The Bivariate Regression:InferenceLecture 4

Aims and Learning ObjectivesBy the end of this session students should be able to:

Understand why we conduct statistical inference

Calculate and interpret interval estimation and hypothesis tests Distinguish between Type I and Type II Errors

Interpret p-Values

4.1 IntroductionIn Lecture 2 we looked at how we calculate point estimates of the regression parameters, andin Lecture 3 under what circumstances these areconsidered to be BLUE (best linear unbiasedestimators) Also, we determined the probability distribution ofOLS estimatorsThe fact that the estimator follows a particularprobability distribution helps us relate the sampleto the population

4.2 Statistical Inference Our goal, therefore, is to use the estimates fromthe sample to infer something about the populationFor our purposes, we assume the sample data wehave is our best and only information about thepopulation

4.2 Statistical Inference How do we decide whether our sample estimatesare close to the population parameters?Remember, each sample from the population givesus different estimates of 1 and 2, resulting in the sampling distributions discussed in Lecture 3^^It is quite possible that one particular estimate comes from an unbiased distribution but is far from the population parameterUse statistical inference by running tests on 2^

4.2 Statistical Inference Recall from Lecture 3, if assumptions A1 to A6hold:We would test the hypothesis H0: 2 = b0 versus H1: 2 b0

We use the (student) t - distribution

Under H0, t has a t Distribution with n2 degrees of freedom. ^^Where

normalt, 10 d.f.t, 5 d.f.Student-t vs. Normal Distribution1. Both are symmetric bell-shaped distributions.2. Student-t distribution has fatter tails than the normal.3. Student-t converges to the normal for infinite sample.4. Student-t conditional on degrees of freedom (df).5. Normal is a good approximation of Student-t for the first few decimal places when df > 30 or so.

Chart1

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0.50.32793866750.33969480110.34580809830.3520653268

0.5250.32318320620.3350617120.34124659030.3475832643

0.550.31829403980.33028262860.3365332640.342943855

0.5750.3132825050.32536727970.33167694570.3381549579

0.60.30815993150.32032549440.32668660330.3332246029

0.6250.3029375960.31516716680.32157131610.3281609686

0.650.29762667730.30990222120.31634024680.3229723597

0.6750.29223821430.30454057830.31100261280.3176671847

0.70.28678306670.29909212260.3055676580.3122539334

0.7250.28127187880.29356667050.30004462570.3067411541

0.750.27571504560.28797394050.29444273170.3011374322

0.7750.27012268250.28232352430.28877113830.2954513673

0.80.2645045980.27662485940.28303892940.2896915528

0.8250.25887026940.27088720410.27725508650.2838665535

0.850.25322882150.26511961350.27142846560.2779848861

0.8750.24758900890.25933091740.26556777550.2720549984

0.90.24195920050.25352970040.25968155710.2660852499

0.9250.23634736730.24772428360.25377816420.2600838934

0.950.23076107350.24192270820.24786574510.2540590565

0.9750.22520746930.2361327210.24195222670.2480187246

10.2196932870.23036176190.23604529820.2419707245

1.050.20880801920.21890508990.22428070110.2298821407

1.10.19815075770.20760570830.21262823270.217852177

1.150.18776040960.19651121850.20113963440.2059362687

1.20.17766952280.18566371040.18986186740.194186055

1.250.16790459780.1750997630.17883695570.1826490854

1.30.15848646720.16485053030.16810191180.171368592

1.350.14943072550.15494190280.15768873750.1603833273

1.40.14074819020.14539473220.14762449440.1497274656

1.450.13244537910.1362251070.13793143570.1394305664

1.50.12452499080.12744466850.12862719180.1295175957

1.550.11698637660.11906095550.11972500010.1200090007

1.60.10982599620.11107776770.11123397270.1109208347

1.650.10303784980.10349553940.10315939150.1022649246

1.70.0966138810.09631171460.09550302460.0940490774

1.750.09054434990.08952111670.08826345470.0862773188

1.80.08481817090.08311630760.08143641560.0789501583

1.850.07942321790.07708793110.0750151270.0720648743

1.90.07434659510.07142503660.06899062580.0656158148

1.950.06957487530.06611538070.06335208660.0595947061

20.06509430730.0611457060.05808712890.0539909665

2.10.05695104110.05216969110.04862238580.043983596

2.20.04980641040.04437963280.04047680630.0354745928

2.30.04355363560.03765554960.03352241060.0283270377

2.40.03809199550.03187946230.02762908060.0223945303

2.50.03332828530.0269387010.022669410.0175283005

2.60.02917753330.02272809740.01852225260.0135829692

2.70.02556318070.01915127520.01507512160.0104209348

2.80.02241689550.01612124150.01222562370.0079154516

2.90.01967814720.01356045690.00988211710.0059525324

30.01729364070.01140053820.00796377480.0044318484

3.10.01521667940.00958171820.00640021660.0032668191

3.20.01340650690.00805215940.00513084850.0023840882

3.30.01182766040.00676719580.00410402420.0017225689

3.40.01044935640.00568855550.00327611780.0012322192

3.50.00924492180.00478360240.00261057330.0008726827

3.60.00819127570.00402461920.002076980.0006119019

3.70.00726846380.00338814760.00165020290.0004247803

3.80.0064592450.00285439160.00130958880.0002919469

3.90.00574872580.00240668640.00103825660.0001986555

40.00512404170.00203103190.00082247310.0001338302

4.10.00457407770.00171568550.00065111370.0000892617

4.20.00408922750.00145081140.00051520140.0000589431

4.30.00366118340.00122817910.00040751620.0000385352

4.40.00328275660.00104090690.00032226970.0000249425

4.50.00294772110.00088324380.00025483330.0000159837

4.60.00265068010.00075038510.00020151460.0000101409

4.70.00238695110.00063831740.00015937460.0000063698

4.80.0021524670.00054368830.0001260780.0000039613

4.90.00194369160.00046369640.0000997720.000002439

50.00175754630.00039600070.0000789890.0000014867

Sample is never a perfect representation of thepopulation from which it is drawn.

In order to quantify the likely magnitude of the error, it is often useful to be able to specifya range of values, within which we could state with reasonable certainty that the population parameter we are estimating should lie.

4.3 Interval Estimation

Standard deviation known95% confidence interval2 - 1.96 sd b2 2 + 1.96 sd Standard deviation unknown, estimated by standard error95% confidence interval2 - t (n-2,2.5%) se(2) b2 2 + t (n-2,2.5%) se(2)

Interval EstimationGeneral Formula:

Pr [sample estimate critical value standard error] = 1-

^^^^^^

1. Determine null and alternative hypotheses. 2. Specify the test statistic and its distribution as if the null hypothesis were true. 3. Select a and determine the rejection region. 4. Calculate the sample value of test statistic. 5. State your conclusion.4.4 Hypothesis Testing

Hypothesis TestingStep 1: State the null hypothesis, H0: the hypothesis we wish to test (e.g. )

Step 2: State the alternative hypothesis, H1, which is true if H0 is false. One-sided (e.g. H1: 2 > b0 or H1: 2 < b0 ) Two-sided (e.g. H1: 2 b0)

Hypothesis TestingStep 3: Select the significance level, , of the test (typically, = 0.1, 0.05 or 0.01).

Step 4: Calculate the test-statistic

Step 5: Calculate the critical values of the distribution

if one-sided; if two-sided.

Hypothesis TestingStep 6: Apply the decision rule

For one-sided test, , if then reject H0. Otherwise do not reject.

For one-sided test, , if then reject H0. Otherwise do not reject.

Two-sided test, if , then reject H0. Otherwise do not reject.

H1:2 < b0H1:2 > b0H1:2 b0

t Distribution

()t0f(t)-tt/2/2red area = rejection region for 2-sided testRejectregionRejectregionRegion of Non-rejectionHypothesis Testing

s.d. of 2 knowndiscrepancy between hypothetical value and sample estimate, in terms of s.d.:5% significance test:reject H0: b2 = b0 ifz > 1.96 or z < -1.96s.d. of 2 not knowndiscrepancy between hypothetical value and sample estimate, in terms of s.e.:5% significance test:reject H0: b2 = b0 ift > tc or t < -tcAccordingly, we refer to the test statistic as a t statistic. In other respects the test procedure is much the same.4Hypothesis Testing

4.5 Type I and Type II errorsType I error: We make the mistake of rejecting the null hypothesis when it is true. = Prob (rejecting H0 when it is true). Type II error: We make the mistake of failing to reject the null hypothesis when it is false. = Prob (failing to reject H0 when it is false).

4.6 p-ValuesA p-value of a test is calculated from the absolutevalue of the t-statisticGeneral Rule: If the p-value is smaller than the chosen value of (significance level) then the test procedure leads to rejection of the null hypothesis (based on a two-sided test)It provides an alternative approach to reporting the significance of regression coefficientsThe p-value reports the probability of falsely rejecting the null hypothesis that 2 = 0 against 2 0. It thereby provides an exact probability of a Type I error