2_Probability and Statistics

It is inconsistent to say that h 0 m 42 and h 0 m 43

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Unformatted text preview: t H 0 against H 1 at the 5% level VER. 9/11/2012. © P. KOLM 64 Remarks Proper language of hypothesis testing Correct usage: “we fail to reject H 0 in favor of H 1 at the 5% significance level” Incorrect usage: “we accept H 0 at the 5% significance level” With the same set of data, there are usually many hypotheses that cannot be rejected. It is inconsistent to say that H 0 : m = 4.2 and H 0 : m = 4.3 are both “accepted,” since only one of these can be true o But it is entirely possible that neither of these hypotheses is rejected. For this reason, we always say “fail to reject H 0 ” rather than “accept H 0 ” Large sample sizes We can invoke the CLT Then hypothesis testing for the population mean (approximately when n > 120 ) is the same whether or not the population distribution is normal. This is because Y - m0 S/ n VER. 9/11/2012. © P. KOLM N (0,1) 65 References Box, G. E. P. (1976). "Science and Statistics." Journal of the American Statistical Sssociation 71(356): 791-799. Geweke, J., J. Horowitz and H. Pesaran (2008). Econometrics. The New Palgrave Dictionary of Economics. S. N. Durlauf and L. E. Blume. Basingstoke, Palgrave Macmillan. Greene, W. H. (2000). Econometric Analysis 4th Edition, Macmillan. Wooldridge, J. M. (2009). Introductory Econometrics: A Modern Approach, South-Western Pub. Footnotes 1 For example: (1) Agents in many financial models are assumed to make decisions are consistent with utility maximization under full information. Temporal effects of their decisions, even those occurring over long periods, are fully internalized. (2) Models return to their equilibrium quickly. (3) Models are often stylized and do not include all important interactions (i.e. other factors, endogeneity, frictions). 2 For an overview and summary, see http://en.wikipedia.org/wiki/Confirmation_bias 3 A population is any entire collection of people, animals, plants or things from which we may collect data. It is the entire group we are interested in, which we wish to describe or draw conclusions about. In order to make any generalizations about a population, a sample, that is meant to be representative of the population, is often studied. For each population there are many possible samples. A sample statistic gives information about a corresponding population parameter. For example, the sample mean for a set of data would give information about the overall population mean. It is important that the investigator carefully and completely defines the population before collecting the sample, including a description of the members to be included. VER. 9/11/2012. © P. KOLM 66 4 Strictly speaking, the expected value exists only for n > 1 . Because the distribution is so “spread out,” the variance does not exist for n > 2 . 5 In Geweke, Horowitz and Pesaran (2008) econometrics is defined as “the application of mathematics and statistical methods to economic data" and described as the branch of economics “that aims to give empirical content to economic relations.” 6 In the statistics literature,...
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