National open university of nigeria introduction to econometrics II eco 356

INTRODUCTION TO ECONOMETRICS II

ECO 306

NOUN
35 the Greek letter that is the equivalent of “r”, and pronounced row, as in row a boat. For variables X and Y it is defined by
2 2
XY
XY
X
Y


 



…[3.24]
If X and Y are independent,
will be equal to 0 because the population covariance will be 0. If there is a positive association between them, then we have

, otherwise will still be positive.
If there is an exact positive linear relationship will assume its maximum value of
1. Similarly, if there is a negative relationship
will be negative, with minimum value of –1. The sample correlation coefficient,

, is defined by replacing the population covariance and variances by their unbiased estimators. We have seen that these maybe obtained by multiplying the sample variances and co-variances by
(
1)
n
n

. Hence, cov(
)
1
var( var )
1 1
XY
n
XY
n
r
n
n
X
Y
n
n






…[3.25]
The factors
(
1)
n
n

could be cancelled out so we can conveniently define the sample correlation by

( )
√ ( ) ( )


…[3.26]
XY
X
Y

 



…[3.27]
Like ρ, r has maximum value 1, which is attained when there is a perfect positive association between the sample values of X and Y (when you plot the scatter diagram, the points lie exactly on an upward-sloping straight line. Similarly, it has minimum value –1, attained when there is a perfect negative association (the points lying exactly on a

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