INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 34
UNIT 3: CORRELATION COEFFICIENT CONTENTS 1.3.1.0 Introduction
1.3.2.0 Objectives
1.3.3.0 Main Content
1.3.3.1 Properties of the regression coefficients
and hypothesis testing 1.3.4.0 Summary
1.3.5.0 Conclusion
1.3.6.0 Tutor-Marked Assignment
1.3.7.0 References/Further Reading
1.3.1.0 INTRODUCTION This unit introduces a statistic called correlation coefficient. The correlation coefficient describes the direction, whether positive or negative and further measures the degree of relationship that exist between two different variables.
1.3.2.0 OBJECTIVE The main objective of this unit is to provide ways for which the studentmay have a simpler understanding of the topic correlation.
1.3.3.0 MAIN CONTENTS Correlation measures the degree of association between two or more variables.
1.3.3.1 Properties of the regression coefficients and hypothesis testing Like
variance and covariance, the correlation coefficient comes in two forms, population and sample. ρ traditionally denotes the population correlation coefficient,
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
XYXYXY
…[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)
nn
. Hence, cov(
)
1
var( var )
1 1
XYnXYnrnnXYnn
…[3.25]
The
factors (
1)
nn
could be cancelled out so we can conveniently define the sample correlation by
( )
√ ( ) ( )
…[3.26]
XYXY
…[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 
