# ECON61001: Econometric Methods

**University of Manchester**

## ECON61001: Econometric Methods

Final Exam

January 2021

Release date/time: 28/01/21, 14.00hrs GMT

Submission deadline: 30/01/21, 14.00hrs GMT

Instructions:

- You must answer all five questions in Section A and two out of the four questions 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. Excellent 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.

c The University of Manchester, 2021.

__A__

- Suppose a researcher is interested in the following linear regression model

y_{i }= x_{i}^{0}β_{0 }+ u_{i}, i = 1,2,…N,

where x_{i }= (1,x_{2,i},x_{3,i},x_{4,i})^{0 }and β_{0 }= (β_{0,1},β_{0,2},β_{0,3},β_{0,4})^{0 }. Given the context, the researcher is able to assume that {u_{i},x_{i}^{0}}^{N}_{i=1 }form a sequence of independently and identically distributed random vectors with E[x_{i}x_{i}^{0}] = Q, a finite, positive definite matrix of constants, E[u_{i}|x_{i}] = 0 and V ar[u_{i}|x_{i}] = σ_{0}^{2}. Therefore, she estimates the model via Ordinary Least Squares (OLS), obtaining the following estimated equation

yˆ_{i }= 1.0213 − 0.0020 x_{2,i }− 0.0208 x_{3,i }+ 0.0095 x_{4,i},

(0.1416) (0.0005) (0.0080) (0.0088)

where the number in parenthesis is the conventional OLS standard error for the coefficient in question. The OLS estimator of σ_{0}^{2 }in this model is σˆ_{N}^{2 }= 0.0081.

Given these results, the researcher concludes that β_{0,4 }= 0 and so decides to estimate the model via OLS with x_{4,i }excluded, obtaining the following estimated equation

yˆ_{i }= 1.1485 − 0.0020x_{2,i }− 0.0211x_{3,i}. (1)

Given that the sample size is N = 108 in both estimations, calculate the OLS estimator of the error variance σ_{0}^{2 }from the estimation in (1). Be sure to carefully explain your calculations. Hint: consider the F-statistic for testing β_{0,4 }=

- [8 marks]
- Suppose it is desired to predict z
_{t }using zˆ_{t }= w_{t}^{0}γ where w_{t }is a vector of observable variables and γ is a vector of constants that needs to be specified. The choice of γ associated with the linear projection of z_{t }on w_{t }is γ_{0 }where E[(z_{t }− w_{t}^{0}γ_{0})w_{t}] = 0.- What optimality property does zˆ
_{t}^{o }= w_{t}^{0}γ_{0 }possess? (Word limit: 50) [1 mark] - Now consider the regression model z
_{t }= w_{t}^{0}γ_{0 }+ v_{t}. Is w_{t }contemporaneously exogenous or strictly exogenous for estimation of γ_{0 }in this model?

- What optimality property does zˆ

Justify your answer. (Word limit: 150) [4 marks]

- Suppose the model in part (b) is dynamically complete. What optimality property does zˆ
_{t}^{o }possess? Briefly justify your answer. (Word limit:

150) [3 marks]

Continued over

__SECTION A continued__

3.(a) Let A be n × n nonsingular symmetric matrix. Show that if A is positive definite then A^{−1 }is positive definite. Hint: AA^{−1 }= I_{n}. [4 marks]

3.(b) Consider the classical linear regression model

y = Xβ_{0 }+ u, (2)

where X is the T×k observable data matrix that is fixed in repeated samples with rank(X) = k, and u is a T × 1 vector with E[u] = 0 and V ar[u] = σ_{0}^{2}I_{T }where σ_{0}^{2 }is an unknown positive finite constant. Let βˆ_{T }be the OLS estimator of β_{0 }based on (2) and βˆ_{R,T }be the Restricted Least Squares (RLS) estimator of β_{0 }based on (2) subject to the restrictions Rβ_{0 }= r where R is a n_{r }× k matrix of specified constants with rank(R) = n_{r }and r is a specified n_{r }× 1 vector of constants. Assuming the restrictions are correct, prove that βˆ_{R,T }is at least as efficient as βˆ_{T}. Hint: you may quote the formula for the variance-covariance matrix of the OLS and RLS estimators without proof; you may also take advantage of the stated result in part (a). [4 marks]

- Consider the linear regression model

y = Xβ_{0 }+ u,

where y and u are T × 1 vectors, X is T × k matrix, and β_{0 }is the k × 1 vector of unknown regression coefficients. Assume that X is fixed in repeated samples with rank(X) = k, and u where σ_{0}^{2 }is an unknown positive constant. Let θˆ_{T }denote the maximum likelihood estimator of the unknown parameter vector θ_{0 }= (β_{0}^{0},σ_{0}^{2})^{0}. Derive the information matrix for this model. Hint: you may state the form of the log likelihood function and score function for this model without proof. [8 marks]

- Consider the model

y_{i }= x_{i}^{0}β_{0 }+ u_{i}, i = 1,2,…,N,

where β_{0 }is the k × 1 vector of unknown regression coefficients, {(x^{0}_{i},u_{i})}^{N}_{i=1 }is a sequence of independently and identically distributed random vectors with E[u_{i}|x_{i}] = 0, V ar[u_{i}|x_{i}] = σ_{0}^{2}, an unknown finite positive constant and E[x_{i}x_{i}^{0}] = Q, a finite positive definite matrix of constants. Let σˆ_{N}^{2 }be the OLS estimator of σ_{0}^{2}. Show that N^{1/2}(σˆ_{N}^{2 }− σ_{0}^{2}) →^{d }N(0, µ_{4 }− σ_{0}^{2}) where µ_{4 }= E[u^{4}_{i }].

Hint: You may assume that:0 N0 −1 PNi=1 xixi0 →p Q; (ii) N−1/2 PiN=1 vi →d N(0,Ω) where v_{i }= (u^{2}_{i }− σ_{0}^{2},x_{i}u_{i}) , and Ω = V ar[v_{i}] is a finite, positive definite (k + 1) × (k + 1) matrix whose elements you must specify as needed to develop your answer. [8 marks]

Continued over

- Consider the regression model

y_{i }= x_{i}^{0}β_{0 }+ u_{i}, i = 1,2,…,N,

where β_{0 }is the k × 1 vector of unknown regression coefficients, {(x^{0}_{i},u_{i})}^{N}_{i=1 }is a sequence of independently and identically distributed random vectors with E[u_{i}|x_{i}] = 0, V ar[u_{i}|x_{i}] = σ_{0}^{2}, an unknown finite positive constant and E[x_{i}x_{i}^{0}] = Q, a finite positive definite matrix of constants. You may further assume that: ( i ) N−1 PNi=1 xixi0 →p Q; (ii) N−1/2 PiN=1 xiui →d N(0,σ02Q).

Let βˆ_{R,N }denote the RLS estimator based on the linear restrictions Rβ = r where R is a n_{r }× k matrix of pre-specified constants with rank equal to n_{r }and r is a n_{r }× 1 vector of pre-specified constants, and let λˆ_{N }be the vector of Lagrange Multipliers associated with this RLS estimation. Assuming Rβ_{0 }= r, answer the following questions.

- Show that N
^{1/2}(βˆ_{R,N }− β_{0}) →^{d }N (0, V_{R }) where

VR = σ02 Q−1 − Q−1R0(RQ−1R0)−1RQ−1 .

Hint: you may quote the formulae for βˆ_{R,N }and βˆ_{N}, the Ordinary Least Squares estimator of β_{0}, without proof. [10 marks]

- A colleague proposes testing H
_{0 }: Rβ_{0 }= r versus H_{1 }: Rβ_{0 }=6 r using the decision rule of the form: reject H_{0 }at the (approximate) 100α% significance level if λˆN0 MNλˆN > cnr(1 − α)

where c_{n}r(1 − α) is the 100(1 − α)^{th }percentile of the χ^{2}_{n}r distribution. However, your colleague is unsure what the matrix M_{N }should be in order that this decision rule has the properties implied by the stated significance level. Provide a suitable choice of M_{N}, being sure to justify your choice carefully. Hint: you may quote without proof: (i) the formulae for λˆ_{N }and βˆ_{N}; (ii) that both the OLS and RLS estimators of σ_{0}^{2 }are consistent under the conditions of the question. [20 marks]

Continued over

7.(a) Let {v_{t}}^{T}_{t=}−_{3 }be a weakly stationary time series process. Consider the following

statistic,

T

ρˆ4,T = Pt=1T v tvt_{2}−4.

Pt=1 _{v t}

Let denote a sequence of independently and identically distributed (i.i.d.) random variables with mean zero and variance σ_{ε}^{2}.

- Assume that v
_{t }= ε_{t}. Show that T^{1/2}ρˆ_{4,T }→^{d }N(0,1). [6 marks] - Assume that vt = θ4vt−4 + εt,

where |θ_{4}| < 1. What is the probability limit of ρˆ_{4,T }as T →∞? Be sure to justify your answer carefully. Hint: v_{t }has the following representation, v_{t }P . [9 marks]

7.(b) A researcher wishes to test the simple efficient-markets hypothesis in the foreign exchange market. Let s_{t }= ln(S_{t}) and f_{t,n }= ln[F_{t,n}], where S_{t }and F_{t,n }are the levels of the spot exchange rate at time t and the n−period forward exchange rate at time t. The simple efficient-markets hypothesis is that f_{t,n }= E[s_{t+n }|I_{t}] where I_{t }is the information set at time t which for the purposes of this question can be taken to be I_{t }= {s_{t},f_{t,n},s_{t}−_{1},f_{t}−_{1,n},s_{t}−_{2},f_{t}−_{2,n},…}. Using daily spot and thirty-day forward exchange rate data for the US dollar UK pound exchange rate, the researcher estimates the model,

yt+n = xt0β0 + ut,n, (3)

where y_{t+n }= s_{t+n }− f_{t,n}, x_{t }is the 3 × 1 vector given by

x_{t}^{0 }= (1, s_{t }− f_{t}−n,n, s_{t}−1 − f_{t}−1−n,n ),

n = 30 and u_{t,n }is the error term. If the simple efficient markets hypothesis holds in this foreign exchange market then E[u_{t,n }|I_{t}] = 0 and the regression coefficients in (3) satisfy a set of restrictions denoted here by g(β_{0}) = 0 where g(·) is n_{g }× 1 vector.

- What is g(β
_{0})? Briefly justify your answer. (Word limit: 75) [4 marks]

Continued over

7.(b) continued

- The researcher tests H
_{0 }: g(β_{0}) = 0 versus H_{1 }: g(β_{0}) =6 0 using the test statistic

^{−}1

S_{T }= Tg(βˆ_{T})^{0 }G(βˆ_{T})Vˆ_{β }G(βˆ_{T})^{0 } g(βˆ_{T }), (4)

where G . Assuming T^{1/2}(βˆ_{T }− β_{0}) →^{d }N(0, V_{β}) and

Vˆ_{β }→^{p }V_{β}, write down a suitable ^{T }decision rule for this test. If S_{T }= 8.2 then what is the outcome of the test? [3 marks]

- Since the y
_{t+n }is a financial variable, the researcher is concerned that the errors may exhibit autoregressive conditional heteroscedasticity and so has calculated Vˆ_{β }in (4) using White’s heteroscedasticity robust estimator. Given this information, do you have any concerns about the test in part (ii)? If so then explain your concerns and how you would modify the test to address these concerns. (Word limit: 350) [8 marks]

Continued over

- Consider the linear regression model

y1,i = γ0y2,i + z10,iδ0 + u1,i = xi0β0 + u1,i,

where x_{i}^{0 }= (y_{2,i},z_{1}^{0}_{,i}), β_{0 }= (γ_{0},δ_{0}^{0 })^{0 }and assume that

y2,i = zi0η0 + u2,i, (5)

where y_{1,i }and y_{2,i }are observable random variables, z_{i }= (z_{1}^{0}_{,i},z_{2}^{0}_{,i})^{0 }is a random vector of observable variables, u_{1,i }and u_{2,i }are the error terms ( unobservable scalar random variables), γ_{0 }is an unknown scalar parameter, and δ_{0}, and η_{0 }are vectors of unknown parameters. Suppose there is a sample of N observations, and let yˆ_{2,i }denote the predicted value of y_{2,i }based on Ordinary Least Squares (OLS) estimation of (5). Define xˆ_{i }= (yˆ_{2,i},z_{1}^{0}_{,i})^{0}. Let X be the N ×k matrix with i^{th }row x_{i}^{0}, Xˆ be the N × k matrix with i^{th }row xˆ^{0}_{i}, Z be the N × q matrix with i^{th }row z_{i}^{0 }and y_{1 }be the N × 1 vector with i^{th }element y_{1,i}. Consider the following three estimators of β_{0}:

- βˆ
_{1 }= (Xˆ^{0}Xˆ)^{−1}Xˆ^{0}y_{1}; • βˆ_{2 }= (Xˆ^{0}X)^{−1}Xˆ^{0}y_{1}; - βˆ3 = {X0Z(Z0Z)−1Z0X }−1 X0Z(Z0Z)−1Z0y1.
- Show that βˆ
_{1 }= βˆ_{2 }= βˆ_{3}. [15 marks] - Let φˆ = (βˆ
^{0},θˆ)^{0 }be the OLS estimator of φ_{0 }= (β_{0}^{0},θ_{0})^{0 }based on the model

y1,i = xi0β0 + θuˆ2,i + “error”

where uˆ_{2,i }is the i^{th }element of uˆ_{2}, the N×1 vector of residual from OLS estimation of (5). Via an application of the Frisch-Waugh-Lovell Theorem or otherwise, show that βˆ = βˆ_{1}.

[15 marks]

Continued over

9.(a) Let {(y_{i},x_{i}^{0})}^{N}_{i=1 }be a sequence of independently and identically distributed ( i.i.d. ) random vectors. Suppose that y_{i }is a dummy variable and so has a sample space of {0,1}. Consider the model

yi = xi0β0 + ui.

- Assume that E[u
_{i}|x_{i}] = 0. Derive the Generalized Least Squares ( GLS ) estimator of β_{0 }in this model? Hint: you may quote the generic formula for the GLS estimator that is, β^{ˆ}_{GLS }= (X^{0}Σ^{−1}X)^{−1}X^{0}Σ^{−1}y, but you must derive

Σ for this model. [6 marks]

- Is your answer to part (i) a feasible or infeasible GLS estimator? If infeasible then suggest a feasible GLS estimator. Do you foresee any potential problems in implementing your proposed Feasible GLS estimator? [4 marks]

(b) Let {V_{i}}^{N}_{i=1 }be a sequence of i.i.d. Bernoulli random variables with P(V_{i }= 1) = θ_{0}. We assume here that θ_{0 }∈ (0,1) and that our sample size is large enough for both outcomes to occur.

- Derive the Wald, Likelihood Ratio and Lagrange Multiplier statistics for testing H
_{0 }: θ_{0 }= θ∗ against H_{1 }: θ_{0 }6= θ∗. Hint: you may quote the form of the log likelihood function, the score equation and the formula for the maximum likelihood estimator for this model without proof. [16 marks] - Given that N = 100 and the sample contains 55 outcomes that are one, use your statistics in part (i) to test the hypothesis H
_{0 }: θ_{0 }= 0.5 against

H_{1 }: θ_{0 }=6 0.5 at the 5% significance level. [4 marks]

END OF EXAMINATION

1 TABLE 1: PERCENTAGE POINTS FOR THE T DISTRIBUTION

- Table 1: Percentage Points for the t distribution

Student’s t Distribution Function for Selected Probabilities
The table provides values of t |
||||||||||

α | 0.750 | 0.800 | 0.900 | 0.950 | 0.975 0.990 | 0.995 | 0.9975 | 0.999 | 0.9995 | |

ν | Values of t_{α,v} |
|||||||||

1 | 1.000 | 1.376 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 | |||

2 | 0.816 | 1.061 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | |||

3 | 0.765 | 0.978 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | |||

4 | 0.741 | 0.941 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | |||

5 | 0.727 | 0.920 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | 4.773 | ||

6 | 0.718 | 0.906 | 1.440 | 1.943 | 2.447 | 3.143 | 3.707 | 4.317 | 5.208 | |

7 | 0.711 | 0.896 | 1.415 | 1.895 | 2.365 | 2.998 | 3.499 | 4.029 | 4.785 | 5.408 |

8 | 0.706 | 0.889 | 1.397 | 1.860 | 2.306 | 2.896 | 3.355 | 3.833 | 4.501 | 5.041 |

9 | 0.703 | 0.883 | 1.383 | 1.833 | 2.262 | 2.821 | 3.250 | 3.690 | 4.297 | 4.781 |

10 | 0.700 | 0.879 | 1.372 | 1.812 | 2.228 | 2.764 | 3.169 | 3.581 | 4.144 | 4.587 |

11 | 0.697 | 0.876 | 1.363 | 1.796 | 2.201 | 2.718 | 3.106 | 3.497 | 4.025 | 4.437 |

12 | 0.695 | 0.873 | 1.356 | 1.782 | 2.179 | 2.681 | 3.055 | 3.428 | 3.930 | 4.318 |

13 | 0.694 | 0.870 | 1.350 | 1.771 | 2.160 | 2.650 | 3.012 | 3.372 | 3.852 | 4.221 |

14 | 0.692 | 0.868 | 1.345 | 1.761 | 2.145 | 2.624 | 2.977 | 3.326 | 3.787 | 4.140 |

15 | 0.691 | 0.866 | 1.341 | 1.753 | 2.131 | 2.602 | 2.947 | 3.286 | 3.733 | 4.073 |

16 | 0.690 | 0.865 | 1.337 | 1.746 | 2.120 | 2.583 | 2.921 | 3.252 | 3.686 | 4.015 |

17 | 0.689 | 0.863 | 1.333 | 1.740 | 2.110 | 2.567 | 2.898 | 3.222 | 3.646 | 3.965 |

18 | 0.688 | 0.862 | 1.330 | 1.734 | 2.101 | 2.552 | 2.878 | 3.197 | 3.610 | 3.922 |

19 | 0.688 | 0.861 | 1.328 | 1.729 | 2.093 | 2.539 | 2.861 | 3.174 | 3.579 | 3.883 |

20 | 0.687 | 0.860 | 1.325 | 1.725 | 2.086 | 2.528 | 2.845 | 3.153 | 3.552 | 3.850 |

21 | 0.686 | 0.859 | 1.323 | 1.721 | 2.080 | 2.518 | 2.831 | 3.135 | 3.527 | 3.819 |

22 | 0.686 | 0.858 | 1.321 | 1.717 | 2.074 | 2.508 | 2.819 | 3.119 | 3.505 | 3.792 |

23 | 0.685 | 0.858 | 1.319 | 1.714 | 2.069 | 2.500 | 2.807 | 3.104 | 3.485 | 3.768 |

24 | 0.685 | 0.857 | 1.318 | 1.711 | 2.064 | 2.492 | 2.797 | 3.091 | 3.467 | 3.745 |

25 | 0.684 | 0.856 | 1.316 | 1.708 | 2.060 | 2.485 | 2.787 | 3.078 | 3.450 | 3.725 |

26 | 0.684 | 0.856 | 1.315 | 1.706 | 2.056 | 2.479 | 2.779 | 3.067 | 3.435 | 3.707 |

27 | 0.684 | 0.855 | 1.314 | 1.703 | 2.052 | 2.473 | 2.771 | 3.057 | 3.421 | 3.690 |

28 | 0.683 | 0.855 | 1.313 | 1.701 | 2.048 | 2.467 | 2.763 | 3.047 | 3.408 | 3.674 |

29 | 0.683 | 0.854 | 1.311 | 1.699 | 2.045 | 2.462 | 2.756 | 3.038 | 3.396 | 3.659 |

30 | 0.683 | 0.854 | 1.310 | 1.697 | 2.042 | 2.457 | 2.750 | 3.030 | 3.385 | 3.646 |

40 | 0.681 | 0.851 | 1.303 | 1.684 | 2.021 | 2.423 | 2.704 | 2.971 | 3.307 | 3.551 |

50 | 0.679 | 0.849 | 1.299 | 1.676 | 2.009 | 2.403 | 2.678 | 2.937 | 3.261 | 3.496 |

60 | 0.679 | 0.848 | 1.296 | 1.671 | 2.000 | 2.390 | 2.660 | 2.915 | 3.232 | 3.460 |

70 | 0.678 | 0.847 | 1.294 | 1.667 | 1.994 | 2.381 | 2.648 | 2.899 | 3.211 | 3.435 |

80 | 0.678 | 0.846 | 1.292 | 1.664 | 1.990 | 2.374 | 2.639 | 2.887 | 3.195 | 3.416 |

90 | 0.677 | 0.846 | 1.291 | 1.662 | 1.987 | 2.368 | 2.632 | 2.878 | 3.183 | 3.402 |

100 | 0.677 | 0.845 | 1.290 | 1.660 | 1.984 | 2.364 | 2.626 | 2.871 | 3.174 | 3.390 |

110 | 0.677 | 0.845 | 1.289 | 1.659 | 1.982 | 2.361 | 2.621 | 2.865 | 3.166 | 3.381 |

120 | 0.677 | 0.845 | 1.289 | 1.658 | 1.980 | 2.358 | 2.617 | 2.860 | 3.160 | 3.373 |

∞ | 0.674 | 0.842 | 1.282 | 1.645 | 1.960 | 2.326 | 2.576 | 2.808 | 3.090 | 3.297 |

- TABLE 2: PERCENTAGE POINTS FOR THE χ
^{2 }DISTRIBUTION 2 Table 2: Percentage Points for the χ^{2 }distribution

The χ^{2 }Distribution Function for Selected Probabilities |
|||||||||||

The table provides values of where Pr( and | |||||||||||

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 χ^{2}_{α,v} |
||||||||||

1 | 0.000 | 0.000 | 0.001 | 0.004 | 0.016 | 0.455 | 2.706 | 3.841 | 5.024 | 6.635 | 7.879 |

2 | 0.010 | 0.020 | 0.051 | 0.103 | 0.211 | 1.386 | 4.605 | 5.991 | 7.378 | 9.210 | 10.60 |

3 | 0.072 | 0.115 | 0.216 | 0.352 | 0.584 | 2.366 | 6.251 | 7.815 | 9.348 | 11.34 | 12.84 |

4 | 0.207 | 0.297 | 0.484 | 0.711 | 1.064 | 3.357 | 7.779 | 9.488 | 11.14 | 13.28 | 14.86 |

5 | 0.412 | 0.554 | 0.831 | 1.145 | 1.610 | 4.351 | 9.236 | 11.07 | 12.83 | 15.09 | 16.75 |

6 | 0.676 | 0.872 | 1.237 | 1.635 | 2.204 | 5.348 | 10.64 | 12.59 | 14.45 | 16.81 | 18.55 |

7 | 0.989 | 1.239 | 1.690 | 2.167 | 2.833 | 6.346 | 12.02 | 14.07 | 16.01 | 18.48 | 20.28 |

8 | 1.344 | 1.646 | 2.180 | 2.733 | 3.490 | 7.344 | 13.36 | 15.51 | 17.53 | 20.09 | 21.95 |

9 | 1.735 | 2.088 | 2.700 | 3.325 | 4.168 | 8.343 | 14.68 | 16.92 | 19.02 | 21.67 | 23.59 |

10 | 2.156 | 2.558 | 3.247 | 3.940 | 4.865 | 9.342 | 15.99 | 18.31 | 20.48 | 23.21 | 25.19 |

11 | 2.603 | 3.053 | 3.816 | 4.575 | 5.578 | 10.34 | 17.28 | 19.68 | 21.92 | 24.72 | 26.76 |

12 | 3.074 | 3.571 | 4.404 | 5.226 | 6.304 | 11.34 | 18.55 | 21.03 | 23.34 | 26.22 | 28.30 |

13 | 3.565 | 4.107 | 5.009 | 5.892 | 7.042 | 12.34 | 19.81 | 22.36 | 24.74 | 27.69 | 29.82 |

14 | 4.075 | 4.660 | 5.629 | 6.571 | 7.790 | 13.34 | 21.06 | 23.68 | 26.12 | 29.14 | 31.32 |

15 | 4.601 | 5.229 | 6.262 | 7.261 | 8.547 | 14.34 | 22.31 | 25.00 | 27.49 | 30.58 | 32.80 |

16 | 5.142 | 5.812 | 6.908 | 7.962 | 9.312 | 15.34 | 23.54 | 26.30 | 28.85 | 32.00 | 34.27 |

17 | 5.697 | 6.408 | 7.564 | 8.672 | 10.09 | 16.34 | 24.77 | 27.59 | 30.19 | 33.41 | 35.72 |

18 | 6.265 | 7.015 | 8.231 | 9.390 | 10.86 | 17.34 | 25.99 | 28.87 | 31.53 | 34.81 | 37.16 |

19 | 6.844 | 7.633 | 8.907 | 10.12 | 11.65 | 18.34 | 27.20 | 30.14 | 32.85 | 36.19 | 38.58 |

20 | 7.434 | 8.260 | 9.591 | 10.85 | 12.44 | 19.34 | 28.41 | 31.41 | 34.17 | 37.57 | 40.00 |

21 | 8.034 | 8.897 | 10.28 | 11.59 | 13.24 | 20.34 | 29.62 | 32.67 | 35.48 | 38.93 | 41.40 |

22 | 8.643 | 9.542 | 10.98 | 12.34 | 14.04 | 21.34 | 30.81 | 33.92 | 36.78 | 40.29 | 42.80 |

23 | 9.260 | 10.20 | 11.69 | 13.09 | 14.85 | 22.34 | 32.01 | 35.17 | 38.08 | 41.64 | 44.18 |

24 | 9.886 | 10.86 | 12.40 | 13.85 | 15.66 | 23.34 | 33.20 | 36.42 | 39.36 | 42.98 | 45.56 |

25 | 10.52 | 11.52 | 13.12 | 14.61 | 16.47 | 24.34 | 34.38 | 37.65 | 40.65 | 44.31 | 46.93 |

26 | 11.16 | 12.20 | 13.84 | 15.38 | 17.29 | 25.34 | 35.56 | 38.89 | 41.92 | 45.64 | 48.29 |

27 | 11.81 | 12.88 | 14.57 | 16.15 | 18.11 | 26.34 | 36.74 | 40.11 | 43.19 | 46.96 | 49.64 |

28 | 12.46 | 13.56 | 15.31 | 16.93 | 18.94 | 27.34 | 37.92 | 41.34 | 44.46 | 48.28 | 50.99 |

29 | 13.12 | 14.26 | 16.05 | 17.71 | 19.77 | 28.34 | 39.09 | 42.56 | 45.72 | 49.59 | 52.34 |

30 | 13.79 | 14.95 | 16.79 | 18.49 | 20.60 | 29.34 | 40.26 | 43.77 | 46.98 | 50.89 | 53.67 |

35 | 17.19 | 18.51 | 20.57 | 22.47 | 24.80 | 34.34 | 46.06 | 49.80 | 53.20 | 57.34 | 60.27 |

40 | 20.71 | 22.16 | 24.43 | 26.51 | 29.05 | 39.34 | 51.81 | 55.76 | 59.34 | 63.69 | 66.77 |

45 | 24.31 | 25.90 | 28.37 | 30.61 | 33.35 | 44.34 | 57.51 | 61.66 | 65.41 | 69.96 | 73.17 |

50 | 27.99 | 29.71 | 32.36 | 34.76 | 37.69 | 49.33 | 63.17 | 67.50 | 71.42 | 76.15 | 79.49 |

60 | 35.53 | 37.48 | 40.48 | 43.19 | 46.46 | 59.33 | 74.40 | 79.08 | 83.30 | 88.30 | 91.95 |

70 | 43.28 | 45.44 | 48.76 | 51.74 | 55.33 | 69.33 | 85.53 | 90.53 | 95.02 | 100.4 | 104.2 |

80 | 51.17 | 53.54 | 57.15 | 60.39 | 64.28 | 79.33 | 96.58 | 101.9 | 106.6 | 112.3 | 116.3 |

90 | 59.20 | 61.75 | 65.65 | 69.13 | 73.29 | 89.33 | 107.6 | 113.1 | 118.1 | 124.1 | 128.3 |

100 | 67.33 | 70.06 | 74.22 | 77.93 | 82.36 | 99.33 | 118.5 | 124.3 | 129.6 | 135.8 | 140.2 |

150 | 109.1 | 112.7 | 118.0 | 122.7 | 128.3 | 149.3 | 172.6 | 179.6 | 185.8 | 193.2 | 198.4 |

200 | 152.2 | 156.4 | 162.7 | 168.3 | 174.8 | 199.3 | 226.0 | 234.0 | 241.1 | 249.4 | 255.3 |

- TABLE 3: UPPER 5% PERCENTAGE POINTS FOR THE F DISTRIBUTION
- Table 3: Upper 5% percentage points for the F distribution

The F Distribution Function for α = 0.05 | ||||||||||||

The table provides values of F_{α,v}_{1,v}_{2 }where Pr(F ≥ F_{α,v}_{1,v}_{2}) = 0.05 and F ∼ F (v_{1},v_{2}) |
||||||||||||

v_{1 }→ |
||||||||||||

v_{2 }↓ |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 |

5 | 6.61 | 5.79 | 5.41 | 5.19 | 5.05 | 4.95 | 4.88 | 4.82 | 4.77 | 4.74 | 4.68 | 4.62 |

6 | 5.99 | 5.14 | 4.76 | 4.53 | 4.39 | 4.28 | 4.21 | 4.15 | 4.10 | 4.06 | 4.00 | 3.94 |

7 | 5.59 | 4.74 | 4.35 | 4.12 | 3.97 | 3.87 | 3.79 | 3.73 | 3.68 | 3.64 | 3.57 | 3.51 |

8 | 5.32 | 4.46 | 4.07 | 3.84 | 3.69 | 3.58 | 3.50 | 3.44 | 3.39 | 3.35 | 3.28 | 3.22 |

9 | 5.12 | 4.26 | 3.86 | 3.63 | 3.48 | 3.37 | 3.29 | 3.23 | 3.18 | 3.14 | 3.07 | 3.01 |

10 | 4.96 | 4.10 | 3.71 | 3.48 | 3.33 | 3.22 | 3.14 | 3.07 | 3.02 | 2.98 | 2.91 | 2.85 |

11 | 4.84 | 3.98 | 3.59 | 3.36 | 3.20 | 3.09 | 3.01 | 2.95 | 2.90 | 2.85 | 2.79 | 2.72 |

12 | 4.75 | 3.89 | 3.49 | 3.26 | 3.11 | 3.00 | 2.91 | 2.85 | 2.80 | 2.75 | 2.69 | 2.62 |

13 | 4.67 | 3.81 | 3.41 | 3.18 | 3.03 | 2.92 | 2.83 | 2.77 | 2.71 | 2.67 | 2.60 | 2.53 |

14 | 4.60 | 3.74 | 3.34 | 3.11 | 2.96 | 2.85 | 2.76 | 2.70 | 2.65 | 2.60 | 2.53 | 2.46 |

15 | 4.54 | 3.68 | 3.29 | 3.06 | 2.90 | 2.79 | 2.71 | 2.64 | 2.59 | 2.54 | 2.48 | 2.40 |

16 | 4.49 | 3.63 | 3.24 | 3.01 | 2.85 | 2.74 | 2.66 | 2.59 | 2.54 | 2.49 | 2.42 | 2.35 |

17 | 4.45 | 3.59 | 3.20 | 2.96 | 2.81 | 2.70 | 2.61 | 2.55 | 2.49 | 2.45 | 2.38 | 2.31 |

18 | 4.41 | 3.55 | 3.16 | 2.93 | 2.77 | 2.66 | 2.58 | 2.51 | 2.46 | 2.41 | 2.34 | 2.27 |

19 | 4.38 | 3.52 | 3.13 | 2.90 | 2.74 | 2.63 | 2.54 | 2.48 | 2.42 | 2.38 | 2.31 | 2.23 |

20 | 4.35 | 3.49 | 3.10 | 2.87 | 2.71 | 2.60 | 2.51 | 2.45 | 2.39 | 2.35 | 2.28 | 2.20 |

21 | 4.32 | 3.47 | 3.07 | 2.84 | 2.68 | 2.57 | 2.49 | 2.42 | 2.37 | 2.32 | 2.25 | 2.18 |

22 | 4.30 | 3.44 | 3.05 | 2.82 | 2.66 | 2.55 | 2.46 | 2.40 | 2.34 | 2.30 | 2.23 | 2.15 |

23 | 4.28 | 3.42 | 3.03 | 2.80 | 2.64 | 2.53 | 2.44 | 2.37 | 2.32 | 2.27 | 2.20 | 2.13 |

24 | 4.26 | 3.40 | 3.01 | 2.78 | 2.62 | 2.51 | 2.42 | 2.36 | 2.30 | 2.25 | 2.18 | 2.11 |

25 | 4.24 | 3.39 | 2.99 | 2.76 | 2.60 | 2.49 | 2.40 | 2.34 | 2.28 | 2.24 | 2.16 | 2.09 |

30 | 4.17 | 3.32 | 2.92 | 2.69 | 2.53 | 2.42 | 2.33 | 2.27 | 2.21 | 2.16 | 2.09 | 2.01 |

35 | 4.12 | 3.27 | 2.87 | 2.64 | 2.49 | 2.37 | 2.29 | 2.22 | 2.16 | 2.11 | 2.04 | 1.96 |

40 | 4.08 | 3.23 | 2.84 | 2.61 | 2.45 | 2.34 | 2.25 | 2.18 | 2.12 | 2.08 | 2.00 | 1.92 |

45 | 4.06 | 3.20 | 2.81 | 2.58 | 2.42 | 2.31 | 2.22 | 2.15 | 2.10 | 2.05 | 1.97 | 1.89 |

50 | 4.03 | 3.18 | 2.79 | 2.56 | 2.40 | 2.29 | 2.20 | 2.13 | 2.07 | 2.03 | 1.95 | 1.87 |

55 | 4.02 | 3.16 | 2.77 | 2.54 | 2.38 | 2.27 | 2.18 | 2.11 | 2.06 | 2.01 | 1.93 | 1.85 |

60 | 4.00 | 3.15 | 2.76 | 2.53 | 2.37 | 2.25 | 2.17 | 2.10 | 2.04 | 1.99 | 1.92 | 1.84 |

70 | 3.98 | 3.13 | 2.74 | 2.50 | 2.35 | 2.23 | 2.14 | 2.07 | 2.02 | 1.97 | 1.89 | 1.81 |

80 | 3.96 | 3.11 | 2.72 | 2.49 | 2.33 | 2.21 | 2.13 | 2.06 | 2.00 | 1.95 | 1.88 | 1.79 |

90 | 3.95 | 3.10 | 2.71 | 2.47 | 2.32 | 2.20 | 2.11 | 2.04 | 1.99 | 1.94 | 1.86 | 1.78 |

100 | 3.94 | 3.09 | 2.70 | 2.46 | 2.31 | 2.19 | 2.10 | 2.03 | 1.97 | 1.93 | 1.85 | 1.77 |

110 | 3.93 | 3.08 | 2.69 | 2.45 | 2.30 | 2.18 | 2.09 | 2.02 | 1.97 | 1.92 | 1.84 | 1.76 |

120 | 3.92 | 3.07 | 2.68 | 2.45 | 2.29 | 2.18 | 2.09 | 2.02 | 1.96 | 1.91 | 1.83 | 1.75 |

150 | 3.90 | 3.06 | 2.66 | 2.43 | 2.27 | 2.16 | 2.07 | 2.00 | 1.94 | 1.89 | 1.82 | 1.73 |

- 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 F_{α,v}_{1,v}_{2 }where Pr(F ≥ F_{α,v}_{1,v}_{2}) = 0.01 and F ∼ F (v_{1},v_{2}) |
||||||||||||

v_{1 }→ |
||||||||||||

v_{2 }↓ |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 | 15 |

5 | 16.3 | 13.3 | 12.1 | 11.4 | 11.0 | 10.7 | 10.5 | 10.3 | 10.2 | 10.1 | 9.89 | 9.72 |

6 | 13.7 | 10.9 | 9.78 | 9.15 | 8.75 | 8.47 | 8.26 | 8.10 | 7.98 | 7.87 | 7.72 | 7.56 |

7 | 12.2 | 9.55 | 8.45 | 7.85 | 7.46 | 7.19 | 6.99 | 6.84 | 6.72 | 6.62 | 6.47 | 6.31 |

8 | 11.3 | 8.65 | 7.59 | 7.01 | 6.63 | 6.37 | 6.18 | 6.03 | 5.91 | 5.81 | 5.67 | 5.52 |

9 | 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 | 10.0 | 7.56 | 6.55 | 5.99 | 5.64 | 5.39 | 5.20 | 5.06 | 4.94 | 4.85 | 4.71 | 4.56 |

11 | 9.65 | 7.21 | 6.22 | 5.67 | 5.32 | 5.07 | 4.89 | 4.74 | 4.63 | 4.54 | 4.40 | 4.25 |

12 | 9.33 | 6.93 | 5.95 | 5.41 | 5.06 | 4.82 | 4.64 | 4.50 | 4.39 | 4.30 | 4.16 | 4.01 |

13 | 9.07 | 6.70 | 5.74 | 5.21 | 4.86 | 4.62 | 4.44 | 4.30 | 4.19 | 4.10 | 3.96 | 3.82 |

14 | 8.86 | 6.51 | 5.56 | 5.04 | 4.69 | 4.46 | 4.28 | 4.14 | 4.03 | 3.94 | 3.80 | 3.66 |

15 | 8.68 | 6.36 | 5.42 | 4.89 | 4.56 | 4.32 | 4.14 | 4.00 | 3.89 | 3.80 | 3.67 | 3.52 |

16 | 8.53 | 6.23 | 5.29 | 4.77 | 4.44 | 4.20 | 4.03 | 3.89 | 3.78 | 3.69 | 3.55 | 3.41 |

17 | 8.40 | 6.11 | 5.18 | 4.67 | 4.34 | 4.10 | 3.93 | 3.79 | 3.68 | 3.59 | 3.46 | 3.31 |

18 | 8.29 | 6.01 | 5.09 | 4.58 | 4.25 | 4.01 | 3.84 | 3.71 | 3.60 | 3.51 | 3.37 | 3.23 |

19 | 8.18 | 5.93 | 5.01 | 4.50 | 4.17 | 3.94 | 3.77 | 3.63 | 3.52 | 3.43 | 3.30 | 3.15 |

20 | 8.10 | 5.85 | 4.94 | 4.43 | 4.10 | 3.87 | 3.70 | 3.56 | 3.46 | 3.37 | 3.23 | 3.09 |

21 | 8.02 | 5.78 | 4.87 | 4.37 | 4.04 | 3.81 | 3.64 | 3.51 | 3.40 | 3.31 | 3.17 | 3.03 |

22 | 7.95 | 5.72 | 4.82 | 4.31 | 3.99 | 3.76 | 3.59 | 3.45 | 3.35 | 3.26 | 3.12 | 2.98 |

23 | 7.88 | 5.66 | 4.76 | 4.26 | 3.94 | 3.71 | 3.54 | 3.41 | 3.30 | 3.21 | 3.07 | 2.93 |

24 | 7.82 | 5.61 | 4.72 | 4.22 | 3.90 | 3.67 | 3.50 | 3.36 | 3.26 | 3.17 | 3.03 | 2.89 |

25 | 7.77 | 5.57 | 4.68 | 4.18 | 3.85 | 3.63 | 3.46 | 3.32 | 3.22 | 3.13 | 2.99 | 2.85 |

30 | 7.56 | 5.39 | 4.51 | 4.02 | 3.70 | 3.47 | 3.30 | 3.17 | 3.07 | 2.98 | 2.84 | 2.70 |

35 | 7.42 | 5.27 | 4.40 | 3.91 | 3.59 | 3.37 | 3.20 | 3.07 | 2.96 | 2.88 | 2.74 | 2.60 |

40 | 7.31 | 5.18 | 4.31 | 3.83 | 3.51 | 3.29 | 3.12 | 2.99 | 2.89 | 2.80 | 2.66 | 2.52 |

45 | 7.23 | 5.11 | 4.25 | 3.77 | 3.45 | 3.23 | 3.07 | 2.94 | 2.83 | 2.74 | 2.61 | 2.46 |

50 | 7.17 | 5.06 | 4.20 | 3.72 | 3.41 | 3.19 | 3.02 | 2.89 | 2.78 | 2.70 | 2.56 | 2.42 |

55 | 7.12 | 5.01 | 4.16 | 3.68 | 3.37 | 3.15 | 2.98 | 2.85 | 2.75 | 2.66 | 2.53 | 2.38 |

60 | 7.08 | 4.98 | 4.13 | 3.65 | 3.34 | 3.12 | 2.95 | 2.82 | 2.72 | 2.63 | 2.50 | 2.35 |

70 | 7.01 | 4.92 | 4.07 | 3.60 | 3.29 | 3.07 | 2.91 | 2.78 | 2.67 | 2.59 | 2.45 | 2.31 |

80 | 6.96 | 4.88 | 4.04 | 3.56 | 3.26 | 3.04 | 2.87 | 2.74 | 2.64 | 2.55 | 2.42 | 2.27 |

90 | 6.93 | 4.85 | 4.01 | 3.53 | 3.23 | 3.01 | 2.84 | 2.72 | 2.61 | 2.52 | 2.39 | 2.24 |

100 | 6.90 | 4.82 | 3.98 | 3.51 | 3.21 | 2.99 | 2.82 | 2.69 | 2.59 | 2.50 | 2.37 | 2.22 |

110 | 6.87 | 4.80 | 3.96 | 3.49 | 3.19 | 2.97 | 2.81 | 2.68 | 2.57 | 2.49 | 2.35 | 2.21 |

120 | 6.85 | 4.79 | 3.95 | 3.48 | 3.17 | 2.96 | 2.79 | 2.66 | 2.56 | 2.47 | 2.34 | 2.19 |

150 | 6.81 | 4.75 | 3.91 | 3.45 | 3.14 | 2.92 | 2.76 | 2.63 | 2.53 | 2.44 | 2.31 | 2.16 |