# Econ 424/Amath 462 Hypothesis Testing in the CER Model

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1 Econ 424/Amath 462 Hypothesis Testing in the CER Model Eric Zivot July 23, 2013

2 Hypothesis Testing 1. Specify hypothesis to be tested 0 : null hypothesis versus. 1 : alternative hypothesis 2. Specify significance level of test level =Pr(Reject 0 0 is true) 3. Construct test statistic, from observed data 4. Use test statistic to evaluate data evidence regarding 0 is big evidence against 0 is small evidence in favor of 0

3 Decide to reject 0 at specified significance level if value of falls in the rejection region rejection region reject 0 Usually the rejection region of is determined by a critical value, such that reject 0 do not reject 0

4 Decision Making and Hypothesis Tests Reality Decision 0 is true 0 is false Reject 0 Type I error No error Do not reject 0 No error Type II error Significance Level of Test level =Pr(Type I error) Pr(Reject 0 0 is true) Goal: Constuct test to have a specified small significance level level =5%or level =1%

5 Power of Test 1 Pr(Type II error) =Pr(Reject 0 0 is false) Goal: Construct test to have high power Problem: Impossible to simultaneously have level 0 and power 1 As level 0 power also 0

6 Hypothesis Testing in CER Model = + =1 ; =1 iid (0 2 ) cov( )= cor( )= cov( )=0 6= for all

7 Test for specific value 0 : = 0 vs. 1 : 6= 0 0 : = 0 vs. 1 : 6= 0 0 : = 0 vs. 1 : 6= 0 Test for sign 0 : =0vs. 1 : 0 or 0 0 : =0vs. 1 : 0 or 0 Test for normal distribution 0 : iid ( 2 ) 1 : not normal

8 Test for no autocorrelation 0 : =corr( )=0, 1 1 : =corr( ) 6= 0for some Test of constant parameters 0 : and areconstantoverentiresample 1 : or changes in some sub-sample

9 Definition: Chi-square random variable and distribution Let 1 be iid (0 1) random variables. Define = Then 2 ( ) = degrees of freedom (d.f.) Properties of 2 ( ) distribution 0 [ ] = 2 ( ) normal as

10 R functions rchisq(): simulate data dchisq(): compute density pchisq(): computecdf qchisq(): compute quantiles

11 Definition: Student s t random variable and distribution with degrees of freedom Properties of distribution: (0 1) 2 ( ) and are independent = q = degrees of freedom (d.f.) [ ]=0 skew( )=0 kurt( )= (0 1) as ( 60)

12 R functions rt(): simulate data dt(): compute density pt(): computecdf qt(): compute quantiles

13 Test for Specific Coefficient Value 1. Test statistic 0 : = 0 vs. 1 : 6= 0 = 0 = ˆ 0 cse(ˆ ) Intuition: If = 0 0 then ˆ 0 and 0 : = 0 should not be rejected If = 0 2, say,thenˆ is more than 2 values of SE(ˆ c ) away from 0 This is very unlikely if = 0 so 0 : = 0 should be rejected.

14 Distribution of t-statistic under 0 Under the assumptions of the CER model, and 0 : = 0 = 0 = ˆ 0 cse(ˆ ) 1 where ˆ = 1 X =1 c SE(ˆ )= ˆ ˆ = v u t 1 1 X =1 1 = Student s t distribution with 1 degrees of freedom (d.f.) ( ˆ ) 2

15 Remarks: 1 is bell-shaped and symmetric about zero (like normal) but with fatter tails than normal d.f. = sample size - number of estimated parameters. In CER model there is one estimated parameter, so df = 1 For 60 1 ' (0 1). Therefore,for 60 = 0 = ˆ 0 cse(ˆ ) ' (0 1)

16 2. Set significance level and determine critical value Pr(Type I error) =5% Test has two-sided alternative so critical value, 025 is determined using where freedom. Pr( )= = = =97 5% quantile of Student-t distribution with 1 degrees of 3. Decision rule: reject 0 : = 0 in favor of 1 : 6= 0 if = 0 975

17 Useful Rule of Thumb: If 60 then and the decision rule is Reject 0 : = 0 at 5% level if 2 = 0

18 4. P-Value of two-sided test significance level at which test is just rejected =Pr( 1 = 0 ) =Pr( 1 = 0 )+Pr( 1 = 0 ) =2 Pr( 1 = 0 ) =2 (1 Pr( 1 = 0 )) Decision rule based on P-Value Reject 0 : = 0 at 5% level if P-Value 5% For 60 P-value =2 Pr( = 0 ) (0 1)

19 Tests based on CLT Let ˆ denote an estimator for. In many cases the CLT justifies the asymptotic normal distribution Consider testing ˆ ( se(ˆ ) 2 ) 0 : = 0 vs. 1 : 6= 0 Result: Under 0 for large sample sizes. = 0 = ˆ 0 cse(ˆ ) (0 1)

20 Example: In the CER model, for large enough the CLT gives and ˆ ( (ˆ ) 2 ) (ˆ )= 2 ˆ ( (ˆ ) 2 ) q 1 2 (ˆ )=

21 Rule-of-thumb Decision Rule Let Pr(Type I error)=5% Then reject 0 : = 0 vs. 1 : 6= 0 at 5% level if = 0 = ˆ 0 cse(ˆ ) 2

22 Relationship Between Hypothesis Tests and Confidence Intervals 0 : = 0 vs. 1 : 6= 0 level =5% 975 = for 60 = 0 = ˆ 0 cse(ˆ ) Reject at 5% level if = 0 2 Approximate 95% confidence interval for ˆ = ±2 cse(ˆ ) =[ˆ 2 cse(ˆ ) ˆ +2 cse(ˆ )] Decision: Reject 0 : = 0 at 5% level if 0 interval. does not lie in 95% confidence

23 Test for Sign 0 : =0vs. 1 : 0 1. Test statistic Intuition: =0 = ˆ cse(ˆ ) If = 0 0 then ˆ 0 and 0 : =0should not be rejected If = 0 0, then this is very unlikely if =0 so 0 : =0vs. 1 : 0 should be rejected.

24 2. Set significance level and determine critical value Pr(Type I error) =5% One-sided critical value is determined using where 1 95 freedom. Pr( 1 05 )= = 1 95 =95%quantile of Student-t distribution with 1 degrees of 3. Decision rule: Reject 0 : =0vs. 1 : 0 at 5% level if =0 1 95

25 Useful Rule of Thumb: If 60 then =1 645 and the decision rule is Reject 0 : =0vs. 1 : 0 at 5% level if = P-Value of test significance level at which test is just rejected =Pr( 1 =0) =Pr( =0) for 60

26 Test for Normal Distribution 0 : iid ( 2 ) 1 : not normal 1. Test statistic (Jarque-Bera statistic) JB = Ã [skew 2 + ( kurt d 3) See R package tseries function jarque.bera.test!

27 Intuition If iid ( 2 ) then [ skew( ) 0 and d kurt( ) 3 so that JB 0 If is not normally distributed then [ skew( ) 6= 0and/or d kurt( ) 6= 3 so that JB 0

28 Distribution of JB under 0 If 0 : iid ( 2 ) is true then JB 2 (2) where 2 (2) denotes a chi-square distribution with 2 degrees of freedom (d.f.).

29 2. Set significance level and determine critical value Critical value is determined using Pr(Type I error) =5% where 2 (2) 95 freedom. Pr( 2 (2) )=0 05 = 2 (2) % quantile of chi-square distribution with 2 degrees of 3. Decision rule: Reject 0 : iid ( 2 ) at 5% level if JB 6

30 4. P-Value of test significance level at which test is just rejected =Pr( 2 (2) JB)

31 Test for No Autocorrelation Recall, the j lag autocorrelation for is Hypotheses to be tested =cor( ) = cov( ) var( ) 0 : =0,forall =1 1 : 6=0for some 1. Estimate using sample autocorrelation ˆ = 1 1 P = +1 ( ˆ )( ˆ ) 1 1 P =1 ( ˆ ) 2

32 Result: Under 0 : =0for all =1 if is large then µ ˆ 0 1 for all 1 SE(ˆ )= 1 2. Test Statistic =0 = and 95% confidence interval ˆ SE(ˆ ) = ˆ ± 2 ˆ 1 = ˆ 1 3. Decision rule Reject 0 : =0at 5% level if =0 = ˆ 2

33 That is, reject if ˆ 2 or ˆ 2 Remark: ThedottedlinesonthesampleACFareatthepoints±2 1

34 Diagnostics for Constant Parameters 0 : is constant over time vs. 1 : changes over time 0 : is constant over time vs. 1 : changes over time 0 : is constant over time vs. 1 : changes over time Remarks Formal test statistics are available but require advanced statistics SeeRpackagestrucchange Informal graphical diagnostics: Rolling estimates of and

35 Rolling Means Idea: compute estimate of over rolling windows of length ˆ ( ) = 1 1 X =0 R function (package zoo) = 1 ( ) rollapply If 0 : is constant is true, then ˆ ( ) should stay fairly constant over different windows. If 0 : is constant is false, then ˆ ( ) should fluctuate across different windows

36 Rolling Variances and Standard Deviations Idea: Compute estimates of 2 and over rolling windows of length 1 X ˆ 2 ( ) = 1 1 q ˆ ( ) = ˆ 2 ( ) =0 ( ˆ ( )) 2 If 0 : is constant is true, then ˆ ( ) should stay fairly constant over different windows. If 0 : is constant is false, then ˆ ( ) should fluctuate across different windows

37 Rolling Covariances and Correlations Idea: Compute estimates of and over rolling windows of length ˆ ( ) = 1 1 ˆ ( ) = 1 X =0 ˆ ( ) ˆ ( )ˆ ( ) ( ˆ ( ))( ˆ ( )) If 0 : is constant is true, then ˆ ( ) should stay fairly constant over different windows. If 0 : is constant is false, then ˆ ( ) should fluctuate across different windows

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