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Hypothesis Testing

Hypothesis Testing. Hypothesis Testing. 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

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Hypothesis Testing

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  1. Hypothesis Testing

  2. 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.

  3. Hypothesis Testing • {y1,y2,…,yT} is a random sample providing information on the (K x 1) parameter vector, Θ where ΘΩ • 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, ΘΩ • Ω0 = {Θ| ΘΩ,R(Θ)=0} • Ω1 = {Θ| ΘΩ, R(Θ)≠0}

  4. Hypothesis Testing • Null Hypothesis: ΘΩ0 (H0) • Alternate Hypothesis: ΘΩ1 (H1) • Hypothesis Testing: • Divide sample space into two portions pertaining to H0 and H1 • The region where we reject H0 referred to as critical regionof the test

  5. 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

  6. 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

  7. Hypothesis Testing Definition of Rejection Region Reject H0 Do Not Reject H0 Reject H0 f(|H0) Prob. rejecting H0 even though true cvL cvU • P(cvL≤  ≤ cvU)=1-Pr(Type I Error)

  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 H0|H0 true} • Pr (Type II Error) = Pr{Accepting H0|H1 true} • Never know with certainty whether you are correct→pos. Pr(Type I Error) • Example of Standard Normal

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

  10. Hypothesis Testing • Example of mean testing • Assume RV is normally distributed: yt~N(b,s2) • H0: b = 1 H1: b ≠ 1 • What is distribution of mean under H0? • Assume s2=10, T=10 • →

  11. Hypothesis Testing β~N(1,1) if H0 True α = 0.05 0.025 0.025 • 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.

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

  13. Hypothesis Testing

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

  15. Hypothesis Testing

  16. Hypothesis Testing • Likelihood Ratio Test:

  17. 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

  18. 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?

  19. 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

  20. Hypothesis Testing • General Likelihood Ratio Test Procedure • Choose probability of Type I error, a (e.g., test sign. level) • Given a, find value of lC that satisfies: P(l >lC | H0 is true) • Evaluate test statistic based on sample information • Reject (fail to reject) null hypothesis if l >lC (<lC)

  21. 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 s2 (on class website)

  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)

  23. 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

  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)

  25. Asymptotic Tests Asymptotic Likelihood Ratio Test 0 l L(l) L() .5LR L(0) L≡ Log-Likelihood Function LR ≡ 2ln(l)=2[L(1)-L(0)] LR~c2J asymptotically (p.851 Greene) Evaluated L(•) at both 1 and 0 l generates unrestricted L(•) max L(0) value obtained under H0

  26. Asymptotic Tests Asymptotic Likelihood Ratio Test • 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 approximate χ2 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.

  27. Asymptotic Tests Impact of Curvature on LR Shows Need For Wald Test • 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 Information Matrix Don’t forget the “–” sign

  28. Asymptotic Tests Impact of Curvature on LR Shows Need For Wald Test 0 l  Max at same point L(l) .5LR0 L() L(0) .5LR1 L1(0) L1() Two samples H0: =0 W=(l-0)2 C(|=l) W=(l-0)2 I(|=l) W~c2J asymptotically Note: Evaluated atl L

  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 H0 not true (e.g., test statistic is larger) • Greene, p. 486-487 gives alternative motivation (careful of notation) • Buse, 153-154

  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 Rj(q) of the form: j=j0, j=1,…k • d()=Ik, • W=(l-0)2 I(|=l)

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

  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 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

  33. Asymptotic Tests Summary of Lagrange Multiplier (Score) Test S() ≡ dL/d 0  L(0) S()=0 S(0) Two samples LA LB L LM= S(0)2 I(0)-1 S(0)=dL/d|=0 LM~c2J asympt. I(0) = -d2L/d2|=0

  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)

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

  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 q, the 3 tests result in same numerical values for large samples

  37. Asymptotic Tests Summary • All test statistics distributed asym. c2 with J d.f. (number of joint hypotheses) • In finite samples W>LR>LM • This implies W more conservative • Example: With s2known, a test of parameter value (e.g., b = b0) results in: • One case where LR=W=LM in finite samples

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

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