Finance reserch report based on provided data

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Data Analysis – 4 Hypothesis Test

FINA305/405

1

Review

2

Review

3

Agenda

Introduction

Distribution

t-statistics, confidence interval and p-value

Type I & II error

Practice in Excel

4

Introduction

We infer/estimate/predict/forecast the underlying true model based on a sample (observations).

So the results on are usually different from the true .

Then, how reliable are ?

5

Introduction

In practice, we are usually interested in 1) whether , 2) then how much effect , which correspond to two major concerns in data analysis:

Statistical significance of : hypothesis test

Economic significance of : the size/magnitude of the estimator

6

Introduction

Population/true/theoretical model

Sample estimation

Q: Can we claim that is different from 0?

Seems that we will make mistake to conclude that is non-zero (positive).

7

Distribution

Distribution

9

Accuracy of

The standard error of the estimated slope:

Decrease with 1) a larger number of observations N (more data points)

2) smaller errors, better model (small SSR)

3) a bigger spread of values of the explanatory variable X (X has a wide range of values, more information)

Copyright © 2005 John Wiley & Sons, Ltd

Copyright © 2005 John Wiley & Sons, Ltd

Copyright © 2005 John Wiley & Sons, Ltd

13

Copyright © 2005 John Wiley & Sons, Ltd

But how small to guarantee the accuracy or reliability of ?

alone cannot tell anything.

Hypothesis testing

Null Hypothesis (): an assumption that a parameter in a statistical model takes a particular value, and is assumed true until experimental evidence suggests otherwise.

Alternative Hypothesis (): expresses the way in which the value of a parameter may deviate from that specified in the null hypothesis, and is favored when the experimental evidence suggests that the null hypothesis is rejected.

Most popular hypothesis test is to test statistical significance: whether there is a statistically significant (different from zero or not, doesn’t care about the exact value) linear relationship between X and Y.

H0: (No effect), H1:

Hypothesis testing

What if we have the following hypothesis:

H0: (No effect)

H1:

Is this still a hypothesis test for significance of ? No

17

34

Inference method 1: t-test

Hypotheses:

H0: (No effect), H1:

e.g. : education has no effect on one’s income: , : education has non-zero effect on one’s income:

Calculate t-statistic:

n-k-1 represents the degrees of freedom associated with the test; when there is one independent variable, k = 1.

18

34

Inference method 1: t-test

19

Inference method 1: t-test

If the test statistic calculated under is unlikely to occur (, then we reject “in favor of ”.

Inference or conclusion: we (do not) reject in favor of at 5% significance level, we find evidence that education has significantly positive effect on income.

Critical region: the null hypothesis is rejected when a calculated value of the test statistic lies within this region.

Significance level (α) : the size of the critical region, which we specify.

Critical value c: t-values which determine the boundary of the critical region. (t-distribution table!)

20

20

21

Inference method 2: confidence interval

Idea: a “point estimator” is seldom equal to the true parameter, we need an “interval estimator” that covers the true parameter for a certain degrees of confidence (eg. confidence level 95%)

Confidence level = 1 – significance level α

Based on , first calculate the confidence interval (CI):

P(-c< t < c) = 95%

c is the critical value given certain significance level

Then use decision rule:

If the estimator obtained is not in the CI, we can reject in favor of at 5% …

22

22

Inference method 3: P-value

It is not efficient to check or memorize critical values in distribution tables

P-value is the area of t-distribution that P(|T|>|)

Decision rule:

If the p-value is smaller than the given significance level, we can reject in favor of at 5% …

23

23

Type I & II error

Type I error: rejecting null when it is true

|) = the significance level

Type II error: failing to reject the null when it is false

P(not Reject |)

We should minimize Type II error or maximize Power of the test: 1 – P(not Reject |)

Set to be a big value

24

24

Why is the estimator statistically insignificant?

Irrelevant

Biased estimator (the causes of biasedness)

Variance of the estimator is too large

Variation of the independent variable

Correlation between the key independent variable and other independent variable

Sample size

25

实际研究中

25

Practice in Excel

http://www.rbnz.govt.nz/statistics/key-graphs/key-graph-house-price-values

Regression (OLS)

Hypothesis test

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XX

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2

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NXX

b

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Very small sample size

-2

-1

0

1

2

3

4

5

00.511.522.533.544.55

X

Y

Chart1

0.41499144
1.7392695
4.3552147
2.5090849
1.4652497
X
Y
Very small sample size
1.9517978
-1.5626615
3.9265792
1.9602345
-1.4335304

TEMPG

X Y
0.41499144 1.9517978
1.7392695 -1.5626615
4.3552147 3.9265792
2.5090849 1.9602345
1.4652497 -1.4335304

-4-2024680123456YX

Large sample size, large error variance

Chart1

0.41499144
1.7392695
4.3552147
2.5090849
1.4652497
4.1631904
5.5978685
1.3950908
3.4648585
2.3840522
4.199944
4.0594848
2.7762218
2.8634311
2.8155518
3.7684032
1.3490345
2.7856822
2.7255748
1.55061
3.8787965
3.1155175
2.76207
4.2452427
3.8256153
3.7775176
3.8198834
4.0253602
3.5608748
4.8679379
3.4025674
4.0876737
3.8725085
1.5141033
3.9760998
0.82039382
2.7245327
3.7305596
1.5424251
2.7324146
2.2295917
1.8473854
1.9383724
3.6736788
4.7562841
3.3332498
2.7674357
4.0709911
1.2858377
4.6738776
2.9666673
3.447162
3.112172
2.3120597
2.8538535
2.0695762
1.6009573
3.9831414
3.4121162
4.536383
3.532158
5.2271265
3.7683971
3.0687003
3.1920888
3.7859784
5.0462328
1.4251468
4.3536434
4.6699201
3.1698032
2.0110589
1.4012837
1.7305831
2.9695773
2.113013
0.0051243003
4.1543706
3.5919939
2.8696234
2.4302331
1.7670064
3.3598998
2.895879
3.6023952
3.9270266
3.7137611
2.5720587
1.716968
3.8845006
2.4301553
2.1208416
3.7741512
4.8933279
1.2330044
2.1467985
3.357127
3.1628424
1.5743457
2.9480853691
X
Y
Large sample size, large error variance
0.57172311
1.3830482
6.1507569
2.6924907
0.26381609
4.6487168
6.7523074
1.1732042
5.9836572
4.9720619
0.95230783
3.1101546
4.6406292
0.91287458
0.58700094
6.4788468
0.88394947
0.37191785
0.58077108
1.9586321
3.488131
-0.3645467
1.1471248
4.8082412
2.8764895
1.6015831
4.1107771
3.8867575
5.3703284
4.4383231
2.4204186
3.3973947
5.9712235
2.86941
4.5833348
1.0657699
3.0239161
1.4114205
-0.48668318
2.6512647
5.2045085
1.6623325
3.7679844
1.3030104
6.3500078
2.4139488
1.9353057
6.7844188
1.6675597
6.180012
2.0568503
2.2000711
-3.2639255
2.3997738
4.7528296
1.4872262
1.0265924
3.0919941
4.2552985
2.792831
4.8472617
4.2609308
2.5109634
1.4737688
4.2949738
1.2434316
4.8544205
5.5463931
6.3619121
5.593296
6.0370803
-2.2954419
3.1940482
0.12002249
0.42630087
0.60232163
-1.6505782
6.4082102
3.7997746
6.0010412
7.4179616
-0.12565026
5.69481
5.6110749
5.3937487
4.6531232
1.6834458
0.1185421
3.3139965
3.02045
3.0558905
0.19213533
5.7968424
3.1694352
-0.78262567
3.8877304
3.8217647
3.1697462
4.4804064
3.2362029373

TEMPG

X Y
0.41499144 0.57172311
1.7392695 1.3830482
4.3552147 6.1507569
2.5090849 2.6924907
1.4652497 0.26381609
4.1631904 4.6487168
5.5978685 6.7523074
1.3950908 1.1732042
3.4648585 5.9836572
2.3840522 4.9720619
4.199944 0.95230783
4.0594848 3.1101546
2.7762218 4.6406292
2.8634311 0.91287458
2.8155518 0.58700094
3.7684032 6.4788468
1.3490345 0.88394947
2.7856822 0.37191785
2.7255748 0.58077108
1.55061 1.9586321
3.8787965 3.488131
3.1155175 -0.3645467
2.76207 1.1471248
4.2452427 4.8082412
3.8256153 2.8764895
3.7775176 1.6015831
3.8198834 4.1107771
4.0253602 3.8867575
3.5608748 5.3703284
4.8679379 4.4383231
3.4025674 2.4204186
4.0876737 3.3973947
3.8725085 5.9712235
1.5141033 2.86941
3.9760998 4.5833348
0.82039382 1.0657699
2.7245327 3.0239161
3.7305596 1.4114205
1.5424251 -0.48668318
2.7324146 2.6512647
2.2295917 5.2045085
1.8473854 1.6623325
1.9383724 3.7679844
3.6736788 1.3030104
4.7562841 6.3500078
3.3332498 2.4139488
2.7674357 1.9353057
4.0709911 6.7844188
1.2858377 1.6675597
4.6738776 6.180012
2.9666673 2.0568503
3.447162 2.2000711
3.112172 -3.2639255
2.3120597 2.3997738
2.8538535 4.7528296
2.0695762 1.4872262
1.6009573 1.0265924
3.9831414 3.0919941
3.4121162 4.2552985
4.536383 2.792831
3.532158 4.8472617
5.2271265 4.2609308
3.7683971 2.5109634
3.0687003 1.4737688
3.1920888 4.2949738
3.7859784 1.2434316
5.0462328 4.8544205
1.4251468 5.5463931
4.3536434 6.3619121
4.6699201 5.593296
3.1698032 6.0370803
2.0110589 -2.2954419
1.4012837 3.1940482
1.7305831 0.12002249
2.9695773 0.42630087
2.113013 0.60232163
0.0051243003 -1.6505782
4.1543706 6.4082102
3.5919939 3.7997746
2.8696234 6.0010412
2.4302331 7.4179616
1.7670064 -0.12565026
3.3598998 5.69481
2.895879 5.6110749
3.6023952 5.3937487
3.9270266 4.6531232
3.7137611 1.6834458
2.5720587 0.1185421
1.716968 3.3139965
3.8845006 3.02045
2.4301553 3.0558905
2.1208416 0.19213533
3.7741512 5.7968424
4.8933279 3.1694352
1.2330044 -0.78262567
2.1467985 3.8877304
3.357127 3.8217647
3.1628424 3.1697462
1.5743457 4.4804064
2.9480853691 3.2362029373

Limited range of X values

-4

-2

0

2

4

6

8

10

00.511.522.533.5

X

Y

Chart1

2.7414991
2.8739269
3.1355215
2.9509085
2.846525
3.116319
3.2597868
2.8395091
3.0464859
2.9384052
3.1199944
3.1059485
2.9776222
2.9863431
2.9815552
3.0768403
2.8349034
2.9785682
2.9725575
2.855061
3.0878797
3.0115517
2.976207
3.1245243
3.0825615
3.0777518
3.0819883
3.102536
3.0560875
3.1867938
3.0402567
3.1087674
3.0872508
2.8514103
3.09761
2.7820394
2.9724533
3.073056
2.8542425
2.9732415
2.9229592
2.8847385
2.8938372
3.0673679
3.1756284
3.033325
2.9767436
3.1070991
2.8285838
3.1673878
2.9966667
3.0447162
3.0112172
2.931206
2.9853853
2.9069576
2.8600957
3.0983141
3.0412116
3.1536383
3.0532158
3.2227127
3.0768397
3.00687
3.0192089
3.0785978
3.2046233
2.8425147
3.1353643
3.166992
3.0169803
2.9011059
2.8401284
2.8730583
2.9969577
2.9113013
2.7005124
3.1154371
3.0591994
2.9869623
2.9430233
2.8767006
3.03599
2.9895879
3.0602395
3.0927027
3.0713761
2.9572059
2.8716968
3.0884501
2.9430155
2.9120842
3.0774151
3.1893328
2.8233004
2.9146798
3.0357127
3.0162842
2.8574346
3.1325516
X
Y
Limited range of X values
2.8982308
2.5177057
4.9310637
3.1343144
1.6450914
3.6018454
4.4142258
2.6176225
5.5652845
5.5264149
-0.12764176
2.1566182
4.8420295
1.0357866
0.75300434
5.7872839
2.3698184
0.56480383
0.82775373
3.2630831
2.6972142
-0.46851243
1.3612618
3.6875227
2.1334357
0.90181734
3.372882
2.9639334
4.8655411
2.757179
2.0581079
2.4184884
5.1859659
4.2067171
3.7048449
3.0274155
3.2718367
0.75391689
0.82513423
2.8920916
5.897876
2.6996857
4.7234492
0.69669948
4.7693521
2.114024
2.1446135
5.8205268
3.2103058
4.6735222
2.0868497
1.7976252
-3.3648803
3.0189201
4.8843615
2.3246076
2.2857308
2.2071668
3.8843939
1.4100862
4.3683195
2.2565169
1.819406
1.4119385
4.1220939
0.53605113
3.0128111
6.963761
5.1436331
4.0903679
5.8842574
-1.4053949
4.6328929
1.2624977
0.45368127
1.4006099
1.0448099
5.3692766
3.2669801
6.1183802
7.9307518
0.98404398
5.3709001
5.7047838
4.851593
3.8187993
1.0410608
0.50368927
4.4687253
2.2243995
3.5687507
0.98337786
5.1001063
1.4654401
0.80767039
4.6556118
3.5003503
3.0231881
5.7634953
4.3718683

TEMPG

X Y
2.7414991 2.8982308
2.8739269 2.5177057
3.1355215 4.9310637
2.9509085 3.1343144
2.846525 1.6450914
3.116319 3.6018454
3.2597868 4.4142258
2.8395091 2.6176225
3.0464859 5.5652845
2.9384052 5.5264149
3.1199944 -0.12764176
3.1059485 2.1566182
2.9776222 4.8420295
2.9863431 1.0357866
2.9815552 0.75300434
3.0768403 5.7872839
2.8349034 2.3698184
2.9785682 0.56480383
2.9725575 0.82775373
2.855061 3.2630831
3.0878797 2.6972142
3.0115517 -0.46851243
2.976207 1.3612618
3.1245243 3.6875227
3.0825615 2.1334357
3.0777518 0.90181734
3.0819883 3.372882
3.102536 2.9639334
3.0560875 4.8655411
3.1867938 2.757179
3.0402567 2.0581079
3.1087674 2.4184884
3.0872508 5.1859659
2.8514103 4.2067171
3.09761 3.7048449
2.7820394 3.0274155
2.9724533 3.2718367
3.073056 0.75391689
2.8542425 0.82513423
2.9732415 2.8920916
2.9229592 5.897876
2.8847385 2.6996857
2.8938372 4.7234492
3.0673679 0.69669948
3.1756284 4.7693521
3.033325 2.114024
2.9767436 2.1446135
3.1070991 5.8205268
2.8285838 3.2103058
3.1673878 4.6735222
2.9966667 2.0868497
3.0447162 1.7976252
3.0112172 -3.3648803
2.931206 3.0189201
2.9853853 4.8843615
2.9069576 2.3246076
2.8600957 2.2857308
3.0983141 2.2071668
3.0412116 3.8843939
3.1536383 1.4100862
3.0532158 4.3683195
3.2227127 2.2565169
3.0768397 1.819406
3.00687 1.4119385
3.0192089 4.1220939
3.0785978 0.53605113
3.2046233 3.0128111
2.8425147 6.963761
3.1353643 5.1436331
3.166992 4.0903679
3.0169803 5.8842574
2.9011059 -1.4053949
2.8401284 4.6328929
2.8730583 1.2624977
2.9969577 0.45368127
2.9113013 1.4006099
2.7005124 1.0448099
3.1154371 5.3692766
3.0591994 3.2669801
2.9869623 6.1183802
2.9430233 7.9307518
2.8767006 0.98404398
3.03599 5.3709001
2.9895879 5.7047838
3.0602395 4.851593
3.0927027 3.8187993
3.0713761 1.0410608
2.9572059 0.50368927
2.8716968 4.4687253
3.0884501 2.2243995
2.9430155 3.5687507
2.9120842 0.98337786
3.0774151 5.1001063
3.1893328 1.4654401
2.8233004 0.80767039
2.9146798 4.6556118
3.0357127 3.5003503
3.0162842 3.0231881
2.8574346 5.7634953
3.1325516 4.3718683

Large sample size, small error variance

-1

0

1

2

3

4

5

6

0123456

X

Y

Chart1

0.41499144
1.7392695
4.3552147
2.5090849
1.4652497
4.1631904
5.5978685
1.3950908
3.4648585
2.3840522
4.199944
4.0594848
2.7762218
2.8634311
2.8155518
3.7684032
1.3490345
2.7856822
2.7255748
1.55061
3.8787965
3.1155175
2.76207
4.2452427
3.8256153
3.7775176
3.8198834
4.0253602
3.5608748
4.8679379
3.4025674
4.0876737
3.8725085
1.5141033
3.9760998
0.82039382
2.7245327
3.7305596
1.5424251
2.7324146
2.2295917
1.8473854
1.9383724
3.6736788
4.7562841
3.3332498
2.7674357
4.0709911
1.2858377
4.6738776
2.9666673
3.447162
3.112172
2.3120597
2.8538535
2.0695762
1.6009573
3.9831414
3.4121162
4.536383
3.532158
5.2271265
3.7683971
3.0687003
3.1920888
3.7859784
5.0462328
1.4251468
4.3536434
4.6699201
3.1698032
2.0110589
1.4012837
1.7305831
2.9695773
2.113013
0.0051243003
4.1543706
3.5919939
2.8696234
2.4302331
1.7670064
3.3598998
2.895879
3.6023952
3.9270266
3.7137611
2.5720587
1.716968
3.8845006
2.4301553
2.1208416
3.7741512
4.8933279
1.2330044
2.1467985
3.357127
3.1628424
1.5743457
4.3255158
X
Y
Large sample size, small error variance
0.42282802
1.7214584
4.4449918
2.5182552
1.405178
4.1874668
5.6555904
1.3839965
3.5907985
2.5134527
4.0375622
4.0120183
2.8694422
2.7659033
2.7041242
3.9039254
1.3257802
2.664994
2.6183346
1.5710111
3.8592632
2.9415143
2.6813228
4.2733927
3.778159
3.6687208
3.8344281
4.0184301
3.6513475
4.8464572
3.35346
4.0531598
3.9774442
1.5818686
4.0064616
0.83266263
2.7395019
3.6146026
1.4409697
2.7283571
2.3783375
1.8381327
2.029853
3.5551453
4.8359703
3.2872848
2.7258292
4.2066625
1.3049238
4.7491844
2.9211765
3.3848075
2.7933671
2.3164454
2.9488023
2.0404587
1.5722391
3.9385841
3.4542753
4.4492054
3.5979132
5.1788167
3.7055254
2.9889537
3.247233
3.658851
5.0366421
1.6312091
4.4540568
4.7160889
3.3131671
1.7957339
1.4909219
1.650055
2.8424135
2.0374784
-0.077660824
4.2670626
3.602383
3.0261943
2.6796195
1.6723736
3.4766453
3.0316388
3.6919629
3.9633314
3.6122454
2.4493829
1.7968194
3.841298
2.4614421
2.0244063
3.8752858
4.8071333
1.1322229
2.2338451
3.3803589
3.1631876
1.7196487
4.3874816

TEMPG

X Y
0.41499144 0.42282802
1.7392695 1.7214584
4.3552147 4.4449918
2.5090849 2.5182552
1.4652497 1.405178
4.1631904 4.1874668
5.5978685 5.6555904
1.3950908 1.3839965
3.4648585 3.5907985
2.3840522 2.5134527
4.199944 4.0375622
4.0594848 4.0120183
2.7762218 2.8694422
2.8634311 2.7659033
2.8155518 2.7041242
3.7684032 3.9039254
1.3490345 1.3257802
2.7856822 2.664994
2.7255748 2.6183346
1.55061 1.5710111
3.8787965 3.8592632
3.1155175 2.9415143
2.76207 2.6813228
4.2452427 4.2733927
3.8256153 3.778159
3.7775176 3.6687208
3.8198834 3.8344281
4.0253602 4.0184301
3.5608748 3.6513475
4.8679379 4.8464572
3.4025674 3.35346
4.0876737 4.0531598
3.8725085 3.9774442
1.5141033 1.5818686
3.9760998 4.0064616
0.82039382 0.83266263
2.7245327 2.7395019
3.7305596 3.6146026
1.5424251 1.4409697
2.7324146 2.7283571
2.2295917 2.3783375
1.8473854 1.8381327
1.9383724 2.029853
3.6736788 3.5551453
4.7562841 4.8359703
3.3332498 3.2872848
2.7674357 2.7258292
4.0709911 4.2066625
1.2858377 1.3049238
4.6738776 4.7491844
2.9666673 2.9211765
3.447162 3.3848075
3.112172 2.7933671
2.3120597 2.3164454
2.8538535 2.9488023
2.0695762 2.0404587
1.6009573 1.5722391
3.9831414 3.9385841
3.4121162 3.4542753
4.536383 4.4492054
3.532158 3.5979132
5.2271265 5.1788167
3.7683971 3.7055254
3.0687003 2.9889537
3.1920888 3.247233
3.7859784 3.658851
5.0462328 5.0366421
1.4251468 1.6312091
4.3536434 4.4540568
4.6699201 4.7160889
3.1698032 3.3131671
2.0110589 1.7957339
1.4012837 1.4909219
1.7305831 1.650055
2.9695773 2.8424135
2.113013 2.0374784
0.0051243003 -0.077660824
4.1543706 4.2670626
3.5919939 3.602383
2.8696234 3.0261943
2.4302331 2.6796195
1.7670064 1.6723736
3.3598998 3.4766453
2.895879 3.0316388
3.6023952 3.6919629
3.9270266 3.9633314
3.7137611 3.6122454
2.5720587 2.4493829
1.716968 1.7968194
3.8845006 3.841298
2.4301553 2.4614421
2.1208416 2.0244063
3.7741512 3.8752858
4.8933279 4.8071333
1.2330044 1.1322229
2.1467985 2.2338451
3.357127 3.3803589
3.1628424 3.1631876
1.5743457 1.7196487
4.3255158 4.3874816