Calculating the Variance of a Pure Random Walk

Question

I am trying to calculate the first two moments of the random walk below:

$y_t$=$y_0$+$\sum_{j=1} ^t u_j$

• Mean: E[$y_t$]=$y_0$

• Variance: E[$y_t ^2$]=t$\sigma^2$

I understand how to obtain the 1st moment because we assume the expectation of the disturbance terms is equal to 0 and the expectation of a constant is the constant.

I do not understand how to obtain the variance. I tried the following formula:

var($y_t$)=E[$y_t ^2$]-$E[y_t]^2$

I incorrectly simplified this in the following steps:

1. $var(y_t)=E[(y_0+\sum u_j)(y_0+\sum u_j)]+y_0 ^2$

2. $var(y_t)=E[y_0 ^2+2\sum u_j+\sum u_j ^2]+y_0 ^2$

3. $var(y_t)=E[y_0]+2E[\sum u_j]+E[(\sum u_j)^2]$

4. $var(y_t)=y_0$

I am hoping someone can explain that form of the variance is obtained.

This information comes from the ninth slide of the following presentation: http://www.ncer.edu.au/events/documents/Lecture1_Intro.pdf