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1 Hypothesis Testing

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2 Greene: App. C:892-897 Statistical Test: Divide parameter space (Ω) into two disjoint sets: Ω 0, Ω 1 Ω 0 ∩ Ω 1 = and Ω 0 Ω 1 =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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3 Hypothesis Testing {y 1,y 2,…,y T } is a random sample providing information on the (K x 1) parameter vector, Θ where Θ Ω R(Θ)=[R 1 (Θ), R 2 (Θ),…R J (Θ)] is a (J x 1) vector of restrictions (e.g., hypotheses) on K parameters, Θ. For this class: R(Θ)=0, Θ Ω Ω 0 = {Θ| Θ Ω, R(Θ)=0} Ω 1 = {Θ| Θ Ω, R(Θ)≠0}

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4 Hypothesis Testing Null Hypothesis: Θ Ω 0 (H 0 ) Alternate Hypothesis: Θ Ω 1 (H 1 ) Hypothesis Testing: Divide sample space into two portions pertaining to H 0 and H 1 The region where we reject H 0 referred to as critical region of the test

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5 Hypothesis Testing Test of whether * 0 or 1 ( * an est. of ) based on a test statistic w/known dist. under H 0 and some other dist. if H 1 true Transform * into test statistic Critical region of hyp. test is the set of values for which H 0 would be rejected (e.g., values of test statistic unlikely to occur if H 0 is true) If test statistic falls into the critical region→evidence that H 0 not true

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6 Hypothesis Testing General Test Procedure Develop a null hypothesis (H o ) that will be maintained until evidence to the contrary Develop an alternate hypothesis (H 1 ) that will be adopted if H 0 not accepted Estimate appropriate test statistic Identify desired critical region Compare calculated test statistic to critical region Reject H 0 if test statistic in critical region

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7 Hypothesis Testing Definition of Rejection Region P(cv L ≤ ≤ cv U )=1-Pr(Type I Error) cv L cv U Do Not Reject H 0 Reject H 0 f( |H 0 ) Prob. rejecting H 0 even though true

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8 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 Error Pr (Type I Error) = Pr{Rejecting H 0 |H 0 true} Pr (Type II Error) = Pr{Accepting H 0 |H 1 true} Never know with certainty whether you are correct→pos. Pr(Type I Error) Example of Standard Normal

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9 Hypothesis Testing Standard Normal Distribution P(-1.96 ≤ z ≤ 1.96)=0.95 α = 0.05 = P(Type I Error) 0.025

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10 Hypothesis Testing Example of mean testing Assume RV is normally distributed: y t ~N( , 2 ) H 0 : = 1 H 1 : ≠ What is distribution of mean under H 0 ? Assume 2 =10, T=10 →→

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11 Hypothesis Testing β~N(1,1) if H 0 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 = 0.05 0.025

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12 Hypothesis Testing Standard Normal Distribution P(-1.96 ≤ z ≤ 1.96)=0.95 α = 0.05 = P(Type I Error) 0.025

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13 Hypothesis Testing

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14 Hypothesis Testing Again, this assumes we know σ P(-t (T-1),α/2 ≤ t ≤ t (T-1),α/2) =1-α

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15 Hypothesis Testing

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16 Hypothesis Testing Likelihood Ratio Test:

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17 Hypothesis Testing Likelihood Ratio Test: Compare value of likelihood function, l(), under the null hypothesis, l(Ω 0 )] 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 H 0

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18 Hypothesis Testing Is this difference in likelihood function values large? Likelihood ratio (λ): λ is a random variable since it depends on y i ’s What are possible values of λ?

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19 Hypothesis Testing Likelihood Ratio Principle Null hypo. defining Ω 0 is rejected if λ > 1 (Why 1?) Need to establish critical level of λ, λ C that is unlikely to occur under H 0 (e.g., is 1.1 far enough away from 1.0)? Reject H 0 if estimated value of λ is greater than λ C λ = 1→Null hypo. does not sign. reduce parameter space H 0 not rejected Result conditional on sample

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20 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 | H 0 is true) Evaluate test statistic based on sample information Reject (fail to reject) null hypothesis if > C (< C )

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21 Hypothesis Testing LR test of mean of Normal Distribution (µ) with not known not known This implies the following test procedures: procedures F-Test t-Test LR test of hypothesized value of 2 (on class website)

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22 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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23 Asymptotic Tests Asymptotic Likelihood Ratio Test y 1,…,y t are iid, E(y t )=β, var(y t )=σ (β*-β)T 1/2 converge in dist to N(0,σ ) As T→∞, use normal pdf to generate LF λ ≡ l * (Ω)/l(Ω 0 ) or l( l )/l( 0 ) l*(Ω) = Max [l( |y 1,…,y T ): Ω] l(Ω 0 ) = Max [l( |y 1,…,y T ): Ω 0 ] Restricted LF given H 0

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24 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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25 Asymptotic Tests Asymptotic Likelihood Ratio Test ll LL LlLl LL.5LR LR ≡ 2ln( )=2[L( 1 )-L( 0 )] LR~ 2 J asymptotically (p.851 Greene) Evaluated L( ) at both 1 and 0 L≡ Log-Likelihood Function l generates unrestricted L() max L( 0 ) value obtained under H 0

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26 Greene defines as: -2[L( 0 )-L( 1 )] Result is the same Buse, p.153, Greene p.484-486 Given H 0 true, LR has an approximate χ 2 dist. with J DF (the number of joint hypotheses) Reject H 0 when 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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27 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 Don’t forget the “–” sign Asymptotic Tests Impact of Curvature on LR Shows Need For Wald Test Information Matrix

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28 Asymptotic Tests Impact of Curvature on LR Shows Need For Wald Test ll LL LlLl LL.5LR 0 LL.5LR 1 L1L1 H 0 : 0 W=( l - 0 ) 2 C( | = l ) W=( l - 0 ) 2 I( | = l ) W~ 2 J asymptotically Note: Evaluated at l Max at same point Two samples L

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29 Asymptotic Tests Impact of Curvature on LR Shows 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 H 0 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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30 Asymptotic Tests Impact of Curvature on LR Shows Need For Wald Test Extending this to J simultaneous hypotheses and k parameters Note that R( ∙ ), d( ∙ ) and I( ∙ ) evaluated at l When R j ( ) of the form: j = j0, j=1,…k d( )=I k, W=( l - 0 ) 2 I( | = l )

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31 Asymptotic Tests Based on the curvature of the log- likelihood function (L) At unrestricted max: Summary of Lagrange Multiplier (Score) Test Score of Likelihood Function

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32 Asymptotic Tests Summary 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 0 will 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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33 Asymptotic Tests Summary of Lagrange Multiplier (Score) Test LL LM~ 2 J asympt. S( 0 ) LBLB LALA S( 0 )=dL/d | = 0 LM= S( 0 ) 2 I( 0 ) -1 I( ) = -d 2 L/d 2 | = 0 S( )=0 S( ) ≡ dL/d Two samples L

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34 Asymptotic Tests Summary 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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35 Asymptotic Tests Summary of Lagrange Multiplier (Score) Test Extending this to multiple parameters Buse, pp. 154-155 Greene, pp.489-490

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36 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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37 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: One case where LR=W=LM in finite samples

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38 Asymptotic Tests Summary Example of asymptotic testsasymptotic tests Buse (pp.155-156) same example but assumes =1

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