INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 114
4.1.3.5 Measurement Errors in the Descriptive Variables) To keep the analysis simple, we will confine it to the simple regression model. Let us suppose that a variable
Y depends
on a variable Z according to the relationship
…[4.06] where
v is a disturbance term with mean 0 and variance
,
distributed independently of Z.
We shall suppose that Z cannot be measured absolutely accurately, and we shall use
X to denote its measured value. In observation is equal to the true value,
,
plus the measurement error,
:
…[4.07] We shall suppose that
w has mean 0 and variance
, that
Z has population variance, and that
w is distributed independently of
Z and
v.
[4.07] into [4.06], will yield
(
)
…[4.08] Two random components are present in [4.08], the original disturbance term
v and the measurement error (multiplied by. Together they form a composite disturbance term,
which we shall call u:
…[4.09] Therefore, [4.08] becomes
…[4.10] You have your data on
Y (which,
for the time being, we shall assume has been measured accurately) and
X, and you unsuspectingly regress
Y on
X. As usual, the regression coefficient
b is given by
( )
( )
( )
( )
…[4.11]
