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mmodels - Advanced Quantitative Research Methodology...

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Advanced Quantitative Research Methodology, Lecture Notes: Multiple Equation Models 1 Gary King http://GKing.Harvard.Edu April 11, 2010 1 c Copyright 2010 Gary King, All Rights Reserved. Gary King http://GKing.Harvard.Edu () Advanced Quantitative Research Methodology, Lecture Notes: April 11, 2010 1 / 1
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Identification Reading: Gary King. Unifying Political Methodology: The Likelihood Theory of Statistical Inference . Ann Arbor: University of Michigan Press, 1998: Chapter 8. Gary King () Multiple Equation Models 2 / 1
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Identification Reading: Gary King. Unifying Political Methodology: The Likelihood Theory of Statistical Inference . Ann Arbor: University of Michigan Press, 1998: Chapter 8. Models that often don’t make sense, even though it is hard to tell. Gary King () Multiple Equation Models 2 / 1
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Example 1: Flat Likelihoods Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Y i f p ( y i | λ i ) Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Y i f p ( y i | λ i ) λ i = 1 + 0 β Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Y i f p ( y i | λ i ) λ i = 1 + 0 β What do we know about β ? Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Y i f p ( y i | λ i ) λ i = 1 + 0 β What do we know about β ? L ( λ | y ) = n i =1 e - λ λ y i y i ! Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Y i f p ( y i | λ i ) λ i = 1 + 0 β What do we know about β ? L ( λ | y ) = n i =1 e - λ λ y i y i ! and the log-likelihood, with (1 + 0 β ) substituted for λ i : Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Y i f p ( y i | λ i ) λ i = 1 + 0 β What do we know about β ? L ( λ | y ) = n i =1 e - λ λ y i y i ! and the log-likelihood, with (1 + 0 β ) substituted for λ i : ln L ( β | y ) = n i =1 {- (0 β + 1) - y i ln(0 β + 1) } Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Y i f p ( y i | λ i ) λ i = 1 + 0 β What do we know about β ? L ( λ | y ) = n i =1 e - λ λ y i y i ! and the log-likelihood, with (1 + 0 β ) substituted for λ i : ln L ( β | y ) = n i =1 {- (0 β + 1) - y i ln(0 β + 1) } = n i =1 - 1 Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods A (dumb) model: Y i f p ( y i | λ i ) λ i = 1 + 0 β What do we know about β ? L ( λ | y ) = n i =1 e - λ λ y i y i ! and the log-likelihood, with (1 + 0 β ) substituted for λ i : ln L ( β | y ) = n i =1 {- (0 β + 1) - y i ln(0 β + 1) } = n i =1 - 1 = - n Gary King () Multiple Equation Models 3 / 1
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Example 1: Flat Likelihoods Gary King () Multiple Equation Models 4 / 1
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Example 1: Flat Likelihoods 1. An identified likelihood has a unique maximum. Gary King () Multiple Equation Models 4 / 1
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Example 1: Flat Likelihoods 1. An identified likelihood has a unique maximum. 2. A likelihood function with a flat region or plateau at the maximum is not identified.
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