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Instructions:
University of Manchester
ECON61001: Econometric Methods
Mock Exam
Release date/time: Submission deadline:
• You must answer all five questions in Section A and two out of the four ques- tions in Section B. If you answer more questions than are required and do not indicate which answers should be ignored, we will mark the requisite number of answers in the order in which they appear in your answer submission: answers beyond that number will not be considered.
• 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.

A
ECON61001
SECTION
1.(a) Let A denote a m × m symmetric matrix. If A is positive definite then what does this imply about any quadratic form involving A? [1 marks]
1.(b) If A is a m × m positive definite matrix then what is rank(A), the rank of A. Justify your answer.
1.(c) Consider
A1 = 􏰒 2 −1 􏰓 and A2 = 􏰒 7 5 􏰓. −1 2 5 3
Verify whether or not A1 and A2 are positive definite, being sure to justify your answer. [4 marks]
2. Let {(yi, x′i)}Ni=1 be a sequence of independently and identically distributed (i.i.d.) random vectors. Suppose that yi is a dummy variable and so has a sample space of {0, 1} with P (yi = 1|xi) = Λ(x′iβ0) where Λ( · ) is the cumulative distribution function of the logistic distribution. Derive:
(a) Derive E[yi|xi], E[yi] and V ar[yi|xi]. [4 marks] (b) Nowsupposethataresearcherestimatesalogitmodelbasedon{(yi,x′i)}Ni=1,
and let xi,j be the jth element of xi and a continuous random variable.
(i) Derive the marginal response of P(yi = 1|xi) to a change in xi,j that is,
∂Λ(x′iβ0)/∂xi,j. [2 marks]
(ii) What is the limit of the marginal response function in your answer to part(i) as the index x′iβ0 tends to infinity? Provide an intuitive explanation for why this must be the case. [2 marks]
3. Let θˆT be an estimator of the unknown parameter θ0. A researcher claims that “if θˆT is an unbiased estimator of θ0 then it must also be a consistent estimator of θ0”. Evaluate this claim, being sure to justify your argument briefly and define any statistical concepts to which you refer. [8 marks]
Continued over
1
[3 marks]

SECTION
4. Consider the following model:
yi =xiβ0+ui,
A
continued
ECON61001
where xi is a scalar observable variable, ui is the unobserved error. Let zi be a q × 1 vector of observable variables. Assume {(xi, ui, zi′)′} is a sequence of independently and identically distributed random vectors.
(a) State the conditions under which zi said to be a valid instrument for xi in this model. Can these conditions be tested? If so then explain how? If not then explain why not? (Word limit: 150 words) [2 marks]
(b) Suppose now that
xi = zi′γ0 + vi,
whereE[zizi′]=Mzz,apositivedefinitematrix,andforwi =(ui,vi)′,E[wi|zi]= 0 and V ar[wi|zi] = Ω0. Provide a set of conditions involving the parameters of the model, β0,γ0, and Ω0 under which xi is an endogenous regressor and zi is a valid instrument for xi in this model, being sure to justify your answer carefully. [6 marks].
5. Let {et} be a univariate white noise process.
(a) Assess which of the following three series are covariance stationary provid-
ing a justification for your answer in each case:
(i) ut = et;
(ii) vt = (−1)t + et;
(iii) wt = (−1)tet.
(b) Assess which of the series in part (a) are strictly stationary providing a justi-
[5 marks] fication for your answer in each case. [3 marks]
Continued over
2
i=1,2,…,n (1)

ECON61001
SECTION
6. Consider the linear regression model
y = Xβ0 + u (2)
whereyisT×1withtth elementyt,XisT×kwithtth rowx′t =[1,x′2,t],uis T × 1 with tth element ut, β0 is a k × 1 vector of unknown parameters. Assume that (2) is the true model for y, X is fixed in repeated samples, rank(X) = k, u ∼ N(0, σ02IT ) for some unknown scalar constant σ02. Let Z be a T ×k matrix that is fixed in repeated samples with rank(Z) = k and assume Z′X is nonsingular. Define β ̃T = (Z′X)−1Z′y, and let β ̃T,i be the ith element of β ̃T .
(a) Show that β ̃T is an unbiased estimator of β0. [4 marks]
(b) Show that V ar[β ̃T ] = σ02(X′Z(Z′Z)−1Z′X)−1. [9 marks]
(c) State the formula for Cov[β ̃T,i,β ̃T,j] in terms of σ02 and (X′Z(Z′Z)−1Z′X)−1. [2 marks]
(d) Showthatβ ̃T ∼N􏰀β0,Var[β ̃T]􏰁. [3marks]
(e) A researcher argues that given the results in parts (a), (b) and (d) there is no reason to prefer inferences based on the OLS estimator of β0 over inferences based on β ̃T . Do you agree? Justify your answer. [4 marks]
(f) Suppose now that X and Z are stochastic with E[u|Z] = 0. Is β ̃T an unbi- ased and/or a consistent estimator of β0? Justify your answer but there is no need to provide a formal analysis of the probability limit of β ̃T . [8 marks]
Continued over
B
3

SECTION
7.(a) Consider the model
B
continued
ECON61001
ut = wt +φwt−2,
t=1,2,…,T, (3)
where φ ̸= 0, and {wt}∞t=−∞ is a sequence of independently and identically dis- tributed random variables with E[wt] = 0 and V ar[wt] = σ2. Let u denote the T ×1 vector with tth element ut. Derive V ar[u] = Ω in terms of φ and σ2. [14 marks]
7.(b) Consider the times series regression model
y t = x ′t β 0 + u t , t = 1 , 2 , . . . , T ( 6 )
where xt = (1, yt−1)′, β0 = (β0,1, β0,2, )′ and {ut} is generated as in part (a). You may assume that yt has the following MA(∞) representation:
􏰈∞ i=0
(i) Evaluate whether yt−1 is contemporaneously exogenous in (6). [10 marks] (ii) Evaluate whether yt−1 is strictly exogenous in (6). [6 marks]
Continued over
yt = μy + where μy is a constant and ψ0,1 ̸= 0.
ψ0,iut−i,
4

ECON61001
SECTION
8.(a) Consider the linear regression model
yi = x′iβ0 + ui (4)
where xi = (1,x′2,i)′ and β0 are k × 1 vectors. Assume {(ui,x′2,i)′, i = 1,2,…N} are independently and identically distributed with: (i) E[xix′i] = Q, a finite, positive definite k × k matrix of constants; (ii) E[ui|xi] = 0; (iii) V ar[ui|xi] = h(xi) > 0. The OLS estimator of β0 is βˆN = (X′X)−1X′y where y is a N × 1 vector with ith element yi, X is a N × k matrix with ith row x′i.
B
continued
Show that
N1/2(βˆN−β0)→d N(0,Vh),
where Vh = Q−1ΩhQ−1, Ωh = E[h(xi)xix′i]. [8 marks] Hint:􏰔you may quote the generic form of the Weak Law of Large Numbers, N−1 Ni=1 zi →p μz, but must verify μz for the specific choices of zt relevant to your a􏰔nswer. Also you may 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.
8.(b) Consider the following model
yt = β0,1 + β0,2×2,t + β0,3×3,t + ut, for t = 1,2,…,T.
LetubetheT×1vectorwithtth elementut andXbetheT×3matrixwithtth row (1, x2,t, x3,t). Suppose that a researcher estimates the model via OLS based on sample of size T = 100, and obtains the fitted equation:
y􏰕 = 0.389 + 0.336x
2,t
+ 1.896x (0.927)
[1.602] {1.421}
3,t
R2 = 0.455 (5)
t
(1.216) (0.234)
[1.362] [0.286] {1.321} {0.321}
where OLS Standard errors in parentheses ( ), White Standard Errors in square brackets [ ] and Newey-West Standard Errors in { }. In parts (i) – (iii) specified on the next page:
• Discuss how to test the hypothesis stated with correct asymptotic size α providing the relevant test statistic and its distribution.
• If more than one test may be formed based on the information in equation (5) state so, providing details on how to perform each test.
• Perform the test given the information in equation (5), discussing the choice of test in the case more than one option is available.
Continued over
5

SECTION
B
continued
ECON61001
8.(b) (i)
for α = 0.05 where V ar(ut|X) = σ2|x2,t| and E[utut−j|X] = 0 for all t and all
H0 :β0,1 =0;
j ̸= 0. [7 marks]
(ii)
for α = 0.01 if ut = εtx1,t where Corr(εt,εt−j) = exp(−|j|) for all t and
(iii)
j.
H0 :β0,2 =β0,3 =0; HA :β0,2 ̸=0and/orβ0,3 ̸=0 for α = 0.1 where u|X ∼ N(0, σ2IT ).
Continued over
[7 marks]
[8 marks]
H0 :β0,2 ≤0; HA :β0,2 >0
6
HA :β0,1 ̸=0

B
continued
ECON61001
SECTION
9. Let {(yi, x′i)}Ni=1 be a sequence of independently and identically distributed (i.i.d.) random vectors. Suppose that yi is a dummy variable and so has a sample space of {0, 1} with P (yi = 1|xi) = Φ(x′iβ0) where Φ( · ) is the cumulative distribution function of the standard normal distribution.
(a) Assume that xi = 1. Show that the maximum likelihood estimator of β0 is βˆ = Φ−1 (y ̄) where y ̄ is the sample mean of y and Φ−1 ( . ) denotes the inverse of the cumulative distribution function of the standard normal distribution that is, if Φ(z) = p then z = Φ−1(p) for any z ∈ (−∞,∞). [12 marks]
(b) A researcher is interested in modeling the probability that a citizen of a US town votes in favour of an increase in the local tax rate to provide additional funding for public schools as a function of certain family and household char- acteristics. Let yesvm be a dummy variable that takes the value one if the citizen votes in favour of the tax increase and the explanatory variables are: loginc, the log of annual household income; ptcon, the log of property taxes paid in the year the vote took place; years the number of years the voter has been living in the community; school, a dummy variable that takes the value one if the voter works in the public school system; and four other control variables denoted x1, x2, x3 and x4 below. Using the Stata output on the next page answer the following questions.
(i) Test whether the amount of property taxes paid by a citizen affects the probability that they vote for the tax increase. Be sure to specify the null and alternative hypothesis, and the decision rule. [4 marks]
(ii) What do the results reveal about how household income affects the probability of voting in favour of the tax increase? Be sure to justify your
answer. [4 marks]
(iii) Fiftynineouttheninetyfivecitizensinthesamplevotedfortheincrease. Use the Likelihood Ratio statistic to test whether the explanatory vari- ables in the model collectively help to explain the probability that a citi- zen votes in favour of the tax increase. Be sure to specify the null and alternative hypotheses, and the decision rule, and to explain how you calculate the test statistic. [10 marks]
Continued over
7

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
9

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
10

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
11

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
12

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