Econometric Theory/Summation and Product Operators

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

${\mathrm{\Sigma }}_{i=1}^n{x}_i=x_1+x_2+\dots +x_n$.

There are some properties concerning the summation operator Σ:

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

2. ${\mathrm{\Sigma }}_{i=1}^{n}k{x}_i=k{\mathrm{\Sigma }}_{i=1}^n{x}_i$, where k is a constant.

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

4. ${\mathrm{\Sigma }}_{i=1}^n(x_i+y_i)={\mathrm{\Sigma }}_{i=1}^n{x}_i+{\mathrm{\Sigma }}_{i=1}^n{y}_i$,

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

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

The double summation operator has the following properties:

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

2. ${\mathrm{\Sigma }}_{i=1}^n{\mathrm{\Sigma }}_{j=1}^m{x}_i{y}_j={\mathrm{\Sigma }}_{i=1}^n{x}_i{\mathrm{\Sigma }}_{j=1}^m{y}_j$.

3. ${\mathrm{\Sigma }}_{i=1}^n{\mathrm{\Sigma }}_{j=1}^m(x_i+y_j)={\mathrm{\Sigma }}_{i=1}^n{x}_i{\mathrm{\Sigma }}_{j=1}^m{x}_{i}j+{\mathrm{\Sigma }}_{i=1}^n{x}_i{\mathrm{\Sigma }}_{j=1}^m{y}_{ij}$.

4. $\begin{array}{cc}{\left[{\mathrm{\Sigma }}_{i=1}^n{x}_i\right]}^2& ={\mathrm{\Sigma }}_{i=1}^n{x_i}^2+2{\mathrm{\Sigma }}_{i=1}^{n-1}{\mathrm{\Sigma }}_{j=i+1}^n{x}_i{x}_j\\ & ={\mathrm{\Sigma }}_{i=1}^n{x_i}^2+2{\mathrm{\Sigma }}_{i<j}{x}_i{x}_j\\ \end{array}$.

Finally, the product operator Π is defined as: ${\mathrm{\Pi }}_{i=1}^n{x}_i=x_1\cdot x_2\cdots x_n$.

Bibliography

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