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Unformatted text preview: CHAPTER 8: HYPOTHESIS TESTING References: Textbook, Chapter 10, Sections 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7, 10.8. 8.1 Hypothesis Testing Example [Finance]: There are two Stocks X and Y: Before making a decision on investment, we would like to know (a) which has a higher expected return; (b) which is more risky? Question (a) can be formulated as the following hypotheses: The null hypothesis H : & X = & Y : The alternative hypothesis H A : & X 6 = & Y : Based on a data, if we can design a test and reject the null hypothesis H and accept H A ; then we can claim that the expected return of X is not equal to the expected return of Y: Question (b) can be formulated as the following hypotheses: The null hypothesis H : 2 X = 2 Y : The alternative hypothesis H A : 2 X 6 = 2 Y : Again, if we reject H and accept H A ; then we can conclude that X and Y are not equally risky. Suppose after doing the above two hypotheses testing, we conclude that X has a higher expected return and a smaller risk, then we will invest on X rather than on Y: Example [Wealth E/ect] : A consumption function is given by Y = + X + W + ; where Y is the consumption, X is the labor income, W is the wealth, and a random disturbance representing other omitted variables (e.g. family structures). The parameters ; and are constants. We are interested in whether the wealth has e/ect on the consumption. For this purpose, we can formulate the hypotheses as H : = 0 : H A : 6 = 0 : If the null hypothesis H is true, then the wealth has no e/ect on the consumption. If H A is true, then there exists a wealth e/ect. 1 Example [Constant Return to Scale]: The CobDouglas production function is speci&ed as F ( L;K ) = cL & K ; where Y is output, L is labor and K is capital. Suppose & + = 1 ; then F ( L;K ) = c ( L ) & ( K ) = a + cL & K = F ( ;K ) ; i.e. increasing both L and K by units will also increase output by units. This is called the constant return to scale (CRS). The null hypothesis H : & + = 1 : The alternative hypothesis is H A : & + 6 = 1 : The null is the CRS, the alternative hypothesis H A says that the production technology is not CRS. Remarks: (i) The null hypothesis H usually formulates some simple economic relationships among population parameters . These relationships may be predicted by the economic theory. (ii) The alternative hypothesis H A is a di/erent hypothesis against the null hypothesis H : In other words, it is the hypothesis that we hope to accept if the null hypothesis H is rejected. 8.2 Tests of the mean of a Normal Distribution when 2 is known Assumptions: (i) The random sample X n = f X 1 ;X 2 ;:::;X n g is from a normal population N ( ; 2 ) ; where is unknown....
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