View source for Econometric Theory/Summation and Product Operators

To sum a series of variables <math> x </math>, the Greek capital letter sigma ''Σ'' is used:

<math>
\Sigma^n_{i=1} x_i = x_1 + x_2 + \ldots + x_n
</math>.

There are some properties concerning the summation operator ''Σ'':

1. <math>
\Sigma^n_{i=1} k = nk 
</math>, where k is a constant.

2. <math>
\Sigma^n_{i=1} k x_i = k \Sigma^n_{i=1} x_i
</math>, where k is a constant.

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

4. <math>
\Sigma^n_{i=1} (x_i + y_i) = \Sigma^n_{i=1} x_i + \Sigma^n_{i=1} y_i
</math>,

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

<math>
\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}
</math>

The double summation operator has the following properties:

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

2. <math>
\Sigma^n_{i=1} \Sigma^m_{j=1} x_i y_j = \Sigma^n_{i=1} x_i \Sigma^m_{j=1} y_j
</math>.

3. <math>
\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}
</math>.

4. <math>
\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}
</math>.

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

==Bibliography==
*{{cite book
 |last = Gujarati
 |first = D.N.
 |authorlink = 
 |year = 2003
 |title = Basic Econometrics, International Edition - 4th ed.
 |others = 
 |publisher = McGraw-Hill Higher Education 
 |isbn = 0-07-112342-3
 |pages = 869–870
}}

{{BookCat}}