Chapter 4: Statistical Hypothesis Testing - Accueil

1 Chapter 4: Statistical Hypothesis TestingChristophe HurlinNovember 20, 2015 Christophe Hurlin ()Advanced Econometrics - Master ESAN ovember 20, 20151 / 225 Section 1 IntroductionChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 20152 / 2251. IntroductionThe outline of this Chapter is the following:Section Hypothesis testingSection in the multiple linear regression modelSubsection Student testSubsection Fisher testSection and InferenceSubsection Likelihood Ratio (LR) testSubsection Wald testSubsection Lagrange Multiplier (LM) testChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 20153 / 2251. IntroductionReferencesAmemiya T. (1985), Advanced Econometrics. Harvard University W. (2007), Econometric Analysis, sixth edition, Pearson - PrenticeHil (recommended)Ruud P., (2000) An introduction to Classical Econometric Theory, OxfordUniversity Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 20154 / 2251.

2 IntroductionNotations:In this Chapter , I will (try ) follow some conventions (y)probability density or mass functionFY(y)cumulative distribution functionPr()probabilityyvectorYmatrixBe careful:in this Chapter , I don t distinguish between a random vector(matrix) and a vector (matrix) of deterministic elements (except in section2). For more appropriate notations, see:Abadir and Magnus (2002), Notation in econometrics: a proposal for astandard, Econometrics Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 20155 / 225 Section 2 Statistical Hypothesis testingChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 20156 / 2252. Statistical Hypothesis testingObjectivesThe objective of this section is to de ne the following concepts:1 Null and alternative hypotheses2 One-sided and two-sided tests3 Rejection region, test statistic and critical value4 Size, power and power function5 Uniformly most powerful (UMP) test6 Neyman Pearson lemma7 Consistent test and unbiased test8p-valueChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 20157 / 2252.

3 Statistical Hypothesis testingIntroduction1A Statistical hypothesistestis a method ofmaking decisionsor arule of decision(as concerned a statement about apopulationparameter) using the data Hypothesis tests de ne a procedure thatcontrols ( xes)the probability of incorrectly decidingthat a default position (nullhypothesis) is incorrect based on how likely it would be for a set ofobservations to occur if the null Hypothesis were Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 20158 / 2252. Statistical Hypothesis testingIntroduction (cont d)In general we distinguish two types of tests:1 Theparametric testsassume that the data have come from a typeof probability distribution and makes inferences about the parametersof the distribution2 Thenon-parametric testsrefer to tests that do not assume the dataor population have any characteristic structure or this course, we only consider the parametric Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 20159 / 2252.

4 Statistical Hypothesis testingIntroduction (cont d)A Statistical test is based on three elements:1A null Hypothesis and an alternative hypothesis2A rejection region based on a test statistic and a critical value3A type I error and a type II errorChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201510 / 2252. Statistical Hypothesis testingIntroduction (cont d)A Statistical test is based on three elements:1A null Hypothesis and an alternative hypothesis2A rejection region based on a test statistic and a critical value3A type I error and a type II errorChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201511 / 2252. Statistical Hypothesis testingDe nition ( Hypothesis )Ahypothesisis a statement about a population parameter. The formaltesting procedure involves a statement of the Hypothesis , usually in termsof a null or maintained Hypothesis and an alternative, conventionallydenoted H0and H1, Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201512 / 2252.

5 Statistical Hypothesis testingIntroduction1 The null Hypothesis refers to a general or default position: that thereis no relationship between two measured phenomena or that apotential medical treatment has no e costs associated to the violation of the null must be higher thanthe cost of a violation of the (Choice of the null Hypothesis )In a credit scoring problem, in general we have: H0:the client is notrisky(acceptance of the loan) versus H1:the client is risky (refusal of theloan).Christophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201513 / 2252. Statistical Hypothesis testingDe nition (Simple and composite hypotheses)Asimple hypothesisspeci es the population distribution completely. Acompositehypothesis does not specify the population (Simple and composite hypotheses)IfX t( ),H0: = 0is a simple Hypothesis . H1: > 0,H1: < 0,and H1: 6= 0are composite Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201514 / 2252.

6 Statistical Hypothesis testingDe nition (One-sided test)Aone-sided testhas the general form:H0: = 0or H0: = 0H1: < 0H1: > 0 Christophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201515 / 2252. Statistical Hypothesis testingDe nition (Two-sided test)Atwo-sided testhas the general form:H0: = 0H1: 6= 0 Christophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201516 / 2252. Statistical Hypothesis testingIntroduction (cont d)A Statistical test is based on three elements:1A null Hypothesis and an alternative hypothesis2A rejection region based on a test statistic and a critical value3A type I error and a type II errorChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201517 / 2252. Statistical Hypothesis testingDe nition (Rejection region)Therejection regionis the set of values of the test statistic (orequivalently the set of samples) for which the null Hypothesis is rejection region is denoted W.

7 For example, a standard rejectionregion W is of the form:W=fx:T(x)>cgor equivalentlyW=fx1,..,xN:T(x1,..,xN)>cgwh erexdenotes a samplefx1,..,xNg,T(x)the realisation of ateststatisticandcthecritical Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201518 / 2252. Statistical Hypothesis testingRemarks1A ( Hypothesis ) test is thus a rule that speci es:1 For which sample values the decision is made to fail to reject H0 astrue;2 For which sample values the decision is made to reject H0 .3 Never say "Accept H1", "fail to reject H1" complement of the rejection region is the non-rejection Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201519 / 2252. Statistical Hypothesis testingRemarkThe rejection region is de ned as to be:W=fx:T(x)|{z}test statistic7c|{z}gcritical valueT(x)is the realisation of the statistic (random variable):T(X)=T(X1,..,XN)The test statisticT(X)has an exact or an asymptotic distributionDunder the null (X) H0 DorT(X)d!

8 H0 DChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201520 / 2252. Statistical Hypothesis testingIntroduction (cont d)A Statistical test is based on three elements:1A null Hypothesis and an alternative hypothesis2A rejection region based on a test statistic and a critical value3A type I error and a type II errorChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201521 / 2252. Statistical Hypothesis testingDecisionFail to reject H0 Reject H0 Truth H0 Correct decisionType I errorH1 Type II errorCorrect decisionChristophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201522 / 2252. Statistical Hypothesis testingDe nition (Size)The probability of a type I error is the (nominal)sizeof the test. This isconventionally denoted and is also called thesigni cance level. =Pr(WjH0)Christophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201523 / 2252.

9 Statistical Hypothesis testingRemarkFor a simple null Hypothesis : =Pr(WjH0)For a composite null Hypothesis : =sup 02H0Pr(WjH0)A test is said to have level if its size is less than or equal to .Christophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201524 / 2252. Statistical Hypothesis testingDe nition (Power)Thepowerof a test is the probability that it will correctly lead torejection of a false null Hypothesis :power=Pr(WjH1)=1 where denotes the probability of type II error, =Pr W H1 andW denotes the non-rejection Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201525 / 2252. Statistical Hypothesis testingExample (Test on the mean)Consider a sequenceX1,.., random variables withXi N m, 2 where 2is known. We want to testH0:m=m0H1:m=m1withm1< econometrician propose the following rule of decision:W=fx:xN<cgwhereXN=N 1 Ni=1 Xidenotes the sample mean andcis a constant(critical value).

10 Question:calculate the size and the power of this Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201526 / 2252. Statistical Hypothesis testingSolutionThe rejection region is W=fx:xN< the null H0:m=m0:XN H0N m0, 2N So, the size of the test is equal to: =Pr(WjH0)=Pr XN<c H0 =Pr XN m0 /pN<c m0 /pN H0 = c m0 /pN Christophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201527 / 2252. Statistical Hypothesis testingSolution (cont d)The rejection region is W=fx:xN< the alternativeH1:m=m1:XN H1N m1, 2N So, the power of the test is equal to:power=Pr(WjH1)=Pr XN m1 /pN<c m1 /pN H1 = c m1 /pN Christophe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201528 / 2252. Statistical Hypothesis testingSolution (cont d)In conclusion: = c m0 /pN =1 power=1 c m1 /pN We have a system of two equations with three parameters: , (or power)and the critical is a trade-o between the probabilities of the errors of type Iand II, and :ifcdecreases, decreases but solution is to impose a size and determine the critical value andthe Hurlin (University of Orl ans)Advanced Econometrics - Master ESAN ovember 20, 201529 / 2252.