THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE 1 In this short sequence we shall decompose a random variable X into its fixed and random components.

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Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Introduction to Econometrics, 5th edition

Presentation transcript:

THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE 1 In this short sequence we shall decompose a random variable X into its fixed and random components. Let the population mean of X be  X. Population mean of X :

The actual value of X in any observation will in general be different from  X. We will call the difference u i, so u i = X i –  X. 2 THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE Population mean of X : Random component In observation i :

Re-arranging this equation, we can decompose X i as the sum of its fixed component,  X, which is the same for all observations, and its random component, u i. 3 THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE Population mean of X: In observation i, the random component is given by Hence X i can be decomposed into fixed and random components: Population mean of X : Random component In observation i : Decomposition of X i :

The expected value of the random component is zero. It does not systematically tend to increase or decrease X. It just makes it deviate from its population mean. 4 THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE Expected value of u i is zero: Population mean of X : Random component In observation i : Decomposition of X i :

The variance of X is equal to the variance of u. This is obvious, since all the variation in X is caused by the variation in u. 5 THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE Population mean of X : Random component In observation i : Decomposition of X i : Variance of X is same as variance of u :

Copyright Christopher Dougherty These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section R.2 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics or the University of London International Programmes distance learning course EC2020 Elements of Econometrics