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L9 More on Hypothesis Testing

# L9 More on Hypothesis Testing - More on Hypothesis...

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More on Hypothesis Testing:An example of some empirical research: An analysis of city size distributions in China introducing The Difference in Means and Associated Tests Sometimes rather than be concerned about the particular value of the mean or variance of a population, or the nature of the underlying population distribution and we may wish to examine whether two population distributions differ, for example we may be interested in comparing the means or variances from two distinct populations. It turns out that with some minor modifications we perform very similar tests to the ones we have already discussed. To exemplify the tests discussed in the previous chapter and to introduce these new tests we will use some recent research on the nature and progress of city sizes in China over the period 1949-1999 (Anderson and Ge (2003)). Some Background: A Theory of City Size Distributions. There is a theory in urban economics (Gabaix (1999)), for which evidence has been found in many countries, which says that when measured by the number of inhabitants city sizes within a country are governed by Zipf’s law Zipf (1949) (the logarithm of the rank of the city size - logarithm of the size relative to the minimum). This means that the distribution of city sizes at any point in time is a Pareto Distribution with a parameter θ = 1. Letting x be the city size and x min be the minimum possible city size, the pdf and cdf of this distribution are given by f(x) = θ (x min /x θ +1 ) and F(x) = 1- (x min /x) θ respectively. The law derives from two ideas, the first is that individual cities start at some minimum size (call it Z 0 ) which is subject to a sequence of mutually independent multiplicative shocks (call the one in the i’th period (1+X i ) where X i is small relative to 1 and to Z 0 ). In essence the shock is related to the number of people that die in, the number of people that are born in, the number of people that immigrate to and the number of people that emigrate from the city in period i, all of which are random events. The logarithm of city size in period I may be written as:

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0 0 1 1 ln( ) ln (1 ) ln I I I i i i i Z Z X Z = = = + = + e where e i = ln(1+X i ). For I sufficiently large (that is after a sufficiently long period of
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L9 More on Hypothesis Testing - More on Hypothesis...

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