faculty.arts.ubc.ca · 2011. 2. 9. · Null and alternative hypotheses I Usually, we have two...

Economics 326Methods of Empirical Research in Economics

Lecture 8: Hypothesis testing (Part 1)

Vadim MarmerUniversity of British Columbia

February 9, 2011

Hypothesis testing

I Hypothesis testing is one of a fundamental problems instatistics.

I A hypothesis is (usually) an assertion about the unknownpopulation parameters such as β1 in Yi = β0 + β1Xi + Ui .

I Using the data, the econometrician has to determine whetheran assertion is true or false.

I Example: Phillips curve:

Unemploymentt = β0 + β1In�ationt + Ut .

In this example, we are interested in testing if β1 = 0 (noPhillips curve) against β1 < 0 (Phillips curve).

Hypothesis testing

Assumptions

I β1 is unknown, and we have to rely on its OLS estimator β̂1.

I We need to know the distribution of β̂1 or of its certainfunctions.

I We will assume that the assumptions of the Normal ClassicalLinear Regression model are satis�ed:

1. Yi = β0 + β1Xi + Ui , i = 1, . . . , n.2. E (Ui jX1, . . . ,Xn) = 0 for all i�s.3. E

�U2i jX1, . . . ,Xn

�= σ2 for all i�s.

4. E�UiUj jX1, . . . ,Xn

�= 0 for all i 6= j .

5. U�s are jointly normally distributed conditional on X�s.

I Recall that, in this case, conditionally on X�s:

β̂1 � N�

β1,Var�

β̂1��, where Var

�β̂1�=

∑ni=1 (Xi � X̄ )

Assumptions

�U2i jX1, . . . ,Xn

4. E�UiUj jX1, . . . ,Xn

�= 0 for all i 6= j .

β̂1 � N�

β1,Var�

�β̂1�=

∑ni=1 (Xi � X̄ )

Assumptions

�U2i jX1, . . . ,Xn

4. E�UiUj jX1, . . . ,Xn

�= 0 for all i 6= j .

β̂1 � N�

β1,Var�

�β̂1�=

∑ni=1 (Xi � X̄ )

Assumptions

�U2i jX1, . . . ,Xn

4. E�UiUj jX1, . . . ,Xn

�= 0 for all i 6= j .

β̂1 � N�

β1,Var�

�β̂1�=

∑ni=1 (Xi � X̄ )

Assumptions

�U2i jX1, . . . ,Xn

4. E�UiUj jX1, . . . ,Xn

�= 0 for all i 6= j .

β̂1 � N�

β1,Var�

�β̂1�=

∑ni=1 (Xi � X̄ )

Null and alternative hypotheses

I Usually, we have two competing hypotheses, and we want todraw a conclusion, based on the data, as to which of thehypotheses is true.

I Null hypothesis , denoted as H0: A hypothesis that is held tobe true, unless the data provides a su¢ cient evidence againstit.

I Alternative hypothesis, denoted as H1: A hypothesis againstwhich the null is tested. It is held to be true if the null isfound false.

I Usually, the econometrician has to carry the "burden ofproof," and the case that he is interested in is stated as H1.

I The econometrician has to prove that his assertion (H1) istrue by showing that the data rejects H0.

I The two hypotheses must be disjoint: it should be the casethat either H0 is true or H1 but never together simultaneously.

I Null hypothesis , denoted as H0:

A hypothesis that is held tobe true, unless the data provides a su¢ cient evidence againstit.

I Alternative hypothesis, denoted as H1:

A hypothesis againstwhich the null is tested. It is held to be true if the null isfound false.

I Alternative hypothesis, denoted as H1: A hypothesis againstwhich the null is tested.

It is held to be true if the null isfound false.

I Usually, the econometrician has to carry the "burden ofproof,"

and the case that he is interested in is stated as H1.I The econometrician has to prove that his assertion (H1) istrue by showing that the data rejects H0.

I The two hypotheses must be disjoint:

it should be the casethat either H0 is true or H1 but never together simultaneously.

Decision rule

I The econometrician has to choose between H0 and H1.

I The decision rule that leads the econometrician to reject ornot to reject H0 is based on a test statistic, which is afunction of the data f(Yi ,Xi ) : i = 1, . . . ng .

I Usually, one rejects H0 if the test statistic falls into a criticalregion. A critical region is constructed by taking into accountthe probability of making a wrong decision.

Decision rule

I The econometrician has to choose between H0 and H1.I The decision rule that leads the econometrician to reject ornot to reject H0 is based on a test statistic, which is afunction of the data f(Yi ,Xi ) : i = 1, . . . ng .

Decision rule

I The econometrician has to choose between H0 and H1.I The decision rule that leads the econometrician to reject ornot to reject H0 is based on a test statistic, which is afunction of the data f(Yi ,Xi ) : i = 1, . . . ng .

Errors

I There are two types of errors that the econometrician canmake:

Decision

H0 H1H0 X Type II errorH1 Type I error X

I Type I error is the error of rejecting H0 when H0 is true.I The probability of Type I error is denoted by α and calledsigni�cance level or size of a test :

P (Type I error) = P (reject H0jH0 is true) = α.

I Type II error is the error of not rejecting H0 when H1 is true.I Power of a test:

1� P (Type II error) = 1� P (Accept H0jH0 is false) .

Errors

Decision

Errors

Decision

I Type I error is the error of rejecting H0 when H0 is true.

I The probability of Type I error is denoted by α and calledsigni�cance level or size of a test :

Errors

Decision

I Type I error is the error of rejecting H0 when H0 is true.I The probability of Type I error is denoted by α and calledsigni�cance level or size of a test

Errors

Decision

I Type II error is the error of not rejecting H0 when H1 is true.

I Power of a test:

Errors

Decision

1� P (Type II error) = 1� P (Accept H0jH0 is false) .5/14

Errors

I The decision rule depends on a test statistic T .

I The real line is split into two regions: acceptance region andrejection region (critical region).

I When T is in the acceptance region, we accept H0 (and riskmaking a Type II error).

I When T is in the rejection (critical) region, we reject H0 (andrisk making a Type I error).

I Unfortunately, the probabilities of Type I and II errors areinversely related. By decreasing the probability of Type I errorα, one makes the critical region smaller, which increases theprobability of the Type II error. Thus it is impossible to makeboth errors arbitrary small.

I By convention, α is chosen to be a small number, for example,α = 0.01, 0.05, or 0.10. (This is in agreement with theeconometrician carrying the burden of proof).

Errors

I The decision rule depends on a test statistic T .I The real line is split into two regions: acceptance region andrejection region (critical region).

Errors

I Unfortunately, the probabilities of Type I and II errors areinversely related.

By decreasing the probability of Type I errorα, one makes the critical region smaller, which increases theprobability of the Type II error. Thus it is impossible to makeboth errors arbitrary small.

Errors

I By convention, α is chosen to be a small number, for example,α = 0.01, 0.05, or 0.10.

(This is in agreement with theeconometrician carrying the burden of proof).

Errors

I The following are the steps of the hypothesis testing:

1. Specify H0 and H1.2. Choose the signi�cance level α.3. De�ne a decision rule (critical region).4. Perform the test using the data: given the data compute thetest statistic and see if it falls into the critical region.

I The decision depends on the signi�cance level α: largervalues of α correspond to bigger critical regions (probability ofType I error is larger).

I It is easier to reject the null for larger values of α.I p-value : Given the data, the smallest signi�cance level atwhich the null can be rejected.

1. Specify H0 and H1.

2. Choose the signi�cance level α.3. De�ne a decision rule (critical region).4. Perform the test using the data: given the data compute thetest statistic and see if it falls into the critical region.

1. Specify H0 and H1.2. Choose the signi�cance level α.

3. De�ne a decision rule (critical region).4. Perform the test using the data: given the data compute thetest statistic and see if it falls into the critical region.

1. Specify H0 and H1.2. Choose the signi�cance level α.3. De�ne a decision rule (critical region).

4. Perform the test using the data: given the data compute thetest statistic and see if it falls into the critical region.

1. Specify H0 and H1.2. Choose the signi�cance level α.3. De�ne a decision rule (critical region).4. Perform the test using the data

: given the data compute thetest statistic and see if it falls into the critical region.

I The decision depends on the signi�cance level α:

largervalues of α correspond to bigger critical regions (probability ofType I error is larger).

I It is easier to reject the null for larger values of α.

I p-value : Given the data, the smallest signi�cance level atwhich the null can be rejected.

Two-sided tests

I For Yi = β0 + β1Xi + Ui , consider testing

H0 : β1 = β1,0,

againstH1 : β1 6= β1,0.

I β1 is the true unknown value of the slope parameter.I β1,0 is a known number speci�ed by the econometrician. (Forexample β1,0 is zero if you want to test β1 = 0).

I Such a test is called two-sided because the alternativehypothesis H1 does not specify in which direction β1 candeviate from the asserted value β1,0.

Two-sided tests

H0 : β1 = β1,0,

Two-sided tests

H0 : β1 = β1,0,

I β1 is the true unknown value of the slope parameter.

I β1,0 is a known number speci�ed by the econometrician. (Forexample β1,0 is zero if you want to test β1 = 0).

Two-sided tests

H0 : β1 = β1,0,

I β1 is the true unknown value of the slope parameter.I β1,0 is a known number speci�ed by the econometrician.

(Forexample β1,0 is zero if you want to test β1 = 0).

Two-sided tests

H0 : β1 = β1,0,

Two-sided tests

H0 : β1 = β1,0,

Two-sided tests

H0 : β1 = β1,0,

Two-sided tests when σ2 is known (infeasible test)

I Suppose for a moment that σ2 is known.

I Consider the following test statistic:

T =β̂1 � β1,0qVar

�β̂1� , where Var �β̂1� = σ2

∑ni=1 (Xi � X̄ )

I Consider the following decision rule (test):

Reject H0 : β1 = β1,0 when jT j > z1�α/2,

where z1�α/2 is the (1� α/2) quantile of the standard normaldistribution (critical value).

I Suppose for a moment that σ2 is known.I Consider the following test statistic:

�β̂1�

, where Var�

β̂1�=

∑ni=1 (Xi � X̄ )

�β̂1� , where Var �β̂1� = σ2

∑ni=1 (Xi � X̄ )

�β̂1� , where Var �β̂1� = σ2

∑ni=1 (Xi � X̄ )

Reject H0 : β1 = β1,0

when jT j > z1�α/2,

�β̂1� , where Var �β̂1� = σ2

∑ni=1 (Xi � X̄ )

Test validity and power

I We need to establish that:

1. The test is valid, where the validity of a test means that it hascorrect size or P (Type I error) = α:

P�jT j > z1�α/2 jβ1 = β1,0

�= α.

2. The test has power: when β1 6= β1,0 (H0 is false), the testrejects H0 with probability that exceeds α:

P�jT j > z1�α/2 jβ1 6= β1,0

�> α.

I We want P�jT j > z1�α/2jβ1 6= β1,0

�to be as large as

possible.I Note that P

�jT j > z1�α/2jβ1 6= β1,0

�depends on the true

value β1.

1. The test is valid, where the validity of a test means that it hascorrect size

or P (Type I error) = α:

P�jT j > z1�α/2 jβ1 = β1,0

�= α.

P�jT j > z1�α/2 jβ1 6= β1,0

�> α.

�jT j > z1�α/2jβ1 6= β1,0

value β1.

1. The test is valid, where the validity of a test means that it hascorrect size or P (Type I error) = α

P�jT j > z1�α/2 jβ1 = β1,0

�= α.

P�jT j > z1�α/2 jβ1 6= β1,0

�> α.

�jT j > z1�α/2jβ1 6= β1,0

value β1.

P�jT j > z1�α/2 jβ1 = β1,0

�= α.

2. The test has power:

when β1 6= β1,0 (H0 is false), the testrejects H0 with probability that exceeds α:

P�jT j > z1�α/2 jβ1 6= β1,0

�> α.

�jT j > z1�α/2jβ1 6= β1,0

value β1.

P�jT j > z1�α/2 jβ1 = β1,0

�= α.

P�jT j > z1�α/2 jβ1 6= β1,0

�> α.

�jT j > z1�α/2jβ1 6= β1,0

value β1.

P�jT j > z1�α/2 jβ1 = β1,0

�= α.

P�jT j > z1�α/2 jβ1 6= β1,0

�> α.

possible.

I Note that P�jT j > z1�α/2jβ1 6= β1,0

value β1.

P�jT j > z1�α/2 jβ1 = β1,0

�= α.

P�jT j > z1�α/2 jβ1 6= β1,0

�> α.

�jT j > z1�α/2jβ1 6= β1,0

value β1.

The distribution of T when σ2 is known (infeasible test)

I Write

�β̂1�

=β̂1 � β1 + β1 � β1,0q

Var�

β̂1�

=β̂1 � β1qVar

�β̂1� + β1 � β1,0q

Var�

β̂1� .

I Under our assumptions and conditionally on X�s:

β̂1 � N�

β1,Var�

β̂1��, or

β̂1 � β1qVar

�β̂1� � N (0, 1) .

I We have that conditionally on X�s: T � N

β1�β1,0qVar(β̂1)

I Write

�β̂1� = β̂1 � β1 + β1 � β1,0q

Var�

β̂1�

=β̂1 � β1qVar

�β̂1� + β1 � β1,0q

Var�

β̂1� .

β̂1 � N�

β1,Var�

β̂1��, or

β̂1 � β1qVar

�β̂1� � N (0, 1) .

I Write

�β̂1� = β̂1 � β1 + β1 � β1,0q

Var�

β̂1�

=β̂1 � β1qVar

�β̂1� + β1 � β1,0q

Var�

β̂1� .

β̂1 � N�

β1,Var�

β̂1��, or

β̂1 � β1qVar

�β̂1� � N (0, 1) .

I Write

�β̂1� = β̂1 � β1 + β1 � β1,0q

Var�

β̂1�

=β̂1 � β1qVar

�β̂1� + β1 � β1,0q

Var�

β̂1� .

β̂1 � N�

β1,Var�

β̂1��,

orβ̂1 � β1qVar

�β̂1� � N (0, 1) .

I Write

�β̂1� = β̂1 � β1 + β1 � β1,0q

Var�

β̂1�

=β̂1 � β1qVar

�β̂1� + β1 � β1,0q

Var�

β̂1� .

β̂1 � N�

β1,Var�

β̂1��, or

β̂1 � β1qVar

�β̂1� � N (0, 1) .

I Write

�β̂1� = β̂1 � β1 + β1 � β1,0q

Var�

β̂1�

=β̂1 � β1qVar

�β̂1� + β1 � β1,0q

Var�

β̂1� .

β̂1 � N�

β1,Var�

β̂1��, or

β̂1 � β1qVar

�β̂1� � N (0, 1) .

I We have that conditionally on X�s:

T � N

I Write

�β̂1� = β̂1 � β1 + β1 � β1,0q

Var�

β̂1�

=β̂1 � β1qVar

�β̂1� + β1 � β1,0q

Var�

β̂1� .

β̂1 � N�

β1,Var�

β̂1��, or

β̂1 � β1qVar

�β̂1� � N (0, 1) .

I We have that conditionally on X�s: T �

I Write

�β̂1� = β̂1 � β1 + β1 � β1,0q

Var�

β̂1�

=β̂1 � β1qVar

�β̂1� + β1 � β1,0q

Var�

β̂1� .

β̂1 � N�

β1,Var�

β̂1��, or

β̂1 � β1qVar

�β̂1� � N (0, 1) .

Size when σ2 is known (infeasible test)

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I When H0 : β1 = β1,0 is true, TH0� N (0, 1).

I We reject H0 when

jT j > z1�α/2 , T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

P (Reject H0jH0 is true) = P (Z > z1�α/2) + P (Z < �z1�α/2)

= α/2+ α/2 = α

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I When H0 : β1 = β1,0 is true

, TH0� N (0, 1).

I We reject H0 when

jT j > z1�α/2 , T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

= α/2+ α/2 = α

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I We reject H0 when

jT j > z1�α/2 , T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

= α/2+ α/2 = α

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I We reject H0 when

jT j > z1�α/2

, T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

= α/2+ α/2 = α

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I We reject H0 when

jT j > z1�α/2 , T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

= α/2+ α/2 = α

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I We reject H0 when

jT j > z1�α/2 , T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

= α/2+ α/2 = α

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I We reject H0 when

jT j > z1�α/2 , T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

P (Reject H0jH0 is true)

= P (Z > z1�α/2) + P (Z < �z1�α/2)

= α/2+ α/2 = α

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I We reject H0 when

jT j > z1�α/2 , T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

= α/2+ α/2 = α

I We have that

T � N

0@ β1 � β1,0qVar

�β̂1� , 11A .

I We reject H0 when

jT j > z1�α/2 , T > z1�α/2 or T < �z1�α/2.

I Let Z � N (0, 1) .

= α/2+ α/2 = α

0CriticalRegion

CriticalRegion

Power of the test when σ2 is known (infeasible test)

I Under H1, β1 � β1,0 6= 0 and the distribution of T is not

centered zero: T � N

I When β1 � β1,0 > 0 :

0CriticalRegion

CriticalRegion

I Rejection probability exceeds α under H1: power increaseswith the distance from H0 (

��β1,0 � β1��) and decreases with

Var�

β̂1�.

Power of the test when σ2 is known (infeasible test)

I Under H1, β1 � β1,0 6= 0 and the distribution of T is not

centered zero: T � N

I When β1 � β1,0 > 0 :

0CriticalRegion

CriticalRegion

I Rejection probability exceeds α under H1: power increaseswith the distance from H0 (

��β1,0 � β1��) and decreases with

Var�

β̂1�.

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