12/10/2007 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 Principles of Econometrics, 3rd Edition Slide 3-2 Assumptions of the Simple Linear Regression Model •SR1. yxe=+ββ12 + •SR2. ⇔ Ee()= 0 Ey()=β12 +β x •SR3. var(ey )=σ2 = var( ) •SR4. cov(eeij , )== cov( yy ij , ) 0 •SR5. The variable x is not random, and must take at least two different values. 2 •SR6. (optional) The values of e are normally distributed about their mean eN~(,)0 σ Principles of Econometrics, 3rd Edition Slide 3-3 1 12/10/2007 The normal distribution of b 2 , the least squares estimator of β, is ⎛⎞σ2 bN~,⎜⎟β 22⎜⎟2 ⎝⎠∑()xxi − A standardized normal random variable is obtained from b 2 by subtracting its mean and dividing by its standard deviation: b22−β ZN= ~0,1() (3.1) 2 2 σ−∑()xxi The standardized random variable Z is normally distributed with mean 0 and variance 1. Principles of Econometrics, 3rd Edition Slide 3-4 PZ()−≤≤=1.96 1.96 .95 ⎛⎞ b −β P⎜⎟−≤1.9622 ≤ 1.96 = .95 ⎜⎟2 xx2 ⎝⎠σ−∑( i ) Pb−σ1.9622 x −≤β≤+σ x22 b 1.96 x −= x .95 ( 222∑∑()ii()) This defines an interval that has probability .95 of containing the parameter β2 . Principles of Econometrics, 3rd Edition Slide 3-5 2 The two endpoints bxx ±σ 1.96 2 − provide an interval estimator. ( 2 ∑()i ) 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. Principles of Econometrics, 3rd Edition Slide 3-6 2 12/10/2007 Replacing σ2 with σˆ 2 creates a random variable t: bbb22−β 22 −β 22 −β tt===~ (2)N − (3.2) 2 2 n se()b2 σ−ˆ ∑()xxi var ()b2 The ratio tb =−β ( 22 ) se ( b 2 ) has a t-distribution with (N – 2) degrees of freedom, which we denote as tt ~ (2) N − . Principles of Econometrics, 3rd Edition Slide 3-7 In general we can say, if assumptions SR1-SR6 hold in the simple linear regression model, then bkk−β ttk==~(N −2) for 1,2 (()3.3) 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. Principles of Econometrics, 3rd Edition Slide 3-8 We can find a “critical value” from a t-distribution such that Pt()(≥= tcc Pt ≤−=α t )2 where α is a probability often taken to be α = .01 or α = .05. The critical value tc for degrees of freedom m is the percentile value t () 12, −α m . Principles of Econometrics, 3rd Edition Slide 3-9 3 12/10/2007 Figure 3.1 Critical Values from a t-distribution Principles of Econometrics, 3rd Edition Slide 3-10 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 Pt()1− cc≤≤ tt=−α (3.4) bkk−β Pt[]1−≤cc ≤ t =−α se(bk ) Pb[kc−≤β≤+=−α t se( b k ) k b kc t se( b k )] 1 (3.5) Principles of Econometrics, 3rd Edition Slide 3-11 For the food expenditure data Pb[22222−≤β≤+= 2.024se( b ) b 2.024se( b )] .95 (3.6) 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 n se(bb22 )=== var( ) 4.38 2.09 Principles of Econometrics, 3rd Edition Slide 3-12 4 12/10/2007 A “95% confidence interval estimate” for β2: bt22±=±cse( b ) 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 5 12/10/2007 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 Principles of Econometrics, 3rd Edition Slide 3-16 The Null Hypothesis The null hypothesis, which is denoted H0 (H-naught), specifies a value for a regression parameter. The null hypothesis is stated H 0 : β= 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 Paired with every null hypothesis is a logical alternative hypothesis, H1, that we will accept if the null hypothesis is rejected. For the null hypothesis H0: βk = c the three possible alternative hypotheses are: Hc1 :β>k Hc1 :β<k Hc1 :β≠k Principles of Econometrics, 3rd Edition Slide 3-18 6 12/10/2007 The Test Statistic tb=−β()kkse( b k ) ~ t(2) N− If the null hypothesis H 0 : β= k c is true, then we can substitute c for βk and it follows that bck − tt= ~ (2)N − (3.7) se(bk ) If the null hypothesis is not true, then the t-statistic in (3.7) does not have a t- distribution with N − 2 degrees of freedom. Principles of Econometrics, 3rd Edition Slide 3-19 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 rejgyjection 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. 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) Principles of Econometrics, 3rd Edition Slide 3-21 7 12/10/2007 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 w it h A lternati ve “Not Equal To” ( ≠) Principles of Econometrics, 3rd Edition Slide 3-22 Figure 3.2 Rejection region for a one-tail test of H0: βk = c against H1: βk > c Principles of Econometrics, 3rd Edition Slide 3-23 When testing the null hypothesis H 0 : β= k c against the alternative hypothesis Hc 1 : β> k , reject the null hypothesis and accept the alternative hypothesis if tt ≥ () 1, −α N − 2 . Principles of Econometrics, 3rd Edition Slide 3-24 8 12/10/2007 Figure 3.3 The rejection region for a one-tail test of H0: βk = c against H1: βk < c Principles of Econometrics, 3rd Edition Slide 3-25 When testing the null hypothesis H 0 : β= k c against the alternative hypothesis Hc 1 : β< k , reject the null hypothesis and accept the alternative hypothesis if tt ≤ () α− ,2N . Principles of Econometrics, 3rd Edition Slide 3-26 Figure 3.4 The rejection region for a two-tail test of H0: βk = c against H1: βk ≠ c Principles of Econometrics, 3rd Edition Slide 3-27 9 12/10/2007 When testing the null hypothesis H 0 : β= k c against the alternative hypothesis Hc 1 : β≠ k , reject the null hypothesis and accept the alternative hypothesis if tt ≤ () α− 2, N 2 or if tt≥ ()12,2−αN − . Principles of Econometrics, 3rd Edition Slide 3-28 STEP-BY-STEP PROCEDURE FOR TESTING HYPOTHESES 1. Determine the null and alternative hypotheses. 2. Specif y th e test stati sti c and i ts di strib uti on if th e null h ypoth esi s i s true. 3. Select α 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 02 :0 β= . The alternative hypothesis is H 12 :0 β> . 2. The test statistic is (3.7). In this case c = 0, so tb= 22se() b ~ t( N−2) 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.