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: XiXE

The actual value of X in any observation will in general be different from X. We will call the difference ui, so ui = Xi – X.

2

THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE

Population mean of X: XiXE

Xii Xu Random component In observation i:

Re-arranging this equation, we can decompose Xi as the sum of its fixed component, X, which is the same for all observations, and its random component, ui.

3

THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE

Population mean of X:

In observation i, the randomcomponent is given by

Hence Xi can be decomposed

into fixed and random components:

Population mean of X: XiXE

Xii Xu

iXi uX

Random component In observation i:

Decomposition of Xi :

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

0

XX

Xi

Xii

EXE

XEuE

Expected value of ui is zero:

Population mean of X: XiXE

Xii Xu

iXi uX

Random component In observation i:

Decomposition of Xi :

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.

2

22

uE

XE XiX

5

THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE

2

22 0

uE

uE iu

Population mean of X: XiXE

Xii Xu

iXi uX

Random component In observation i:

Decomposition of Xi :

Variance of X is same as variance of u:

Copyright Christopher Dougherty 2012.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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2012.10.29