University of Manchester
ECON61001: Econometric Methods
Mid-Term Exam
Release date/time: 27/11/20, 9.00hrs Greenwich Mean Time (GMT) Submission deadline: 30/11/20, 9.00hrs GMT
Instructions:
• You must answer all three questions.
• Your answers could be typed or hand-written (and scanned to a single pdf file that can be submitted) or a combination of a typed answer with included images of algebra or figures.
• Where relevant, questions include word limits. These are limits, not targets. Ex- cellent answers can be shorter than the word limit. If you go beyond the word limit the additional text will be ignored. Where a question includes a word limit you HAVE to include a word count for your answer (excluding formulae). You could use https://wordcounter.net to obtain word counts.
• Candidates are advised that the examiners attach considerable importance to the clarity with which answers are expressed.
• You must correctly enter your registration number and the course code on your answer.
ECON61001
1. Two researchers are interested in whether or not the mean household income is the same in both the North and the South of England. Suppose they obtain a random sample of N observations on the income of households in England at a particular moment in time. Let yi be the income of the ith household and Ri be a dummy variable that indicates the region in which the household lives as follows:
Ri = 1, if ith household lives in the North, = 0, if ith household lives in the South.
Researcher A considers inference based on the regression model
yi = x′iβ0 + ui, (1)
where x′i = (1, Ri), β0 = (β0,1, β0,2)′. Let βˆN denote the OLS estimator of β0 based on (1).
Researcher B considers inference based on the regression model
y i = w i′ γ 0 + u i , ( 2 )
where wi′ = (1 − Ri, Ri), γ0 = (γ0,1, γ0,2)′. Let γˆN denote the OLS estimator of γ0 based on (2).
Assume that ui is independent of Ri and that {ui; i = 1,2,…,N} is a sequence of independently and identically distributed random variables with a normal dis- tribution with mean equal to zero and variance equal to σ02, an unknown positive constant.
(a) Show that:
βˆN= y ̄s , y ̄ n − y ̄ s
where y ̄n, y ̄s are the sample mean incomes for households living, respec- tively, in the North and the South of England. [10 marks]
(b) Under the conditions above, it can be shown that N1/2(βˆN − β0) →d N(0, Vβ) and N1/2(γˆN − γ0) →d N(0, Vγ). What is the relationship between Vγ and
Vβ ? Be sure to justify your answer. [10 marks]
Continued over
1
ECON61001
2. Consider the linear regression model
y = Xβ0 + u,
where X is 20 × 4 matrix that is fixed in repeated samples with full column rank,
and u ∼ N(0σ02I20) where σ02 is an unknown positive constant.
(a) Let βˆT,3 be the OLS estimator of β0,3, the third element of β0. Consider the following two inference procedures relating to β0,3 based on βˆT ,3:
• 100(1 − α)% confidence interval for β0,3;
• the two-sided hypotheses test of H0 : β0,3 = 1.
Show that the null hypothesis of this two-sided test is rejected at the 100α% significance level if and only if one is not in the 100(1 − α)% confidence interval for β0,3. [5 marks]
(b) Propose a 95% confidence interval for σ02 and show that it possesses the stated coverage rate. Note: you may quote any relevant distributional result from lecture notes without proof. [10 marks]
3. Consider the linear regression model
yi = x′iβ0 + ui, i = 1,2,…,N,
where β0 is the k × 1 vector of unknown regression coefficients, {(x′i,ui)}Ni=1 is a sequence of independent and identically distributed random vectors with E[ui|xi] = 0, V ar[ui|xi] = σ02, an unknown positive constant, and E[xix′i] = Q, a finite positive definite matrix of constants. Let βˆR,N denote the RLS estimator based on the linear restrictions Rβ = r where where R is a nr × k matrix of pre- specified constants with rank equal to nr and r is a nr × 1 vector of pre-specified constants that is,
βˆR,N = βˆN − (X′X)−1R′{R(X′X)−1R′}−1(RβˆN − r),
where βˆN is the OLS estimator of β0 and X is the N × k matrix with ith row x′i.
Show that βˆR,N is a consistent estimator for β0. [15 marks]
Note: (i) You may quote the formula for the OLS estimator without proof; (ii) you may quote the generic form of both the Weak Law of Large Numbers,
N −1 Ni=1 zi →p μz , but must verify μz for the specific choices of zi relevant to your answer; you may also quote the generic form of the Central Limit Theorem, N−1/2 Ni=1(zi − μz) →d N(0, Ω) but must verify μz and Ω for the specific choices of zi relevant to your answer.
END OF EXAMINATION
2
1 TABLE 1: PERCENTAGE POINTS FOR THE T DISTRIBUTION
1 Table 1: Percentage Points for the t distribution
Student’s t Distribution Function for Selected Probabilities
The table provides values of tα,v where Pr(T ≤ tα,v) = α and T ∼ tv
α
0.750 0.800 0.900 0.950 0.975 0.990 0.995 0.9975 0.999 0.9995
ν
Values of tα,v
1 2 3 4 5 6 7 8 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100 110 120
∞
1.000 1.376 3.078 6.314 12.706 31.821 63.657 0.816 1.061 1.886 2.920 4.303 6.965 9.925 0.765 0.978 1.638 2.353 3.182 4.541 5.841 0.741 0.941 1.533 2.132 2.776 3.747 4.604 0.727 0.920 1.476 2.015 2.571 3.365 4.032 0.718 0.906 1.440 1.943 2.447 3.143 3.707 0.711 0.896 1.415 1.895 2.365 2.998 3.499 0.706 0.889 1.397 1.860 2.306 2.896 3.355 0.703 0.883 1.383 1.833 2.262 2.821 3.250 0.700 0.879 1.372 1.812 2.228 2.764 3.169 0.697 0.876 1.363 1.796 2.201 2.718 3.106 0.695 0.873 1.356 1.782 2.179 2.681 3.055 0.694 0.870 1.350 1.771 2.160 2.650 3.012 0.692 0.868 1.345 1.761 2.145 2.624 2.977 0.691 0.866 1.341 1.753 2.131 2.602 2.947 0.690 0.865 1.337 1.746 2.120 2.583 2.921 0.689 0.863 1.333 1.740 2.110 2.567 2.898 0.688 0.862 1.330 1.734 2.101 2.552 2.878 0.688 0.861 1.328 1.729 2.093 2.539 2.861 0.687 0.860 1.325 1.725 2.086 2.528 2.845 0.686 0.859 1.323 1.721 2.080 2.518 2.831 0.686 0.858 1.321 1.717 2.074 2.508 2.819 0.685 0.858 1.319 1.714 2.069 2.500 2.807 0.685 0.857 1.318 1.711 2.064 2.492 2.797 0.684 0.856 1.316 1.708 2.060 2.485 2.787 0.684 0.856 1.315 1.706 2.056 2.479 2.779 0.684 0.855 1.314 1.703 2.052 2.473 2.771 0.683 0.855 1.313 1.701 2.048 2.467 2.763 0.683 0.854 1.311 1.699 2.045 2.462 2.756 0.683 0.854 1.310 1.697 2.042 2.457 2.750 0.681 0.851 1.303 1.684 2.021 2.423 2.704 0.679 0.849 1.299 1.676 2.009 2.403 2.678 0.679 0.848 1.296 1.671 2.000 2.390 2.660 0.678 0.847 1.294 1.667 1.994 2.381 2.648 0.678 0.846 1.292 1.664 1.990 2.374 2.639 0.677 0.846 1.291 1.662 1.987 2.368 2.632 0.677 0.845 1.290 1.660 1.984 2.364 2.626 0.677 0.845 1.289 1.659 1.982 2.361 2.621 0.677 0.845 1.289 1.658 1.980 2.358 2.617 0.674 0.842 1.282 1.645 1.960 2.326 2.576
4.773
4.317 5.208 4.029 4.785 3.833 4.501 3.690 4.297 3.581 4.144 3.497 4.025 3.428 3.930 3.372 3.852 3.326 3.787 3.286 3.733 3.252 3.686 3.222 3.646 3.197 3.610 3.174 3.579 3.153 3.552 3.135 3.527 3.119 3.505 3.104 3.485 3.091 3.467 3.078 3.450 3.067 3.435 3.057 3.421 3.047 3.408 3.038 3.396 3.030 3.385 2.971 3.307 2.937 3.261 2.915 3.232 2.899 3.211 2.887 3.195 2.878 3.183 2.871 3.174 2.865 3.166 2.860 3.160 2.808 3.090
5.408 5.041 4.781 4.587 4.437 4.318 4.221 4.140 4.073 4.015 3.965 3.922 3.883 3.850 3.819 3.792 3.768 3.745 3.725 3.707 3.690 3.674 3.659 3.646 3.551 3.496 3.460 3.435 3.416 3.402 3.390 3.381 3.373 3.297
3
2 TABLE 2: PERCENTAGE POINTS FOR THE χ2 DISTRIBUTION 2 Table 2: Percentage Points for the χ2 distribution
The χ2 Distribution Function for Selected Probabilities
The table provides values of χ2α,v where Pr(χ2 ≤ χ2α,v) = α and χ2 ∼ χ2v
α
0.005 0.01 0.025 0.05 0.1 0.5 0.9 0.95 0.975 0.99 0.995
v
Values of χ2α,v
1 2 3 4 5 6 7 8 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
45
50
50
70
80
90
100 150 200
0.000 0.000 0.001 0.004 0.016 0.455 2.706 3.841 5.024 6.635 7.879 0.010 0.020 0.051 0.103 0.211 1.386 4.605 5.991 7.378 9.210 10.60 0.072 0.115 0.216 0.352 0.584 2.366 6.251 7.815 9.348 11.34 12.84 0.207 0.297 0.484 0.711 1.064 3.357 7.779 9.488 11.14 13.28 14.86 0.412 0.554 0.831 1.145 1.610 4.351 9.236 11.07 12.83 15.09 16.75 0.676 0.872 1.237 1.635 2.204 5.348 10.64 12.59 14.45 16.81 18.55 0.989 1.239 1.690 2.167 2.833 6.346 12.02 14.07 16.01 18.48 20.28 1.344 1.646 2.180 2.733 3.490 7.344 13.36 15.51 17.53 20.09 21.95 1.735 2.088 2.700 3.325 4.168 8.343 14.68 16.92 19.02 21.67 23.59 2.156 2.558 3.247 3.940 4.865 9.342 15.99 18.31 20.48 23.21 25.19 2.603 3.053 3.816 4.575 5.578 10.34 17.28 19.68 21.92 24.72 26.76 3.074 3.571 4.404 5.226 6.304 11.34 18.55 21.03 23.34 26.22 28.30 3.565 4.107 5.009 5.892 7.042 12.34 19.81 22.36 24.74 27.69 29.82 4.075 4.660 5.629 6.571 7.790 13.34 21.06 23.68 26.12 29.14 31.32 4.601 5.229 6.262 7.261 8.547 14.34 22.31 25.00 27.49 30.58 32.80 5.142 5.812 6.908 7.962 9.312 15.34 23.54 26.30 28.85 32.00 34.27 5.697 6.408 7.564 8.672 10.09 16.34 24.77 27.59 30.19 33.41 35.72 6.265 7.015 8.231 9.390 10.86 17.34 25.99 28.87 31.53 34.81 37.16 6.844 7.633 8.907 10.12 11.65 18.34 27.20 30.14 32.85 36.19 38.58 7.434 8.260 9.591 10.85 12.44 19.34 28.41 31.41 34.17 37.57 40.00 8.034 8.897 10.28 11.59 13.24 20.34 29.62 32.67 35.48 38.93 41.40 8.643 9.542 10.98 12.34 14.04 21.34 30.81 33.92 36.78 40.29 42.80 9.260 10.20 11.69 13.09 14.85 22.34 32.01 35.17 38.08 41.64 44.18 9.886 10.86 12.40 13.85 15.66 23.34 33.20 36.42 39.36 42.98 45.56 10.52 11.52 13.12 14.61 16.47 24.34 34.38 37.65 40.65 44.31 46.93 11.16 12.20 13.84 15.38 17.29 25.34 35.56 38.89 41.92 45.64 48.29 11.81 12.88 14.57 16.15 18.11 26.34 36.74 40.11 43.19 46.96 49.64 12.46 13.56 15.31 16.93 18.94 27.34 37.92 41.34 44.46 48.28 50.99 13.12 14.26 16.05 17.71 19.77 28.34 39.09 42.56 45.72 49.59 52.34 13.79 14.95 16.79 18.49 20.60 29.34 40.26 43.77 46.98 50.89 53.67 17.19 18.51 20.57 22.47 24.80 34.34 46.06 49.80 53.20 57.34 60.27 20.71 22.16 24.43 26.51 29.05 39.34 51.81 55.76 59.34 63.69 66.77 24.31 25.90 28.37 30.61 33.35 44.34 57.51 61.66 65.41 69.96 73.17 27.99 29.71 32.36 34.76 37.69 49.33 63.17 67.50 71.42 76.15 79.49 27.99 29.71 32.36 34.76 37.69 49.33 63.17 67.50 71.42 76.15 79.49 43.28 45.44 48.76 51.74 55.33 69.33 85.53 90.53 95.02 100.4 104.2 51.17 53.54 57.15 60.39 64.28 79.33 96.58 101.9 106.6 112.3 116.3 59.20 61.75 65.65 69.13 73.29 89.33 107.6 113.1 118.1 124.1 128.3 67.33 70.06 74.22 77.93 82.36 99.33 118.5 124.3 129.6 135.8 140.2 109.1 112.7 118.0 122.7 128.3 149.3 172.6 179.6 185.8 193.2 198.4 152.2 156.4 162.7 168.3 174.8 199.3 226.0 234.0 241.1 249.4 255.3
4
3 TABLE 3: UPPER 5% PERCENTAGE POINTS FOR THE F DISTRIBUTION
3 Table 3: Upper 5% percentage points for the F distribution
The F Distribution Function for α = 0.05
The table provides values of Fα,v1,v2 where Pr(F ≥ Fα,v1,v2 ) = 0.05 and F ∼ F (v1, v2)
v1 →
v2 ↓
1 2 3 4 5 6 7 8 9 10 12 15
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
55
60
70
80
90
100
110
120
150
6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 2.79 2.72 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 2.60 2.53 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 2.53 2.46 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.42 2.35 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 2.38 2.31 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.34 2.27 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 2.31 2.23 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.25 2.18 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 2.23 2.15 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 2.20 2.13 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 2.16 2.09 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 4.12 3.27 2.87 2.64 2.49 2.37 2.29 2.22 2.16 2.11 2.04 1.96 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 4.06 3.20 2.81 2.58 2.42 2.31 2.22 2.15 2.10 2.05 1.97 1.89 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 1.95 1.87 4.02 3.16 2.77 2.54 2.38 2.27 2.18 2.11 2.06 2.01 1.93 1.85 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 3.98 3.13 2.74 2.50 2.35 2.23 2.14 2.07 2.02 1.97 1.89 1.81 3.96 3.11 2.72 2.49 2.33 2.21 2.13 2.06 2.00 1.95 1.88 1.79 3.95 3.10 2.71 2.47 2.32 2.20 2.11 2.04 1.99 1.94 1.86 1.78 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97 1.93 1.85 1.77 3.93 3.08 2.69 2.45 2.30 2.18 2.09 2.02 1.97 1.92 1.84 1.76 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91 1.83 1.75 3.90 3.06 2.66 2.43 2.27 2.16 2.07 2.00 1.94 1.89 1.82 1.73
5
4 TABLE 4: UPPER 1% PERCENTAGE POINTS FOR THE F DISTRIBUTION
4 Table 4: Upper 1% percentage points for the F distribution
The F Distribution Function for α = 0.01
The table provides values of Fα,v1,v2 where Pr(F ≥ Fα,v1,v2 ) = 0.01 and F ∼ F (v1, v2)
v1 →
v2 ↓
1 2 3 4 5 6 7 8 9 10 12 15
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
30
35
40
45
50
55
60
70
80
90
100
110
120
150
16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.2 10.1 9.89 9.72 13.7 10.9 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 12.2 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 11.3 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 10.6 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 10.0 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54 4.40 4.25 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10 3.96 3.82 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03 3.94 3.80 3.66 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.55 3.41 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.46 3.31 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51 3.37 3.23 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.30 3.15 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3.31 3.17 3.03 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.12 2.98 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.07 2.93 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 3.03 2.89 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22 3.13 2.99 2.85 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.84 2.70 7.42 5.27 4.40 3.91 3.59 3.37 3.20 3.07 2.96 2.88 2.74 2.60 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.66 2.52 7.23 5.11 4.25 3.77 3.45 3.23 3.07 2.94 2.83 2.74 2.61 2.46 7.17 5.06 4.20 3.72 3.41 3.19 3.02 2.89 2.78 2.70 2.56 2.42 7.12 5.01 4.16 3.68 3.37 3.15 2.98 2.85 2.75 2.66 2.53 2.38 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.50 2.35 7.01 4.92 4.07 3.60 3.29 3.07 2.91 2.78 2.67 2.59 2.45 2.31 6.96 4.88 4.04 3.56 3.26 3.04 2.87 2.74 2.64 2.55 2.42 2.27 6.93 4.85 4.01 3.53 3.23 3.01 2.84 2.72 2.61 2.52 2.39 2.24 6.90 4.82 3.98 3.51 3.21 2.99 2.82 2.69 2.59 2.50 2.37 2.22 6.87 4.80 3.96 3.49 3.19 2.97 2.81 2.68 2.57 2.49 2.35 2.21 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 2.34 2.19 6.81 4.75 3.91 3.45 3.14 2.92 2.76 2.63 2.53 2.44 2.31 2.16
6