Econometric Theory/Summation and Product Operators

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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[edit]

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