If y i is an indirect effect on itself it can blow up

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If Y i is an indirect effect on itself, it can blow up . In the example above, we have Y 1 = γ 1 , 1 X 1 + β 1 , 2 Y 2 + ζ 1 Y 2 = γ 2 , 2 X 2 + γ 2 , 3 X 3 + β 2 , 1 Y 1 + ζ 2 . Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 42 / 54
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Indirect and direct effects We can then think of the effect of Y 2 on itself as happening in stages. First, Y 2 has a direct effect on Y 1 by β 1 , 2 . Then, Y 1 has a direct effect on Y 2 by β 2 , 1 , giving an effect from Y 2 to itself by β 1 , 2 × β 2 , 1 in two steps. Another such loop gives an effect from Y 2 to itself by ( β 1 , 2 × β 2 , 1 ) 2 in four steps. The total effect so far is β 1 , 2 × β 2 , 1 + ( β 1 , 2 × β 2 , 1 ) 2 In 2 K steps, Y 2 has a total effect on itself by K X k = 1 ( β 1 , 2 × β 2 , 1 ) k . Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 43 / 54
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Indirect and direct effects However, the loop is infinite , so that the total effect from Y 2 to itself is k = 1 ( β 1 , 2 × β 2 , 1 ) k . If we do not have | β 1 , 2 × β 2 , 1 | < 1, this sum is infinite , and the system is said to explode. We do not wish this to be allowed, and consider such models faulty. Path models in matrix formulation: Y = BY + Γ ξ + ζ . Explosion occurs if the (possibly complex) eigenvalues of B are outside the unit circle on the complex plane. A simple sufficient condition, is that the largest eigenvalue of BB t is less than one (this eigenvalue is always a real number). Stata will print the largest eigenvalue of BB t by using estat stable after SEM-estimation. It should be less than one. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 44 / 54
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Contents 1 Confirmatory factor analysis Stata: Inputting data based on empirical covariance matrices Inference, goodness of fit and modification indices An empirical example: Mental abilities 2 More on Path models Indirect and direct effects A brief empirical example from sociology On R 2 Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 45 / 54
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A brief empirical example from sociology A published paper in sociology investigated the union sentiment of southern non-union textile workers for 173 workers. At the time, southern textile workers typically were portrayed as anti-union and extremely submissive towards managers, but there was little empirical evidence to support such an assertion. The focus was in understanding what influenced sentiment towards unions and submissiveness to managers, as well as support for labor activism. The scientists measured the following variables for each individuals. 1 Y 1 = Submissiveness to managers 2 Y 2 = Support for labor activism 3 Y 3 = Sentiment towards unions 4 X 1 = Logarithm of years in textile mill 5 X 2 = Age Statistical problem: To explain Y 1 , Y 2 , Y 3 through their interrelation and through the exogenous variables X 1 , X 2 . Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 46 / 54
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A brief empirical example from sociology 1 Y 1 = Submissiveness to managers 2 Y 2 = Support for labor activism 3 Y 3 = Sentiment towards unions 4 X 1 = Logarithm of years in textile mill 5 X 2 = Age The sociologiests’ theories supported the following relations among the variables Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 47 / 54
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