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Section A

You must answer one question from this section.

You must use ONE WHITE answer sheet per question. To provide an

answer that exceeds the space on the answer sheet, please raise your

hand to request a YELLOW answer sheet.

1. Consider the simple regression with a constant

Yi = ↵ + Xi + ui, i = 1, . . . , n. (1)

1.1. Verify the following numerical properties for the OLS estimator:

(i)

Xn

i=1

uˆi = 0, (ii)

Xn

i=1

uˆiYˆi = 0, (iii)

Xn

i=1

Yˆi = Xn

i=1

Yi

where ˆui = Yi Yˆi, Yˆi = ˆ↵ + ˆXi, and ˆ↵ and ˆ denote the OLS

estimators of ↵ and , respectively. [20%]
1.2. Set = 0 in model (1), that is, consider the model

Yi = ↵ + ui, i = 1, . . . , n. (2)

Assume that ui ⇠ i.i.d.(0, 2), that is, the errors ui are independent

and identically distributed across i, with mean zero and variance 2.

Show that the OLS estimator of ↵, ˆ↵, is equal to Y¯ = n1 Pn

i=1 Yi

and the variance of ˆ↵ is 2/n. [20%]
1.3. Verify which of the numerical properties described in sub-question

1.1. hold for the OLS estimator of model (2). [10%]
2. Let F (FEMALE) be a variable which takes the value “0” for male and

“1” for female. Similarly, the variable M (MALE) takes on the value “1”

for male and “0” for female. Y denotes the Earnings.

For a given parameter ✓, let ˆ✓ denote its OLS estimator.

2.1. Derive the OLS estimators of ↵F and ↵M for the regression model

Yi = ↵F Fi + ↵MMi + ui, i = 1,…,n (3)

Show that ˆ↵F = Y¯F , the average of the Yi’s only for females, and

↵ˆM = Y¯M, the average of the Yi’s only for males. [20%]
Continued overleaf

2.2. Consider the regression

Yi = ↵ + Fi + ui, i = 1, . . . , n. (4)

Substitute M = 1 F in (3) and show that ↵ = ↵M and =

↵F ↵M. [10%]
2.3. Derive the OLS estimators of ↵ and for model (4). Show that

↵ˆ = ˆ↵M and ˆ = ˆ↵F ↵ˆM, where ˆ↵F and ˆ↵M have been derived in

sub-question 2.1. [20%]
Section B

You must answer one question from this section.

You must use ONE WHITE answer sheet per question. To provide an

answer that exceeds the space on the answer sheet, please raise your

hand to request a YELLOW answer sheet.

3. Consider the following wage-determination equation for the British economy for the period 1950-1969:

W

ct = 8.582 + 0.364 P Ft + 0.004 P Ft1 2.560 Ut

(1.129) (0.080) (0.072) (0.658)

(5)

where

W= wages and salaries per employee.

PF= prices of final output at factor cost.

U= unemployment in Great Britain as a percentage of the total number

of employees in Great Britain.

t= time, measured in years.

The figures in parenthesis are standard errors. The R2 of the regression

is 0.873.

3.1. Interpret briefly the preceding equation. [12.5%]
3.2. What is the rationale for the introduction of P Ft1?. [12.5%]
Continued overleaf

3.3. Should the variable P Ft1 be dropped from the model? Explain

your answer [12.5%]
3.4. Suppose that you are asked to estimate the elasticity of wages and

salaries per employee with respect to the price of final output. How

would you modify the regression model (5) to accomplish your task?

[12.5%]
4. Consider the Cobb-Douglas production function

Y = 1L2K3 (6)

where Y= output, L = labour input, and K = capital input.

Dividing equation (6) through by K, we get

(Y/K) = 1 (L/K)

2 K2+31 (7)

Taking the natural log of (7) and adding the error term, we obtain

ln (Y/K) = 0 + 2 ln (L/K)+(2 + 3 1)ln (K) + u (8)

where 0 = ln(1).

4.1. Suppose you had data to run the regression (8). How would you

test the hypothesis that there are constant returns to scale, that is,

(2 + 3) = 1? [17%]
4.2. If there are constant returns to scale, how would you interpret regression (8)? [16%]
4.3. Suppose we divide (6) by L rather than by K. Assuming constant

returns to scale, how would you interpret this regression? [17%]
END OF QUESTION PAPER

APPENDIX D: STATISTICAL TABLES 961

TABLE D.2 PERCENTAGE POINTS OF THE t DISTRIBUTION

Example

Pr (t > 2.086) = 0.025

Pr (t > 1.725) = 0.05 for df = 20

Pr (|t| > 1.725) = 0.10

Pr 0.25 0.10 0.05 0.025 0.01 0.005 0.001

df 0.50 0.20 0.10 0.05 0.02 0.010 0.002

1 1.000 3.078 6.314 12.706 31.821 63.657 318.31

2 0.816 1.886 2.920 4.303 6.965 9.925 22.327

3 0.765 1.638 2.353 3.182 4.541 5.841 10.214

4 0.741 1.533 2.132 2.776 3.747 4.604 7.173

5 0.727 1.476 2.015 2.571 3.365 4.032 5.893

6 0.718 1.440 1.943 2.447 3.143 3.707 5.208

7 0.711 1.415 1.895 2.365 2.998 3.499 4.785

8 0.706 1.397 1.860 2.306 2.896 3.355 4.501

9 0.703 1.383 1.833 2.262 2.821 3.250 4.297

10 0.700 1.372 1.812 2.228 2.764 3.169 4.144

11 0.697 1.363 1.796 2.201 2.718 3.106 4.025

12 0.695 1.356 1.782 2.179 2.681 3.055 3.930

13 0.694 1.350 1.771 2.160 2.650 3.012 3.852

14 0.692 1.345 1.761 2.145 2.624 2.977 3.787

15 0.691 1.341 1.753 2.131 2.602 2.947 3.733

16 0.690 1.337 1.746 2.120 2.583 2.921 3.686

17 0.689 1.333 1.740 2.110 2.567 2.898 3.646

18 0.688 1.330 1.734 2.101 2.552 2.878 3.610

19 0.688 1.328 1.729 2.093 2.539 2.861 3.579

20 0.687 1.325 1.725 2.086 2.528 2.845 3.552

21 0.686 1.323 1.721 2.080 2.518 2.831 3.527

22 0.686 1.321 1.717 2.074 2.508 2.819 3.505

23 0.685 1.319 1.714 2.069 2.500 2.807 3.485

24 0.685 1.318 1.711 2.064 2.492 2.797 3.467

25 0.684 1.316 1.708 2.060 2.485 2.787 3.450

26 0.684 1.315 1.706 2.056 2.479 2.779 3.435

27 0.684 1.314 1.703 2.052 2.473 2.771 3.421

28 0.683 1.313 1.701 2.048 2.467 2.763 3.408

29 0.683 1.311 1.699 2.045 2.462 2.756 3.396

30 0.683 1.310 1.697 2.042 2.457 2.750 3.385

40 0.681 1.303 1.684 2.021 2.423 2.704 3.307

60 0.679 1.296 1.671 2.000 2.390 2.660 3.232

120 0.677 1.289 1.658 1.980 2.358 2.617 3.160

∞ 0.674 1.282 1.645 1.960 2.326 2.576 3.090

Note: The smaller probability shown at the head of each column is the area in one tail; the larger probability

is the area in both tails.

Source: From E. S. Pearson and H. O. Hartley, eds., Biometrika Tables for Statisticians, vol. 1, 3d ed., table 12,

Cambridge University Press, New York, 1966. Reproduced by permission of the editors and trustees of Biometrika.

0 1.725

0.05

t

5

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