2.1 — Data 101 & Descriptive Statistics — Appendix
The Summation Operator
Many elementary propositions in econometrics (and statistics) involve the use of the sums of numbers. Mathematicians often use the summation operator (the greek letter
Let
- The term beneath
is known as the “index,” which tells us where to begin our adding (at the 1st individual term, )- Note other letters, such as
, or may be used (especially if is defined elsewhere)
- Note other letters, such as
- The term above
is the total number of terms we should add - Essentially, read
as “add up all the individual observations from the 1st to the final .”
Useful Properties of Summation Operators
Rule 1: The summation of a constant
Proof:
Rule 2: The summation of a sum of two random variables is equal to the sum of their summations:
Proof:
Rule 3: The summation of constant over
Proof:
Combining these 3 rules: for the sum of a linear combination of a random variable (
Proof: left to you as an exercise!
Advanced: Useful Properties for Regression
There are some additional properties of summations that may not be immediately obvious, but will be quite essential in proving properties of linear regressions.
Using the properties above, we can describe the mean, variance, and covariance of random variables.1
First, define the mean of a sequence
Second, the variance of
Third, the covariance of
Rule 4: The sum of the deviations of observations of
Proof:
Rule 5: The squared deviations of
Proof:
Rule 6: The following summations involving
Proof:
equivalently:
Footnotes
For more beyond the mere definition, see the appendix on Covariance and Correlation↩︎