Econometrics 1: A Review of Statistical Concepts 1.3 - Covariance and Correlation

Question 1

Why would economists use covariance & correlation over mean & variance?

Answer 1

The mean and variance are properties of a single random variable. Economists are usually
interested in the relationship between two or more variables

Question 2

Describe ‘covariance’

Answer 2

Covariance: Covariance is a measure of how two variables say, 𝑋 and 𝑌, are associated with each
other, i.e. how they co-vary. This is given by 𝐸[(𝑋 − 𝜇bottom right𝑋)(𝑌 − 𝜇bottom right𝑌)] = ∫lower bound ‘x’ ∫lower bound ‘y’(𝑥 − 𝜇bottom right𝑋)(𝑦 − 𝜇bottom right𝑌) 𝑓(𝑥, 𝑦) 𝑑𝑦𝑑𝑥 ≡ 𝑐𝑜𝑣(𝑋, 𝑌)
where 𝜇bottom right𝑋 and 𝜇bottom right𝑌 are the means of 𝑋 and 𝑌 respectively and 𝑓(𝑥, 𝑦) is the joint density function (depicting the probabilities associated with 𝑋 taking on certain values while 𝑌 takes on certain values). The value of the covariance can be positive or negative. When positive it implies that as
the values assumed by 𝑋 increase/decrease, then the values of the 𝑌 variable also
increase/decrease. If the covariance is negative it means that as the values that the 𝑋 variable
assumes increase/decrease, then the values of the 𝑌 variable decrease/increase, i.e. they move in
the opposite direction.
For example, it is recognised that violent crime rates increase during hot weather, because there is
more scope for interaction between people. Hence one would expect a positive covariance
between violent crime and temperatures. Other economic examples; there is a positive covariance
between the amount people earn and how much they consume, there is a negative covariance
between price and demand for goods