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Hypothesis Testing

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Hypothesis Testing

Greene: App. C:892-897

Statistical Test: Divide parameter space (Ω) into two disjoint sets: Ω0, Ω1

Ω0 ∩ Ω1= and Ω0 Ω1=Ω

Based on sample evidence does estimated parameter (*) and therefore the true parameter fall into one of these sets? We answer this question using a statistical test.

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Hypothesis Testing

y1,y2,…,yT is a random sample providing information on the (K x 1) parameter vector, Θ where ΘΩ

R(Θ)=[R1(Θ), R2(Θ),…RJ(Θ)] is a (J x 1) vector of restrictions (e.g., hypotheses) on K parameters, Θ.

For this class: R(Θ)=0, ΘΩ Ω0 = Θ| ΘΩ, R(Θ)=0 Ω1 = Θ| ΘΩ, R(Θ)≠0

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Hypothesis Testing

Null Hypothesis: ΘΩ0 (H0)

Alternate Hypothesis: ΘΩ1 (H1)

Hypothesis Testing: Divide sample space into two

portions pertaining to H0 and H1

The region where we reject H0 referred to as critical region

of the test

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Hypothesis Testing

Test of whether * 0 or 1 (* an est. of ) based on a test statistic w/known dist. under H0 and some other dist. if H1 true Transform * into test statistic Critical region of hyp. test is the set of values for which H0

would be rejected (e.g., values of test statistic unlikely to occur if H0 is true) If test statistic falls into the critical region→evidence that H0 not true

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Hypothesis Testing

General Test Procedure Develop a null hypothesis (Ho)

that will be maintained until evidence to the contrary

Develop an alternate hypothesis (H1) that will be adopted if H0 not accepted

Estimate appropriate test statistic

Identify desired critical region Compare calculated test

statistic to critical region Reject H0 if test statistic in

critical region

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Hypothesis Testing

Definition of Rejection Region

P(cvL ≤ ≤ cvU)=1-Pr(Type I Error)

cvL cvU

Do Not Reject H0Reject H0 Reject H0

f(|H0)

Prob. rejecting H0 even though true

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Hypothesis Testing

Defining the Critical Region Select a region that identifies

parameter values that are unlikely to occur if the null hypothesis is true

Value of Type I ErrorPr (Type I Error) = PrRejecting H0|H0 truePr (Type II Error) = PrAccepting H0|H1 true

Never know with certainty whether you are correct→pos. Pr(Type I Error)

Example of Standard Normal

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Hypothesis Testing

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Standard Normal Distribution

P(-1.96 ≤ z ≤ 1.96)=0.95

α = 0.05= P(Type I Error)0.025 0.025

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Hypothesis Testing

Example of mean testing Assume RV is normally

distributed: yt~N(,2) H0: = 1 H1: ≠ What is distribution of mean

under H0?2σ

β~N 1,T

æ ö÷ç ÷ç ÷ç ÷çè ø

Assume 2=10, T=10→ ( ) 0β~N 1,1 if H true

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Hypothesis Testing

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

β~N(1,1) if H0 True

P(-0.96 ≤ β ≤ 2.96)=0.95 P(-1.96 ≤ z ≤ 1.96)=0.95 (e.g, transform dist. of β into RV with std. normal dist.

α = 0.050.025 0.025

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Hypothesis Testing

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Standard Normal Distribution

P(-1.96 ≤ z ≤ 1.96)=0.95

α = 0.05= P(Type I Error)

0.025 0.025

0 0

ˆ~ (0,1)

/if normally dist. and H true (e.g., = )

oz NT

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Hypothesis Testing

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Hypothesis Testing

Again, this assumes we know σ

P(-t(T-1),α/2 ≤ t ≤ t(T-1),α/2)=1-α

1

0 0

ˆ~

ˆ /if normally dist. and H true (e.g., = )

oTt t

T

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Hypothesis Testing

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Hypothesis Testing

Likelihood Ratio Test:

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Hypothesis Testing

Likelihood Ratio Test: Compare value of likelihood function, l(•), under the null hypothesis, l(Ω0)] vs. value with unrestricted parameter choice [l*(Ω)] Null hyp. could reduce set of

parameter values. What does this do to the max.

likelihood function value? If the two resulting max. LF

values are close enough→can not reject H0

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Hypothesis Testing

Is this difference in likelihood function values large?

Likelihood ratio (λ):

λ is a random variable since it depends on yi’s

What are possible values of λ?

0 0

LF* Ω *

LF Ω

l

l

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Hypothesis Testing

Likelihood Ratio Principle Null hypo. defining Ω0 is

rejected if λ > 1 (Why 1?) Need to establish critical level

of λ, λC that is unlikely to occur under H0 (e.g., is 1.1 far enough away from 1.0)? Reject H0 if estimated value of

λ is greater than λC λ = 1→Null hypo. does not

sign. reduce parameter space

H0 not rejected Result conditional on sample

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Hypothesis Testing

General Likelihood Ratio Test Procedure

Choose probability of Type I error, (e.g., test sign. level)

Given , find value of C that

satisfies: P(> C | H0 is

true) Evaluate test statistic based on

sample information

Reject (fail to reject) null hypothesis if > C (<C)

0

*l

l

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Hypothesis Testing

LR test of mean of Normal Distribution (µ) with s2not known

This implies the following test procedures:

F-Test t-Test

LR test of hypothesized value of 2 (on class website)

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Asymptotic Tests

Previous tests based on finite samples Use asymptotic tests when

appropriate finite sample test statistic is unavailable

Three tests commonly used: Asymptotic Likelihood Ratio Wald Test Lagrangian Multiplier (Score)

Test Greene p.484-492 Buse article (on website)

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Asymptotic Tests

Asymptotic Likelihood Ratio Test y1,…,yt are iid, E(yt)=β, var(yt)=σ

(β*-β)T1/2 converge in dist to N(0,σ) As T→∞, use normal pdf to generate LF

λ ≡ l*(Ω)/l(Ω0) or l(l)/l(0) l*(Ω) = Max[l(|y1,…,yT):Ω] l(Ω0) = Max[l(|y1,…,yT):Ω0]

Restricted LF given H0

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Asymptotic Tests

Asymptotic Likelihood Ratio (LR) LR ≡ 2ln(λ) = 2[L*()-L(0)] L() = lnl() LR~χ

J asymptotically where J is the number of joint null hypothesis (restrictions)

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Asymptotic Tests

Asymptotic Likelihood Ratio Test

l

L

Ll L

.5LR

LR ≡ 2ln()=2[L(1)-L(0)]LR~2

J asymptotically (p.851 Greene)Evaluated L(•) at both 1 and 0

L≡ Log-LikelihoodFunction

l generates unrestricted L(•) max L(0) value obtained under H0

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Greene defines as: -2[L(0)-L(1)] Result is the same Buse, p.153, Greene p.484-486

Given H0 true, LR has an approximateχ2 dist. with J DF (the number of joint hypotheses) Reject H0when LR > χ

c where χ

c is the predefined critical value of the dist. given J DF.

Asymptotic Tests

Asymptotic Likelihood Ratio Test

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Suppose consists of 1 element Have 2 samples generating different estimates of the LF with same value of that max. the LF

0.5LR will depend on Distance between l and 0(+) The curvature of the LF (+) C() represents LF curvature

2

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l

d LC

dL

Don’t forget the “–” sign

Asymptotic TestsImpact of Curvature on LRShows Need For Wald Test

InformationMatrix

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Asymptotic TestsImpact of Curvature on LRShows Need For Wald Test

l

L

Ll

L.5LR0

L.5LR1

L1

H0: 0 W=(l-0)2 C(|=l)W=(l-0)2 I(|=l) W~2

J asymptoticallyNote: Evaluated at l

Max at same point

Two samples

L

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Asymptotic TestsImpact of Curvature on LRShows Need For Wald Test

The above weights the squared distance, (l - 0)2 by the curvature of the LF instead of using the differences as in LR test

Two sets of data may produce the same (l - 0)2 value but give diff. LR values because

of curvature The more curvature, the more

likely H0 not true (e.g., test statistic is larger)

Greene, p. 486-487 gives alternative motivation (careful of notation)

Buse, 153-154

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Asymptotic TestsImpact of Curvature on LRShows Need For Wald Test

Extending this to J simultaneous hypotheses and k parameters

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

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l lll l

2l J

l

d RIR dWJ x k J x1k x k1x J k x J

Rd ; W ~ assymp.

|

--é ù¢¢ Q QQQ Qê ú= ê ú

ê úë û¶ Q

Q = c¶Q Q=Q

Note that R(∙), d(∙) and I(∙) evaluated at l

When Rj() of the form: j=j0,j=1,…k

d()=Ik, W=(l-0)2 I(|=l)

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Asymptotic Tests

Based on the curvature of the log-likelihood function (L) At unrestricted max:

Summary of Lagrange Multiplier (Score) Test

log0

d LS

d

Score of LikelihoodFunction

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Asymptotic TestsSummary of Lagrange Multiplier (Score) Test

How much does S() depart from 0 when evaluated at the hypothesized value?

Weight squared slope by curvature The greater the curvature, the closer 0will be to the max. value Weight by C()-1→smaller test statistic the more curvature Small values of test statistic, LM, will be generated if the value of L(0) is close to the max. value, L(l), e.g. slope closer to 0

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Asymptotic TestsSummary of Lagrange Multiplier (Score) Test

L

LM~2J asympt.

S(0)

LB

LA

S(0)=dL/d|=0LM= S(0)2 I(0)-1

I() = -d2L/d2|=0

S()=0

S() ≡ dL/d

Two samples

L

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Asymptotic TestsSummary of Lagrange Multiplier (Score) Test

Small values of test statistic, LM, should be generated when

L(∙) has greater curvature when evaluated at 0 The test statistic is smaller when 0 nearer the value that

generates maximum LF value (e.g. S(0) is closer to zero)

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Asymptotic TestsSummary of Lagrange Multiplier (Score) Test

Extending this to multiple parameters

( )( )

( )( )

( )( )

1000

2J

SISLMk x1k x k1x k

LM ~

-¢ qqq=

c

Buse, pp. 154-155

Greene, pp.489-490

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Asymptotic Tests Summary

LR, W, LM differ in type of information required

LR requires both restricted and unrestricted parameter estimates W requires only unrestricted

estimates LM requires only restricted

estimates If log-likelihood quadratic with respect to the 3 tests result in same numerical values for large samples

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Asymptotic Tests Summary

All test statistics distributed asym. 2 with J d.f. (number of joint

hypotheses) In finite samples W > LR > LM This implies W more conservative Example: With 2 known, a test of

parameter value (e.g., 0) results in:

( )202

ˆLR W LM

T

b- b= = = æ ös ÷ç ÷÷çè ø

One case where LR=W=LM in finite samples

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Asymptotic Tests Summary

Example of asymptotic tests

Buse (pp.155-156) same example but assumes =1