1031) ( ) ( ) MODEL B ( ) ( ) (0.3702) (6.2339) (0.1018) (a) [10%] Using Model A, give an interpretation of the estimated coefficient of the variable grad. (b) [15%] Using Model A, test, at the 5% level of significance, whether attending grad school lowers a CEO s salary relative to the salary which results from attending college only. Explain why it is that the test you perform allows this hypothesis to be examined. (c) [10%] Using Model A, what is the nature of the estimated relationship between salary and age? (d) [20%] Using Model A, at what age is CEO salary at a maximum? (e) [15%] Using Model A, test at the 5% level of significance whether there is a nonlinear relationship between salary and age. (f) [30%] Using methods for testing joint hypotheses, examine whether there is evidence of an effect on CEO salary due to either attendance at college or attendance at graduate school. Use a 10% significance level for this test.

5 5 7. In February 1983 a law was passed to make compulsory the wearing of seat belts by car drivers. The following table shows the results of a regression attempting to explain variation in the numbers of drivers killed on UK roads. The data are monthly time series from January 1969 to December Model 1: OLS estimates using the 192 observations 1969: :12 Dependent variable: DriversKilled coefficient std. error const l_kms l_petrolprice law Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared The variables are defined as: DriversKilled - The number of drivers killed in that month; l_kms - the natural logarithm of the number of kilometers driven on the UK road system that month; l_petrolprice - the natural logarithm of the price of petrol (price is measured in pounds sterling per litre); law is a variable taking the value 1 if the month is February 1983 or later. (a) [25%] According to these results, how many lives are saved each month as a result of this law? (b) [25%] Test, at the 5% level of significance, whether the law has led to a reduction in the number of drivers killed. (c) [25%] Give an interpretation of the estimated coefficient of the variable l_kms. (d) [25%] Give an interpretation of the estimated coefficient of the variable l_petrolprice. TURN OVER

6 6 8. A study to explain the determinants of athletes body-fat used a data sample of 89 Australian athletes whose discipline is either rowing, running 400 metres, or basketball. The results are tabulated below. The variables are defined as: Bfat - the weight of the athlete s body fat divided by total body weight, expressed as a percentage; Ht - the athlete s height in centimetres; Wt - the athlete s weight in kilograms; Rower - dummy variable to indicate that the athlete s discipline is Rowing; Run400m - dummy variable to indicate that the athlete s discipline is the 400 metre running event; Female: a dummy variable to indicate that the athlete s gender is female. Model 1: OLS estimates using the 89 observations 1-89 Dependent variable: Bfat coefficient std. error const Ht Wt Rower Run400m Female Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared (a) [20%] Give an interpretation of the estimated coefficient of the variable Female. (b) [20%] Test, at the 5% level, whether a Rower s body-fat percentage is significantly greater than that of a Basketball player, given the athlete s height, gender and weight. Explain why it is that the test you perform allows this hypothesis to be examined. (c) [25%] An additional variable HF was constructed by multiplying Ht by Female, and is included in a second regression (Model 2), the results of which are tabulated below. Model 2: OLS estimates using the 89 observations 1-89 Dependent variable: Bfat coefficient std. error const Ht Wt Rower Run400m Female HF Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared Give an interpretation the numerical estimate of the coefficient of the variable HF (d) [35%] A further regression was carried out, which differed from the specification of Model 2 only in that the variables Female and HF were omitted. The RSS from this regression was Use this information to test whether an athlete s gender affects his or her body fat. END