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54 Pages

### ch03Hypothesistesting

Course: STAT 2550, Spring 2010
School: Memorial University
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Word Count: 2216

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6002 ECON Econometrics Memorial University of Newfoundland Adapted from Vera Tabakovas notes 3.1 3.2 3.3 3.4 3.5 Interval Estimation Hypothesis Tests Rejection Regions for Specific Alternatives Examples of Hypothesis Tests The p-value y = 1 + 2 x + e E (e) = 0 E ( y ) = 1 + 2 x var( e) = 2 = var( y ) cov(ei , e j ) = cov( yi , y j ) = 0 e ~ N (0, 2 ) Principles of Econometrics, 3rd Edition Slide 3-3...

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6002 ECON Econometrics Memorial University of Newfoundland Adapted from Vera Tabakovas notes 3.1 3.2 3.3 3.4 3.5 Interval Estimation Hypothesis Tests Rejection Regions for Specific Alternatives Examples of Hypothesis Tests The p-value y = 1 + 2 x + e E (e) = 0 E ( y ) = 1 + 2 x var( e) = 2 = var( y ) cov(ei , e j ) = cov( yi , y j ) = 0 e ~ N (0, 2 ) Principles of Econometrics, 3rd Edition Slide 3-3 The normal distribution of b2 , the least squares estimator of , is 2 b2 ~ N 2 , 2 ( x x ) i A standardized normal random variable is obtained from b2 by subtracting its mean and dividing by its standard deviation: Z= b2 2 2 ( xi x ) 2 ~ N ( 0,1) The standardized random variable Z is normally distributed with mean 0 and variance 1. P ( 1.96 Z 1.96 ) = .95 b2 2 1.96 P 1.96 = .95 2 2 ( xi x ) P b2 1.96 ( 2 ( xi x ) 2 2 b2 + 1.96 2 ( xi x ) 2 ) = .95 This defines an interval that has probability .95 of containing the parameter 2 . The 2 two endpoints b2 1.96 ( ( xi x ) 2 ) provide an interval estimator. In repeated sampling 95% of the intervals constructed this way will contain the true value of the parameter 2. This easy derivation of an interval estimator is based on the assumption SR6 and that we know the variance of the error term 2. Replacing 2 with 2 creates a random variable t: t= b2 2 2 ( xi x ) 2 b2 2 b2 2 = = ~ t( N 2) var ( b ) se ( b2 ) 2 The ratio t = ( b2 2 ) se ( b2 ) has a t-distribution with (N 2) degrees of freedom, which we denote as t ~ t( N 2) . In general we can say, if assumptions SR1-SR6 hold in the simple linear regression model, then bk k t= ~ t( N 2 ) for k = 1, 2 se ( bk ) The t-distribution is a bell shaped curve centered at zero. It looks like the standard normal distribution, except it is more spread out, with a larger variance and thicker tails. The shape of the t-distribution is controlled by a single parameter called the degrees of freedom, often abbreviated as df. We can find a critical value from a t-distribution such that P ( t t c ) = P ( t t c ) = 2 where is a probability often taken to be = .01 or = .05. The critical value tc for degrees of freedom m is the percentile value t( 1 2, m ) . Figure 3.1 Critical Values from a t-distribution Principles of Econometrics, 3rd Edition Slide 3-10 Each shaded tail area contains /2 of the probability, so that 1 of the probability is contained in the center portion. Consequently, we can make the probability statement P(tc t tc ) = 1 bk k P[tc tc ] = 1 se(bk ) P[bk tc se(bk ) k bk + tc se(bk )] = 1 Principles of Econometrics, 3rd Edition Slide 3-11 For the food expenditure data P[b2 2.024se(b2 ) 2 b2 + 2.024se(b2 )] = .95 The critical value tc = 2.024, which is appropriate for = .05 and 38 degrees of freedom. To construct an interval estimate for 2 we use the least squares estimate b2 = 10.21 and its standard error se(b2 ) = var(b2 ) = 4.38 = 2.09 Principles of Econometrics, 3rd Edition Slide 3-12 A 95% confidence interval estimate for 2: b2 tc se(b2 ) = 10.21 2.024(2.09)=[5.97,14.45] When the procedure we used is applied to many random samples of data from the same population, then 95% of all the interval estimates constructed using this procedure will contain the true parameter. Principles of Econometrics, 3rd Edition Slide 3-13 Principles of Econometrics, 3rd Edition Slide 3-14 Principles of Econometrics, 3rd Edition Slide 3-15 Components of Hypothesis Tests 1. 2. 3. 4. 5. A null hypothesis, H0 An alternative hypothesis, H1 A test statistic A rejection region A conclusion Principles of Econometrics, 3rd Edition Slide 3-16 The Null Hypothesis parameter. The null hypothesis, which is denoted H0 (H-naught), specifies a value for a regression The null hypothesis is stated H 0 : k = c, where c is a constant, and is an important value in the context of a specific regression model. Principles of Econometrics, 3rd Edition Slide 3-17 The Alternative Hypothesis accept if the null hypothesis is rejected. Paired with every null hypothesis is a logical alternative hypothesis, H1, that we will For the null hypothesis H0: k = c the three possible alternative hypotheses are: H1 : k > c H1 : k < c H1 : k c Principles of Econometrics, 3rd Edition Slide 3-18 The Test Statistic t = ( bk k ) se(bk ) ~ t( N 2) If the null hypothesis follows that H 0 : k = c is true, then we can substitute c for and it k bk c t= ~ t( N 2) se(bk ) If the null hypothesis is not true, then the t-statistic in (3.7) does not have a tdistribution with N 2 degrees of freedom. Principles of Econometrics, 3rd Edition Slide 3-19 The Rejection Region the test statistic that leads to rejection of the null hypothesis. It is possible to construct a rejection region only if we have: a test statistic whose distribution is known when the null hypothesis is true an alternative hypothesis a level of significance The rejection region depends on the form of the alternative. It is the range of values of The level of significance is usually chosen to be .01, .05 or .10. Principles of Econometrics, 3rd Edition Slide 3-20 A Conclusion We make a correct decision if: The null hypothesis is false and we decide to reject it. The null hypothesis is true and we decide not to reject it. Our decision is incorrect if: The null hypothesis is true and we decide to reject it (a Type I error) The null hypothesis is false and we decide not to reject it (a Type II error or beta error) Principles of Econometrics, 3rd Edition Slide 3-21 3.3.1. One-tail Tests with Alternative Greater Than (>) 3.3.2. One-tail Tests with Alternative Less Than (<) 3.3.3. Two-tail Tests with Alternative Not Equal To () Principles of Econometrics, 3rd Edition Slide 3-22 Figure 3.2 Rejection region for a one-tail test of H0: k = c against H1: k > c Principles of Econometrics, 3rd Edition Slide 3-23 When testing the null hypothesis H 0 : k = c against the alternative hypothesis H1 : k > c , reject the null hypothesis and accept the alternative hypothesis if t t( 1 , N 2) . Principles of Econometrics, 3rd Edition Slide 3-24 Figure 3.3 The rejection region for a one-tail test of H0: k = c against H1: k < c Principles of Econometrics, 3rd Edition Slide 3-25 When testing the null hypothesis H 0 : k = c against the alternative hypothesis H1 : k < c , reject the null hypothesis and accept the alternative hypothesis if t t( , N 2) . Principles of Econometrics, 3rd Edition Slide 3-26 Figure 3.4 The rejection region for a two-tail test of H0: k = c against H1: k c Principles of Econometrics, 3rd Edition Slide 3-27 When testing the null hypothesis H 0 : k = c against the alternative hypothesis H1 : k c , reject the null hypothesis and accept the alternative hypothesis if t t( 2, N 2) or if t t( 1 2, N 2) . Principles of Econometrics, 3rd Edition Slide 3-28 STEP-BY-STEP PROCEDURE FOR TESTING HYPOTHESES 1.Determine the null and alternative hypotheses. 2.Specify the test statistic and its distribution if the null hypothesis is true. 3.Select and determine the rejection region. 4.Calculate the sample value of the test statistic. 5.State your conclusion. Principles of Econometrics, 3rd Edition Slide 3-29 3.4.1a One-tail Test of Significance 1. The null hypothesis is H 0 : 2 = 0 . The alternative hypothesis is H1 : 2 > 0 . 2. The test statistic is (3.7). In this case c 0, = so t = b2 se ( b2 ) ~ t( N 2) if the null hypothesis is true. 3. Let us select = .05. The critical value for the right-tail rejection region is the 95th percentile of the t-distribution with N 2 = 38 degrees of freedom, t(95,38) = 1.686. Thus we will reject the null hypothesis if the calculated value of t 1.686. If t < 1.686, we will not reject the null hypothesis. Principles of Econometrics, 3rd Edition Slide 3-30 1. Using the food expenditure data, we found that b2 = 10.21 with standard error se(b2) = 2.09. The value of the test statistic is b2 10.21 t= = = 4.88 se ( b2 ) 2.09 1. Since t = 4.88 > 1.686, we reject the null hypothesis that 2 = 0 and accept the alternative that 2 > 0. That is, we reject the hypothesis that there is no relationship between income and food expenditure, and conclude that there is a statistically significant positive relationship between household income and food expenditure. Principles of Econometrics, 3rd Edition Slide 3-31 3.4.1b One-tail Test of an Economic Hypothesis 1. The null hypothesis is H 0 : 2 5.5 . The alternative hypothesis is H1 : 2 > 5.5 . 2. The test statistic t = ( b2 5.5 ) se ( b2 ) ~ t( N 2) if the null hypothesis is true. 3. Let us select = .01. The critical value for the right-tail rejection region is the 99th percentile of the t-distribution with N 2 = 38 degrees of freedom, t(99,38) = 2.429. We will reject the null hypothesis if the calculated value of t 2.429. If t < 2.429, we will not reject the null hypothesis. Principles of Econometrics, 3rd Edition Slide 3-32 1. Using the food expenditure data, b2 = 10.21 with standard error se(b2) = 2.09. The value of the test statistic is t= b2 5.5 10.21 5.5 = = 2.25 se ( b2 ) 2.09 1. Since t = 2.25 < 2.429 we do not reject the null hypothesis that 2 5.5. We are not able to conclude that the new supermarket will be profitable and will not begin construction. Principles of Econometrics, 3rd Edition Slide 3-33 1. The null hypothesis is H 0 : 2 15 . The alternative hypothesis is H1 : 2 < 15 . The test statistic t = ( b2 15 ) se ( b2 ) ~ t( N 2) if the null hypothesis is true. 2. 3. Let us select = .05. The critical value for the left-tail rejection region is the 5th percentile of the t-distribution with N 2 = 38 degrees of freedom, t(05,38) = -1.686. We will reject the null hypothesis if the calculated value of t 1.686. If t >1.686, we will not reject the null hypothesis. Principles of Econometrics, 3rd Edition Slide 3-34 1. Using the food expenditure data, b2 = 10.21 with standard error se(b2) = 2.09. The value of the test statistic is t= b2 15 10.21 15 = = 2.29 se ( b2 ) 2.09 1. Since t = 2.29 < 1.686 we reject the null hypothesis that 2 15 and accept the alternative that 2 < 15 . We conclude that households spend less than \$15 from each additional \$100 income on food. Principles of Econometrics, 3rd Edition Slide 3-35 3.4.3a Two-tail Test of an Economic Hypothesis 1. The null hypothesis is H 0 : 2 = 7.5 . The alternative hypothesis is H1 : 2 7.5 . 2. The test statistic t = ( b2 7.5 ) se ( b2 ) ~ t( N 2) if the null hypothesis is true. 3. Let us select = .05. The critical values for this two-tail test are the 2.5percentile t(.025,38) = 2.024 and the 97.5-percentile t(.975,38) = 2.024 . Thus we will reject the null hypothesis if the calculated value of t 2.024 or if t 2.024. If 2.024 < t < 2.024, we will not reject the null hypothesis. Principles of Econometrics, 3rd Edition Slide 3-36 1. Using the food expenditure data, b2 = 10.21 with standard error se(b2) = 2.09. The value of the test statistic is t= b2 7.5 10.21 7.5 = = 1.29 se ( b2 ) 2.09 1. Since 2.204 < t = 1.29 < 2.204 we do not reject the null hypothesis that 2 = 7.5. The sample data are consistent with the conjecture households will spend an additional \$7.50 per additional \$100 income on food. Principles of Econometrics, 3rd Edition Slide 3-37 3.4.3b Two-tail Test of Significance 1. The null hypothesis is H 0 : 2 = 0 . The alternative hypothesis is H1 : 2 0 . 2. The test statistic t = b2 se ( b2 ) ~ t( N 2) if the null hypothesis is true. 3. Let us select = .05. The critical values for this two-tail test are the 2.5percentile t(.025,38) = 2.024 and the 97.5-percentile t(.975,38) = 2.024 . Thus we will reject the null hypothesis if the calculated value of t 2.024 or if t 2.024. If 2.024 < t < 2.024, we will not reject the null hypothesis. Principles of Econometrics, 3rd Edition Slide 3-38 1. Using the food expenditure data, b2 = 10.21 with standard error se(b2) = 2.09. The value of the test statistic is t= b2 10.21 = = 4.88 se ( b2 ) 2.09 1. Since t = 4.88 > 2.204 we reject the null hypothesis that 2 = 0 and conclude that there is a statistically significant relationship between income and food expenditure. Principles of Econometrics, 3rd Edition Slide 3-39 Principles of Econometrics, 3rd Edition Slide 3-40 p-value rule: Reject the null hypothesis when the p-value is less than, or equal to, the level of significance . That is, if p then reject H0. If p > then do not reject H0. Principles of Econometrics, 3rd Edition Slide 3-41 If t is the calculated value of the t-statistic, then: if H1: K > c, p = probability to the right of t if H1: K < c, p = probability to the left of t if H1: K c, p = sum of probabilities to the right of |t| and to the left of |t| Principles of Econometrics, 3rd Edition Slide 3-42 Recall section 3.4.1b: The null hypothesis is H0: 2 5.5. The alternative hypothesis is H1: 2 > 5.5. t= b2 5.5 10.21 5.5 = = 2.25 se ( b2 ) 2.09 If FX(x) is the cdf for a random variable X, then for any value x=c the cumulative probability is P [ X c ] = FX ( c ) p = P t(38) 2.25 = 1 P t(38) 2.25 = 1 .9848 = .0152 . Principles of Econometrics, 3rd Edition Slide 3-43 Figure 3.5 The p-value for a right tail test Principles of Econometrics, 3rd Edition Slide 3-44 Recall section 3.4.2: The null hypothesis is H0: 2 15. The alternative hypothesis is H1: 2 < 15. t= b2 15 10.21 15 = = 2.29 se ( b2 ) 2.09 P t( 38) 2.29 = .0139 Principles of Econometrics, 3rd Edition Slide 3-45 Figure 3.6 The p-value for a left tail test Principles of Econometrics, 3rd Edition Slide 3-46 Recall section 3.4.3a: The null hypothesis is H0: 2 = 7.5. The alternative hypothesis is H1: 2 7.5. t= b2 7.5 10.21 7.5 = = 1.29 se ( b2 ) 2.09 p = P t( 38) 1.29 + P t( 38) 1.29 = .2033 Principles of Econometrics, 3rd Edition Slide 3-47 Figure 3.7 The p-value for a two-tail test Principles of Econometrics, 3rd Edition Slide 3-48 Recall section 3.4.3b: The null hypothesis is H0: 2 = 0. The alternative hypothesis is H1: 2 0 p = P t( 38) 4.88 + P t( 38) 4.88 = 0.0000 Slide 3-49 Principles of Econometrics, 3rd Edition Slide 3-50 Principles of Econometrics, 3rd Edition Slide 3-51 2 b2 ~ N 2 , ( xi x ) 2 Z= 2 b2 2 var(b2 ) 2 ~ N (0,1) 2 2 (3A.1) ei e1 e2 eN = + + L + ~ (2N ) ei2 = ( N 2)2 V= 2 2 Principles of Econometrics, 3rd Edition (3A.2) (3A.3) Slide 3-52 ( N 2) 2 V= ~ (2N 2) 2 (3A.4) t= Z V ( N 2) b2 2 = ( b2 2 ) 2 ( xi x ) 2 (3A.5) ( N 2)2 2 N 2 b2 2 b2 2 = = ~ t( N 2) var(b ) se(b2 ) 2 = 2 ( xi x ) 2 Principles of Econometrics, 3rd Edition Slide 3-53 b2 1 t= ~ t( N 2) se(b2 ) b2 c 1 c ~ N ,1 var(b ) var(b2 ) 2 var ( b2 ) = 2 (3B.1) where ( xi x ) 2 Principles of Econometrics, 3rd Edition Slide 3-54
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UGA - BIO - 1108
Biology, 8e (Campbell) Chapter 16 The Molecular Basis of Inheritance Multiple-Choice Questions 1) For a couple of decades, biologists knew the nucleus contained DNA and proteins. The prevailing opinion was that the genetic material was proteins, and not D
University of Phoenix - IT - 205
Foundations of Business Intelligence: Databases and Information ManagementCHAPTER5STUDENT LEARNING OBJECTIVESAfter completing this chapter, you will be able to answer the following questions:1.How does a relational database organize data and how doe
UGA - BIO - 1108
Biology, 8e (Campbell) Chapter 17 From Gene to Protein Multiple-Choice Questions 1) Garrod hypothesized that &quot;inborn errors of metabolism&quot; such as alkaptonuria occur because A) genes dictate the production of specific enzymes, and affected individuals hav
University of Phoenix - IT - 205
Telecommunications, the Internet, and Wireless TechnologyCHAPTER6STUDENT LEARNING OBJECTIVESAfter completing this chapter, you will be able to answer the following questions:1.What are the principal components of telecommunications networks and key
UGA - BIO - 1108
Biology, 8e (Campbell) Chapter 18 Regulation of Gene Expression Multiple-Choice Questions 1) What does the operon model attempt to explain? A) the coordinated control of gene expression in bacteria B) bacterial resistance to antibiotics C) how genes move
University of Phoenix - IT - 205
Improving Decision Making and Managing Knowledge10CHAPTERISBN: 0-558-30397-8STUDENT LEARNING OBJECTIVESAfter completing this chapter, you will be able to answer the following questions:1.What are the different types of decisions, and how does the d