1. 1 Hypothesis Testing

2. 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.

3. 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}

4. 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

5. 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

6. 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

7. 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

8. 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

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

10. 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 →→

11. 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

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

13. 13 Hypothesis Testing

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

15. 15 Hypothesis Testing

16. 16 Hypothesis Testing  Likelihood Ratio Test:

17. 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

18. 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?

19. 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

20. 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 )

21. 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)

22. 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. 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

24. 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. 25 Asymptotic Tests Asymptotic Likelihood Ratio Test  ll LL LlLl L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-Likelihood Function  l generates unrestricted L() max L(  0 ) value obtained under H 0

26. 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

27. 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

28. 28 Asymptotic Tests Impact of Curvature on LR Shows Need For Wald Test  ll LL LlLl LL.5LR 0 LL.5LR 1 L1L1 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

29. 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

30. 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 )

31. 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

32. 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

33. 33 Asymptotic Tests Summary of Lagrange Multiplier (Score) Test  LL 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

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

36. 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

37. 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

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