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_{i=1} \Sigma^n_{j=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 .

      Bibliography

      • Gujarati, D.N. (2003). Basic Econometrics, International Edition - 4th ed.. McGraw-Hill Higher Education. pp. 869-870. ISBN 0-07-112342-3. 
      Last modified on 24 May 2009, at 21:54