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319lec10_2010

# 319lec10_2010 - CHAPTER 8 HYPOTHESIS TESTING References...

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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 0 : ° 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 0 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 0 : ± 2 X = ± 2 Y : The alternative hypothesis H A : ± 2 X 6 = ± 2 Y : Again, if we reject H 0 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 = ² + ³X + ´W + µ; where Y is the consumption, X is the labor income, W is the wealth, and µ a random disturbance representing other omitted variables (e.g. family structures). The parameters ²; ³ 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 : ´ = 0 : H A : ´ 6 = 0 : If the null hypothesis H 0 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

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Example [Constant Return to Scale]: The Cob-Douglas production function is speci°ed as F ( L; K ) = cL ° K ± ; where Y is output, L is labor and K is capital. Suppose ² + ³ = 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 0 : ² + ³ = 1 : The alternative hypothesis is H A : ² + ³ 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 0 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 0 : In other words, it is the hypothesis that we hope to accept if the null hypothesis H 0 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. (ii) ± 2 is known. Hypotheses: H 0 : ° = ° 0 : H A : ° 6 = ° 0 : where ° 0 is a given real number (e.g. ° 0 = 5) : Question: How to design a test for these hypotheses?
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319lec10_2010 - CHAPTER 8 HYPOTHESIS TESTING References...

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