Econ 620Null hypothesis vs. alternative hypothesisSuppose that we have data y =(y1, ···,yn) and the data is generated by the true probability distributionPθ0, from a family of probability distribution Pθindexed by θ ∈ Θ. We can partition the parameter space Θinto two subsets, Θ0and ΘA. Now, consider the following two hypothesis;H0; θ0∈ Θ0HA; θ0∈ ΘAH0is called the null hypothesis and HAis called the alternative hypothesis. The union of null andalternative hypothesis defines a hypothesis H ∈ Θ=Θ0∪ ΘAcalled the maintained hypothesis.• A hypothesis is called simple if it completely specify the probability distribution and otherwise com-posite.Example 1 Suppose that we observe data y =(y1, ···,yn) . If we are willing to assume that the data set isa random sample from N (θ, 10) where θ ∈{1, 2} . We want to check which value of θ is consistent with data.Then, we can formulate the hypotheses;H0; θ =1 HA; θ =2Here, both null and alternative hypotheses are simple since mean and variance are sufficient to completelyspecify a normal distribution. The maintained hypothesis in this case is that H; θ ∈{1, 2} . If we assume thatthe data set is a random sample from N (θ, 10) where θ = R. We can formulate the following hypotheses;H0; θ =1 HA; θ =1The null hypothesis is simple but the alternative hypothesis is composite since the alternative hypothesisincludes infinite numbers of normal distributions. The maintained hypothesis in this case is that H; θ = R.Alternatively, we can formulate different hypotheses such asH0; θ ≥ 1 HA; θ<1In this example, both null and alternative hypotheses are composite and again the maintained hypothesis isthat H; θ = R.The null hypothesisNull hypotheses can arise for consideration in a number of different ways, the main ones being as follows;• H0may corresponds to the prediction of some scientific(economic) theory or to some model of thesystem thought quite likely to be true or nearly so.• H0may represent some simple set of circumstances which, in the absence of evidence to the contrary,we wish to assume holds. For example, the null hypothesis might assert the ineffectiveness of newly-developed medicine for AIDS. We want to play safe by assuming ineffectiveness unless we can find asignificant evidence against our presumption.• H0may assert complete absence of structure in some sense. So long as the data are consistent with thenull hypothesis it can not be justified to claim that the data provide clear evidence in favor of someparticular kind of structure. Testing joint significance of slope coefficients in linear regression model isan example.1Type I and type II errorsThere are four possible cases once we take an action in testing hypotheses - correctly accept the null, correctlyreject the null, wrongly accept the null and wrongly reject the null. We don’t have any concern with thefirst two cases. A good test should avoid or minimize the possibilities of the last two cases.• Type I error is the probability that we reject the null hypothesis when it is true;α ≡ P [reject H0| H0is true]• Type II error is the probability that we do not reject the null hypothesis when it is not true;β ≡ P [do not reject H0| HAis true]• We call α,the probability that we reject the null hypothesis when it is true, size of the test• We call (1 − β) , the probability that we reject the null hypothesis when the alternative hypothesis istrue, the power of the test.The ultimate goal in designing a test statistic is to minimize the size and maximize the power as much aspossible. Unfortunately, it is quite easy to prove that we can not design a test which has both the minimumsize and the maximum power. Here is an intuitive example why it is the case. Consider minimum size first.What is the test which has the minimum size? It is a test with which we always accept the null hypothesisno matter what we observe from the data. Since we always accept the null hypothesis, the size of the test is0 which is the minimum possible size ; P [reject H0| H0is true]=0− note that we never reject. However,what is the power of this test? It is a pathetic test as far as power is concerned - power of the test is 0; 1−P [do not reject H0| HAis true]=1− 1=0. Now consider the opposite case - the test with which wealways reject the null hypothesis. Power is great - it is 1, maximum possible power. But, the size of the testalso is 1-again pathetic.We have to compromise between the two conflicting goals. Convention in testing procedure is to fix thesize at a arbitrary prespecified level and search for a test which maximizes the power - even this is impossiblein most cases.• Critical region of a test is the area where we commit the type I error.Test procedure and test statisticThe next question naturally arising is that how we can actually test the hypotheses. An obvious answeris that the test, whatever it is, should be based on the observed data. The data set itself is too lousy todetermine the plausibility of the null hypothesis. We need a kind of summary measure of the data, whichshould be handy but retain relevant information on the true data generating process.• Let t = t (y) be a function of the observations and let T = t (Y ) be the corresponding random variable.We call T a test statistic for the testing of H0if(i) the distribution of T when H0is true is knwon, at least approximately(asymptotically).(ii) the larger the value of t the stronger the evidence of departure from H0of the typeit is required to testAfter getting the distribution of test statistic under the null hypothesis, we now need a decision rule todetermine whether the null hypothesis is consistent with the observed data. For given observations y we cancalculate t = tobs= t (y) , say, and identify a critical region for a given size of the test. If the value of thetest statistic falls into the critical region, we reject the null hypothesis.To sum up the test procedure;1. Set up the null and alternative hypotheses22. Design a test statistic which has some good properties3. Find the distribution of test statistic under the null hypothesis - exact or asymptotic distribution4. Identify the critical region for the null distribution with a given size of the test5. Calculate the test statistic using the observed data6. Check whether or not the value of the test statistic falls into the critical