Interval Estimation and Hypothesis Testing - Applied Econometrics | ECON 446, Exams of Introduction to Econometrics

Download Interval Estimation and Hypothesis Testing - Applied Econometrics | ECON 446 and more Exams Introduction to Econometrics in PDF only on Docsity! Interval Estimation and Hypothesis Testing Prepared by Vera Tabakova, East Carolina University 3.1 Interval Estimation 3.2 Hypothesis Tests 3.3 Rejection Regions for Specific Alternatives 3.4 Examples of Hypothesis Tests 3.5 The p-value This defines an interval that has probability .95 of containing the parameter β2 . ( )1.96 1.96 .95P Z− ≤ ≤ = ( ) 2 2 22 1.96 1.96 .95 i bP x x ⎛ ⎞−β⎜ ⎟− ≤ ≤ = ⎜ ⎟σ −⎝ ⎠∑ ( ) ( )( )2 22 22 2 21.96 1.96 .95i iP b x x b x x− σ − ≤ β ≤ + σ − =∑ ∑ The two endpoints provide an interval estimator. In repeated sampling 95% of the intervals constructed this way will contain the true value of the parameter β2. This easy derivation of an interval estimator is based on the assumption SR6 and that we know the variance of the error term σ2. ( )( )222 1.96 ib x x± σ −∑ Replacing σ2 with creates a random variable t: The ratio has a t-distribution with (N – 2) degrees of freedom, which we denote as . 2σ̂ ( ) ( ) ( ) 2 2 2 2 2 2 ( 2)22 2 2 ~ seˆ var N i b b bt t bx x b − −β −β −β = = = σ −∑ ( ) ( )2 2 2set b b= −β ( 2)~ Nt − Figure 3.1 Critical Values from a t-distribution Slide 3-10Principles of Econometrics, 3rd Edition Each shaded “tail” area contains α/2 of the probability, so that 1–α of the probability is contained in the center portion. Consequently, we can make the probability statement Slide 3-11Principles of Econometrics, 3rd Edition ( ) 1c cP t t t− ≤ ≤ = −α [ ] 1 se( ) k k c c k bP t t b −β − ≤ ≤ = −α [ se( ) se( )] 1k c k k k c kP b t b b t b− ≤ β ≤ + = −α For the food expenditure data The critical value tc = 2.024, which is appropriate for α = .05 and 38 degrees of freedom. To construct an interval estimate for β2 we use the least squares estimate b2 = 10.21 and its standard error Slide 3-12Principles of Econometrics, 3rd Edition 2 2 2 2 2[ 2.024se( ) 2.024se( )] .95P b b b b− ≤ β ≤ + = 2 2se( ) var( ) 4.38 2.09b b= = = 3.1.4 The Repeated Sampling Context Table 3.2 Interval Estimates from 10 Random Samples Sample b, — t,se(b,) b, + t,se(b)) by — t,se(b2) by + t,se(b2) 49.54 213.85 2.52 10.44 —9,83 124.32 7.65 14.12 28.56 179.26 4.51 11.77 —20.96 113.97 8.65 15.15 0.93 167.53 5.27 13.30 —66.04 119.30 9.08 18.02 —0.63 129.05 7.31 14.06 19.19 140.13 6.85 12.68 38.32 156.29 5.21 10.89 20.69 171.23 4.14 11.40 eae DIAM) Nee LOLA e OTK C WB eXe TCI Components of Hypothesis Tests 1. A null hypothesis, H0 2. An alternative hypothesis, H1 3. A test statistic 4. A rejection region 5. A conclusion Slide 3-16Principles of Econometrics, 3rd Edition The Null Hypothesis The null hypothesis, which is denoted H0 (H-naught), specifies a value for a regression parameter. The null hypothesis is stated , where c is a constant, and is an important value in the context of a specific regression model. Slide 3-17Principles of Econometrics, 3rd Edition 0 : kH cβ = The Rejection Region The rejection region depends on the form of the alternative. It is the range of values of 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 level of significance α is usually chosen to be .01, .05 or .10. Slide 3-20Principles of Econometrics, 3rd Edition 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) Slide 3-21Principles of Econometrics, 3rd Edition 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” (≠) Slide 3-22Principles of Econometrics, 3rd Edition Figure 3.3 The rejection region for a one-tail test of H0: βk = c against H1: βk < c Slide 3-25Principles of Econometrics, 3rd Edition Slide 3-26Principles of Econometrics, 3rd Edition When testing the null hypothesis against the alternative hypothesis , reject the null hypothesis and accept the alternative hypothesis if . 0 : kH cβ = 1 : kH cβ < ( ), 2Nt t α −≤ Figure 3.4 The rejection region for a two-tail test of H0: βk = c against H1: βk ≠ c Slide 3-27Principles of Econometrics, 3rd Edition 3.4.1a One-tail Test of Significance 1. The null hypothesis is . The alternative hypothesis is . 2. The test statistic is (3.7). In this case c = 0, so if the null hypothesis is true. 3. Let us select α = .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. Slide 3-30Principles of Econometrics, 3rd Edition 0 2: 0H β = 1 2: 0H β > ( ) ( )2 2 2se ~ Nt b b t −= 4. 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 5. Since t = 4.88 > 1.686, we reject the null hypothesis that β2 = 0 and accept the alternative that β2 > 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. Slide 3-31Principles of Econometrics, 3rd Edition ( ) 2 2 10.21 4.88 se 2.09 bt b = = = 3.4.1b One-tail Test of an Economic Hypothesis 1. The null hypothesis is . The alternative hypothesis is . 2. The test statistic if the null hypothesis is true. 3. Let us select α = .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. Slide 3-32Principles of Econometrics, 3rd Edition 0 2: 5.5H β ≤ 1 2: 5.5H β > ( ) ( ) ( )2 2 25.5 se ~ Nt b b t −= − 4. Using the food expenditure data, b2 = 10.21 with standard error se(b2) = 2.09. The value of the test statistic is 5. Since t = –2.29 < –1.686 we reject the null hypothesis that β2 ≥ 15 and accept the alternative that β2 < 15 . We conclude that households spend less than $15 from each additional $100 income on food. Slide 3-35Principles of Econometrics, 3rd Edition ( ) 2 2 15 10.21 15 2.29 se 2.09 bt b − − = = = − 3.4.3a Two-tail Test of an Economic Hypothesis 1. The null hypothesis is . The alternative hypothesis is . 2. The test statistic if the null hypothesis is true. 3. Let us select α = .05. The critical values for this two-tail test are the 2.5- percentile 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. Slide 3-36Principles of Econometrics, 3rd Edition 0 2: 7.5H β = 1 2: 7.5H β ≠ ( ) ( ) ( )2 2 27.5 se ~ Nt b b t −= − 4. Using the food expenditure data, b2 = 10.21 with standard error se(b2) = 2.09. The value of the test statistic is 5. Since –2.204 < t = 1.29 < 2.204 we do not reject the null hypothesis that β2 = 7.5. The sample data are consistent with the conjecture households will spend an additional $7.50 per additional $100 income on food. Slide 3-37Principles of Econometrics, 3rd Edition ( ) 2 2 7.5 10.21 7.5 1.29 se 2.09 bt b − − = = = Coefficient Variable SiceMaheueye 83.41600 a ERR eatee ee er OTs 2.093264 eae DIAM) Nee LOLA e OTK C WB eXe TCI Slide 3-40 Slide 3-41Principles of Econometrics, 3rd Edition p-value rule: Reject the null hypothesis when the p-value is less than, or equal to, the level of significance α. That is, if p ≤ α then reject H0. If p > α then do not reject H0. If t is the calculated value of the t-statistic, then: ▪ if H1: βK > c, p = probability to the right of t ▪ if H1: βK < c, p = probability to the left of t ▪ if H1: βK ≠ c, p = sum of probabilities to the right of |t| and to the left of – |t| Slide 3-42Principles of Econometrics, 3rd Edition Recall section 3.4.2: ▪ The null hypothesis is H0: β2 ≥ 15. The alternative hypothesis is H1: β2 < 15. ▪ Slide 3-45Principles of Econometrics, 3rd Edition ( ) 2 2 15 10.21 15 2.29 se 2.09 bt b − − = = = − ( )38 2.29 .0139P t⎡ ⎤≤ − =⎣ ⎦ Figure 3.6 The p-value for a left tail test Slide 3-46Principles of Econometrics, 3rd Edition Recall section 3.4.3a: ▪ The null hypothesis is H0: β2 = 7.5. The alternative hypothesis is H1: β2 ≠ 7.5. ▪ Slide 3-47Principles of Econometrics, 3rd Edition ( ) 2 2 7.5 10.21 7.5 1.29 se 2.09 bt b − − = = = ( ) ( )38 381.29 1.29 .2033p P t P t⎡ ⎤ ⎡ ⎤= ≥ + ≤ − =⎣ ⎦ ⎣ ⎦ alternative hypothesis = rejection region ereba(elaele poeta attic ROME atibele tiles critical value Cate Lat jae ola Cloke waren Coan two-tail tests hypotheses OMe eosceg J eygeemisemayebil sd eye PIT ecg inference pUlteraveeleyaberstaCeyey level of significance null hypothesis CTEM Cott point estimates probability value p-value eae DIAM) Nee LOLA e OTK C WB eXe TCI NIC Eel) a Appendix 3A Derivation of the ¢-distribution ERP-W ihe Io MDs Chele rie Me Mio (emblem ell eae DIAM) Nee LOLA e OTK C WB eXe TCI A/T Eey | Slide 3-52Principles of Econometrics, 3rd Edition (3A.1) (3A.2) (3A.3) 2 2 2 2~ , ( )i b N x x ⎛ ⎞σ β⎜ ⎟ −⎝ ⎠∑ 2 2 2 ~ (0,1) var( ) bZ N b −β = 2 22 2 21 2 ( )~i N N e ee e⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + + + χ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟σ σ σ σ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ∑ L 2 2 2 2 ˆ ˆ( 2)ie NV − σ= = σ σ ∑