Econometric Theory/Summation and Product Operators

To sum a series of variables $x$, the Greek capital letter sigma Σ is used:

$\Sigma _{i=1}^{n}x_{i}=x_{1}+x_{2}+\ldots +x_{n}$.

There are some properties concerning the summation operator Σ:

1. $\Sigma _{i=1}^{n}k=nk$, where k is a constant.

2. $\Sigma _{i=1}^{n}kx_{i}=k\Sigma _{i=1}^{n}x_{i}$, where k is a constant.

3. $\Sigma _{i=1}^{n}(a+bx_{i})=na+b\Sigma _{i=1}^{n}x_{i}$, where a and b are constants. This is a result of rules 1 and 2 above.

4. $\Sigma _{i=1}^{n}(x_{i}+y_{i})=\Sigma _{i=1}^{n}x_{i}+\Sigma _{i=1}^{n}y_{i}$,

The double summation operator is used to sum up twice for the same variable:

${\begin{aligned}\Sigma _{i=1}^{n}\Sigma _{j=1}^{m}x_{ij}&=\Sigma _{i=1}^{n}(x_{i1}+x_{i2}+\ldots +x_{im})\\&=(x_{11}+x_{21}+\ldots +x_{n1})+(x_{12}+x_{22}+\ldots +x_{n2})+\ldots +(x_{1m}+x_{2m}+\ldots +x_{nm})\\\end{aligned}}$

The double summation operator has the following properties:

1. $\Sigma _{i=1}^{n}\Sigma _{j=1}^{m}x_{ij}=\Sigma _{j=1}^{m}\Sigma _{i=1}^{n}x_{ij}$. The order of the summation signs is interchangeable.

2. $\Sigma _{i=1}^{n}\Sigma _{j=1}^{m}x_{i}y_{j}=\Sigma _{i=1}^{n}x_{i}\Sigma _{j=1}^{m}y_{j}$.

3. $\Sigma _{i=1}^{n}\Sigma _{j=1}^{m}(x_{i}+y_{j})=\Sigma _{i=1}^{n}x_{i}\Sigma _{j=1}^{m}x_{i}j+\Sigma _{i=1}^{n}x_{i}\Sigma _{j=1}^{m}y_{ij}$.

4. ${\begin{aligned}\left[\Sigma _{i=1}^{n}x_{i}\right]^{2}&=\Sigma _{i=1}^{n}{x_{i}}^{2}+2\Sigma _{i=1}^{n-1}\Sigma _{j=i+1}^{n}x_{i}x_{j}\\&=\Sigma _{i=1}^{n}{x_{i}}^{2}+2\Sigma _{i<j}x_{i}x_{j}\\\end{aligned}}$.

Finally, the product operator Π is defined as: $\Pi _{i=1}^{n}x_{i}=x_{1}\cdot x_{2}\cdots x_{n}$.

Bibliography

• Gujarati, D.N. (2003). Basic Econometrics, International Edition - 4th ed. McGraw-Hill Higher Education. pp. 869–870. ISBN 0-07-112342-3.