The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

spurious correlation.
This term, due to Karl Pearson,
refers to the situation where correlation is found to be present between the ratios of vari-
ables even though the original variables are uncorrelated or random.
41
Thus, in the model
Y
i
=
β
1
+
β
2
X
i
+
u
i
, 
Y
and 
X
may not be correlated but in the transformed model
Y
i
/
X
i
=
β
1
(1
/
X
i
)
+
β
2
, 
Y
i
/
X
i
and 1
/
X
i
are often found to be correlated.
4. When 
σ
2
i
are not directly known and are estimated from one or more of the trans-
formations that we have discussed earlier, all our testing procedures using the 
t
tests,
F
tests, etc., are, 
strictly speaking, valid only in large samples. 
Therefore, one has to be
careful in interpreting the results based on the various transformations in small or finite
samples.
42
11.7
Concluding Examples
In concluding our discussion of heteroscedasticity we present three examples illustrating
the main points made in this chapter.
39
However, as a practical matter, one may plot 
ˆ
u
2
i
against each variable and decide which 
X
variable
may be used for transforming the data. (See Fig. 11.9.)
40
Sometimes we can use ln (
Y
i
+
k
) or ln (
X
i
+
k
), where 
k
is a positive number chosen in such a way
that all the values of 
Y
and 
X
become positive.
41
For example, if 
X
1
, 
X
2
, and 
X
3
are mutually uncorrelated 
r
12
=
r
13
=
r
23
=
0 and we find that the
(values of the) ratios 
X
1
/
X
3
and 
X
2
/
X
3
are correlated, then there is spurious correlation. “More gener-
ally, correlation may be described as spurious if it is induced by the method of handling the data and
is not present in the original material.” M. G. Kendall and W. R. Buckland, 
A Dictionary of Statistical
Terms, 
Hafner Publishing, New York, 1972, p. 143.
42
For further details, see George G. Judge et al., op. cit., Section 14.4, pp. 415–420.
guj75772_ch11.qxd 12/08/2008 07:04 PM Page 395

---