Monash University Semester One Examination Period 2014 Faculty Of Business and Economics EXAM CODES:
ETC1010
TITLE OF PAPER:
DATA MODELLING
EXAM DURATION:
2 hours writing time
READING TIME:
10 minutes
THIS THIS PAPER IS FOR STUDENTS STUDENTS STUDYING STUDYING AT:( tic k w here app licable)
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Page 1 of 9
Question 1 (10 marks) 1 John is 20 years old and wants to start saving for his superannuation. He would like to have
accumulated a pot of money of $500,000 with which to buy his pension at age 65. (a)
(b)
If he can achieve a return of 6% p.a. effective and, assuming that John invests the same amount at the end of each year until age 65, how much should John invest at the end of the year? Assume that John, now aged 65, has accumulated $500,000 in his superannuation fund and he is now going to draw down $6,000 at the start of each month from this pot of money. Assuming he continues to receive a return of 6% p.a. effective. How long will John be able to continue drawing down on his funds for?
(5 marks)
(5 marks)
Question 2 (14 marks) 2 (a) Using the mortality table provided on the next page, calculate the following
(i) The probability that a person currently aged 50 will survive until (2 marks) age 65 (2 marks) (ii) The probability that a person currently aged 60 will die before age 65 (3 marks) (iii) The probability that a person currently aged 60 will die between ages 65 and 80 (b)
Describe the shape of a mortality curve (i) (ii) (iii)
If plotted with age as the x axis If plotted with time as the x axis What information does (ii) give us about future life expectancy?
Question 3 (6 marks) 3 (a) List the assumptions of the chain ladder method
(2 marks) (2 marks) (3 marks)
(2 marks)
The cumulative cost of incurred claims (in $000), from a particular group of policies are shown in the tables given below. We can assume the claims are fully run-off at the end of the development year 2.
The total amount of claims that have been paid to date is $1,701 (b) Use the chain ladder method to estimate the outstanding claim reserve in respect of accident years 2007 and 2008.
(4 marks)
Page 2 of 9
Appendix Mortality table for question 2 Age x
q_x
l_x
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
0.0074 0.0074 0.0075 0.0075 0.0076 0.0077 0.0079 0.0081 0.0084 0.0087 0.0091 0.0097 0.0104 0.0112 0.0122 0.0134 0.0149 0.0166 0.0186 0.0209 0.0236 0.0266 0.0300 0.0338 0.0381 0.0427 0.0478 0.0534 0.0593 0.0658 0.0726 0.0799 0.0875 0.0956 0.1041 0.1130 0.1222 0.1319 0.1419 0.1524 0.1650 0.1787 0.1928 0.2072 0.2219 0.2368
100,000 99,257 98,515 97,773 97,028 96,278 95,518 94,745 93,954 93,137 92,287 91,394 90,448 89,436 88,345 87,159 85,862 84,437 82,866 81,132 79,219 77,110 74,794 72,263 69,511 66,540 63,357 59,976 56,417 52,707 48,881 44,977 41,040 37,116 33,252 29,495 25,890 22,475 19,285 16,346 13,650 11,211 9,050 7,175 5,582 4,260
Page 3 of 9
Question 4 (15 marks)
In a study of the effects of case mix on hospital costs in the text book Economic Analysis for the Health Service Industry, North-Holland 1967, Chapter 1, M. S. Feldstein considered the following model:
Where for each hospital the regressand C is the average cost per case treated and the value of each regressor gives the proportion of cases treated in each treatment category where the categories are: M= Medical P= Pediatrics GS= General Surgery E= Enterology TO= Traumatic & Orthopedic Surgery OS= Other Surgery G= Gynecology Ob= Obstetrics N.B.
Where other is the proportion of cases treated a miscellaneous collection of other treatment categories.
Data on 177 hospitals yielded the following least squares regression results where denotes the standard error of the regression coefficient . The Coefficient of Determination was 0.3076. The F statistic for the overall significance was 9.33 (The Excel function value “F.INV(0.99,8,168)=2.6817”.).
Page 4 of 9
Regressor
s.e.(b)
AC
M
44.97 18.89
114.48
P
-44.54 28.51
24.97
GS
-36.81 14.88
32.7
E
-54.26 16.52
15.25
TO
-29.82 17.18
39.69
OS
28.51 20.27
98.02
G
-10.79 21.47
58.72
Ob
-34.63 16.34
34.88
Constant
69.51
69.51
1. Each of the regressor variables is obviously nominal. Are they also measured on a ratio scale? Briefly explain your answer. (2 Marks) 2. Which of the eight regressor variables has a coefficient that is not statistically
significantly different from zero? (The Excel function value “T.INV(0.975,168)=1.9742”.) It is suggested that the variables that do not have significant coefficients should be deleted from the relationship. Is this suggestion well founded? If so, why? If not, why not? (8 Marks)
3. How have the values in the column labelled AC been calculated? What is the appropriate interpretation to be attached to these numbers? (5 Marks)
Page 5 of 9
Question 5 (15 marks)
In a study of the United Kingdom brick, pottery, glass and cement industry for the period 1961 to 1981 (inclusive), R.L. Thomas { Introductory Econometrics: Theory and Applications Longman 1985, pp 244-246} obtained the following OLS regression results:
̂
Where Q= index of production at constant factor cost, K= gross capital stock at replacement cost, H= hours worked per employee, and the figures in parentheses are the coefficient standard errors. 1. Do the estimated coefficients suggest that the industry was experiencing increasing or decreasing returns to scale over these two decades? Justify your answer. (3Marks)
2. Verify that each partial regression slope coefficient is statistically different from zero by conducting an appropriate one sided hypothesis test at the 5% level of significance. (The Excel function value “T.INV(0.95,18)=1.7341”.) (3 Marks)
To allow for changes in technology over time the model was re-estimated to give
̂ Where T= time in years – used here as a proxy for technological change. 3. What interpretation should be placed on the value 0.0272 obtained for the estimated coefficient on the variable T? (2 Marks)
4. Verify that the coefficient of log K is now statistically insignificant at the 5% level. (Show your working using the Excel function value “T.INV(0.05,17) = 2.5669”.) (3 Marks)
5. Given that the correlation coefficient between T and log K is 0.98, account for the insignificance of the regressor log K in this second model? (4 Marks)
Page 6 of 9
Question 6 (20 marks)
is a cubic function with domain D = [-3,3]. Some of the values of are in the table below: The function
x
-3 -29
-2 12
-1 23
0 16
1 3
2 -4
3 7
It is also known that and and (where ’ denotes derivative and ’’ denotes second derivative.) (a) (b) (c) (d)
have a local minimum on D? If so at what value(s) of x? Does have a local maximum on D? If so at what value(s) of x? Does have a global minimum on D? If so at what value(s) of x? Does have a global maximum on D? If so at what value(s) of x? Does
(3 marks each)
is a continuous function on D = [-3,3] with also continuous on D. Must have a global maximum on D? Must a global maximum be at a turning
Suppose (e)
point?
(6 marks)
(f)
Does the function
|| have a stationary point at x = 0?
(2 marks)
Page 7 of 9
Question 7 (10 marks)
A farmer has a long brick wall on his property and a small amount of fencing. He wishes to build a rectangular paddock for his pets using the brick wall as one side. He has 12m of fence. The rectangular paddock will have dimensions
a
b and
therefore area
A
ab .
Use the Lagrangian method to find the values of a and b that maximise area A subject to the constraint 2a + b 12 . State the value of that area.
Page 8 of 9
Question 8 (10 marks)
Recall the Samsung television example from tutorials.
The corresponding multiple linear regression was given by
(1) With the residuals or errors given by In cells H42, I42 and J42 (not shown above) excel’s SUM function was used to calculate the sum of , the sum of and the sum of absolute error respectively
∑|| . Note MAD
∑|| .
Solver was used to minimise the value in objective cell H42. (a) From the output above, determine the cell address of the estimates of the three beta coefficients. (3 marks)
Solver was also used to minimise the value in objective cell J42. (b) State with reasons why you prefer the estimates of the beta coefficients in model (1) from minimising cell H42 or cell J42? (5 marks)
(c) Which of (a) and (b) corresponds to OLS regression estimates? Why? (2 marks)
END OF EXAMINATION Page 9 of 9
