2.4 STOCHASTIC SPECIFICATION OF PRF ( Damodar N. Gujarati )

It clear from figure 2.1 that , as family income increases, family consumption expenditure on the average increases,too. But What about the consumption expenditure of an individual family in relation to its (fixed) level of income? It is obvious from Table 2.1 an Figure 2.14 that an individual family's consumption expenditure does not necessarily increase as the income level increases. For example , from Table2.1 we observe that corresponding to  the income level of $100 there is one family whose consumption  expenditure of $65 is less than the consumption expenditure of two families whose weekly income is only $80. But notice that average consumption  expenditure of families with a weekly income of $100 is greater than the average consumption expenditure  of families with a weekly income of $80 ($77 versus $65).
What, then, can we say about the relationship between an individual family's consumption expenditure an a given level of income? We see from Figure 2.1 that, given the income level of X1, an individual family's consumption expenditure is clustered around the average consumption  of all families at that X1,that is, around its conditional expectation, Therefore.we can express the deviation of an individula Yi around its expected value as follow:

ui = Yi  - E ( Y \ Xi )

or 

                                                   Yi =  E( Y | Xi ) + ui                                                   

(2.4.1)

____________________
8.See App.A for a brief discussion of the properties of the expectation operator E. Note that E(Y|Xi), once the value of Xi is fixed, is a constant.
9.As a matter of fact, in the method of least squares to be developed in chap.3 it is assumed e3xplicitly that  E = (ui | Xi ) = 0
 See Sec.3.2.