Appendix IV – Tables of Critical Values

Table IV1: Critical values of the two-sided tk test at the 5% and 10% significance levels

k

5%

10%

k

5%

10%

k

5%

10%

і

12.704

6.313

11

2.201

1.796

21

2.080

1.721

2

4.303

2.920

12

2.179

1.782

22

2.074

1.717

3

3.183

2.353

13

2.160

1.771

23

2.069

1.714

4

2.776

2.132

14

2.145

1.761

24

2.064

1.711

5

2.571

2.015

15

2.131

1.753

25

2.059

1.708

6

2.447

1.943

16

2.120

1.746

26

2.056

1.706

7

2.365

1.895

17

2.110

1.740

27

2.052

1.703

8

2.306

1.859

18

2.101

1.734

28

2.048

1.701

9

2.262

1.833

19

2.093

1.729

29

2.045

1.699

10

2.228

1.813

20

2.086

1.725

30

2.042

1.697

Table IV2: Critical values of the right-sided tk test at the 5% and 10% significance levels

k

5%

10%

k

5%

10%

k

5%

10%

1

6.313

3.078

11

1.796

1.363

21

1.721

1.323

2

2.920

1.886

12

1.782

1.356

22

1.717

1.321

3

2.353

1.638

13

1.771

1.350

23

1.714

1.319

4

2.132

1.533

14

1.761

1.345

24

1.711

1.318

5

2.015

1.476

15

1.753

1.341

25

1.708

1.316

6

1.943

1.440

16

1.746

1.337

26

1.706

1.315

7

1.895

1.415

17

1.740

1.333

27

1.703

1.314

8

1.859

1.397

18

1.734

1.330

28

1.701

1.313

9

1.833

1.383

19

1.729

1.328

29

1.699

1.311

10

1.813

1.372

20

1.725

1.325

30

1.697

1.310

Note: For k >30 the critical values of the tk test are approximately equal to the critical values of the standard normal test in Table IV3.

Table IV3: Critical values of the N(0, 1) test

Significance levels:

5%

10%

Two-sided:

1.960

1.645

Right-sided:

1.645

1.282

Table IV.4: Critical values of the test at the 5% and 10% significance levels

k

5%

10%

k

5%

10%

k

5%

10%

1

3.841

2.705

11

19.675

17.275

21

32.671

29.615

2

5.991

4.605

12

21.026

18.549

22

33.925

30.814

3

7.815

6.251

13

22.362

19.812

23

35.172

32.007

4

9.488

7.780

14

23.684

21.064

24

36.414

33.196

5

11.071

9.237

15

24.995

22.307

25

37.653

34.381

6

12.591

10.645

16

26.296

23.541

26

38.885

35.563

7

14.067

12.017

17

27.588

24.769

27

40.114

36.741

8

15.507

13.361

18

28.869

25.990

28

41.336

37.916

9

16.919

14.683

19

30.144

27.204

29

42.557

39.088

10

18.307

15.987

20

31.410

28.412

30

43.772

40.256

Note: Because the Xk test is used to test parameter restrictions with the degrees of freedom k equal to the number of restrictions, it is unlikely that you will need the critical values of the x2 test for k> 30.

Подпись: 308

mk 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

161.

199.

216.

225.

230.

234.

237.

239.

241.

242.

243.

244.

245.

245.

246.

2

18.5

19.0

19.2

19.2

19.3

19.3

19.4

19.4

19.4

19.4

19.4

19.4

19.4

19.4

19.4

3

10.1

9.55

9.28

9.12

9.01

8.94

8.89

8.84

8.81

8.79

8.76

8.75

8.73

8.72

8.70

4

7.71

6.94

6.59

6.39

6.26

6.16

6.09

6.04

6.00

5.96

5.94

5.91

5.89

5.87

5.86

5

6.61

5.79

5.41

5.19

5.05

4.95

4.88

4.82

4.77

4.73

4.70

4.68

4.66

4.64

4.62

6

5.99

5.14

4.76

4.53

4.39

4.28

4.21

4.15

4.10

4.06

4.03

4.00

3.98

3.96

3.94

7

5.59

4.74

4.35

4.12

3.97

3.87

3.79

3.73

3.68

3.64

3.60

3.57

3.55

3.53

3.51

8

5.32

4.46

4.07

3.84

3.69

3.58

3.50

3.44

3.39

3.35

3.31

3.28

3.26

3.24

3.22

9

5.12

4.26

3.86

3.63

3.48

3.37

3.29

3.23

3.18

3.14

3.10

3.07

3.05

3.03

3.01

10

4.96

4.10

3.71

3.48

3.33

3.22

3.14

3.07

3.02

2.98

2.94

2.91

2.89

2.86

2.84

11

4.84

3.98

3.59

3.36

3.20

3.09

3.01

2.95

2.90

2.85

2.82

2.79

2.76

2.74

2.72

12

4.75

3.89

3.49

3.26

3.11

3.00

2.91

2.85

2.80

2.75

2.72

2.69

2.66

2.64

2.62

13

4.67

3.81

3.41

3.18

3.03

2.92

2.83

2.77

2.71

2.67

2.63

2.60

2.58

2.55

2.53

14

4.60

3.74

3.34

3.11

2.96

2.85

2.76

2.70

2.65

2.60

2.57

2.53

2.51

2.48

2.46

15

4.54

3.68

3.29

3.06

2.90

2.79

2.71

2.64

2.59

2.54

2.51

2.48

2.45

2.42

2.40

16

4.49

3.63

3.24

3.01

2.85

2.74

2.66

2.59

2.54

2.49

2.46

2.42

2.40

2.37

2.35

17

4.45

3.59

3.20

2.96

2.81

2.70

2.61

2.55

2.49

2.45

2.41

2.38

2.35

2.33

2.31

18

4.41

3.55

3.16

2.93

2.77

2.66

2.58

2.51

2.46

2.41

2.37

2.34

2.31

2.29

2.27

19

4.38

3.52

3.13

2.90

2.74

2.63

2.54

2.48

2.42

2.38

2.34

2.31

2.28

2.26

2.23

20

4.35

3.49

3.10

2.87

2.71

2.60

2.51

2.45

2.39

2.35

2.31

2.28

2.25

2.22

2.20

21

4.32

3.47

3.07

2.84

2.68

2.57

2.49

2.42

2.37

2.32

2.28

2.25

2.22

2.20

2.18

22

4.30

3.44

3.05

2.82

2.66

2.55

2.46

2.40

2.34

2.30

2.26

2.23

2.20

2.17

2.15

23

4.28

3.42

3.03

2.80

2.64

2.53

2.44

2.37

2.32

2.27

2.24

2.20

2.18

2.15

2.13

24

4.26

3.40

3.01

2.78

2.62

2.51

2.42

2.35

2.30

2.25

2.22

2.18

2.15

2.13

2.11

25

4.24

3.39

2.99

2.76

2.60

2.49

2.40

2.34

2.28

2.24

2.20

2.16

2.14

2.11

2.09

 

26

4.22

3.37

2.98

2.74

2.59

2.47

2.39

27

4.21

3.35

2.96

2.73

2.57

2.46

2.37

28

4.20

3.34

2.95

2.71

2.56

2.45

2.36

29

4.18

3.33

2.93

2.70

2.55

2.43

2.35

ЗО

4.17

3.32

2.92

2.69

2.53

2.42

2.33

40

4.08

3.23

2.84

2.61

2.45

2.34

2.25

50

4.03

3.18

2.79

2.56

2.40

2.29

2.20

60

4.00

3.15

2.76

2.53

2.37

2.25

2.17

70

3.98

3.13

2.74

2.50

2.35

2.23

2.14

80

3.96

3.11

2.72

2.49

2.33

2.21

2.13

90

3.95

ЗЛО

2.71

2.47

2.32

2.20

2.11

100

3.94

3.09

2.70

2.46

2.31

2.19

2.10

mk

16

17

18

19

20

21

22

1

247.

247.

247.

248.

248.

248.

249.

2

19.4

19.4

19.4

19.4

19.4

19.5

19.5

3

8.69

8.68

8.67

8.67

8.66

8.65

8.65

4

5.84

5.83

5.82

5.81

5.80

5.79

5.79

5

4.60

4.59

4.58

4.57

4.56

4.55

4.54

6

3.92

3.91

3.90

3.88

3.87

3.86

3.86

7

3.49

3.48

3.47

3.46

3.44

3.43

3.43

8

3.20

3.19

3.17

3.16

3.15

3.14

3.13

9

2.99

2.97

2.96

2.95

2.94

2.93

2.92

10

2.83

2.81

2.80

2.79

2.77

2.76

2.75

11

2.70

2.69

2.67

2.66

2.65

2.64

2.63

12

2.60

2.58

2.57

2.56

2.54

2.53

2.52

13

2.52

2.50

2.48

2.47

2.46

2.45

2.44

 

2.32

2.27

2.22

2.18

2.15

2.12

2.09

2.07

2.31

2.25

2.20

2.17

2.13

2.10

2.08

2.06

2.29

2.24

2.19

2.15

2.12

2.09

2.06

2.04

2.28

2.22

2.18

2.14

2.10

2.08

2.05

2.03

2.27

2.21

2.16

2.13

2.09

2.06

2.04

2.01

2.18

2.12

2.08

2.04

2.00

1.97

1.95

1.92

2.13

2.07

2.03

1.99

1.95

1.92

1.89

1.87

2.10

2.04

1.99

1.95

1.92

1.89

1.86

1.84

2.07

2.02

1.97

1.93

1.89

1.86

1.84

1.81

2.06

2.00

1.95

1.91

1.88

1.84

1.82

1.79

2.04

1.99

1.94

1.90

1.86

1.83

1.80

1.78

2.03

1.97

1.93

1.89

1.85

1.82

1.79

1.77

23

24

25

26

27

28

29

ЗО

249.

249.

249.

250.

250.

250.

250.

250.

19.5

19.5

19.5

19.5

19.5

19.5

19.5

19.5

8.64

8.64

8.63

8.63

8.63

8.62

8.62

8.62

5.78

5.77

5.77

5.76

5.76

5.75

5.75

5.75

4.53

4.53

4.52

4.52

4.51

4.51

4.50

4.50

3.85

3.84

3.83

3.83

3.82

3.82

3.81

3.81

3.42

3.41

3.40

3.40

3.39

3.39

3.38

3.38

3.12

3.12

3.11

3.10

3.10

3.09

3.08

3.08

2.91

2.90

2.89

2.89

2.88

2.87

2.87

2.86

2.75

2.74

2.73

2.72

2.72

2.71

2.70

2.70

2.62

2.61

2.60

2.59

2.59

2.58

2.58

2.57

2.51

2.51

2.50

2.49

2.48

2.48

2.47

2.47

2.43

2.42

2.41

2.40

2.40

2.39

2.39

2.38

 

Подпись: 309

Подпись: 310

Table IV5 (continued)

mk

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

14

2.44

2.43

2.41

2.40

2.39

2.38

2.37

2.36

2.35

2.34

2.33

2.33

2.32

2.31

2.31

15

2.38

2.37

2.35

2.34

2.33

2.32

2.31

2.30

2.29

2.28

2.27

2.27

2.26

2.25

2.25

16

2.33

2.32

2.30

2.29

2.28

2.26

2.25

2.24

2.24

2.23

2.22

2.21

2.21

2.20

2.19

17

2.29

2.27

2.26

2.24

2.23

2.22

2.21

2.20

2.19

2.18

2.17

2.17

2.16

2.15

2.15

18

2.25

2.23

2.22

2.20

2.19

2.18

2.17

2.16

2.15

2.14

2.13

2.13

2.12

2.11

2.11

19

2.21

2.20

2.18

2.17

2.16

2.14

2.13

2.12

2.11

2.11

2.10

2.09

2.08

2.08

2.07

20

2.18

2.17

2.15

2.14

2.12

2.11

2.10

2.09

2.08

2.07

2.07

2.06

2.05

2.05

2.04

21

2.16

2.14

2.12

2.11

2.10

2.08

2.07

2.06

2.05

2.05

2.04

2.03

2.02

2.02

2.01

22

2.13

2.11

2.10

2.08

2.07

2.06

2.05

2.04

2.03

2.02

2.01

2.00

2.00

1.99

1.98

23

2.11

2.09

2.08

2.06

2.05

2.04

2.02

2.01

2.01

2.00

1.99

1.98

1.97

1.97

1.96

24

2.09

2.07

2.05

2.04

2.03

2.01

2.00

1.99

1.98

1.97

1.97

1.96

1.95

1.95

1.94

25

2.07

2.05

2.04

2.02

2.01

2.00

1.98

1.97

1.96

1.96

1.95

1.94

1.93

1.93

1.92

26

2.05

2.03

2.02

2.00

1.99

1.98

1.97

1.96

1.95

1.94

1.93

1.92

1.91

1.91

1.90

27

2.04

2.02

2.00

1.99

1.97

1.96

1.95

1.94

1.93

1.92

1.91

1.90

1.90

1.89

1.88

28

2.02

2.00

1.99

1.97

1.96

1.95

1.93

1.92

1.91

1.91

1.90

1.89

1.88

1.88

1.87

29

2.01

1.99

1.97

1.96

1.94

1.93

1.92

1.91

1.90

1.89

1.88

1.88

1.87

1.86

1.85

30

1.99

1.98

1.96

1.95

1.93

1.92

1.91

1.90

1.89

1.88

1.87

1.86

1.85

1.85

1.84

40

1.90

1.89

1.87

1.85

1.84

1.83

1.81

1.80

1.79

1.78

1.77

1.77

1.76

1.75

1.74

50

1.85

1.83

1.81

1.80

1.78

1.77

1.76

1.75

1.74

1.73

1.72

1.71

1.70

1.69

1.69

60

1.82

1.80

1.78

1.76

1.75

1.73

1.72

1.71

1.70

1.69

1.68

1.67

1.66

1.66

1.65

70

1.79

1.77

1.75

1.74

1.72

1.71

1.70

1.68

1.67

1.66

1.65

1.65

1.64

1.63

1.62

80

1.77

1.75

1.73

1.72

1.70

1.69

1.68

1.67

1.65

1.64

1.63

1.63

1.62

1.61

1.60

90

1.76

1.74

1.72

1.70

1.69

1.67

1.66

1.65

1.64

1.63

1.62

1.61

1.60

1.59

1.59

100

1.75

1.73

1.71

1.69

1.68

1.66

1.65

1.64

1.63

1.62

1.61

1.60

1.59

1.58

1.57

 

mk

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1

39.9

49.5

53.6

55.8

57.2

58.2

58.9

59.4

59.8

60.2

60.5

60.7

60.9

61.1

61.2

2

8.53

9.00

9.16

9.24

9.29

9.33

9.35

9.37

9.38

9.39

9.40

9.41

9.41

9.42

9.42

3

5.54

5.46

5.39

5.34

5.31

5.28

5.27

5.25

5.24

5.23

5.22

5.22

5.21

5.20

5.20

4

4.54

4.32

4.19

4.11

4.05

4.01

3.98

3.95

3.94

3.92

3.91

3.90

3.89

3.88

3.87

5

4.06

3.78

3.62

3.52

3.45

3.40

3.37

3.34

3.32

3.30

3.28

3.27

3.26

3.25

3.24

6

3.78

3.46

3.29

3.18

3.11

3.05

3.01

2.98

2.96

2.94

2.92

2.90

2.89

2.88

2.87

7

3.59

3.26

3.07

2.96

2.88

2.83

2.78

2.75

2.72

2.70

2.68

2.67

2.65

2.64

2.63

8

3.46

3.11

2.92

2.81

2.73

2.67

2.62

2.59

2.56

2.54

2.52

2.50

2.49

2.48

2.46

9

3.36

3.01

2.81

2.69

2.61

2.55

2.51

2.47

2.44

2.42

2.40

2.38

2.36

2.35

2.34

10

3.29

2.92

2.73

2.61

2.52

2.46

2.41

2.38

2.35

2.32

2.30

2.28

2.27

2.26

2.24

11

3.23

2.86

2.66

2.54

2.45

2.39

2.34

2.30

2.27

2.25

2.23

2.21

2.19

2.18

2.17

12

3.18

2.81

2.61

2.48

2.39

2.33

2.28

2.24

2.21

2.19

2.17

2.15

2.13

2.12

2.10

13

3.14

2.76

2.56

2.43

2.35

2.28

2.23

2.20

2.16

2.14

2.12

2.10

2.08

2.07

2.05

14

3.10

2.73

2.52

2.39

2.31

2.24

2.19

2.15

2.12

2.10

2.07

2.05

2.04

2.02

2.01

15

3.07

2.70

2.49

2.36

2.27

2.21

2.16

2.12

2.09

2.06

2.04

2.02

2.00

1.99

1.97

16

3.05

2.67

2.46

2.33

2.24

2.18

2.13

2.09

2.06

2.03

2.01

1.99

1.97

1.95

1.94

17

3.03

2.64

2.44

2.31

2.22

2.15

2.10

2.06

2.03

2.00

1.98

1.96

1.94

1.93

1.91

18

3.01

2.62

2.42

2.29

2.20

2.13

2.08

2.04

2.00

1.98

1.95

1.93

1.92

1.90

1.89

19

2.99

2.61

2.40

2.27

2.18

2.11

2.06

2.02

1.98

1.96

1.93

1.91

1.89

1.88

1.86

20

2.97

2.59

2.38

2.25

2.16

2.09

2.04

2.00

1.96

1.94

1.91

1.89

1.87

1.86

1.84

21

2.96

2.57

2.36

2.23

2.14

2.08

2.02

1.98

1.95

1.92

1.90

1.88

1.86

1.84

1.83

22

2.95

2.56

2.35

2.22

2.13

2.06

2.01

1.97

1.93

1.90

1.88

1.86

1.84

1.83

1.81

23

2.94

2.55

2.34

2.21

2.11

2.05

1.99

1.95

1.92

1.89

1.87

1.84

1.83

1.81

1.80

24

2.93

2.54

2.33

2.19

2.10

2.04

1.98

1.94

1.91

1.88

1.85

1.83

1.81

1.80

1.78

25

2.92

2.53

2.32

2.18

2.09

2.02

1.97

1.93

1.89

1.87

1.84

1.82

1.80

1.79

1.77

Table IV6 (continued)

mk

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

26

2.91

2.52

2.31

2.17

2.08

2.01

1.96

1.92

1.88

1.85

1.83

1.81

1.79

1.77

1.76

27

2.90

2.51

2.30

2.17

2.07

2.00

1.95

1.91

1.87

1.85

1.82

1.80

1.78

1.76

1.75

28

2.89

2.50

2.29

2.16

2.06

2.00

1.94

1.90

1.87

1.84

1.81

1.79

1.77

1.75

1.74

29

2.89

2.50

2.28

2.15

2.06

1.99

1.93

1.89

1.86

1.83

1.80

1.78

1.76

1.75

1.73

30

2.88

2.49

2.28

2.14

2.05

1.98

1.93

1.88

1.85

1.82

1.79

1.77

1.75

1.74

1.72

40

2.84

2.44

2.23

2.09

2.00

1.93

1.87

1.83

1.79

1.76

1.74

1.71

1.70

1.68

1.66

50

2.81

2.41

2.20

2.06

1.97

1.90

1.84

1.80

1.76

1.73

1.70

1.68

1.66

1.64

1.63

60

2.79

2.39

2.18

2.04

1.95

1.87

1.82

1.77

1.74

1.71

1.68

1.66

1.64

1.62

1.60

70

2.78

2.38

2.16

2.03

1.93

1.86

1.80

1.76

1.72

1.69

1.66

1.64

1.62

1.60

1.59

80

2.77

2.37

2.15

2.02

1.92

1.85

1.79

1.75

1.71

1.68

1.65

1.63

1.61

1.59

1.57

90

2.76

2.36

2.15

2.01

1.91

1.84

1.78

1.74

1.70

1.67

1.64

1.62

1.60

1.58

1.56

100

2.76

2.36

2.14

2.00

1.91

1.83

1.78

1.73

1.69

1.66

1.64

1.61

1.59

1.57

1.56

mk

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

1

61.4

61.5

61.6

61.7

61.7

61.8

61.9

61.9

62.0

62.1

62.1

62.2

62.2

62.2

62.3

2

9.43

9.43

9.44

9.44

9.44

9.44

9.45

9.45

9.45

9.45

9.45

9.45

9.46

9.46

9.46

3

5.20

5.19

5.19

5.19

5.18

5.18

5.18

5.18

5.18

5.18

5.17

5.17

5.17

5.17

5.17

4

3.86

3.86

3.85

3.85

3.84

3.84

3.84

3.83

3.83

3.83

3.83

3.82

3.82

3.82

3.82

5

3.23

3.22

3.22

3.21

3.21

3.20

3.20

3.19

3.19

3.19

3.18

3.18

3.18

3.18

3.17

6

2.86

2.85

2.85

2.84

2.84

2.83

2.83

2.82

2.82

2.81

2.81

2.81

2.81

2.80

2.80

7

2.62

2.61

2.61

2.60

2.59

2.59

2.58

2.58

2.58

2.57

2.57

2.56

2.56

2.56

2.56

8

2.45

2.45

2.44

2.43

2.42

2.42

2.41

2.41

2.40

2.40

2.40

2.39

2.39

2.39

2.38

9

2.33

2.32

2.31

2.30

2.30

2.29

2.29

2.28

2.28

2.27

2.27

2.26

2.26

2.26

2.25

10

2.23

2.22

2.22

2.21

2.20

2.19

2.19

2.18

2.18

2.17

2.17

2.17

2.16

2.16

2.16

11

2.16

2.15

2.14

2.13

2.12

2.12

2.11

2.11

2.10

2.10

2.09

2.09

2.08

2.08

2.08

 

Подпись: 312

12

2.09

2.08

2.08

2.07

2.06

2.05

2.05

2.04

2.04

2.03

2.03

2.02

2.02

2.01

2.01

13

2.04

2.03

2.02

2.01

2.01

2.00

1.99

1.99

1.98

1.98

1.97

1.97

1.96

1.96

1.96

14

2.00

1.99

1.98

1.97

1.96

1.96

1.95

1.94

1.94

1.93

1.93

1.92

1.92

1.92

1.91

15

1.96

1.95

1.94

1.93

1.92

1.92

1.91

1.90

1.90

1.89

1.89

1.88

1.88

1.88

1.87

16

1.93

1.92

1.91

1.90

1.89

1.88

1.88

1.87

1.87

1.86

1.86

1.85

1.85

1.84

1.84

17

1.90

1.89

1.88

1.87

1.86

1.85

1.85

1.84

1.84

1.83

1.83

1.82

1.82

1.81

1.81

18

1.87

1.86

1.85

1.84

1.84

1.83

1.82

1.82

1.81

1.80

1.80

1.80

1.79

1.79

1.78

19

1.85

1.84

1.83

1.82

1.81

1.81

1.80

1.79

1.79

1.78

1.78

1.77

1.77

1.76

1.76

20

1.83

1.82

1.81

1.80

1.79

1.79

1.78

1.77

1.77

1.76

1.76

1.75

1.75

1.74

1.74

21

1.81

1.80

1.79

1.78

1.78

1.77

1.76

1.75

1.75

1.74

1.74

1.73

1.73

1.72

1.72

22

1.80

1.79

1.78

1.77

1.76

1.75

1.74

1.74

1.73

1.73

1.72

1.72

1.71

1.71

1.70

23

1.78

1.77

1.76

1.75

1.74

1.74

1.73

1.72

1.72

1.71

1.70

1.70

1.70

1.69

1.69

24

1.77

1.76

1.75

1.74

1.73

1.72

1.71

1.71

1.70

1.70

1.69

1.69

1.68

1.68

1.67

25

1.76

1.75

1.74

1.73

1.72

1.71

1.70

1.70

1.69

1.68

1.68

1.67

1.67

1.66

1.66

26

1.75

1.73

1.72

1.71

1.71

1.70

1.69

1.68

1.68

1.67

1.67

1.66

1.66

1.65

1.65

27

1.74

1.72

1.71

1.70

1.70

1.69

1.68

1.67

1.67

1.66

1.65

1.65

1.64

1.64

1.64

28

1.73

1.71

1.70

1.69

1.69

1.68

1.67

1.66

1.66

1.65

1.64

1.64

1.63

1.63

1.63

29

1.72

1.71

1.69

1.68

1.68

1.67

1.66

1.65

1.65

1.64

1.63

1.63

1.62

1.62

1.62

30

1.71

1.70

1.69

1.68

1.67

1.66

1.65

1.64

1.64

1.63

1.63

1.62

1.62

1.61

1.61

40

1.65

1.64

1.62

1.61

1.61

1.60

1.59

1.58

1.57

1.57

1.56

1.56

1.55

1.55

1.54

50

1.61

1.60

1.59

1.58

1.57

1.56

1.55

1.54

1.54

1.53

1.52

1.52

1.51

1.51

1.50

60

1.59

1.58

1.56

1.55

1.54

1.53

1.53

1.52

1.51

1.50

1.50

1.49

1.49

1.48

1.48

70

1.57

1.56

1.55

1.54

1.53

1.52

1.51

1.50

1.49

1.49

1.48

1.47

1.47

1.46

1.46

80

1.56

1.55

1.53

1.52

1.51

1.50

1.49

1.49

1.48

1.47

1.47

1.46

1.45

1.45

1.44

90

1.55

1.54

1.52

1.51

1.50

1.49

1.48

1.48

1.47

1.46

1.45

1.45

1.44

1.44

1.43

100

1.54

1.53

1.52

1.50

1.49

1.48

1.48

1.47

1.46

1.45

1.45

1.44

1.43

1.43

1.42

Notes: For m > 100 the critical values of the Fk, m test are approximately equal to the critical values of the xl test divided by k. Because the Fk, m test is used to test parameter restrictions with the degrees of freedom к equal to the number of restrictions, it is unlikely that you will need the critical values of the Fk, m test for к > 30.

 

Подпись: 313

[1] In the spring of 2000, the Texas Lottery changed the rules. The number of balls was increased to fifty-four to create a larger jackpot. The official reason for this change was to make playing the lotto more attractive because a higher jackpot makes the lotto game more exciting. Of course, the actual intent was to boost the lotto revenues!

[2] Under the new rules (see Note 1), this probability is 1 /25,827,165.

[3] These binomial numbers can be computed using the “Tools ^ Discrete distribution tools” menu of EasyReg International, the free econometrics software package de­veloped by the author. EasyReg International can be downloaded from Web page http://econ. la. psu. edu/~hbierens/EASYREG. HTM

[4] Note that the latter phrase is superfluous because ^ C ^ signifies that every element of ^ is included in ^, which is clearly true, and 0C2 is true because 0C0UQ = ^.

[5] Also called afield.

[6] Also called a a – field or a Borel field.

[7] In the sequel we will denote the probability of an event involving random variables or vectors X as P (“expression involving X’) without referring to the corresponding set in &. For example, for random variables X and Y defined on a common probability space [&, &, P}, the shorthand notation P(X > Y) should be interpreted as P([ш є & : X(ш) > Y(ш)}).

[8] See also Appendix 1.A.

[9] The notation /g(x)dд(x) is somewhat odd because дф) has no meaning. It would be better to denote the integral involved by f g(x)/x(dx) (which some authors do), where dx represents a Borel set. The current notation, however, is the most common and is therefore adopted here too.

[10] Because oo — oo is not defined.

[11] Again, the notation /X(ofdPff is odd because P(ш) has no meaning. Some authors use the notation f X(ш)P(da>), where dш represents a set in &. The former notation is the most common and is therefore adopted.

[12] Here and in the sequel the notations P(Y = y |X = x), P(Y = y and X = x), P(X = x), and similar notations involving inequalities are merely shorthand notations for the probabilities P({ю є ^ : Y(ю) = y}|{&> є ^ : X(o)) = x}), P({&> є ^ : Y(&>) = y}0

{&> є ^ : X(ю) = x}), and P({ю є ^ : X(ю) = x}), respectively.

[13] The t distribution was discovered by W. S. Gosset, who published the result under the pseudonym Student. The reason for this was that his employer, an Irish brewery, did not want its competitors to know that statistical methods were being used.

[14] To distinguish the variance of a random variable from the variance matrix of a random vector, the latter will be denoted by Var with capital V

[15] The capital C in Cov indicates that this is a covariance matrix rather than a covariance of two random variables.

[16] Recall that “argmin” stands for the argument for which the function involved takes a minimum.

[17] The OLS estimator is called “ordinary” to distinguish it from the nonlinear least-squares estimator. See Chapter 6 for the latter.

[18] Let Xbe n – variate standard normally distributed, and let A be a nonstochastic n x k matrix with rank k < n. The projection of X on the column space of A is a vector p such that the following two conditions hold:

(1) p is a linear combination of the columns of A;

(2) the distance between X and p, ||X — p\ = ^(X — p)T(X — p), is minimal.

(a) Show that p = A(ATA)—1ATX.

(b) Is it possible to write down the density of p? If yes, do it. If no, why not?

(c) Show that ||p||2 = pTp has a x2 distribution. Determine the de­grees of freedom involved.

(d) Show that ||X — p ||2 has a x2 distribution. Determine the degrees of freedom involved.

(e) Show that | p| and | X — p| are independent.

[19] Prove Theorem 5.13.

[20] Show that (5.11) is true for в in an open set & if d2 fn(x 19)/(d9)2 is, for each x, continuous on & and f sup9e&|d2 fn(x|9)/(d9)2|dx < ж. Hint: Use the mean-value theorem and the dominated convergence theorem.

[21] Law of cosines: Consider a triangle ABC, let у be the angle between the legs C ^ A and

C ^ B, and denote the lengths of the legs opposite to the points A, B, and C by а, в, and Y, respectively. Then у2 = а2 + в2 – 2ав cos(y).

[23] In writing a matrix product it is from now on implicitly assumed that the matrices involved are conformable.

[24] 0 0 0 10 0.5 0 1/

‘2 4 2

0 0 0 , 0 3 0/

[25] + x2

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