Economics 140A Confidence Intervals and Hypothesis Testing


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1 Ecoomics 140A Cofidece Itervals ad Hypothesis Testig Obtaiig a estimate of a parameter is ot the al purpose of statistical iferece because it is highly ulikely that the populatio value of a parameter is equal to the estimate. We wish istead to kow how close the populatio value is likely to be to a estimate. To determie this, we must estimate a iterval, which uses more of the iformatio i the data (the variace of the estimator) tha does poit estimatio. Costructio of a Co dece Iterval Let fy i g i=1 be a sequece of idepedet idetically distributed N (; 2 ) radom variables. Hece so (Diagram desity) Y N From tabulated values of N (0; 1) ; 2 ; Y = p N (0; 1) : P 1:96 Y = p 1:96! = :95: Do the algebra oe step at a time o the board. Step 1 P 1:96p Y 1:96 p = :95: Step 2 P Y 1:96 p Y + 1:96 p = :95: Step 3 P Y 1:96p Y + 1:96p = :95:
2 Thus, the radom iterval Y 1:96 p ; Y + 1:96 p cotais 95 percet of the time. Remark: Note the iterval is radom ad is xed. It is clearer to state 95 percet of all itervals cotai rather tha falls i the iterval 95 percet of the time. For a give sample, we have the estimate y. If we replace the estimator with the estimate we have y 1:96 p ; y + 1:96 p ; which is a xed iterval. Such a iterval either cotais or does ot, so we caot refer to the probability that the iterval cotais. To overcome the di  culty, we itroduce the word co dece, which has the same practical cootatio as probability. We say the co dece that lies i y 1:96p ; y + 1:96p is.95 or a 95 percet co dece iterval for is y 1:96 p ; y + 1:96 p. While co dece ad probability are similar, they are ot idetical. The lik If there is a high probability that a estimator lies ear the populatio parameter, the there is a high degree of co dece that the populatio parameter lies ear the estimate. Co dece satis es several probability axioms: i) the co dece for ay iterval is oegative 2
3 ii) the co dece for the etire parameter space is 1 Key di erece: To derive the probability iterval, we begi with a probability distributio, which uiquely gives the probability for ay iterval. For co dece, we begi with co dece iterval but caot uiquely determie a co dece desity ad so caot assig co dece to other itervals from the iformatio cotaied i oe iterval. Hypothesis Testig Statistical tests are ot as de itive as mathematical proofs, because sample data are subject to samplig error. If we caot be absolutely sure a theory is true or false, what do we do? We could make probability statemets such as Based o the available data, there is a 90 percet probability that the theory is true. Bayesias are willig to make such statemets, but most of us believe a theory is either true or false, rather tha true 90 percet of the time. Rather tha estimate the probability that a theory is true based o the observed data, we calculate the probability that we would observe such data if the theory were true. If this probability is low, the the data are ot cosistet with the theory ad we therefore reject it. This is proof by statistical cotradictio. Notice, too, that for a theory ot to be rejected, it eed oly be cosistet with the data. Such a coclusio is relatively weak, as may other theories may be cosistet with the data. For proof by statistical cotradictio, we rst make a assumptio, called the ull hypothesis, about the populatio from which the sample is draw. Typically, the ull hypothesis is a straw assumptio that we aticipate rejectig. (The term ull hypothesis arises because i early developmet, the value put forward to reject was zero.) The alterative hypothesis describes the populatio if the ull hypothesis is ot true. Remark: There are two geeral classes of assumptios: 1) cocers the form of the probability distributio from which the populatio is draw (i.e. ormal 3
4 vs. logormal); 2) cocers the parameters from a give probability distributio. We focus o 2). Remark: The ull ad alterative hypotheses ca be simple or composite. A simple hypothesis speci es the values of all parameters of the populatio distributio. The major theorem o hypothesis testig is prove oly for simple hypotheses, although virtually all applicatios are of composite hypotheses. Example: Let fy i g i=1 be a sequece of idepedet idetically distributed N (; 1) radom variables. (The oly ukow parameter is.) Simple ull ad alterative hypotheses are: H 0 : = 0 ad H 1 : = :5. To see how easily composite hypotheses arise, ote H 1 : 6= 0 is composite, as would be both hypotheses above if the variace were ukow. I what follows we cosider the simple ull ad composite alterative H 0 : = 0 ad H 1 : 6= 0: The alterative hypothesis is two sided. If, before seeig the data, we could rule out that the populatio mea is positive (or egative), the alterative hypothesis would be oe sided (H 1 : < 0). Oce we have speci ed our ull ad alterative hypotheses, we collect ad examie our sample data. As we are cocered with the value of, the populatio mea, we calculate the mea of our radom sample, because this is what we would use to estimate the value of the populatio mea. The probability that we obtai a sample mea idetically equal to the value i H 0 is zero, so it would be uwise to reject H 0 simply because the estimate does ot equal the hypothesized value. The farther the sample mea is from the hypothesized value, the more persuasive is the evidece agaist the ull hypothesis. How far is far eough to be statistically persuasive? Recall Y 1= p N (0; 1) ; (remember, the variace is 1 here) so P Y 1:96p 1 ; Y + 1:96 1 p cotais is 95 percet. (We could choose other critical values to obtai a di eret sigi cace level.) 4
5 Uder the ull hypothesis = 0, so if the ull hypothesis is true, 95 percet of all samples will yield a iterval Y 1:96 p 1 ; Y + 1:96 p 1 that cotais 0. Thus we check to see if the estimated co dece iterval cotais 0. If it does ot, we coclude that the observed di erece betwee y ad H 0 is too large to be attributed to chace aloe ad is statistically sigi cat. If the estimated co dece iterval does cotai 0, we caot reject H 0. Not rejectig a ull hypothesis is ot at all the same as provig the ull hypothesis to be true. A urejected ull hypothesis is but oe of may parameter values that are cosistet with the data. Thus we must be careful to say the data do ot reject the ull hypothesis rather tha the data allow us to accept the ull hypothesis to be true. Give the iheret ucertaity of statistical proof by cotradictio, what types of mistakes are we subject to? First, because the probability that our radom iterval cotais the true parameter value is less tha 1, we could reject the ull hypothesis eve though the ull hypothesis is true. Because we reject the ull hypothesis if the estimated co dece iterval does ot cotai the hypothesized value P (reject H 0 jh 0 is true) = 5 percet, which is termed the size of the test. Secod, it is possible that our radom iterval cotais the hypothesized value eve though the hypothesized value is false. Clearly, such a outcome is more likely if the populatio value is close to the hypothesized value. P (fail to reject H 0 jh 0 is false) = 1 power We ca reduce the size of the test by selectig a large iterval (diagram) (a 99 percet co dece iterval reduces the size to 1 percet) Yet the co dece iterval is ow loger, so we are more likely to fail to reject the ull whe it is false. Such a result is the heart of the NeymaPeasrso lemma that states we caot simultaeously reduce both errors. (diagram power curve) 5
6 Additioal Material The above calculatios all assume that the variace is kow. I practice, the variace is ukow. We replace the kow variace with a estimator of the variace, S 2. It ca be show that ( 1) S ad that p Y is idepedet of ( We the call o a theorem, which states that 1) S2 : 2 Y q Z 1 If Y N (0; 1), Z 2 t 1. 1, ad Y is idepedet of Z, the Thus p ( Y ) p p 1s = 1 p Y s t 1 : 6
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