Econometric Theory/Summation and Product Operators

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

$$ \Sigma^n_{i=1} x_i = x_1 + x_2 + \ldots + x_n $$.

There are some properties concerning the summation operator Σ:

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

2. $$ \Sigma^n_{i=1} k x_i = k \Sigma^n_{i=1} x_i $$, where k is a constant.

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

4. $$ \Sigma^n_{i=1} (x_i + y_i) = \Sigma^n_{i=1} x_i + \Sigma^n_{i=1} y_i $$,

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

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

The double summation operator has the following properties:

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

2. $$ \Sigma^n_{i=1} \Sigma^m_{j=1} x_i y_j = \Sigma^n_{i=1} x_i \Sigma^m_{j=1} y_j $$.

3. $$ \Sigma^n_{i=1} \Sigma^m_{j=1} (x_i + y_j) = \Sigma^n_{i=1} x_i \Sigma^m_{j=1} x_ij + \Sigma^n_{i=1} x_i \Sigma^m_{j=1} y_{ij} $$.

4. $$ \begin{align} \left [ \Sigma^n_{i=1} x_i \right ]^2 & = \Sigma^n_{i=1} {x_i}^2 + 2 \Sigma^{n-1}_{i=1} \Sigma^n_{j=i+1} x_i x_j \\ & = \Sigma^n_{i=1} {x_i}^2 + 2 \Sigma_{i < j} x_i x_j \\ \end{align} $$.

Finally, the product operator Π is defined as: $$ \Pi^n_{i=1} x_i = x_1 \cdot x_2 \cdots x_n $$.